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Last year, we wrote about our experiences of a 4K shoot we carried out for stock footage library Pond5. Since then, we’ve done a number of other shoots using a fully 4K workflow, so I thought it would be a good time to share our thoughts on how these went, and our practical experiences storing, working with and outputting at this resolution. These apply to small-to-medium-sized production companies like ours; if you work at a large facility with hundreds of staff, the basics will still apply but you can probably add a zero or two onto the complexity of anything I mention here.
It’s also worth noting that although most people refer to “4K” as being double the resolution of HD, it’s not (quite). 4K technically refers to a frame that’s 4096 x 2160 pixels, whereas something that’s double the size of HD – 3840 x 2160 pixels – is 4K UHD. Since there’s not much practical difference in the pixel count, and most people say “4K” when they mean UHD, I’ve used 4K as a catch-all term to mean both types of ultra high-definition.
## Background, or: why 4K?
There have been plenty of musings about whether 4K is here to stay, or whether it’s “just another 3D” (or – controversially – VR; check back in a few years to see if VR was a flash in the pan, or if it stuck). While the human eye hasn’t been upgraded in millennia, and VHS worked well enough for years for everyone to have entire libraries of 80s action movies which are now gathering dust in charity shops, 4K is now being pushed at us on all sides. Most TVs are now 4K-capable (via HDMI 1.4+, although only HDMI 2.0+ is capable of carrying 4K at more than 24 frames per second), although you’ll probably be hard pressed to spot the difference in most cases. We recently bought a 65” 4K TV for use at a tradeshow, and decided to shoot a variety of content in 4K to go alongside our regular showreel material, which is all in 1080p: nobody (yet), apart from Pond5, has asked us to shoot or finish anything in 4K. A technical glitch on the day meant that the reel we played on the stand was actually in 1080p. I looked closely to see anyone wrinkle their noses or complain it looked grainy, but nobody did. (If you were there, and thought our reel looked pixellated, please let me know; we’ll dig out a prize for you to win.)
## The elements of working in 4K
Gone are the days when changing your workflow meant upgrading all your existing tape bays. We’ve been tapeless since 2009, and I haven’t missed tape for a second. However, even though we’re now only talking about large files becoming even larger files, there are several things to think about when committing to a 4K workflow.
### 1. The file size
4k footage takes up four times the disk space of 2k.
Footage shot in 1080p is 1920 pixels by 1080, or just over 2 megapixels in size. 4K UHD footage is double this – in both dimensions – giving 8.3 megapixels per frame, or 4 times the actual pixel count of HD. True 4K is slightly more than this at 8.85 megapixels per frame. What this means in practice is that to shoot either 4K or UHD properly, at a bitrate high enough for you to notice the difference, you have to quadruple the data rate you would use for HD. So if you’d shoot HD at 50Mbps, you’ll need to shoot UHD at 200Mbps. I personally feel 50Mbps is the bare minimum that HD footage should be acquired in, unless you’re shooting something very static like an interview, so in most cases we would shoot HD at around 100Mbps or more, depending on the content. (Basically: the more movement per frame, the higher the bitrate you need. A classic case where you need a really high bitrate would be for something like a confetti cannon, which is often used as part of the finale at a gig. You can see on this video that the picture looks fine until the confetti pops out, at which point it starts looking pretty horrible.)
What all this boils down to is that both 4K and UHD involve much larger amounts of storage than HD does. Our Pond5 shoot resulted in two hours of footage totalling 479GB. This is fairly typical.
### 2. Data transfer speeds
If you thought your edit machine was slow before…
Once you’re able to store all this data, the next challenge is to get it from wherever it’s being stored to whatever you’re editing it with. If you’re a one-person-band, using a desktop or laptop computer and an external drive, then you’re probably fine with either USB3 or Thunderbolt. If you’re in a multi-user environment, though, this probably means storing the footage on a server somewhere.
The most common way to be able to shunt this amount of footage from A to B is via ethernet: you’ll need at least 1GbE throughout, which includes all switches and other network hardware you might be using. The network will work at the speed of the slowest component, so for example if you’re using a cheap switch that bills itself as 1Gb, it probably isn’t – that 1Gb is likely to be shared between all the ports, so if anyone else is doing anything on the network while you’re trying to work with your footage, everything will slow down. We use a Netgear GS724Tv3 which works fine with the whole team hammering away at it at once.
If your network isn’t up to it, you can also copy all the footage you need onto your local disk(s) and work with it from there. For best results, use SSDs if you can, rather than “spinning” disks: a decent 512GB SSD will give you a far better transfer rate than a 7200rpm SATA HD. Typically this will mean around 400-500MB/sec for an SSD, versus 100MB/sec for a non-solid-state disk.
### 3. Actually playing back the footage
Press the spacebar, and wait.
This is where it can get tricky. Whether or not a given computer will be able to play back footage at 4K depends on a number of things, and they might not be immediately obvious. We were unable to even work with the footage we shot last September, for example, until we upgraded the driver for the network card on the laptop we were using. Suddenly, hey presto – the footage played back fine, instead of jerkily or not at all.
Apart from the CPU you’re using, though – and we’ll assume you’re not trying to edit 4K footage on an old Pentium or something – the most important component will probably be your graphics card. Adobe Premiere, which is what we use, uses CUDA acceleration to speed things up, but this does depend on the graphics card you have. We use both nVidia Quadro (K4000 and K5000) and GeForce (GTX980M) cards, and the GeForce ones don’t seem any slower than the much more expensive Quadros. It’s also worth noting that even though a card might say it can support a certain number of screens at 4K, in practice it might puff and wheeze if they’re all on at once. When we first got our 4K screen, we hooked it up to the fastest machine we’ve got: a 20-core/40-thread PC with SSDs, a Quadro K5000 graphics card, and 64GB of RAM. This machine was already running two screens – one at 2560 x 1440 and one at 1920 x 1080 – but it struggled with a loop of 4K content played full-screen from YouTube, which it wouldn’t play back without juddering. This illustrates the point that it isn’t necessarily the speed that you can get data to the edit machine which will cause the problem, as the footage was buffering from YouTube without any trouble. The solution was to ditch one (or, even better, both) the other screens, after which the 4K content played back fine. We now use this TV hooked up to a 4-core / 16GB laptop with a mid-range graphics card, and it plays most low-bitrate content without complaining.
### 4. Working with the footage
Intermediate codecs are crucial.
So, you’ve got everything set up, and you can play back 4K in your editing program without your computer throwing a wobbly. You’ll now probably need to decide on an intermediate codec for any times that you need to export something to a third-party application (like After Effects) and then reimport it.
We don’t use Final Cut or Avid, which both transcode footage at the ingest stage to ProRes or DNxHD respectively; this step is less relevant if you use either of those packages. Premiere, however, doesn’t transcode anything: you throw any footage you want to use into a timeline, and Premiere will make the best fist it can of playing it back. While this saves time when starting a project, it can often lead to confusion at the point when you want to round-trip footage via After Effects, Mocha, or some other compositing or fx package.
Our workhorse cameras, currently a pair of Sony FS7s, can output 4K at up to 10-bit 4:2:2. If you want to add something in to a shot and then grade it later, you therefore need to be able to keep the full 10-bit colour gamut. You could output a shot as an uncompressed TIFF or OpenEXR sequence, work with it as individual frames, and then reimport these into Premiere, but they’ll play pretty slowly as they’ll be uncompressed. A better option is to use a codec like MXF OP1a, which gives you 10-bit 4:2:2 options and is, as far as I can tell, exactly what comes out of the camera – so you shouldn’t lose any quality, at least not perceptibly.
### 5. Outputting
The most useable output codec for 4K footage is currently H.264.
The options you have when outputting 4K are a lot more obvious than they are when you’re working with it. H.264 MP4 files work absolutely fine for most things: they’re 8-bit, but if you’re putting them on YouTube or Vimeo, they’ll get retranscoded anyway. If you’re delivering to broadcast, each network or station will have its own, highly detailed, delivery guidelines, which they’ll have sent you along with their invoice for the airtime.
The bitrate you need for H.264 will depend, as I’ve already mentioned, on how complex the action in the footage is, but we find a good rule of thumb (for most 25p content) is:
$bitrate = {width * height}/{250}$
So for UHD footage, a rough bitrate would be 3840 x 2160 / 250 = 35Mbit. This is an arbitrary formula, but it’s a good starting point: render out the busiest-looking section from your edit at this bitrate and see if it looks dreadful. If it does, double the bitrate and try again.
### Conclusion
Using a fully 4K workflow is an interesting challenge. It’ll certainly get easier, but for us, and for now, the extra hassle involved in dealing with it really outweighs the benefits of using it. (This is why we offer it as an option at +25% of the cost of HD.) However, if the content you’re shooting is tremendously detailed – such as night-time timelapses shot somewhere with no light pollution, or landscapes, or aerial photography – it can be a useful option to offer. If you’ve worked with 4K and want to share your experiences, whether good or bad, leave us a comment below.
0118 324 3500 |
# Nand2Tetris Code Module
I'm working through the Assembler assignment in the Nand2Tetris course (chapter 6). The suggested implementation contains 4 modules: Main, Parser, Code, SyntaxTree. Today I wanted to get some feedback on my implementation of the Code module.
The book specifies a general API in order to allow students to use whichever programming language they want; I choose Python 2.7. The Code module takes a line of a simplified dialect of Assembly code called Hack and produces binary.
The Code module only deals with a single class of instruction in Hack language: The C-Instruction. This instruction takes the form dest=comp;jump, where either dest= OR ;jump may be optionally omitted. comp is one of 28 valid expressions. dest is the destination to store the result of the computation: it can be any permutation of the three registers available (A, D, M). jump is a condition which, if true, moves control flow the address in instruction memory currently stored in register A.
The resulting opcode takes the form: 111a cccc ccdd djjj, where a and cccc cc correspond to the comp, ddd corresponds to the dest, and jjj corresponds to the jump. For more details, see Chapters 4 and 6 of the textbook.
Please review on style, implementation (especially those terribly ugly lookup tables), etc.
#!/usr/bin/python
# hack code translator
# Sample conversion:
# HACK Instruction: M = 1
# 111
# a = 0
# cccccc = 1111 11
# ddd = 001
# jjj = 000
# BINARY Opcode: 1110 1111 1100 1000
jumpTable = { 'null': '000',
'JGT' : '001',
'JEQ' : '010',
'JEQ' : '010',
'JGE' : '011',
'JLT' : '100',
'JNE' : '101',
'JLE' : '110',
'JMP' : '111' }
compTable = { '0' : '101010',
'1' : '111111',
'-1' : '111010',
'D' : '001100',
'A' : '110000',
'!D' : '001101',
'!A' : '110001',
'-D' : '001111',
'-A' : '110011',
'D+1' : '011111',
'A+1' : '110111',
'D-1' : '001110',
'A-1' : '110010',
'D+A' : '000010',
'D-A' : '010011',
'A-D' : '000111',
'D&A' : '000000',
'D|A' : '010101' }
def dest(srcStr):
# Of the three portions of the opcodes,
# this is the only one that makes obvious logical sense
# No point in a lookup table
# Any combination of A, D, M
# in that order
# TODO: bitops this
bits = ['0', '0', '0'] # 0 by default
if 'A' in srcStr:
bits[0] = '1'
if 'D' in srcStr:
bits[1] = '1'
if 'M' in srcStr:
bits[2] = '1'
return ''.join(bits)
def comp(srcStr):
aBit = '0'
if 'M' in srcStr:
if 'A' in srcStr:
raise SyntaxError
else:
aBit = '1'
nStr = srcStr.replace('M', 'A') # this way we use only one lookup table
else:
nStr = srcStr[:] # non-destructive copy
return aBit + compTable[nStr]
def jump(srcStr):
return jumpTable[srcStr]
1. The values in your lookup tables are strings of binary digits, for example '111010'. It would be closer to the way that real assemblers work if you represented these as numbers in binary, for example 0b111010. I presume you use string concatenation to assemble the instructions at present:
instruction = '111' + comp + dest + jump
With numeric values for the instruction fields, you'll need to use bitwise operations instead:
instruction = 0b111 << 13 | comp << 6 | dest << 3 | jump
2. There's no error handling. It's likely that assembly programs will contain mistakes, for example:
M=-1;JGR // oops: typo for JGT
The user would like to get an error message, preferably one that gives the file and line number and an explanation of what went wrong:
prog.asm(17): bad jump field "JGR"
But as far I can see from the code you posted, what will happen here is that Python will raise a KeyError in jump, which is much less useful.
(Possibly there's error handling in the rest of the code that you didn't show us, but I can only comment on what I see here.)
3. It's even worse for the dest field: this doesn't seem to produce any errors at all, so if I write:
BAD=1;JEQ // Oops: typo for MAD
AD=1;JEQ
and I never get told about my mistake.
4. The logic for lookup of the comp field is needlessly complex. It would be simplest just to make a 7-bit lookup table (including the a bit). Trying to modify the instruction wastes time, and it makes it harder to produce good error messages (outputting the modified instruction will confuse the user, so you have to hang on to the unmodified instruction for use in error messages).
If you don't want to write out the whole 7-bit lookup table, then write code to generate it:
fullCompTable = compTable.copy()
for k, v in compTable.items():
if 'A' in k:
fullCompTable[k.replace('A', 'M')] = 1 << 6 | v
and run this once, when the assembler starts up.
5. The comment for dest says "No point in a lookup table" but that's not true. A lookup table will be faster than the computation in dest. And it only needs 15 entries. Again, if you don't want to write it out by hand, you could compute it:
from itertools import permutations
destTable = {
''.join('MDA'[i] for i in p): sum(1<<i for i in p)
for n in range(1, 4)
for p in permutations(range(3), n)
}
but this is longer than writing it out in full:
destTable = {
'A': 4, 'AD': 6, 'ADM': 7, 'AM': 5, 'AMD': 7, 'D': 2, 'DA': 6, 'DAM': 7,
'DM': 3, 'DMA': 7, 'M': 1, 'MA': 5, 'MAD': 7, 'MD': 3, 'MDA': 7
}
6. There's no need to take a copy here:
nStr = srcStr[:] # non-destructive copy
Just write:
nStr = srcStr
• Maybe generating the jump table (dictionary) on the fly is better, typing it all out a typo may easily slip in. On the other hand it is easier to write it by hand... which of the two do you suggest? May 29 '15 at 18:01
• @Caridorc: I don't see any way to compute jumpTable, so it has to be written out. May 29 '15 at 18:04
• sorry I meant the destTable May 29 '15 at 18:04
• @Caridorc: You could do both: that is, compute it and copy the result into the source code. (That's what I did.) May 29 '15 at 18:06
• Nice solution, so that after the computation you can inspect it and check if it is all right :) May 29 '15 at 18:09 |
# Percentages
Here is everything you need to know about percentages for GCSE maths (Edexcel, AQA and OCR). You’ll learn how to find the percentage of an amount and calculate with percentage multipliers.
You will also work out how to increase and decrease a number by a percentage, percentage change and reverse percentages.
Look out for the percentages worksheets and exam questions at the end.
## What are percentages?
A percentage is a number which is expressed as a fraction of 100
Percent means “number of parts per hundred” and the symbol we use for percent is the percent sign %.
E.g.
$43\%=\frac{43}{100}=0.43$
$1\%=\frac{1}{100}=0.01$
### What are percentages?
There are different types of percentage questions.
• Percentage of an amount
This is where we are asked to find a certain percentage of an amount.
E.g. Find 25% of £32
This is the same as finding a ¼ of £32.
$32\div4=8$
Step-by-step guide: Percentage of an amount
• Percentage conversions
Percentages, fractions and decimals can all be used to represent part of a whole.
For example,
It is useful to be able to convert between percentages, fractions and decimals.
To change a percentage into a fraction, write the percentage number as the numerator of a fraction and 100 as the denominator, and then simplify the resulting fraction if possible.
E.g.
42\%= \frac{42}{100} = \frac{21}{50}
To change a fraction into a percentage, there are two methods.
First see if it is possible to write an equivalent fraction with a denominator of 100; then the numerator will be the percentage.
Eg. \frac{7}{20} = \frac{7\times5}{20\times5}=\frac{35}{100}=35\%
A second method is to carry out the division represented in the fraction and then multiply by 100.
E.g.
\frac{1}{8} = 1 \div 8 =0.125
0.125 \times 100 = 12.5
\frac{1}{8} = 12.5 \%
To change a percentage into a decimal, divide the percentage number by 100.
Eg.
73\% = 73 \div 100 = 0.73
To change a decimal into a percentage, multiply the decimal by 100.
E.g. 0.29
0.29 \times 100 = 29
29\%
• Percentage multipliers
This is where we can find a decimal number and use it as a multiplier to make calculating percentages more efficient.
E.g. Find 41% of £800
We can use 0.41 as a multiplier to find the amount needed.
$41\%=\frac{41}{100}=0.41$
$800\times0.41=328$
Step-by-step guide: Percentage multipliers
• Percentage as operator
In order to calculate a percentage of an amount, a percentage increase or a percentage decrease we can use a percentage multiplier. To do this we change the percentage that we want into a decimal, and then multiply the amount by that decimal to calculate the answer.
For example,
34% of 58.
Here we want 34% which as a decimal is 0.34
Therefore the calculation is
58 \times 0.34 = 19.72
For example,
Increase 78 by 15%
Here we want an increase of 15% which means in total we want 115%, which as a decimal is 1.15
Therefore the calculation is
78 \times 1.15 = 89.7
For example,
Decrease 45 by 20%
Here we want a decrease of 20% which means in total we want 80%, which as a decimal is 0.8
Therefore the calculation is 45 \times 0.8 = 36
• Percentage increase
This is where we are asked to increase (make bigger) a value by a certain amount.
E.g. Increase 40g by 10%.
We can find 10% and add it on.
10% of 40 is 4
$40+4=44$
Step-by-step guide: Percentage increase
• Percentage decrease
This is where we are asked to decrease (make smaller) a value by a certain amount.
E.g. Decrease 60kg by 10%.
We can find 10% and subtract it.
10% of 60 is 6
$60-6=54$
Step-by-step guide: Percentage decrease
• Percentage change
When values change, we can express this change as a percentage of the original value.
E.g. Work out the percentage change of 26kg from 25kg.
The actual change from 25 to 26 is 1.
$\frac{1}{25}=\frac{4}{100}=4\%$
Step-by-step guide: Percentage change
• Reverse percentages
This is where we are given a certain percentage of a number and we have to find the original number.
E.g. 20% of a number is 6, what is the number?
$20\%=\frac{1}{5}$
To find the original number we need to find 100% or one whole.
$5\times6=30$
Step-by-step guide: Reverse percentages
• Percentage calculations
Let’s look at different methods that can be used to perform percentage calculations.
For example,
What is 40% of 70?
Method 1: The one percent method
Find 1% first by dividing the amount by 100 and then multiply the amount by the percent you want.
\frac{70}{100}\times 40 = 28
Method 2: The decimal multiplier method
Write the percent you want as a decimal and then multiply the amount by this decimal.
40\%=0.4
70 \times 0.4=28
Method 3: Using equivalent fractions
Write the percent you want as a fraction in simplest form and then multiply the amount by this fraction.
40\% = \frac{40}{100} =\frac{4}{10} =\frac{2}{5}
70 \times \frac{2}{5} = \frac{70\times 2}{5} = \frac{140}{5} = 28
Method 4: Building up an answer from simple percentages you know
Using simple percentages you can “
build up the answer to the question.
For this question if you know 10% of 70 then you can multiply this answer by 4 to find 40%.
10\% of 70 = 7
40\% of 70 = 7 \times 4 = 28
Note, methods 1 and 2 lend themselves best to questions where you are allowed to use a calculator. Methods 3 and 4 are most helpful when calculators are not permitted.
## Equation to percentage
We could be asked to solve problems where we need to form and solve equations to find answers as percentages. We often see this when dealing with interest and depreciation calculations.
For example,
A car purchased for £36000 depreciates by x% each year. If after 3 years, the car has a value of £15187.50, what percentage does the car depreciate by annually?
If we were solving this as a depreciation question with the percentage value known and the car value unknown we would use the calculation
36000 \times \text { multiplier }^{3}=15187.50
We need to work backwards to find the multiplier. Essentially solving the equation, for ease let’s call the multiplier, x.
36000 \times x^{3}=15187.50
x^{3}=\frac{15187.50}{36000}
x^{3}=0.421875
x=\sqrt[3]{0.421875}
x=0.75
The depreciation multiplier used for this calculation was 0.75, this tells us that the final car value of £15187.50 was 75% of its original value. 100% – 75% = 25%, so the value of the car depreciated by 25% each year.
## Percentages examples
### Example 1: percentage of an amount
Find 23% of £160.
1. Write down what you have and what you are trying to find.
100% is £160.
We need to work out 23% of £160.
2 Work out what you need.
$23\% = 20\%+3\%$
$10\% = 16, 1\% = 1.6$
$(2\times16)+(3\times1.6)=32+4.6=36.8$
3 Write down the final answer.
23% of £160 is £36.80
### Example 2: percentage multipliers
Find 39% of £4700
100% is £4700
We want to find 39% of £4700
We can write the percentage as a decimal number.
$39\%=0.39$
This gives the percentage in decimal form which is the percentage multiplier.
$4700\times0.39=1833$
39% of £4700 is £1833
### Example 3: percentage increase
Increase £200 by 30%
100% is £200
We want to find 130% of £200
$100\%+30\%=130\%$
10% is 20
$200\div10=20$
30% is 60
$3\times20=60$
So 130% is
$200 + 60 = 260$
OR
You could also use the decimal multiplier.
$100+30=130$
$130\%=\frac{130}{100}=1.30$
$200\times1.3=260$
£200 increased by 30% is £260
### Example 4: percentage decrease
Decrease 700g by 20%
100% is 700 g
We want to find 80% of 700 g
$100\%-20\%=80\%$
10% is 70
$700\div10=70$
20% is 140
$2\times70=140$
So 80% is
$700-140=560$
OR
You could use the decimal multiplier.
$100-20=80$
$80\%=\frac{80}{100}=0.80$
$700\times0.80=560$
700g decreased by 20% is 560 g.
### Example 5: percentage change
The price of a t-shirt increased from £10 to £12. Work out the percentage change.
The original amount is £10.
We need to find the change as a percentage of the original amount.
The actual change is
$12-10=2$
The actual change written as a fraction of the original is
$\frac{2}{10}$
The actual change is the numerator (top number).
The original number is the denominator (bottom number).
Convert the fraction to an equivalent fraction where the denominator is 100
$\frac{2}{10}=\frac{20}{100}=20\%$
OR
Write the actual change as a fraction of the original amount and multiply by 100.
$\frac{2}{10}\times100=20$
The percentage change of £12 from £10 is 20%
### Example 6: reverse percentages
The price of a coat is £40 after a 20% price cut. Find the original price.
$100\%-20\%=80\%$
£40 represents 80% of the original price.
So 80% = £40
We need the original price which is 100%.
80% is 40
We can work out 10%
$40\div8=4$
Then we can work out 100%
$5\times 10 = 50$
OR
We can make an equation using x as the original number. Use the decimal multiplier and the new number.
\begin{aligned} x\times0.80&=40\\\\ x&=40\div0.80\\\\ x&=50 \end{aligned}
The price after the cut is £40. The original price was £50
### Common misconceptions
• Money needs two digits for the pence
E.g. Find 34% of £620
$620\times0.34=210.8$
• The decimal multiplier to work out a percentage increase can be greater than 1
E.g. To increase 50km by 3% we can work 103% of 50
$50\times1.03 =51.5$
• Percentages can be greater than 100%
E.g. Calculate the percentage change from 200 to 450.
The actual change is 450-200=250
$\frac{250}{200}\times100=125$
So the percentage change from 200 to 450 is 125%
### How to work out percentage practice questions
1. Find 40\% of \pounds 300 .
\pounds 1200
\pounds 120
\pounds 7.50
\pounds 12
10\% of \pounds 300 is \pounds 30 . We can multiply this by 4 to get 40\% .
2. Calculate 12.4\% of \pounds 3000 .
\pounds 3372
\pounds 37200
\pounds 372
\pounds 3720
As a multiplier, 12.4\% is 0.124 . To get the answer we can calculate 0.124\times3000
3. Increase 600m by 30\% .
630m
780m
180m
420m
30\% of 600 is 180 . We can add this on to the original amount to find the quantity after the increase.
4. Decrease 80 kg by 5\% .
75kg
95kg
76kg
84kg
5\% of 80 is 4 . We can subtract this from the original amount to find the quantity after the decrease.
5. Calculate the percentage change from 400 kg to 600 kg .
50\%
200\%
100\%
150\%
The actual increase is given by
600 − 400 = 200
The percentage increase is then given by
\frac{200}{400}\times100=50\%
6. A book costs \pounds 4 after a price reduction of 20\% . What was the original price?
\pounds 4.20
\pounds 5.00
\pounds 20.00
\pounds 4.25
\pounds 4 represents 80\% of the original amount, which means 10\% of the original amount is \pounds 0.50 .
Multiplying by ten to get 100\% means the original price was \pounds 5 .
### Percentages GCSE exam questions
1. Work out 90\% of \pounds 70.
(2 marks)
10\% is 7 , so 90\% is 9 × 7
(1)
9\times7 = 63
(1)
2. Charlotte invests \pounds 3000 for 4 years.
She gets a simple interest rate of 2\% per year.
Work out the total interest Charlotte gets.
(3 marks)
3000\times0.02=
(1)
4\times60
(1)
= \pounds 240
(1)
3. Last year Ron paid \pounds 450 for his car insurance.
This year he has to pay \pounds 603 for his car insurance.
Work out the percentage increase in his car insurance.
(3 marks)
The change is
603-450=
(1)
\frac{153}{450}\times 100
(1)
= 34\%
(1)
## Learning checklist
You have now learned how to:
• How to work out the percentage of a number
• How to calculate percentage change
• How to work out a percentage increase or decrease
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# Understanding Your Customer Base: Churn
Churn is a means to understanding change in a business’s customer base due to loss of customers. Since keeping customers is generally cheaper than gaining new customers, then preventing the loss of customers is a profitable endeavor. The purpose of measuring churn, and similar metrics, is to evaluate actions taken by the business to retain customers or to identify weak points in the business.
Churn rate is a measure of churn in a given time period. Churn rate can be defined in multiple ways; here we’ll define churn rate as the ratio of customers lost in a given time period compared to the number of customers that could have been lost.
We can look at the extremes to understand churn rate better,
1. If the churn rate is 0, then we have not lost any customers in the time period
2. If the churn rate is 1, then we have lost all our customers in the time period
Below we can see the population at risk for each time period as green smiley faces, with red X’s over the population lost, i.e. the churned customers. On the lower axis, we display the calculated churn rate for each time period, $\frac{\text{customers lost}}{\text{customers at risk to be lost}}.$
#### Basic consideration for tracking churn rate:
1. Define the moment a person becomes a customer (or user)
2. Define when a customer has churned and is no longer a customer
• deciding when a customer has churned can be subtle in some environments, such as eCommerce
3. Determine the appropriate time period
• the smallest gradient of time to track flux of customers; if you choose weeks it’s possible to aggregate up from weeks to months or years, but not down from weeks to days.
###### A Simple Example
Consider a subscription based business, where the customer pays in advance of receiving the service. Online services, phone companies, gyms and many other companies work in this way.
1. A person becomes a customer on the date the service agreement is signed, and the first payment is received.
2. The lost of a customer is set at the next billing date after the customer cancels the subscription, or the immediately after no payment is received.
3. Since the billing cycle is monthly, a monthly time frame makes sense to record churn. A monthly period would imply there is a set billing date that is the same for all customers and likely a prorating of the first payment.
###### A More Subtle Example
A Swedish grocery store is an example of a commerce based business. It’s easy to understand grocery stores have repeat customers; it may be you buy all your groceries as the same store.
1. A person becomes a customer when they sign up for a loyalty card.
2. The customer is lost when there is zero spending on the loyalty card for a set amount of time. This amount of time can only be set by knowing and understanding your customer base, and your plan of action for preventing churn. A typical time period is 90 days.
3. The time period for recording churn in this case is dependent on the set amount of time for zero spending. If 90 days is the limit for zero spending observing churn monthly is reasonable, however it may be advisable to consider shorter time periods to understand the effect of churn prevention programs.
##### Summary
Churn is flux in a customer base due to loss. Churn rate is a measure of the amount of churn in a predetermined time period. Calculating and recording churn rate is a fundamental steps in measuring, understanding, and reducing churn. Churn rate in this format is lets us look back and evaluate past actions, which leads to the question how do we look forward?
###### And Beyond
Looking forward with churn rate, we can calculate the expected churn rate for time period. Once we have the expected value of churn we can determine when churn is too high, or churn is reducing. One simple way to approach this is to use at averages and t-test, but a more accurate way is to apply some basic Bayesian analysis.
Going beyond churn rate, we can start look at calculating, measuring, and leveraging insight from tenure, hazard, and survival analysis. In future posts, we’ll review the fundamentals and some basic insights that can be gained form tenure, hazard and survival analysis. These three will give us insights on how and when churn is happening in the customers life cycle. |
# NAG FL Interfaces18asf (bessel_i0_real_vector)
## ▸▿ Contents
Settings help
FL Name Style:
FL Specification Language:
## 1Purpose
s18asf returns an array of values of the modified Bessel function ${I}_{0}\left(x\right)$.
## 2Specification
Fortran Interface
Subroutine s18asf ( n, x, f,
Integer, Intent (In) :: n Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: ivalid(n) Real (Kind=nag_wp), Intent (In) :: x(n) Real (Kind=nag_wp), Intent (Out) :: f(n)
#include <nag.h>
void s18asf_ (const Integer *n, const double x[], double f[], Integer ivalid[], Integer *ifail)
The routine may be called by the names s18asf or nagf_specfun_bessel_i0_real_vector.
## 3Description
s18asf evaluates an approximation to the modified Bessel function of the first kind ${I}_{0}\left({x}_{i}\right)$ for an array of arguments ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$.
Note: ${I}_{0}\left(-x\right)={I}_{0}\left(x\right)$, so the approximation need only consider $x\ge 0$.
The routine is based on three Chebyshev expansions:
For $0,
$I0(x)=ex∑′r=0arTr(t), where t = 2 (x4) -1.$
For $4,
$I0(x)=ex∑′r=0brTr(t), where t=x-84.$
For $x>12$,
$I0(x)=exx ∑′r=0crTr(t), where t=2(12x) -1.$
For small $x$, ${I}_{0}\left(x\right)\simeq 1$. This approximation is used when $x$ is sufficiently small for the result to be correct to machine precision.
For large $x$, the routine must fail because of the danger of overflow in calculating ${e}^{x}$.
## 4References
NIST Digital Library of Mathematical Functions
## 5Arguments
1: $\mathbf{n}$Integer Input
On entry: $n$, the number of points.
Constraint: ${\mathbf{n}}\ge 0$.
2: $\mathbf{x}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Input
On entry: the argument ${x}_{\mathit{i}}$ of the function, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
3: $\mathbf{f}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Output
On exit: ${I}_{0}\left({x}_{i}\right)$, the function values.
4: $\mathbf{ivalid}\left({\mathbf{n}}\right)$Integer array Output
On exit: ${\mathbf{ivalid}}\left(\mathit{i}\right)$ contains the error code for ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
${\mathbf{ivalid}}\left(i\right)=0$
No error.
${\mathbf{ivalid}}\left(i\right)=1$
${x}_{i}$ is too large. ${\mathbf{f}}\left(\mathit{i}\right)$ contains the approximate value of ${I}_{0}\left({x}_{i}\right)$ at the nearest valid argument. The threshold value is the same as for ${\mathbf{ifail}}={\mathbf{1}}$ in s18aef , as defined in the Users' Note for your implementation.
5: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).
## 6Error Indicators and Warnings
If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, at least one value of x was invalid.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=-99$
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
## 7Accuracy
Let $\delta$ and $\epsilon$ be the relative errors in the argument and result respectively.
If $\delta$ is somewhat larger than the machine precision (i.e., if $\delta$ is due to data errors etc.), then $\epsilon$ and $\delta$ are approximately related by:
$ε≃ | x I1(x) I0 (x) |δ.$
Figure 1 shows the behaviour of the error amplification factor
$| xI1(x) I0(x) |.$
However, if $\delta$ is of the same order as machine precision, then rounding errors could make $\epsilon$ slightly larger than the above relation predicts.
For small $x$ the amplification factor is approximately $\frac{{x}^{2}}{2}$, which implies strong attenuation of the error, but in general $\epsilon$ can never be less than the machine precision.
For large $x$, $\epsilon \simeq x\delta$ and we have strong amplification of errors. However, for quite moderate values of $x$ ($x>\stackrel{^}{x}$, the threshold value), the routine must fail because ${I}_{0}\left(x\right)$ would overflow; hence in practice the loss of accuracy for $x$ close to $\stackrel{^}{x}$ is not excessive and the errors will be dominated by those of the standard function exp.
## 8Parallelism and Performance
s18asf is not threaded in any implementation.
None.
## 10Example
This example reads values of x from a file, evaluates the function at each value of ${x}_{i}$ and prints the results.
### 10.1Program Text
Program Text (s18asfe.f90)
### 10.2Program Data
Program Data (s18asfe.d)
### 10.3Program Results
Program Results (s18asfe.r) |
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## Site Tag Cloud
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• CommentRowNumber1.
• CommentAuthorTim_Porter
• CommentTimeApr 25th 2012
I have split off ordinal sum from the entry on joins as it is needed in other entries as well. I have not revised the entry just doing a cut and paste, so it needs more work!
1. I added a note here on Lawvere’s definition of the ordinal sum of categories, from “Ordinal sums and equational doctrines”.
2. Previous edit said there were isomorphisms from $[i] \oplus [j]$ to $[j] \oplus [i]$, this was false and has been corrected.
Anonymous
• CommentRowNumber4.
• CommentAuthorTim_Porter
• CommentTimeJan 3rd 2019
• (edited Jan 3rd 2019)
@ Anonymous #3 ???? There are no natural isomorphisms but for finite ordinals, $[i]\oplus [j]$ is the same ordinal as $[j]\oplus [i]$ as both are $[i+j+1]$, so is that what you ment?
• CommentRowNumber5.
• CommentAuthorMike Shulman
• CommentTimeJan 3rd 2019
Further clarified that addition of finite ordinals is symmetric, but not infinite ones. Actually this page needs some more serious work in clarifying the finite/infinite distinction; it reads kind of as if whoever wrote it thought that “ordinal” meant “finite ordinal”. (Not that I think anyone actually thought that, I’m just saying the wording is confusing.)
• CommentRowNumber6.
• CommentAuthorTodd_Trimble
• CommentTimeJan 3rd 2019
Wait a minute: anonymous must have been saying that there is no symmetry isomorphism (even for finite ordinals) for the ordinal sum monoidal product that is natural with respect to ordinal maps. That’s a correct statement! The current page suggests that $(\Delta_a, +)$ carries symmetric monoidal structure, but that’s wrong.
• CommentRowNumber7.
• CommentAuthorTim_Porter
• CommentTimeJan 3rd 2019
• (edited Jan 3rd 2019)
I have tried to correct the entry a bit. There is probably a lot more that should be done however as it still is largely centred on finite ordinals and the application of ordinal sum in that context.
• CommentRowNumber8.
• CommentAuthorMike Shulman
• CommentTimeJan 3rd 2019
Thanks Todd; that’s probably a better guess as to what Anonymous had in mind.
• CommentRowNumber9.
• CommentAuthorTodd_Trimble
• CommentTimeMay 2nd 2020
Added a redirect for ordinal sum of categories. |
# How do you simplify 3 (4 + 5s ) - 12 + ( - 3s )?
Apr 7, 2018
$12 s$
#### Explanation:
$3 \left(4 + 5 s\right) - 12 + \left(- 3 s\right)$
$= 12 + 15 s - 12 - 3 s$
$= \left(12 - 12\right) + \left(15 s - 3 s\right)$
$= 0 + 12 s$
$= 12 s$
Apr 7, 2018
The answer is $12 s$.
#### Explanation:
Firstly you have to open the bracket $3 \left(4 + 5 s\right)$ it will be $12 + 15 s$ and the second bracket $+ \left(- 3 s\right)$ will be $- 3 s$. So your equation will be $12 + 15 s - 12 - 3 s$ then collect like terms, it will be $12 - 12 + 15 s - 3 s$, then it will be $12 s$.
Hope it helps.😀😀 |
It is currently 19 Feb 2018, 02:22
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# What is the greatest value of x such that 8^x is a factor of 16! ?
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What is the greatest value of x such that 8^x is a factor of 16! ?
A. 2
B. 3
C. 5
D. 6
E. 8
[Reveal] Spoiler: OA
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Re: What is the greatest value of x such that 8^x is a factor of 16! ? [#permalink]
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First we'll find power of $$2$$ in $$16!$$
$$[\frac{16}{2}] + [\frac{16}{2^2}] + [\frac{16}{2^3}] + [\frac{16}{2^4}] = 8 + 4 + 2 + 1 = 15$$
and we have:
$$2^{15} = (2^3)^5 = 8^5$$
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Re: What is the greatest value of x such that 8^x is a factor of 16! ? [#permalink]
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14 Dec 2016, 07:02
Bunuel wrote:
What is the greatest value of x such that 8^x is a factor of 16! ?
A. 2
B. 3
C. 5
D. 6
E. 8
8 = 2^3
Highest power of 2 in 16! is 15
16/2 = 8
8/2 = 4
4/2 = 2
2/2 = 1
So, The highest power of 8 in 16! will be 15/3 = 5
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Re: What is the greatest value of x such that 8^x is a factor of 16! ? [#permalink]
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15 Dec 2016, 17:15
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Expert's post
Bunuel wrote:
What is the greatest value of x such that 8^x is a factor of 16! ?
A. 2
B. 3
C. 5
D. 6
E. 8
Since 8 = 2^3, we are actually trying to determine the greatest value of x such that 2^(3x) is a factor of 16!.
Let’s first determine the number of factors of 2 within 18!. To do that, we can use the following shortcut in which we divide 18 by 2, and then divide the quotient of 18/2 by 2 and continue this process until we can no longer get a nonzero integer as the quotient.
16/2 = 8
8/2 = 4
4/2 = 2
2/2 = 1
Since 1/2 does not produce a nonzero quotient, we can stop.
The final step is to add up our quotients; that sum represents the number of factors of 2 within 16!.
Thus, there are 8 + 4 + 2 + 1 = 15 factors of 2 within 16!
However, we are not asked for the number of factors of 2; instead we are asked for the number of factors of 8. We see that 15 factors of 2 will produce 5 factors of 8.
Note: To clarify the final answer, note that the 16 factors of 2 can be expressed as 2^16. We now must break this number 2^16 into as many factors of 8 as possible; thus, we will have
2^16 = 2^3 x 2^3 x 2^3 x 2^3 x 2^3 x 2^1
2^16 = 8 x 8 x 8 x 8 x 8 x 2
2^16 = 8^5 x 2
Note that we can get only 5 factors of 8 out of 2^16; there is a “leftover” 2 that cannot be used.
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What is the greatest value of x such that 8^x is a factor of 16! ? [#permalink]
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16 Dec 2016, 04:13
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Bunuel wrote:
What is the greatest value of x such that 8^x is a factor of 16! ?
A. 2
B. 3
C. 5
D. 6
E. 8
Since, we need to find the greatest value of x such that $$8^x$$ is a factor of $$16!$$, it is same as finding the highest power of 8 in $$16!$$
In any question where we need to find the highest power of a number which can divide a factorial (product of first n natural numbers) or in other words, is a factor of the given factorial, all we need to do is
1. Prime factorize the number whose highest power is to be found.
2. Find the highest power of each of the prime factors in the factorial
3. Calculate how many such numbers (whose highest power is to be found) can be created using the highest power of each of its prime factors.
Let's apply the above steps to solve this question and then we will look at a couple of questions where we can apply this learning.
We need to find the highest power of 8, so let's begin by doing
Step-1: Prime factorization of 8.
$$8 = 2 * 2 * 2 = 2^3$$
Step-2: Find the highest power of each of the prime factors in the factorial
Since 8 has only one prime factor 2, we need to find the highest power of 2 in 16!. Now how do we do so? There are 2 ways based on the same principle.
Method 1
Keep dividing the factorial successively by 2 and keep adding the quotient till you don't have anything left to divide. Remember, successive division means dividing the quotient obtained at each step by the same divisor by which we start the division.
So, let's do it quickly.
$$\frac {16}{2} = 8$$
$$\frac {8}{2} = 4$$
$$\frac {4}{2} = 2$$
$$\frac {2}{2} = 1$$
$$\frac {1}{2} = 0$$
As there is nothing left to divide, let's add the quotients to find the highest power of 2 in $$16!$$
Sum of quotients $$= 8+4+2+1+0 = 15$$
Method 2
Divide 16 by consecutive powers of 2, till you get 0 as a quotient and add all quotients. This method is based on the same principle as Method 1.
So, we have highest power of 2 in 16! = $$\frac {16}{2} + \frac {16}{2^2} +\frac {16}{2^3}+\frac {16}{2^4} = 8+4+2+1 = 15$$
So, we can conclude that the highest power of 2 in $$16!$$ is $$2^{15}$$
Step-3: Calculate how many such numbers can be created using the highest power of each of its prime factors
Let's try to figure how many 8's we can create using $$2^{15}$$.
Since, $$8 = 2^3$$, we can write $$2^{15}$$ as $$(2^{3})^5$$.
Or, in simple terms $$2^{15} = 8^5$$
Hence, the greatest value of x such that $$8^x$$ is a factor of 16! is 5. Hence, answer is choice C.
Let me post a couple of questions on similar lines where we can use the method discussed in this post to solve this type of questions very quickly.
Question 1: What is the greatest value of x such that $$15^x$$ completely divides 300! ?
A. 20
B. 54
C. 74
D. 148
E. 222
Question 2: What is the greatest value of "a" such that $$45^a$$ completely divides 300! ?
A. 8
B. 36
C. 74
D. 148
E. 222
Detailed solutions will be posted soon. Use the method highlighted in this post to solve the above questions. All the best.
To practise ten 700+ Level Number Properties Questions attempt the The E-GMAT Number Properties Knockout
Regards,
Piyush
e-GMAT
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Re: What is the greatest value of x such that 8^x is a factor of 16! ? [#permalink]
### Show Tags
16 Dec 2016, 09:11
Bunuel wrote:
What is the greatest value of x such that 8^x is a factor of 16! ?
A. 2
B. 3
C. 5
D. 6
E. 8
8^x can be simplified to 2^3x
largest power of 2 in 16!= 16/2 + 16/4 + 16/8 + 16/16= 8+4+2+1=15
8^x= 2^15
x= 15/3=5
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Re: What is the greatest value of x such that 8^x is a factor of 16! ? [#permalink]
### Show Tags
25 Dec 2016, 06:10
1
KUDOS
Expert's post
EgmatQuantExpert wrote:
Bunuel wrote:
What is the greatest value of x such that 8^x is a factor of 16! ?
A. 2
B. 3
C. 5
D. 6
E. 8
Since, we need to find the greatest value of x such that $$8^x$$ is a factor of $$16!$$, it is same as finding the highest power of 8 in $$16!$$
In any question where we need to find the highest power of a number which can divide a factorial (product of first n natural numbers) or in other words, is a factor of the given factorial, all we need to do is
1. Prime factorize the number whose highest power is to be found.
2. Find the highest power of each of the prime factors in the factorial
3. Calculate how many such numbers (whose highest power is to be found) can be created using the highest power of each of its prime factors.
Let's apply the above steps to solve this question and then we will look at a couple of questions where we can apply this learning.
We need to find the highest power of 8, so let's begin by doing
Step-1: Prime factorization of 8.
$$8 = 2 * 2 * 2 = 2^3$$
Step-2: Find the highest power of each of the prime factors in the factorial
Since 8 has only one prime factor 2, we need to find the highest power of 2 in 16!. Now how do we do so? There are 2 ways based on the same principle.
Method 1
Keep dividing the factorial successively by 2 and keep adding the quotient till you don't have anything left to divide. Remember, successive division means dividing the quotient obtained at each step by the same divisor by which we start the division.
So, let's do it quickly.
$$\frac {16}{2} = 8$$
$$\frac {8}{2} = 4$$
$$\frac {4}{2} = 2$$
$$\frac {2}{2} = 1$$
$$\frac {1}{2} = 0$$
As there is nothing left to divide, let's add the quotients to find the highest power of 2 in $$16!$$
Sum of quotients $$= 8+4+2+1+0 = 15$$
Method 2
Divide 16 by consecutive powers of 2, till you get 0 as a quotient and add all quotients. This method is based on the same principle as Method 1.
So, we have highest power of 2 in 16! = $$\frac {16}{2} + \frac {16}{2^2} +\frac {16}{2^3}+\frac {16}{2^4} = 8+4+2+1 = 15$$
So, we can conclude that the highest power of 2 in $$16!$$ is $$2^{15}$$
Step-3: Calculate how many such numbers can be created using the highest power of each of its prime factors
Let's try to figure how many 8's we can create using $$2^{15}$$.
Since, $$8 = 2^3$$, we can write $$2^{15}$$ as $$(2^{3})^5$$.
Or, in simple terms $$2^{15} = 8^5$$
Hence, the greatest value of x such that $$8^x$$ is a factor of 16! is 5. Hence, answer is choice C.
Let me post a couple of questions on similar lines where we can use the method discussed in this post to solve this type of questions very quickly.
Question 1: What is the greatest value of x such that $$15^x$$ completely divides 300! ?
A. 20
B. 54
C. 74
D. 148
E. 222
Question 2: What is the greatest value of "a" such that $$45^a$$ completely divides 300! ?
A. 8
B. 36
C. 74
D. 148
E. 222
Detailed solutions will be posted soon. Use the method highlighted in this post to solve the above questions. All the best.
To practise ten 700+ Level Number Properties Questions attempt the The E-GMAT Number Properties Knockout
Regards,
Piyush
e-GMAT
Alright so let's look at the detailed solution of the first question and the answer of the second question. Once you go through the above post and this solution, you should have a strong understanding of this approach. Let's use the simple 3-step approach to solve questions similar to the above question.
Step 1. Prime factorize the number whose highest power is to be found.
Step 2. Find the highest power of each of the prime factors in the factorial
Step 3. Calculate how many such numbers (whose highest power is to be found) can be created using the highest power of each of its prime factors.
Let's apply the above steps to solve this question and then we will look at a couple of questions where we can apply this learning.
We need to find the highest power of 15, so let's begin by doing
Step-1: Prime factorization of 15.
$$15 = 3*5 = 3^1*5^1$$
Step-2: Find the highest power of each of the prime factors in the factorial
Since 15 has two prime factors 3 and 5, we need to find the highest power of 3 and 5 separately in 300!. Now how do we do so? There are 2 ways based on the same principle.
Note: The question can be solved by finding the highest power of 5 alone. However, we don't advocate using this shortcut until you're 100% confident about these questions. The reason is there are certain complicated factors that you need to keep in mind to use this shortcut without making errors and the advantage gained is not big enough to take that risk. Hence, let's find the highest powers of both the prime factors 3 and 5.
Method 1
Keep dividing the factorial successively by 3 and keep adding the quotient till you don't have anything left to divide. Remember, successive division means dividing the quotient obtained at each step by the same divisor by which we start the division.
So, let's do it quickly.
$$\frac {300}{3} = 100$$
$$\frac {100}{3} = 33$$
$$\frac {33}{3} = 11$$
$$\frac {11}{3} = 3$$
$$\frac {3}{3} = 1$$
$$\frac {1}{3} = 0$$
As there is nothing left to divide, let's add the quotients to find the highest power of 3 in $$300!$$
Sum of quotients $$= 100+33+11+3+1+0 = 148$$
Similarly, let's find the highest power of 5 in 300! by using the same method.
$$\frac {300}{5} = 60$$
$$\frac {60}{5} = 12$$
$$\frac {12}{5} = 2$$
$$\frac {2}{5} = 0$$
As there is nothing left to divide, let's add the quotients to find the highest power of 5 in $$300!$$
Sum of quotients $$= 60+12+2+0 = 74$$
Method 2
Divide 300 by consecutive powers of 3, till you get 0 as a quotient and add all quotients. This method is based on the same principle as Method 1.
So, we have highest power of 3 in 300! = $$\frac {300}{3} + \frac {300}{3^2} +\frac {300}{3^3}+\frac {300}{3^4}+\frac {300}{3^5} = 100+33+11+3+1 = 148$$
So, we can conclude that the highest power of 3 in $$300!$$ is $$3^{148}$$
Following a similar process to find the highest power of 5 in 300!, we get that the highest power is $$5^{74}$$
Step-3: Calculate how many such numbers can be created using the highest power of each of its prime factors
Let's try to figure how many 15's we can create using $$3^{148}$$ and $$5^{74}$$.
Since, $$15 = 3^1*5^1$$, we can write $$300!$$ as $$300!=3^{148}*5^{74}*k = (3*5)^{74} * 3^{74}*k =15^{74}*3^{74}*k$$, where k is a positive integer
Or, in simple terms the highest power of 15 in 300 is $$15^{74}$$.
If use the same approach to solve Q2, you will understand that $$45 = 3^2*5^1$$.
The highest power of 3 in 300! is $$3^{148}$$ and that of 5 is $$5^{74}$$. Therefore, the highest power of 45 or $$3^2*5^1$$ in 300! is 74. Hence, the answer is choice C.
Regards,
Piyush
e-GMAT
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Re: What is the greatest value of x such that 8^x is a factor of 16! ? [#permalink]
### Show Tags
29 Mar 2017, 10:12
The key to this question is to first visualize what is going on
we have 16! = 16*15*14*....*3*2*1.
We need to find that max value of exponent x such that 8^x divides 16!.
let us see if we multiply 2 with 4 alone that will have one multiple of 8. We got first group of three 2s. We need to see how many groups can we construct of three 2s.
If we start summing the exponent of 2 in every even number we will get 15. In 2 we have just 1, in 4 we got 2, in 8 we got 3 and so on. Now since 15/3 = 5. Since we are looking for batch of three 2s. 5 should be the answer.
Re: What is the greatest value of x such that 8^x is a factor of 16! ? [#permalink] 29 Mar 2017, 10:12
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## Jefferies, Brian R. F.
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Author ID: jefferies.brian-r-f Published as: Jefferies, Brian; Jefferies, B.; Jefferies, B. R. F.; Jefferies, Brian R. F. Further Spellings: Jefferies, Brian Raymond Frederick External Links: MGP · ResearchGate · IdRef
Documents Indexed: 90 Publications since 1984, including 3 Books 4 Contributions as Editor Reviewing Activity: 1 Review Co-Authors: 22 Co-Authors with 38 Joint Publications 462 Co-Co-Authors
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### Co-Authors
55 single-authored 8 Johnson, Gerald William 8 Ricker, Werner Joseph 7 Okada, Susumu 5 McIntosh, Alan 2 Brzeźniak, Zdzisław 2 Doust, Ian 2 Nielsen, Lance 2 Piskarëv, S. I. 2 Straub, Bernd-Michael 1 Bass, Ludvik 1 Bracken, Anthony J. 1 Fremlin, David H. 1 García-Raffi, Luis M. 1 Gaudry, Garth Ian 1 Hillman, Jonathan Arthur 1 Holmåker, Kjell 1 Ichinose, Takashi 1 Kim, Byoung Soo 1 Li, Chun 1 Picton-Warlow, James 1 Rodríguez-Piazza, Luis 1 Rothnie, Paul
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### Serials
5 Bulletin of the Australian Mathematical Society 5 Journal of Functional Analysis 4 Integral Equations and Operator Theory 3 Studia Mathematica 3 Journal of the Korean Mathematical Society 3 Proceedings of the American Mathematical Society 3 Publications of the Research Institute for Mathematical Sciences, Kyoto University 3 Proceedings of the Centre for Mathematical Analysis, Australian National University 2 Journal of Mathematical Analysis and Applications 2 Journal of Mathematical Physics 2 Linear and Multilinear Algebra 2 Mathematical Proceedings of the Cambridge Philosophical Society 2 Rocky Mountain Journal of Mathematics 2 Illinois Journal of Mathematics 2 Nagoya Mathematical Journal 2 Acta Applicandae Mathematicae 2 Journal of the Australian Mathematical Society. Series A 2 Advances in Applied Clifford Algebras 2 Mathematical Physics, Analysis and Geometry 2 Journal of the Australian Mathematical Society 1 Journal d’Analyse Mathématique 1 Journal of the Australian Mathematical Society, Series B 1 Mathematical Notes 1 Revue Roumaine de Mathématiques Pures et Appliquées 1 Archiv der Mathematik 1 Commentationes Mathematicae Universitatis Carolinae 1 Dissertationes Mathematicae 1 Hokkaido Mathematical Journal 1 Indiana University Mathematics Journal 1 Quaestiones Mathematicae 1 Journal of Physics A: Mathematical and General 1 Linear Algebra and its Applications 1 Indagationes Mathematicae. New Series 1 Russian Journal of Mathematical Physics 1 Infinite Dimensional Analysis, Quantum Probability and Related Topics 1 Journal of the Indonesian Mathematical Society 1 Lecture Notes in Mathematics 1 Mathematics and its Applications (Dordrecht) 1 Integration: Mathematical Theory and Applications 1 Complex Analysis and Operator Theory 1 Mathematics 1 Journal of Operators 1 Proceedings of the Centre for Mathematics and its Applications, Australian National University
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### Fields
65 Operator theory (47-XX) 47 Functional analysis (46-XX) 35 Measure and integration (28-XX) 21 Quantum theory (81-XX) 18 Probability theory and stochastic processes (60-XX) 13 Functions of a complex variable (30-XX) 10 Partial differential equations (35-XX) 6 Linear and multilinear algebra; matrix theory (15-XX) 5 Integral equations (45-XX) 4 General and overarching topics; collections (00-XX) 4 Harmonic analysis on Euclidean spaces (42-XX) 3 Global analysis, analysis on manifolds (58-XX) 2 Ordinary differential equations (34-XX) 2 Abstract harmonic analysis (43-XX) 1 Biology and other natural sciences (92-XX)
### Citations contained in zbMATH Open
51 Publications have been cited 270 times in 132 Documents Cited by Year
Spectral properties of noncommuting operators. Zbl 1056.47002
Jefferies, Brian
2004
The Weyl calculus and Clifford analysis. Zbl 0915.47015
Jefferies, Brian; McIntosh, Alan
1998
Feynman’s operational calculus for families of noncommuting operators: tensor products, ordered supports, and the disentangling of an exponential factor. Zbl 1030.47008
Jefferies, B.; Johnson, G. W.
2001
Feynman’s operational calculi for noncommuting operators: definitions and elementary properties. Zbl 1186.81082
Jefferies, B.; Johnson, G. W.
2001
Feynman’s operational calculi for time dependent noncommuting operators. Zbl 0984.47013
Jefferies, Brian; Johnson, G. W.; Nielsen, Lance
2001
The monogenic functional calculus. Zbl 0971.47013
Jefferies, Brian; McIntosh, Alan; Picton-Warlow, James
1999
Feynman’s operational calculi for noncommuting operators: spectral theory. Zbl 1096.47502
Jefferies, B.; Johnson, G. W.
2002
Feynman’s operational calculi for noncommuting operators: The monogenic calculus. Zbl 1096.47501
Jefferies, B.; Johnson, G. W.
2001
Weakly integrable semigroups on locally convex spaces. Zbl 0589.47043
Jefferies, Brian
1986
Bilinear integration in tensor products. Zbl 0936.46035
1998
Integration with respect to closable set functions. Zbl 0589.46034
Jefferies, Brian
1986
Evolution processes and the Feynman-Kac formula. Zbl 0844.60027
Jefferies, Brian
1996
The Weyl calculus for hermitian matrices. Zbl 0848.47014
Jefferies, Brian
1996
Jefferies, Brian
1985
Feynman’s operational calculi: Methods for iterative disentangling. Zbl 1156.81403
Jefferies, B.; Johnson, G. W.; Kim, B. S.
2006
The generation of weakly integrable semigroups. Zbl 0621.47037
Jefferies, Brian
1987
Vector-valued multipliers: Convolution with operator-valued measures. Zbl 0966.46023
Gaudry, G. I.; Jefferies, B. R. F.; Ricker, W. J.
2000
Commutativity for systems of $$(2\times 2)$$ selfadjoint matrices. Zbl 0796.15015
Jefferies, B. R. F.; Ricker, W. J.
1993
Integration with respect to vector valued Radon polymeasures. Zbl 0796.28009
Jefferies, Brian; Ricker, Werner J.
1994
An application of bilinear integration to quantum scattering. Zbl 1308.81167
García-Raffi, L. M.; Jefferies, B.
2014
Some recent applications of bilinear integration. Zbl 1248.28015
Jefferies, Brian
2010
Integro-differential equations for the self-organisation of liver zones by competitive exclusion of cell-types. Zbl 0628.92015
Bass, L.; Bracken, A. J.; Holmåker, K.; Jefferies, B. R. F.
1987
Differential properties of the numerical range map of pairs of matrices. Zbl 0885.15017
Hillman, J. A.; Jefferies, B. R. F.; Ricker, W. J.; Straub, B.
1997
An indecomposable Daniell integral. Zbl 0634.28006
Jefferies, B. R. F.; Fremlin, D. H.
1987
Exponential bounds for noncommuting systems of matrices. Zbl 0985.47014
Jefferies, Brian
2001
Feynman’s operational calculi: spectral theory for noncommuting self-adjoint operators. Zbl 1180.47013
Jefferies, Brian; Johnson, Gerald W.; Nielsen, Lance
2007
On the additivity of unbounded set functions. Zbl 0738.28007
Jefferies, Brian
1992
Lacunas in the support of the Weyl calculus for two Hermitian matrices. Zbl 1059.47016
Jefferies, Brian; Straub, Bernd
2003
The CLR inequality for dominated semigroups. Zbl 1359.47040
Jefferies, Brian
2014
Semigroups and diffusion processes. Zbl 0626.60072
Jefferies, Brian
1987
A process associated with the radially symmetric Dirac equation. Zbl 0804.47042
Jefferies, Brian
1994
An operator bound related to Feynman-Kac formulae. Zbl 0817.47012
Jefferies, Brian
1994
Characterization of one-dimensional point interactions for the Schrödinger operator by means of boundary conditions. Zbl 0998.81024
Brzeźniak, Zdzisław; Jefferies, Brian
2001
Bilinear integration with positive vector measures. Zbl 1048.28008
Jefferies, Brian; Rothnie, Paul
2003
The propagator of the radial Dirac equation. Zbl 1060.81020
Ichinose, Takashi; Jefferies, Brian
2002
Processes associated with evolution equations. Zbl 0715.47028
Jefferies, Brian
1990
An operator bound related to regular operators. Zbl 0871.47031
1996
The Weyl calculus and a singular integral in $$L^ 1(\mathbb{R})$$. Zbl 0820.47054
Jefferies, Brian R. F.; Ricker, Werner J.
1993
Semicompact functionals. Zbl 0626.28005
Jefferies, Brian
1987
Collectively measurable sets and abstract Wiener spaces. Zbl 0506.46029
Jefferies, Brian
1984
Pettis integrals with singular kernels. Zbl 0609.46022
Jefferies, Brian
1986
Pettis integral operators and resolvents. Zbl 0609.46023
Jefferies, Brian
1986
Remarks on the Feynman representation. Zbl 0614.47037
Jefferies, Brian
1985
Conditional expectation for operator-valued measures and functions. Zbl 0568.46034
Jefferies, Brian
1984
Pettis integrals and singular integral operators. Zbl 0931.47031
1994
Traceability of positive integral operators. Zbl 1341.47024
Jefferies, Brian
2016
The Feynman representation for the Dirac propagator with a radially symmetric potential. Zbl 0858.47024
Jefferies, Brian
1996
Singular bilinear integrals in quantum physics. Zbl 1330.81093
Jefferies, Brian
2015
The monogenic functional calculus. Zbl 1334.47019
Jefferies, Brian
2015
Dominated semigroups of operators and evolution processes. Zbl 1075.47026
2004
Advances and applications of the Feynman integral. Zbl 1095.28014
Jefferies, Brian
2004
Traceability of positive integral operators. Zbl 1341.47024
Jefferies, Brian
2016
Singular bilinear integrals in quantum physics. Zbl 1330.81093
Jefferies, Brian
2015
The monogenic functional calculus. Zbl 1334.47019
Jefferies, Brian
2015
An application of bilinear integration to quantum scattering. Zbl 1308.81167
García-Raffi, L. M.; Jefferies, B.
2014
The CLR inequality for dominated semigroups. Zbl 1359.47040
Jefferies, Brian
2014
Some recent applications of bilinear integration. Zbl 1248.28015
Jefferies, Brian
2010
Feynman’s operational calculi: spectral theory for noncommuting self-adjoint operators. Zbl 1180.47013
Jefferies, Brian; Johnson, Gerald W.; Nielsen, Lance
2007
Feynman’s operational calculi: Methods for iterative disentangling. Zbl 1156.81403
Jefferies, B.; Johnson, G. W.; Kim, B. S.
2006
Spectral properties of noncommuting operators. Zbl 1056.47002
Jefferies, Brian
2004
Dominated semigroups of operators and evolution processes. Zbl 1075.47026
2004
Advances and applications of the Feynman integral. Zbl 1095.28014
Jefferies, Brian
2004
Lacunas in the support of the Weyl calculus for two Hermitian matrices. Zbl 1059.47016
Jefferies, Brian; Straub, Bernd
2003
Bilinear integration with positive vector measures. Zbl 1048.28008
Jefferies, Brian; Rothnie, Paul
2003
Feynman’s operational calculi for noncommuting operators: spectral theory. Zbl 1096.47502
Jefferies, B.; Johnson, G. W.
2002
The propagator of the radial Dirac equation. Zbl 1060.81020
Ichinose, Takashi; Jefferies, Brian
2002
Feynman’s operational calculus for families of noncommuting operators: tensor products, ordered supports, and the disentangling of an exponential factor. Zbl 1030.47008
Jefferies, B.; Johnson, G. W.
2001
Feynman’s operational calculi for noncommuting operators: definitions and elementary properties. Zbl 1186.81082
Jefferies, B.; Johnson, G. W.
2001
Feynman’s operational calculi for time dependent noncommuting operators. Zbl 0984.47013
Jefferies, Brian; Johnson, G. W.; Nielsen, Lance
2001
Feynman’s operational calculi for noncommuting operators: The monogenic calculus. Zbl 1096.47501
Jefferies, B.; Johnson, G. W.
2001
Exponential bounds for noncommuting systems of matrices. Zbl 0985.47014
Jefferies, Brian
2001
Characterization of one-dimensional point interactions for the Schrödinger operator by means of boundary conditions. Zbl 0998.81024
Brzeźniak, Zdzisław; Jefferies, Brian
2001
Vector-valued multipliers: Convolution with operator-valued measures. Zbl 0966.46023
Gaudry, G. I.; Jefferies, B. R. F.; Ricker, W. J.
2000
The monogenic functional calculus. Zbl 0971.47013
Jefferies, Brian; McIntosh, Alan; Picton-Warlow, James
1999
The Weyl calculus and Clifford analysis. Zbl 0915.47015
Jefferies, Brian; McIntosh, Alan
1998
Bilinear integration in tensor products. Zbl 0936.46035
1998
Differential properties of the numerical range map of pairs of matrices. Zbl 0885.15017
Hillman, J. A.; Jefferies, B. R. F.; Ricker, W. J.; Straub, B.
1997
Evolution processes and the Feynman-Kac formula. Zbl 0844.60027
Jefferies, Brian
1996
The Weyl calculus for hermitian matrices. Zbl 0848.47014
Jefferies, Brian
1996
An operator bound related to regular operators. Zbl 0871.47031
1996
The Feynman representation for the Dirac propagator with a radially symmetric potential. Zbl 0858.47024
Jefferies, Brian
1996
Integration with respect to vector valued Radon polymeasures. Zbl 0796.28009
Jefferies, Brian; Ricker, Werner J.
1994
A process associated with the radially symmetric Dirac equation. Zbl 0804.47042
Jefferies, Brian
1994
An operator bound related to Feynman-Kac formulae. Zbl 0817.47012
Jefferies, Brian
1994
Pettis integrals and singular integral operators. Zbl 0931.47031
1994
Commutativity for systems of $$(2\times 2)$$ selfadjoint matrices. Zbl 0796.15015
Jefferies, B. R. F.; Ricker, W. J.
1993
The Weyl calculus and a singular integral in $$L^ 1(\mathbb{R})$$. Zbl 0820.47054
Jefferies, Brian R. F.; Ricker, Werner J.
1993
On the additivity of unbounded set functions. Zbl 0738.28007
Jefferies, Brian
1992
Processes associated with evolution equations. Zbl 0715.47028
Jefferies, Brian
1990
The generation of weakly integrable semigroups. Zbl 0621.47037
Jefferies, Brian
1987
Integro-differential equations for the self-organisation of liver zones by competitive exclusion of cell-types. Zbl 0628.92015
Bass, L.; Bracken, A. J.; Holmåker, K.; Jefferies, B. R. F.
1987
An indecomposable Daniell integral. Zbl 0634.28006
Jefferies, B. R. F.; Fremlin, D. H.
1987
Semigroups and diffusion processes. Zbl 0626.60072
Jefferies, Brian
1987
Semicompact functionals. Zbl 0626.28005
Jefferies, Brian
1987
Weakly integrable semigroups on locally convex spaces. Zbl 0589.47043
Jefferies, Brian
1986
Integration with respect to closable set functions. Zbl 0589.46034
Jefferies, Brian
1986
Pettis integrals with singular kernels. Zbl 0609.46022
Jefferies, Brian
1986
Pettis integral operators and resolvents. Zbl 0609.46023
Jefferies, Brian
1986
Jefferies, Brian
1985
Remarks on the Feynman representation. Zbl 0614.47037
Jefferies, Brian
1985
Collectively measurable sets and abstract Wiener spaces. Zbl 0506.46029
Jefferies, Brian
1984
Conditional expectation for operator-valued measures and functions. Zbl 0568.46034
Jefferies, Brian
1984
all top 5
### Cited by 120 Authors
38 Jefferies, Brian R. F. 21 Colombo, Fabrizio 17 Sabadini, Irene 11 Nielsen, Lance 6 Ricker, Werner Joseph 5 Johnson, Gerald William 4 Kim, Byoung Soo 4 Struppa, Daniele Carlo 4 Yang, Rongwei 3 Alpay, Daniel Aron 3 Kimsey, David Patrick 3 Pinton, Stefano 3 Qian, Tao 2 Blasco, Oscar 2 Boulabiar, Karim Mohamed 2 Calabuig, José Manuel 2 Dobrakov, Ivan 2 Gantner, Jonathan 2 Ichinose, Takashi 2 Kumar, Nageswaran Shravan 2 Kumar, Vishvesh 2 Okada, Susumu 2 Sarma, Ritumoni 2 Sorrentino, Alfonso 2 Straub, Bernd-Michael 2 Villanueva, Ignacio 1 Aharonov, Yakir 1 Ahn, Byung Moo 1 Albeverio, Sergio A. 1 Almeida Carvalho, Suélen 1 Arfaoui, Sabrine 1 Ball, Richard N. 1 Basu, Santwana 1 Baur, Franziska 1 Behrndt, Jussi 1 Ben Mabrouk, Anouar 1 Bjon, Sten 1 Bogachev, Vladimir Igorevich 1 Brzeźniak, Zdzisław 1 Cade, Patrick 1 Cangiotti, N. 1 Cerejeiras, Paula 1 Cerrai, Sandra 1 Chakraborty, N. D. 1 Chang, Kun Soo 1 Chaurasia, Praveen Kumar 1 Chetouani, Lyazid 1 Clément, Philippe J. E. 1 Coine, Clement 1 De Martino, Antonino 1 Dieckmann, O. 1 Dong, Baohua 1 Dooley, Anthony Haynes 1 Douglas, Ronald George 1 Dubin, Daniel A. 1 Ebaid, Abdelhalim 1 Eydenberg, Michael S. 1 Federico, Salvatore 1 Ferreira, José Claudinei 1 Fink, Arlington Michael 1 Fremlin, David H. 1 García-Raffi, Luis M. 1 Gentili, Graziano 1 Gesztesy, Fritz 1 Gill, Tepper L. 1 Goldys, Beniamin 1 Gong, Yafang 1 Gutkin, Eugene 1 Gyllenberg, Mats 1 Hafsi, Hamza 1 Hajji, Rawaa 1 Haouat, Salah 1 Heijmans, Henk J. A. M. 1 Hillman, Jonathan Arthur 1 Jiménez Fernández, Eduardo 1 Jonckheere, Edmond A. 1 Juan, M. A. 1 Kähler, Uwe 1 Kaloshin, Vadim Yu. 1 Karow, Michael 1 Kiukas, Jukka 1 Klempnauer, Stefan 1 Kouche, Mahiéddine 1 Kreĭn, Selim Grigor’evich 1 Kunze, Markus Christian 1 Lakos, Gyula 1 Lapidus, Michel L. 1 Lávička, Roman 1 Le Merdy, Christian 1 Luna-Elizarrarás, María Elena 1 Mazzucchi, Sonia 1 Pang, Michael M. H. 1 Park, Yeonhee 1 Pavone, Christopher M. 1 Pellonpää, Juha-Pekka 1 Piskarëv, S. I. 1 Portal, Pierre 1 Raffoul, Raed W. 1 Rodríguez Ruiz, José 1 Rosestolato, Mauro ...and 20 more Authors
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### Cited in 65 Serials
8 Journal of Functional Analysis 6 Czechoslovak Mathematical Journal 6 Acta Applicandae Mathematicae 6 Advances in Applied Clifford Algebras 5 Proceedings of the American Mathematical Society 4 Bulletin of the Australian Mathematical Society 4 Journal of Mathematical Analysis and Applications 4 Journal of Mathematical Physics 4 Mathematical Physics, Analysis and Geometry 3 Rocky Mountain Journal of Mathematics 3 Integral Equations and Operator Theory 3 Publications of the Research Institute for Mathematical Sciences, Kyoto University 3 Semigroup Forum 3 The Journal of Geometric Analysis 3 Linear Algebra and its Applications 2 Mathematical Methods in the Applied Sciences 2 Journal of Geometry and Physics 2 Annali di Matematica Pura ed Applicata. Serie Quarta 2 Archiv der Mathematik 2 Journal of Differential Equations 2 Nagoya Mathematical Journal 2 Expositiones Mathematicae 2 Indagationes Mathematicae. New Series 2 Russian Journal of Mathematical Physics 2 Positivity 2 Journal of the Australian Mathematical Society 2 Complex Analysis and Operator Theory 1 Journal d’Analyse Mathématique 1 Mathematical Proceedings of the Cambridge Philosophical Society 1 Nonlinearity 1 Studia Mathematica 1 Acta Mathematica Vietnamica 1 Advances in Mathematics 1 Annales de l’Institut Fourier 1 Functiones et Approximatio. Commentarii Mathematici 1 Illinois Journal of Mathematics 1 International Journal of Mathematics and Mathematical Sciences 1 Journal of Soviet Mathematics 1 Mathematische Nachrichten 1 Transactions of the American Mathematical Society 1 Topology and its Applications 1 Zeitschrift für Analysis und ihre Anwendungen 1 Proceedings of the Indian Academy of Sciences. Mathematical Sciences 1 Journal of Mathematical Sciences (New York) 1 Annales Mathématiques Blaise Pascal 1 The New York Journal of Mathematics 1 Discrete and Continuous Dynamical Systems 1 Honam Mathematical Journal 1 Infinite Dimensional Analysis, Quantum Probability and Related Topics 1 Annals of Mathematics. Second Series 1 Acta Mathematica Sinica. English Series 1 Quantitative Finance 1 Comptes Rendus. Mathématique. Académie des Sciences, Paris 1 Computational Methods and Function Theory 1 Computational & Mathematical Methods in Medicine 1 Journal of Topology and Analysis 1 Science China. Mathematics 1 Kyoto Journal of Mathematics 1 Annals of Functional Analysis 1 Analysis and Mathematical Physics 1 Mathematics 1 Concrete Operators 1 Nonlinear Analysis. Theory, Methods & Applications 1 Journal of Operators 1 Proceedings of the American Mathematical Society. Series B
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### Cited in 26 Fields
86 Operator theory (47-XX) 54 Functional analysis (46-XX) 25 Functions of a complex variable (30-XX) 25 Quantum theory (81-XX) 24 Measure and integration (28-XX) 16 Probability theory and stochastic processes (60-XX) 9 Partial differential equations (35-XX) 6 Abstract harmonic analysis (43-XX) 6 Integral transforms, operational calculus (44-XX) 5 Harmonic analysis on Euclidean spaces (42-XX) 4 Linear and multilinear algebra; matrix theory (15-XX) 4 Integral equations (45-XX) 4 Global analysis, analysis on manifolds (58-XX) 3 Several complex variables and analytic spaces (32-XX) 3 Dynamical systems and ergodic theory (37-XX) 2 Differential geometry (53-XX) 2 Biology and other natural sciences (92-XX) 1 Order, lattices, ordered algebraic structures (06-XX) 1 Associative rings and algebras (16-XX) 1 Topological groups, Lie groups (22-XX) 1 Special functions (33-XX) 1 Ordinary differential equations (34-XX) 1 General topology (54-XX) 1 Mechanics of particles and systems (70-XX) 1 Optics, electromagnetic theory (78-XX) 1 Game theory, economics, finance, and other social and behavioral sciences (91-XX) |
## anonymous 4 years ago 2nd question: Find lim (sin2xcot4x) X->0.
1. anonymous
are you allowed to use l'hopital's rule?
2. anonymous
assuming this is $\lim_{x\rightarrow 0}\sin(2x)\cot(4x)$ $\lim_{x\rightarrow 0}\frac{\sin(2x)\cos(4x)}{\sin(4x)}$
1/2
4. anonymous
in any case the answer is $\frac{1}{2}$
5. anonymous
but if you need to show your work your answer will depend on what you are allowed to use. l'hopital's rule is the simplest, otherwise it will be some work
6. anonymous
i dont know the hospital rule that you have said :(
7. anonymous
you mean you do not know it or you are not allowed to use it? i assume this is calc class, so hve you covered deriviatives yet or are you just starting out?
|dw:1328531016861:dw|
without using l'hopital rule
10. anonymous
any computation as a long the answer will be the same in a long method
11. anonymous
i dont know it sorry :(
12. anonymous
there is a nice neat answer above. i think you can also use $\frac{sin(2x)\cos(4x)}{\sin(4x)}=\frac{\sin(2x)(\cos^2(2x)-\sin^2(2x))}{2\sin(2x)\cos(2x)}$
13. anonymous
here i used the double angle formula for sine and cosine
14. anonymous
can i post the 3rd question too?
15. anonymous
then cancel the sin(2x) top and bottom, get $\frac{\cos^2(2x)-\sin^2(2x)}{2\cos(2x)}$ replace x by zero, get $\frac{1}{2}$
16. anonymous
no limit to the amount you can post
17. anonymous
yes i will write your solution :)
18. anonymous
if f(x)=tanx-x and g(x)=x^3, evaluate the limit of f(x) over g(x) as x approaches 0. -3rd question.
19. anonymous
@nenadmatematica if you can do this without l'hopital or power series i will be impressed
haha I just wanted to ask you the same thing :D ....
21. anonymous
lol well, i guess we cannot give an elementary reason for this. the answer is $\frac{1}{3}$ but i cannot think of a gimmick to simplify this expression. are you sure you have not covered l'hopital? because i am stumped. in particular you have a trig fuction combined in combination with x and x^3 so there is no simple trig identity that will change the form of this for you
22. anonymous
@nenadmatematika tahnks for helping us too :)
well you're welcome....I agree with satellite that this example is very convenient for using L'Hopital rule.....I can't think of any other way now :D
24. anonymous
i really dont know but if both of you wants to use L'hopital rule. then i will agree to both of you
25. anonymous
4th question: Evaluate the lim x^2-16 over x+4 as x->4. |
Home > Relative Error > Relative Error In Accuracy Formula
# Relative Error In Accuracy Formula
## Contents
Chemistry Homework Help Worked Chemistry Problems Absolute Error and Relative Error Calculation Examples of Error Calculations Absolute and experimental error are two types of error in measurements. While this measurement is much more precise than the original estimate, how do you know that it is accurate, and how confident are you that this measurement represents the true value In order to calculate relative error, you must calculate the absolute error as well. In fact, it is reasonable to use the standard deviation as the uncertainty associated with this single new measurement. navigate here
Incidental energy/material loss, such as the little fluid left in the beaker after pouring, changes in temperature due to the environment, etc. Absolute, Relative and Percentage Error The Absolute Error is the difference between the actual and measured value But ... When you compute this area, the calculator might report a value of 254.4690049 m2. The smooth curve superimposed on the histogram is the gaussian or normal distribution predicted by theory for measurements involving random errors.
## Relative Error Formula
In plain English: The absolute error is the difference between the measured value and the actual value. (The absolute error will have the same unit label as the measured quantity.) Relative Absolute, Relative and Percentage Error The Absolute Error is the difference between the actual and measured value But ... Absolute error and relative error are two types of experimental error. Well, we just want the size (the absolute value) of the difference.
when measuring we don't know the actual value! The error in measurement is a mathematical way to show the uncertainty in the measurement. Generalizations These definitions can be extended to the case when v {\displaystyle v} and v approx {\displaystyle v_{\text{approx}}} are n-dimensional vectors, by replacing the absolute value with an n-norm.[1] Examples As Absolute Error Formula Chemistry For example, in 20 of the measurements, the value was in the range 9.5 to 10.5, and most of the readings were close to the mean value of 10.5.
Then each deviation is given by δxi = xi − x, for i = 1, 2, , N. Given some value v and its approximation vapprox, the absolute error is ϵ = | v − v approx | , {\displaystyle \epsilon =|v-v_{\text{approx}}|\ ,} where the vertical bars denote This packet is an overview of the terms Accuracy and Precision, and the difference between them. The percent error is the relative error expressed in terms of per 100.
Zellmer Chem 102 February 9, 1999 Over 10,635,000 live tutoring sessions served! Absolute Error Definition This value is clearly below the range of values found on the first balance, and under normal circumstances, you might not care, but you want to be fair to your friend. Lag time and hysteresis (systematic) — Some measuring devices require time to reach equilibrium, and taking a measurement before the instrument is stable will result in a measurement that is too The cost increases exponentially with the amount of precision required, so the potential benefit of this precision must be weighed against the extra cost.
## Relative Error Definition
How to Calculate the Relative Error? read this article Co-authors: 14 Updated: Views:245,735 75% of people told us that this article helped them. Relative Error Formula In the previous example, we find the standard error is 0.05 cm, where we have divided the standard deviation of 0.12 by 5. Relative Error Chemistry Precision indicates the quality of the measurement, without any guarantee that the measurement is "correct." Accuracy, on the other hand, assumes that there is an ideal value, and tells how far
Since the measurement was made to the nearest tenth, the greatest possible error will be half of one tenth, or 0.05. 2. check over here Van Loan (1996). In both of these cases, the uncertainty is greater than the smallest divisions marked on the measuring tool (likely 1 mm and 0.05 mm respectively). We can write out the formula for the standard deviation as follows. Absolute And Relative Error In Numerical Methods
If the uncertainty ranges do not overlap, then the measurements are said to be discrepant (they do not agree). Anne Marie Helmenstine, Ph.D., About.com,http://chemistry.about.com/od/chemistryquickreview/a/experror.htm. It would be unethical to arbitrarily inflate the uncertainty range just to make a measurement agree with an expected value. http://wapgw.org/relative-error/relative-error-formula-physics.php Tolerance intervals: Error in measurement may be represented by a tolerance interval (margin of error).
No ... Type Of Error In Measurement Consider, as another example, the measurement of the width of a piece of paper using a meter stick. Hysteresis is most commonly associated with materials that become magnetized when a changing magnetic field is applied.
## You measure the book and find it to be 75 mm.
To continue the example of measuring between two trees: Your Absolute Error was 2 feet, and the Actual Value was 20 feet. 2ft20ft{\displaystyle {\frac {2ft}{20ft}}} Relative Error =.1feet{\displaystyle =.1feet}[7] 2 Multiply ed. Types of Errors Measurement errors may be classified as either random or systematic, depending on how the measurement was obtained (an instrument could cause a random error in one situation and Absolute Error Formula Physics It is also a good idea to check the zero reading throughout the experiment.
Sign In Forgot your Password? Examples: 1. This tells you what percentage of the final measurement you messed up by. weblink The term human error should also be avoided in error analysis discussions because it is too general to be useful.
This simple equation tells you how far off you were in comparison to the overall measurement. you didn't measure it wrong ... So how do we express the uncertainty in our average value? Note, however that this doesn't make sense when giving percentages, as your error is not 10% of 2 feet.
For example, suppose you measure an angle to be: θ = 25° ± 1° and you needed to find f = cos θ, then: ( 35 ) fmax = cos(26°) = Start a free trial now. Failure to zero a device will result in a constant error that is more significant for smaller measured values than for larger ones. In this case, some expenses may be fixed, while others may be uncertain, and the range of these uncertain terms could be used to predict the upper and lower bounds on |
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# Help
0
134
4
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A rectangle is divided into four small rectangles as shown. The areas of three of the four small rectangles are labeled in the diagram. What is the area of the remaining small rectangle?
Oct 7, 2018
#1
+625
+2
Okay, lets work this one through.
The rectangle with area 3 is sharing its length with the rectangle with area 4. Let this value be $$x$$
The rectangle with area 3 is sharing its width with the rectangle with area 5. Let this value be $$y$$
The rectangle with area 4 is sharing its width with the rectangle with unknown area. Let this value be $$z$$
The rectangle with area 5 is sharing its length with the rectangle with unknown area. Let this value be $$w$$
If we set up and equation for each rectangle, we get the following:
Bottom left rectangle: $$xy = 3$$
Bottom right rectangle: $$wy = 5$$
Top left rectangle: $$xz = 4$$
Top right rectangle: $$wz = ?$$
First off, I am going to solve for the $$x$$ in the first equation. We get $$x = \frac{3}{y}$$.
Next, I am going to use the second equation to solve for $$y$$. We get $$y = \frac{5}{w}$$.
I am going to use the second variable we solved for and plug it into the first, like this: $$x = \frac{3}{\frac{5}{w}}$$
If we simplify this eqution, we get: $$x = 3\div \frac{5}{w} = 3(\frac{w}{5}) = \frac{3w}{5}$$
At this point, we know we're getting close to the answer, but we're not done yet.
If we use the value we just got for $$x$$ and plug it into the third equation, look what we get:
$$xz = 4 , (\frac{3w}{5})(z) = 4 , (3w)(z) = 20 , wz = \frac{20}{3}$$
Wait...isn't $$wz$$ the unknown area?
Therefore, the answer is $$\boxed {\frac{20}{3}}$$, or $$\boxed {6.\overline {6}}$$
.
Oct 8, 2018
#2
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Excellent, Knockout !!!!
BTW....welcome aboard !!!!
CPhill Oct 8, 2018
#3
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CPhill is right. This is excellent. Not your math, but your amazing skills for shoving 4 variables of bullshitt in a 1 variable bag.
If you were in Texas and wanted to go to Mexico, you first go to Russia, and then to Hong Kong, then swing over to equatorial Africa then to Mexico. That is the scenic route.
For the direct rout, just divide 4/3 and multiply by 5. Or divide 5/3 and multiply by 4.
Guest Oct 8, 2018
#4
+464
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Thank you so much Knock Out and Guest for helping me
Oct 8, 2018 |
# Factor out constant terms with square roots
I would like to use FactorTerms to factor out constant numerical terms out of an expression, this works as follows:
2 x^2 + 2 y^2 + 4 x y // FactorTerms
(* = 2 (x^2 + 2 x y + y^2) *)
However, this doesn't seem to work if there are square roots:
Sqrt[2] x^2 + Sqrt[2] y^2 + 2 Sqrt[2] x y // FactorTerms
(* = Sqrt[2] x^2 + 2 Sqrt[2] x y + Sqrt[2] y^2 *)
I'm not sure whether that's the way it's supposed to behave or a bug. If that's the desired behaviour, then what is the correct way of factoring out constant numerical terms containing square roots?
\$Version gives 11.2.0 for Mac OS X x86 (64-bit) (September 11, 2017).
It seems it works when you specify the variables in your term:
FactorTerms[Sqrt[2] x^2 + Sqrt[2] y^2 + 2 Sqrt[2] x y, {x, y}]
(* Sqrt[2] (x^2 + 2 x y + y^2) *)
To include JM's answer to your comment: If you have several variables and don't want to type them by hand, you can use
FactorTerms[poly, Variables[poly]]
to use them automatically.
• This works indeed, thanks! Even though it can be quite clumsy for a large number of variables. The description in the documentation does sound like it should do it also without specifying the variables explicitly. – Stan Mar 5 '18 at 11:29
• @Stan, in that case, you could do FactorTerms[poly, Variables[poly]] so that you don't have to bother with manually getting the variables. – J. M. is in limbo Mar 5 '18 at 11:54
Factor[Sqrt[2] x^2 + Sqrt[2] y^2 + 2 Sqrt[2] x y ]
(* Sqrt[2] (x + y)^2*)
works too!
• Hi Ulrich, thanks for the answer, but I explicitly do not want that x^2 + 2 x y + y^2 is transformed to (x + y)^2. I only want to factor out constant numerical terms without performing other possible factorisations. – Stan Mar 5 '18 at 11:37 |
# bojjenclon
## tl;dr
I'm a Calculating... year old nerd with a passion for programming. I graduated from Stetson University in 2017, with a B.S. in Computer Information Systems (CIS).
## FAQ
### How did you start programming?
The origin of my interest in programming can potentially be traced back to a couple of different (but ultimately similar) places. Regardless of what you consider the "starting location" of my interest, I was around 14 or 15 when said interest cropped up. I used to mess around with a program called RPG Maker XP, which could be used to make GameBoy-style games without any coding knowledge. However, if you wanted your game to be unique or to have features beyond what the program came with, you needed to be able to extend it with the software's varient of Ruby. This is probably where the spark began, but it wasn't until I started using GameMaker that I actually tried to learn to code seriously. From there it was simply a matter of realizing that relying on pre-built systems such as RMXP or GM was incredibly limiting. I wanted more, and I knew I'd have to learn to code properly in order to reach my newfound goals. I followed tutorials online and taught myself as much as I could, and my passion for coding grew as a I did so.
### Have you always wanted to code for a living?
Since I didn't really start programming until I was about 14/15, I certainly didn't always want to be a programmer. That isn't to say I wasn't interested in general computing concepts; I've always love computers and the various tasks they can perform. Despite this, when I was young my goal was to become a scientist one day. I didn't know much about the various "types" of scientists (chemists, biologists, physists, etc.), I just knew I wanted to do something in that realm. After all, Bill Nye the Science guy was practically my hero. The middle school science fair was when I realized that science as a career just wasn't going to work. To put it bluntly, I hated the science fair. Fortunately, it was soon after the fair that I began experimenting with coding. After taking a couple of programming classes in highschool (one of which being AP Computer Science), I knew I wanted to code professionally. |
# Getting to Philosophy¶
Click here to run this chapter on Colab
# Getting to Philosophy¶
The goal of this notebook is to develop a Web crawler that tests the “Getting to Philosophy” conjecture. As explained on this Wikipedia page:
Clicking on the first link in the main text of an English Wikipedia article, and then repeating the process for subsequent articles, usually leads to the Philosophy article. In February 2016, this was true for 97% of all articles in Wikipedia…
More specifically, the link can’t be in parentheses or italics, and it can’t be an external link, a link to the current page, or a link to a non-existent page.
We’ll use the urllib library to download Wikipedia pages and BeautifulSoup to parse HTML text and navigate the Document Object Model (DOM).
Before we start working with Wikipedia pages, let’s warm up with a minimal HTML document, which I’ve adapted from the BeautifulSoup documentation.
html_doc = """
<body>
<p class="title"><b>The Dormouse's story</b></p>
<p class="story">Once upon a time there were three little sisters; and their names were
(<a href="http://example.com/elsie" class="sister" id="link1">Elsie</a>),
<i><a href="http://example.com/lacie" class="sister" id="link2">Lacie</a> and</i>
<a href="http://example.com/tillie" class="sister" id="link3">Tillie</a>;
and they lived at the bottom of a well.</p>
<p class="story">...</p>
"""
This document contains three links, but the first one is in parentheses and the second is in italics, so the third is the link we would follow to get to philosophy.
Here’s how we parse this document and make a BeautifulSoup object.
from bs4 import BeautifulSoup
soup = BeautifulSoup(html_doc)
type(soup)
bs4.BeautifulSoup
To iterate through the elements in the DOM, we can write our own implementation of depth first search, like this:
def iterative_DFS(root):
stack = [root]
while(stack):
element = stack.pop()
yield element
children = getattr(element, "contents", [])
stack.extend(reversed(children))
For example, we can iterate through the elements and print all NavigableString elements:
from bs4 import NavigableString
for element in iterative_DFS(soup):
if isinstance(element, NavigableString):
print(element.string, end='')
The Dormouse's story
The Dormouse's story
Once upon a time there were three little sisters; and their names were
(Elsie),
Lacie and
Tillie;
and they lived at the bottom of a well.
...
But we can also use descendants, which does the same thing.
for element in soup.descendants:
if isinstance(element, NavigableString):
print(element.string, end='')
The Dormouse's story
The Dormouse's story
Once upon a time there were three little sisters; and their names were
(Elsie),
Lacie and
Tillie;
and they lived at the bottom of a well.
...
## Checking for Parentheses¶
One theory of software development suggests you should tackle the hardest problem first, because it will drive the design. Then you can figure out how to handle the easier problems.
For “Getting to Philosophy”, one of the harder problems is to figure out whether a link is in parentheses. If you have a link, you could work your way outward looking for enclosing parentheses, but in a tree, that could get complicated.
The alternative I chose is to iterate through the text in order, counting open and close parentheses, and yield links only if they are not enclosed.
from bs4 import Tag
paren_stack = []
for element in root.descendants:
if isinstance(element, NavigableString):
for char in element.string:
if char == '(':
paren_stack.append(char)
if char == ')':
paren_stack.pop()
if isinstance(element, Tag) and element.name == "a":
if len(paren_stack) == 0:
yield element
Now we can iterate through the links that are not in parentheses.
for link in link_generator(soup):
<a class="sister" href="http://example.com/lacie" id="link2">Lacie</a>
<a class="sister" href="http://example.com/tillie" id="link3">Tillie</a>
## Checking for Italics¶
To see whether a link is in italics, we can:
1. If its parent is a Tag with name a, it’s in italics.
2. Otherwise we have to check the parent of the parent, and so on.
3. If we get to the root without finding an italics tag, it’s not in italics.
For example, here’s the first link from link_generator.
link = next(link_generator(soup))
<a class="sister" href="http://example.com/lacie" id="link2">Lacie</a>
Its parent is an italics tag.
parent = link.parent
isinstance(parent, Tag)
True
parent.name
'i'
Exercise: Write a function called in_italics that takes an element and returns True if it is in italics.
Then write a more general function called in_bad_element that takes an element and returns True if:
• The element or one of its ancestors has a “bad” tag name, like i, or
• The element or one of its ancestors is a div whose class attribute contains a “bad” class name.
We will need the general version of this function to exclude invalid links on Wikipedia pages.
Actual Wikipedia pages are more complicated that the simple example, so it will take some effort to understand their structure and make sure we select the right “first link”.
from os.path import basename, exists
filename = basename(url)
if not exists(filename):
from urllib.request import urlretrieve
local, _ = urlretrieve(url, filename)
url = "https://en.wikipedia.org/wiki/Python_(programming_language)"
Now we can parse it and make soup.
filename = basename(url)
fp = open(filename)
soup2 = BeautifulSoup(fp)
If you use a web browser to view this page, and use the Inspect Element tool to explore the structure, you’ll see that the body of the article is in a div element with the class name mw-body-content.
We can use find to get this element, and we’ll use it as the root for our searches.
root = soup2.find(class_='mw-body-content')
Exercise: Write a generator function called valid_link_generator that uses link_generator to find links that are not in parentheses; then it should filter out links that are not valid, including links that are in italics, links to external pages, etc.
Test your function with a few different pages until it reliably finds the “first link” that seems most consistent with the spirit of the rules.
## WikiFetcher¶
When you write a Web crawler, it is easy to download too many pages too fast, which might violate the terms of service for the server you are downloading from. To avoid that, we’ll use an object called WikiFetcher that does two things:
1. It encapsulates the code for downloading and parsing web pages.
2. It measures the time between requests and, if we don’t leave enough time between requests, it sleeps until a reasonable interval has elapsed. By default, the interval is one second.
Here’s the definition of WikiFetcher:
from urllib.request import urlopen
from bs4 import BeautifulSoup
from time import time, sleep
class WikiFetcher:
next_request_time = None
min_interval = 1 # second
def fetch_wikipedia(self, url):
self.sleep_if_needed()
fp = urlopen(url)
soup = BeautifulSoup(fp, 'html.parser')
return soup
def sleep_if_needed(self):
if self.next_request_time:
sleep_time = self.next_request_time - time()
if sleep_time > 0:
sleep(sleep_time)
self.next_request_time = time() + self.min_interval
fetch_wikipedia takes a URL as a String and returns a BeautifulSoup object that represents the contents of the page.
sleep_if_needed checks the time since the last request and sleeps if the elapsed time is less than min_interval.
Here’s an example that demonstrates how it’s used:
wf = WikiFetcher()
url = "https://en.wikipedia.org/wiki/Python_(programming_language)"
print(time())
wf.fetch_wikipedia(url)
print(time())
wf.fetch_wikipedia(url)
print(time())
1640031013.2612915
1640031013.7938814
1640031014.7832372
If things have gone according to plan, the three timestamps should be no less than 1 second apart, which is consistent with the terms in Wikipedia’s robots.txt:
Friendly, low-speed bots are welcome viewing article pages, but not dynamically-generated pages please.
Exercise: Now let’s pull it all together. Write a function called get_to_philosophy that takes as a parameter the URL of a Wikipedia page. It should:
1. Use the WikiFetcher object we just created to download and parse the page.
2. Traverse the resulting BeautifulSoup object to find the first valid link according to the spirit of the rules.
3. If the page has no links, or if the first link is a page we have already seen, the program should indicate failure and exit.
4. If the link matches the URL of the Wikipedia page on philosophy, the program should indicate success and exit.
5. Otherwise it should go back to Step 1 (although you might want to put a limit on the number of hops).
The program should build a list of the URLs it visits and display the results at the end (whether it succeeds or fails).
Since the links you find are relative, you might find the urljoin function helpful:
from urllib.parse import urljoin
url = "https://en.wikipedia.org/wiki/Python_(programming_language)"
relative_path = "/wiki/Interpreted_language"
urljoin(url, relative_path)
'https://en.wikipedia.org/wiki/Interpreted_language'
get_to_philosophy(url)
https://en.wikipedia.org/wiki/Python_(programming_language)
https://en.wikipedia.org/wiki/Interpreted_language
https://en.wikipedia.org/wiki/Computer_science
https://en.wikipedia.org/wiki/Computation
https://en.wikipedia.org/wiki/Calculation
https://en.wikipedia.org/wiki/Arithmetic
https://en.wikipedia.org/wiki/Mathematics
https://en.wikipedia.org/wiki/Epistemology
https://en.wikipedia.org/wiki/Outline_of_philosophy
Got there in 9 steps!
['https://en.wikipedia.org/wiki/Python_(programming_language)',
'https://en.wikipedia.org/wiki/Interpreted_language',
'https://en.wikipedia.org/wiki/Computer_science',
'https://en.wikipedia.org/wiki/Computation',
'https://en.wikipedia.org/wiki/Calculation',
'https://en.wikipedia.org/wiki/Arithmetic',
'https://en.wikipedia.org/wiki/Mathematics',
'https://en.wikipedia.org/wiki/Epistemology',
'https://en.wikipedia.org/wiki/Outline_of_philosophy'] |
# 2. (10) Express the given equations without using AND gates (a) X = A-B 3. (10) Using a similar...
2. (10) Express the given equations without using AND gates (a) X = A-B 3. (10) Using a similar method to when we proved NAND was a complete logic set, prove that NOR is also a complete set. (a) Construct a NOT gate using only NOR gates (b) Construct an AND gate using only NOR gates (c) Construct an OR gate using only NOR gates
Attachments: |
# Crosses, no Noughts
Everyone realizes that Tic Tac Toe is a solved game. However, the Misère version of only-Xs provides an interesting alternative.
In this version of the game, both players play Xs onto the board and try to avoid making three in a row. If you'd like to see more about this, Numberphile has a nice video about this concept.
## Given a board of Misère Crosses, play an optimal move.
A board is three lines of three characters each, which are X or . Thus:
X X
X
XX
is a valid board. You may take this in any convenient format, so long as your input and output use the same format. Formats include (but are not limited to): A multi-line string (with optional trailing newline); A 2D array of characters which are X or ; a 1D flattened array of boolean values representing if each position has been played.
An optimal move is one that guarantees you will win by continuing to play optimally or prolongs your loss for as long as possible and is defined by the following rules:
• Avoid making three in a row.
• If you go first, play in the middle.
• If the only occupied space is the middle, play in any of the remaining spaces.
• If the middle square is not occupied and an outer square is, play opposite your opponent's last play.
• If the middle square is occupied and an outer square is, play a "knights move" (opposite, one over) away from a previous move that does not cause you to lose.
• If no remaining squares are left where you won't lose, play in any of the remaining squares.
[NOTE: this has been proved to be non-optimal in one case but you should use this algorithm anyway.]
You may assume that all of your previous moves were optimal. Thus, the first example board is not a valid input. Your opponent's moves may or may not be optimal.
If the game has ended (i.e. a three in a row has been made), behavior is undefined.
As this is , the shortest answer in bytes wins!
One possible path, using only optimal moves, is this:
[ ] [ ] [X ] [X ] [X ] [X ] [XX ]
[ ]->[ X ]->[ X ]->[ XX]->[ XX]->[ XX]->[ XX]
[ ] [ ] [ ] [ ] [ X ] [XX ] [XX ]
Here are possible inputs originating from the opponent using non-optimal moves:
(Note that only the left-side boards on this list are valid inputs.)
[X ] [X ]
[ ]->[ ]
[ ] [ X]
[XX ] [XX ]
[ ]->[ ]
[ X] [ XX]
[XX ] [XX ]
[X ]->[X X]
[ XX] [ XX]
• Related – Sp3000 Mar 28 '16 at 4:17
• What are the input and output formats? I'm assuming a board taken as an array or string? However this does not provide info on the last move, hence my next question. – Level River St Mar 28 '16 at 12:30
• The strategy "play opposite your opponent's last play" assumes either knowledge of your opponent's move history, or that you have previously followed this strategy, i.e have not inherited a board such as .XX\nX..\nX.. for example. Do we have to consider inheriting boards such as this? – Level River St Mar 28 '16 at 12:30
• @LevelRiverSt As written, "You may assume that all of your previous moves were optimal," so that board would be invalid input. You can take input in whatever format you like, but a multi line string such as your example there would be the "default": I just don't want to restrict anyone to having to parse the String when the move logic is the point of the challenge. – CAD97 Mar 28 '16 at 12:56
# Ruby, Rev B 121 bytes
Submission is the anonymous function, minus the f=. Shown in test program to illustrate use.
f=->n{["~mK)\7","}uYwQO"][l=n%2].bytes{|t|9.times{|i|(m=n|1<<i)==n||8.times{|j|m/2*257>>j&255==126-t&&t+j%2!=119&&l=m}}}
l}
puts g=f[gets.to_i]
puts
[7,6,5,
8,0,4,
1,2,3].each{|i|print g>>i&1; puts if i/3==1}
2 bytes saved by making the centre square the least significant bit instead of most significant bit (remove by /2 instead of %256.) Remaining savings by a reorganization of the table of acceptable moves. Organizing as centre square free/occupied instead of by total number of X's allows for a simpler test. Also, now there are only 2 strings in the array so the %w{string1 string2} syntax is abandoned in favour of the ["string1","string2"] syntax. This enables a nonprintable character \7 to be included, which in turn enables a simpler encoding to be used: 126-t instead of (36-t)%120.
# Ruby, Rev A 143 bytes
->n{l=r=("%b"%n).sum%8
%w{\$ %5 - I+Wy Q S#}[r].bytes{|t|9.times{|i|(m=n|1<<i)==n||8.times{|j|m%256*257>>j&255==(t-36)%120&&t+j%2!=43&&l=m}}}
l}
This is an anonymous function. The input / output format was left open, so I've gone for a 9-bit binary number. the 512's bit represents the centre, with the remaining bits spiralling round it (the 1's bit is considered to be a corner.)
There are far more possible inputs than acceptable outputs, so the algorithm is to try all moves, and find one that fits an acceptable output pattern. The acceptable output patterns for each number of X's are hardcoded.
The information about the centre square is stripped off and the remaining 8 bits are multiplied by 257 to duplicate them. This pattern is then rotated past the acceptable patterns by rightshifting.
The loop is not exited when a pattern is found, so the returned pattern will be the LAST acceptable pattern found. For this reason, preferable patterns (where there is a preference) come later in the list.
Given the 'Knights move' strategy it is of little importance whether a pattern is rotated by 45 degrees or not. The ungolfed version follows the knights move strategy and therefore does not need to differentiate between corner squares and edge squares: three in a row is to be avoided anyway.
However, I found that this is not always the best strategy, as there is the following trick. If your opponent goes first and takes the centre he should win. But on his second move he makes the error of allowing you to make a 2x2 square you should take it, as this allows you to force him to make three in a row. This is implemented in the golfed version. A little extra code is needed in this one instance to distinguish between three X's in a corner (force opponent to lose) and 3 X's along one edge (immediate suicide.)
Ungolfed in test program
The ungolfed version follows the logic expressed in the question.
In the golfed version the table is slightly modified to [[0],[1,17],[9],[37,7,51,85],[45],[47,119]] to implement the slightly different behaviour for the case r=3. It is then compressed to printable ASCII (requiring decoding (t-36)%120). An additional bit of logic is required to differentiate between three X's in a corner and three X's along an edge in the case of the table entry 7: &&t+j%2!=43
f=->n{l=r=("%b"%n).sum%8 #convert input to text, take character checksum to count 1's(ASCII 49.)
#0 is ASCII 48, so %8 removes unwanted checksum bloat of 48 per char.
#l must be initialised here for scoping reasons.
[[0],[1,17],[9],[11,13,37,51,85],[45],[47,119]][r].each{|t| #according to r, find the list of acceptable perimeter bitmaps, and search for a solution.
9.times{|i|(m=n|1<<i)==n|| #OR 1<<i with input. if result == n, existing X overwritten, no good.
#ELSE new X is in vacant square, good. So..
8.times{|j|m%256*257>>j&255==t&&l=m}} #%256 to strip off middle square. *257 to duplicate bitmap.
#rightshift, see if pattern matches t. If so, write to l
}
l} #return l (the last acceptable solution found) as the answer.
#call function and pretty print output (not part of submission)
puts g=f[gets.to_i]
puts
[6,7,0,
5,8,1,
4,3,2].each{|i|print g>>i&1; puts if i<3}
Output of test program
This what happens when the computer plays itself.
C:\Users\steve>ruby tictac.rb
0
256
000
010
000
C:\Users\steve>ruby tictac.rb
256
384
010
010
000
C:\Users\steve>ruby tictac.rb
384
400
010
010
100
C:\Users\steve>ruby tictac.rb
400
404
010
010
101
C:\Users\steve>ruby tictac.rb
404
436
010
110
101
C:\Users\steve>ruby tictac.rb
436
444
010
110
111
GAME ANALYSIS PLAYING FIRST
This is actually very simple and linear.
When playing first, the middle square will always be the first square occupied.
### r=0
... binary representation 0
.X.
...
### r=2
X.. binary representation 1001=9
.XX
...
### r=4
X.. binary representation 101101=45
.XX
XX.
There is only one way (up to symmetry) to have five X's including the middle square on the board without the game being over. There is an X in the middle square, one on each diagonal (at 90 degrees to each other) and one on each horizontal/vertical centreline (at 90 degrees to each other.) As an entire edge cannot be occupied the above is the only arrangement possible. Other player must lose on next move.
GAME ANALYSIS PLAYING SECOND
Play is quite different depending if the other player chooses the middle square.
### r=1
middle square occupied
.X. X.. binary representation 1
.X. .X.
... ...
middle square free
X.. .X. binary representation 10001=17
... ...
..X .X.
### r=3
Middle square occupied, if other player plays adjacent to your last X Playing a knight's move away as below is supported in the ungolfed version
XX. .XX binary representation 1011=11
.X. XX. or mirror image 1101=13
X.. ...
However the above is NOT the best move and is not supported in the golfed version. The best move is as follows, forcing a win on the next turn:
XX. binary representation 111=7. XXX
XX. Only to be used where j is odd. .X.
... Even j would look like image to right. ...
Middle square occupied, if other player plays at 90 or 135 degrees to your last X (play knight's move away.)
X.X .X. binary representation 100101=37
.X. .XX
.X. X..
Middle square free
X.X .X. XX. binary representations:
... X.X ... 1010101=85 (first two)
X.X .X. .XX and 110011=51 (last one)
### r=5
middle square occupied. For the reasons stated above in r=4, there are four possible moves, all of which lose. only one is supported: 101111=47.
middle square free. There is only one possible board up to symmetry, as follows. Other player must lose on next move, so there is no need to support r>5.
XX. binary representation 1110111=119
X.X
.XX
• This is a marvelous answer! I thought I had checked all cases for the optimal moe, but I guess I missed one. The spec'll stay the same for simplicity, though. (Really this is amazing thank you for doing this and this is so well explained! I left the I/O lose so people could do something amazing like this.) – CAD97 Mar 30 '16 at 22:34
• Thanks, it was an interesting challenge. I've managed to golf quite a bit more out of it now. – Level River St Mar 31 '16 at 12:16 |
## Social Security as PonziJanuary 2, 2009
Posted by A Texan In Grad School in Economic Theories, Federal Debt.
Tags: , ,
The recent Madoff scandal has brought renewed attention to the Ponzian nature of Social Security. But, is Socially Security really a Ponzi scheme? Michael Mandel at Business Week’s Economics Unbound says that because technology advances more than population does, Social Security is not a Ponzi scheme. The key to any debate is to define the terms. So let’s look at the definition of Ponzi Scheme:
An investment swindle in which high profits are promised from fictitious sources and early investors are paid off with funds raised from later ones.
That’s according to The American Heritage Dictionary. From this definition, it seems that Social Security is a Ponzi scheme. Generation A pays in, then Generation B pays in while A receives, so on and so on. But, interestingly I noticed that some dictionaries have a slightly tweaked definition of a Ponzi Scheme:
an investment swindle in which some early investors are paid off with money put up by later ones in order to encourage more and bigger risk
That’s from Merriam-Webster. The intriguing part of this definition is the idea that a Ponzi scheme has the goal of encouraging more and bigger risk. Who is taking these risks and what they are is ambiguous. Social Security encourages some risks because people don’t feel as much a need to save for their own retirement because Uncle Sam will pay them to be old. But I don’t think this is the kind of risk Merriam & Webster have in mind. So, under this (I believe poor) definition, Social Security is not a Ponzi scheme.
But, ultimately what seperates Social Security from a Ponzi scheme is the ability of the government to always make good on its promises. The government can always raise taxes or print more money. The government could even stop forcing people to pay in to Social Security while still paying people. Obviously this would have perverse effects on inflation and debt, but it’s possible. Social Security does not require that people pay in, so that it can pay out to others, at least in nominal terms. Therefore Social Security is not a Ponzi scheme. But, that isn’t exactly a good thing.
## More on Stimulus and ImportsDecember 7, 2008
Posted by A Texan In Grad School in Economic Theories, Federal Debt.
Apparently BW beat Rodrik et. al. to discussing the difficulties of orchestrating a domestically effective fiscal stimulus. Business Week claims that,
The financial crisis was caused, in large part, by U.S. consumers borrowing trillions of dollars from the rest of the world to buy imported cars, clothes, and gasoline, even as jobs slipped overseas. As long as the U.S. is running a big trade deficit and borrowing from abroad, a fundamental cause of the crisis remains.
Which is a bit misleading. While our extremely leveraged consumption played a role in the propagation of the crisis, I personally think that it was not a cause. Furthermore, our current problem is not so much one of too much existing debt, but the inability to get new debt.
So let’s think about all this. The problem with a fiscal stimulus is that it sends too much money abroad to other countries that manufacture our goods. This means fewer jobs created here. But, one must never forget the relationship between the current account and the capital account. If our stimulus is spent on goods from abroad, then our current account deficit will increase, but our capital account surplus will also increase. This is because all the foreigners who have our dollars have to do something with them. That is, either buy our goods or invest in our economy. This will increase liquidity.
Now, does this mean that a stimulus is definitely the best policy? Not necessarily. Keynesian multipliers and stimulus are not a free lunch. While foreigners will be investing money in our economy, there will also be an increase in government debt to fund the stimulus. This will cause crowding-out of private investment. Also if the government is selling more debt that puts upward pressure on interest rates. Perhaps what we need is a stimulus funded by federal lands.
## Multiple ProblemsDecember 6, 2008
Posted by A Texan In Grad School in Economic Theories, Federal Debt.
Here’s an interesting question on trade and Keynesian fiscal policy inspired by a post by Dani Rodrik:
The Keynesian multiplier is:
$\frac{1}{1-c(1-t)+m}$
where c is the marginal propesnity to consume, t is the tax rate, and m is the marginal propensity to import. It is clear from this formula that m is inversely proportional to the multiplier. So, if we decrease m, we will maximize the multiplier, in terms of m. Therefore, raising import tariffs such that m=0 will give us high growth and employment for the same amount of government spending. Also we eliminate our current account deficit. Seems good all around…
Now, how can this be square with Ricardian theories of comparative advantage? How can eliminating trade actually increase economic growth?
## Reagan Revolution, RIPNovember 23, 2008
Posted by A Texan In Grad School in Federal Debt. |
# Percentage Questions
FACTS AND FORMULAE FOR PERCENTAGE QUESTIONS
I.Concept of Percentage : By a certain percent , we mean that many hundredths. Thus x percent means x hundredths, written as x%.
To express x% as a fraction : We have , x% = x/100.
Thus, 20% = 20/100 = 1/5;
48% = 48/100 = 12/25, etc.
To express a/b as a percent : We have, $\frac{a}{b}=\left(\frac{a}{b}×100\right)%$ .
Thus, $\frac{1}{4}=\left(\frac{1}{4}×100\right)%=25%$
II. If the price of a commodity increases by R%, then the reduction in consumption so as not to increase the expenditure is $\left[\frac{R}{\left(100+R\right)}×100\right]%$
If the price of the commodity decreases by R%,then the increase in consumption so as to decrease the expenditure is $\left[\frac{R}{\left(100-R\right)}×100\right]%$
III. Results on Population : Let the population of the town be P now and suppose it increases at the rate of R% per annum, then :
1. Population after n years = $P{\left(1+\frac{R}{100}\right)}^{n}$
2. Population n years ago = $\frac{P}{{\left(1+\frac{R}{100}\right)}^{n}}$
IV. Results on Depreciation : Let the present value of a machine be P. Suppose it depreciates at the rate R% per annum. Then,
1. Value of the machine after n years = $P{\left(1-\frac{R}{100}\right)}^{n}$
2. Value of the machine n years ago = $\frac{P}{{\left(1-\frac{R}{100}\right)}^{n}}$
V. If A is R% more than B, then B is less than A by
$\left[\frac{R}{\left(100+R\right)}×100\right]%$
If A is R% less than B , then B is more than A by
$\left[\frac{R}{\left(100-R\right)}×100\right]%$
Q:
A bag contains 600 coins of 25 p denomination and 1200 coins of 50 p denomination. If 12% of 25 p coins and 24% of 50 p coins are removed, the percentage of money removed from the bag is nearly :
A) 21.6 % B) 15.3 % C) 14.6 % D) 12.5 %
Explanation:
Total money = Rs.[600*(25/100)+1200*(50/100)]= Rs. 750.
25 paise coins removed = Rs. (600*12/100) = 72.
50 paise coins removed = Rs. (1200*24/100)= 288.
Money removed =Rs.(72*25/100+288*50/100) = Rs.162.
Required percentage = (162/750*100)% = 21.6 %.
63 27736
Q:
How many litres of pure acid are there in 8 litres of a 20% solution ?
A) 1.4 B) 1.5 C) 1.6 D) 1.7
Explanation:
Quantity of pure acid = 20% of 8 litres = ((20/100)*8) litres = 1.6 litres.
44 27083
Q:
Two numbers are 20% and 30% less than the third number . How much percent is the second number less than first?
Let the third number be 100.
Then, the first number = 100 - 20 = 80 and second number = 100 - 30 = 70.
Difference between the first and second number = 80-70 = 10
$\inline&space;\therefore$ Required percentage = % = 100/8 = 12.5%
26540
Q:
A student multiplied a number by 2/5 instead of 5/2. What is the percentage error in evaluation ?
A) 52% B) 64% C) 84% D) 77%
Explanation:
Let the number be 'x'
Then, according to the given data,
$52x-25x52xx100$
$2125x100$
= 84%
18 25966
Q:
The sum of the number of boys and girls in a school is 150. if the number of boys is x, then the number of girls becomes x% of the total number of students. The number of boys is :
A) 60 B) 70 C) 80 D) 90
Explanation:
We have : x + (x% of 150 )= 150
=> x+(x/100)*150] = 150
=>
=> x = (150*2)/5 = 60
47 25786
Q:
In some quantity of ghee, 60% is pure ghee and 40% is vanaspati. If 10 kg of pure ghee is added, then the strength of vanaspati ghee becomes 20%. The original quantity was :
A) 10 kg B) 15 kg C) 20 kg D) 25 kg
Explanation:
Let the original quantity be x kg. Vanaspati ghee in x kg = (40x / 100 )kg = (2x / 5) kg.
Now, (2x/5)/(x + 10) = 20/100
=> 2x / (5x + 50) = 1/5
=> 5x = 50
=> x = 10.
56 25783
Q:
The diference of two numbers is 20% of the larger number, if the smaller number is 20, then the larger number is :
A) 15 B) 25 C) 35 D) 45
Explanation:
Let the large number be x.
Then x - 20 = 20x/100
=> x - x/5 = 20 => x = 25.
59 25531
Q:
The difference between a number and its two-fifth is 510. What is 10% of that number ?
A) 75 B) 85 C) 95 D) 105
Explanation:
Let the number be x. Then, x-(2/5)x = 510
=> 3x/5 =510
=>x =[510 * ( 5/3)] =850
10 % 0f 850 = 85. |
<meta http-equiv="refresh" content="1; url=/nojavascript/"> Triangle Sum Theorem ( Assessments ) | Geometry | CK-12 Foundation
# Triangle Sum Theorem
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Practice Triangle Sum Theorem
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Find The Measure of the Third Angle
Teacher Contributed
The measures of two angles of a triangle are 70$70^\circ$ and 45$45^\circ$. What is the measure of the third angle?
qid: 100158 |
# Definition:Weakly Locally Connected at Point/Definition 2
## Definition
Let $T = \struct {S, \tau}$ be a topological space.
Let $x \in S$.
The space $T$ is weakly locally connected at $x$ if and only if every open neighborhood $U$ of $x$ contains an open neighborhood $V$ of $x$ such that every two points of $V$ lie in some connected subset of $U$.
## Also known as
If $T$ is weakly locally connected at $x$, it is also said to be connected im kleinen at $x$.
Some sources refer to a space which is weakly locally connected at $x$ as locally connected at $x$. |
<meta http-equiv="refresh" content="1; url=/nojavascript/"> Whole Number Division ( Read ) | Arithmetic | CK-12 Foundation
You are viewing an older version of this Concept. Go to the latest version.
# Whole Number Division
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Practice Whole Number Division
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Whole Number Division
Did you figure out how many buckets of seafood Jonah will need?
Now that he knows how many pounds of seafood is needed, he will need to figure out how many buckets he needs to order. The seafood comes in 25 pound buckets. We know from the last Concept that Jonah will need to order 3,311 pounds of seafood. That will be enough to feed 43 seals for one week.
How many buckets should he order? Given that it comes in 25 pound buckets, will there be any seafood left over? This Concept will show you how to divide whole numbers. It is exactly what you will need to solve this problem.
### Guidance
You have learned how to add, subtract and multiply. The last operation that we will learn is division .
First, let’s talk about what the word “division” actually means. To divide means to split up into groups. Since multiplication means to add groups of things together, division is the opposite of multiplication.
$72 \div 9 = \underline{\;\;\;\;\;\;\;\;\;\;}$
In this problem, 72 is the number being divided, it is the dividend . 9 is the number doing the dividing, it is the divisor . We can complete this problem by thinking of our multiplication facts and working backwards. Ask yourself "What number multiplied by 9 equals 72?" If you said "8", you're right! 9 x 8 = 72, so 72 can be split into 8 groups of 9.
The answer to a division problem is called the quotient.
Sometimes, a number won’t divide evenly. When this happens, we have a remainder.
$15 \div 2 =\underline{\;\;\;\;\;\;\;\;\;\;}$
Hmmm. This is tricky, fifteen is not an even number. There will be a remainder here.
We can use an “ $r$ ” to show that there is a remainder. We can also divide larger numbers. We can use a division box to do this.
$8 \overline{)825 \;}$
Here we have a one digit divisor, 8, and a three digit dividend, 825. We need to figure out how many 8’s there are in 825. To do this, we divide the divisor 8 into each digit of the dividend.
$& 8 \overline{)825 \;} \qquad How \ many \ 8's \ are \ there \ in \ 8?''\\& \qquad \qquad \ \ The \ answer \ is \ 1.$
We put the 1 on top of the division box above the 8.
$& \overset{\ 1}{8\overline{ ) 825}}\\& \underline{-8} \Bigg \downarrow\\& \quad 02$
We multiply 1 by 8 and subtract our result from the dividend. Then we can bring down the next number in the dividend. Then, we need to look at the next digit in the dividend. “How many 8’s are there in 2?” The answer is 0.
We put a 0 into the answer next to the 1. $& \overset{\ 10}{8\overline{ ) 825}}\\& \underline{-8} \;\; \Bigg \downarrow\\& \quad \ 025$
Because we couldn’t divide 8 into 2, now we can bring down the next number, 5, and use the two numbers together: 25
“How many 8’s are in 25?” The answer is 3 with a remainder of 1. We can add this into our answer.
$& \overset{\ 103r1}{8\overline{ ) 825 \;}}\\& \ \underline{ -8 \ \ }\\& \ \ \ 025\\& \ \ \underline{-24}\\& \qquad 1$ We can check our work by multiplying the answer by the divisor.
$& \qquad 103\\& \ \underline {\times \quad \ \ 8 \ }\\& \qquad 824 + r \ \text{of} \ 1 = 825$
Our answer checks out.
Let’s look at a problem with a two-digit divisor.
$& \overset{\ \hspace{2 mm} 2}{12\overline{ ) 2448}} && How \ many \ 12&squot;s \ are \ in \ 2? \ None.&squot;&squot;\\& \ \underline{-24} \Bigg \downarrow && How \ many \ 12&squot;s \ are \ in \ 24? \ Two. \ So \ fill \ that \ in.&squot;&squot;\\& \qquad \ 4 && \ Now \ bring \ down \ the \ "4".\\\\& \overset{\ \hspace{4 mm} 20}{12\overline{) 2448}} && How \ many \ 12&squot;s \ are \ in \ 4? \ None, \ so \ we \ add \ a \ zero \ to \ the \ answer.&squot;&squot;\\&& &How \ many \ 12&squot;s \ are \ in \ 48?&squot;&squot;\\&& &Four\\&& &There \ is \ not \ a \ remainder \ this \ time \ because \ 48 \ divides \ exactly \ by \ 12.\\\\&\overset{\ \hspace{6 mm} 204}{12\overline{ ) 2448}}$
We check our work by multiplying: $204 \times 12$ .
$& \qquad \quad 204\\& \ \underline {\times \qquad \ 12}\\& \qquad \quad 408\\& \ \underline {+ \quad \ 2040}\\& \qquad \ 2448$
Our answer checks out.
We can apply these same steps to any division problem even if the divisor has two or three digits. We work through each value of the divisor with each value of the dividend. We can check our work by multiplying our answer by the divisor.
Now let's practice by dividing whole numbers
#### Example A
$4\overline{ ) 469 \;}$
Solution: 117 r 1
#### Example B
$18\overline{ ) 3678 \;}$
Solution: 204 r 6
#### Example C
$20\overline{ ) 5020 \;}$
Solution: 251
Now back to Jonah and the buckets of seafood.
If the seafood comes in 25 lb. buckets, how many buckets will he need?
To complete this problem, we need to divide the number of pounds of seafood by the number of pounds in a bucket. Notice, that we divide pounds by pounds. The items we are dividing have to be the same.
Let’s set up the problem.
$& \overset{\ \ \ \hspace{2 mm} 132}{25\overline{) 3311 \;}}\\& \ \ \underline{-25}\\& \quad \ \ 81\\& \quad \underline{-75 \ }\\& \qquad \ 61\\& \quad \ \ \underline{-50}\\& \qquad \ \ 11$
Uh oh, we have a remainder. This means that we are missing 11 pounds of fish. One seal will not have enough to eat if Jonah only orders 132 buckets. Therefore, Jonah needs to order 133 buckets. There will be extra fish, but all the seals will eat.
### Vocabulary
Dividend
the number being divided
Divisor
the number doing the dividing
Quotient
the answer to a division problem
Remainder
the value left over if the divisor does not divide evenly into the dividend
### Guided Practice
Here is a problem for you to solve on your own.
$25 \overline{)3075 \;}$
Next, we divide twenty- five into 3075.
Division
### Video Review
Here are a few videos for review.
### Practice
Directions: Use what you have learned to solve each problem.
1. $12 \div 6 = \underline{\;\;\;\;\;\;\;\;\;}$
2. $13 \div 4 = \underline{\;\;\;\;\;\;\;\;\;}$
3. $132 \div 7 = \underline{\;\;\;\;\;\;\;\;\;}$
4. $124 \div 4 = \underline{\;\;\;\;\;\;\;\;\;}$
5. $130 \div 5 = \underline{\;\;\;\;\;\;\;\;\;}$
6. $216 \div 6 = \underline{\;\;\;\;\;\;\;\;\;}$
7. $1161 \div 43 = \underline{\;\;\;\;\;\;\;\;\;}$
8. $400 \div 16 = \underline{\;\;\;\;\;\;\;\;\;}$
9. $1827 \div 21 = \underline{\;\;\;\;\;\;\;\;\;}$
10. $1244 \div 40 = \underline{\;\;\;\;\;\;\;\;\;}$
11. $248 \div 18 = \underline{\;\;\;\;\;\;\;\;\;}$
12. $3264 \div 16 = \underline{\;\;\;\;\;\;\;\;\;}$
13. $4440 \div 20 = \underline{\;\;\;\;\;\;\;\;\;}$
14. $7380 \div 123 = \underline{\;\;\;\;\;\;\;\;\;}$
15. $102000 \div 200 = \underline{\;\;\;\;\;\;\;\;\;}$
16. $10976 \div 98 = \underline{\;\;\;\;\;\;\;\;\;}$ |
## Monday, 15 May 2017
### Through time & space
I've continued to fill in the data from the polls and re-run the model for the next UK general election. I think the dynamic element is interesting in principle, mainly because of how the data from the most recent polls could be weighed differently than those further in the past.
Roberto had done an amazing job, building on Linzer's work and using a rather complex model to account for the fact that the polls are temporally correlated and, as you get closer to election day, the historical data are much less informative. This time, I have done something much simpler and somewhat more arbitrary, simply based on discounting the polls depending on how distant they are from "today".
This is the results given by my model in the period from May 1st to May 12th $-$ at every day, I've only included the polls available at that time and discounted using a 10% rate, assuming modern life really runs very fast (which it reasonably does...). Not much is really changing and the predictions in terms of the number of seats won by the parties in England, Wales and Scotland seems fairly stable $-$ Labour is probably gaining a couple of seats, but the story is basically unchanged.
The other interesting thing (which I had done here and here too) is to analyse the predicted geographical distribution of the votes/seats. Now, however, I'm taking full advantage of the probabilistic nature of the model and not only am I plotting on the map the "most likely outcome" (assigning a colour to each constituency, depending on who's predicted to win it). In the graph below, I've also computed the probability that the party most likely to win a given seat actually does so (based on the simulations from the posterior distributions of the vote shares, as explained here) $-$ I've shaded the colours so that lighter constituencies are more uncertain (i.e. the win may be more marginal).
There aren't very many marginal seats (according to the model) and most of the times, the chance of a party winning a constituency exceeds 0.6 (which is fairly high, as it would mean a swing of over 10% from the prediction to overturn this).
This is also the split across different regions $-$ again, not many open battlefields, I think. In London, Hornsey and Wood Green is predicted to go Labour but with a probability of only 54%, while Tooting is predicted to go Tory (with a chance of 58%). |
# Math Help - Determine whether the equation is a linear equation..Help!
1. ## Determine whether the equation is a linear equation..Help!
1. $\frac{x}{2} = 10 + \frac{2y}{3}$
2. $7n - 8m = 4 - 2m$
2. Originally Posted by Phresh
1. $\frac{x}{2} = 10 + \frac{2y}{3}$
2. $7n - 8m = 4 - 2m$
I guess both are linear, because
$\frac{x}{2} = 10 + \frac{2y}{3}$
$y = \frac{3}{2}(\frac{x}{2}-10)$
and
$7n - 8m = 4 - 2m$
6m = 4 - 7n
$m = \frac{4-7n}{6}$
This is a straight line, too |
# Problem understanding Green's function equality in Messiah QM II
## Main Question or Discussion Point
Hi,
It's about green's function in the book Messiah - Quantum Mechanics II - Chapter 16.3.2
(see http://books.google.de/books?id=OJ1XQ5hnINwC&pg=PA200&lpg=PA202&ots=NWr6A89Mkt&dq=messiah+quantenmechanik+kapitel+16.3&hl=de). The book actually is in german, but I guess that doesn't matter understanding the formulas.
I don't understand the last equality between eq. (16.60) & (16.61). Why is there in the first term's nominator a 1 (instead of "z - H_o - lV")? Can somebody help me out?
Cheers
Tobe
Related Quantum Physics News on Phys.org
George Jones
Staff Emeritus
$$\frac{1}{z-H_0} \left( z - H_0 -\lambda V \right) \frac{1}{z - H_0 -\lambda V} = \frac{1}{z-H_0}.$$ |
# Math Help - Solve 3 variables with 2 equations
1. ## Solve 3 variables with 2 equations
Here is the problem:
Total sales are $5125 and units sold are 1653. Units can be sold either on sale or not on sale. The price on sale or feature sale price (FSP) was$2.25. How many units were sold on sale?
I can derive 2 formulas, but I have 3 variables to solve. Here's how far I have gotten:
Regular price = RSP
Units sold on Sale = Promoted volume (PV)
Units sold at regular = non-promoted volume (NPV)
Total volume = TV
Equation 1: TV = PV + NPV
Equation 2: Total Sales = (RSP x NPV) + (FSP x PV) or $5125 = (RSP x NPV) + 2.25PV Given these two equations (or any other equations that you can think of), how can I solve for PV? Any help will be greatly appreciated. 2. Hi abenedet, Reduce the unknowns to 2 and write two equations. There is an easy solution. bjh 3. That's where I am stuck. How do I reduce it to two unknowns? If I can get it to that point, I think I can take it from there. Thanks! 4. Code: 1652 @$2.25 = $3,717 1 @$1,408.00 = $1,408 ==== ===== 1653$5,125
Hmmm...who bought the one not on sale?
5. Hi abenedet,
Let x= units sold at sale price $2.25 1653-x = units sold at regular price Let y= regular price Equation 1 equates sales by units to$5125
Equation 2 defines y
bjh
6. Ya; per BJ: 2.25x + y(1653 - x) = 5125
But you can have a solution for any value of x < 1653.
Example: x = 1000
2.25(1000) + y(1653 - 1000) = 5125
y = 4.4027565...
Only solution that has "integer results" is x=1652 and y=1408.
7. Hi Wilmer,
My solution was wrong since it only works ( not exact integers)when the sales are equal.Thanks for the correction.
bjh
8. Thanks everyone for the responses.
Is there another way to solve this without fixing x to be an exact #? That is, is there any other formula that can be derived to find the exact solution?
Thanks again
9. Originally Posted by abenedet
Is there another way to solve this without fixing x to be an exact #? That is, is there any other formula that can be derived to find the exact solution?
NO.
We have 1 equation: 2.25x + y(1653 - x) = 5125;
equation has 2 variables: x and y.
With 2 variables, you need 2 equations for an exact solution.
Like, to illustrate with something simple:
a + b = 5
1 + 4 = 5
2 + 3 = 5
3 + 2 = 5
4 + 1 = 5
5 + 0 = 5
6 - 1 = 5
and so on... |
# Metavision Designer RAW to Video Sample¶
The sample in <install-prefix>/share/metavision/designer/core/samples/metavision_raw_to_video.py shows how to generate an AVI video from a RAW file.
## Expected Output¶
Metavision RAW to Video sample generates an AVI file and by default saves it to the same directory and with the same name as the input RAW file.
## How to start¶
To start the sample based on recorded data, provide the full path to a RAW file (here, we use the file from Metavision Dataset):
Linux
python3 /usr/share/metavision/designer/core/samples/metavision_raw_to_video.py -i spinner.raw
Windows
python "C:\Program Files\Prophesee\share\metavision\designer\core\samples\metavision_raw_to_video.py" -i spinner.raw
python3 /usr/share/metavision/designer/core/samples/metavision_raw_to_video.py -h
python "C:\Program Files\Prophesee\share\metavision\designer\core\samples\metavision_raw_to_video.py" -h |
Corpus ID: 212736956
# An arithmetic enrichment of B\'ezout's Theorem
@article{McKean2020AnAE,
title={An arithmetic enrichment of B\'ezout's Theorem},
author={S. McKean},
journal={arXiv: Algebraic Geometry},
year={2020}
}
• S. McKean
• Published 2020
• Mathematics
• arXiv: Algebraic Geometry
• The classical version of Bezout's Theorem gives an integer-valued count of the intersection points of hypersurfaces in projective space over an algebraically closed field. Using work of Kass and Wickelgren, we prove a version of Bezout's Theorem over any perfect field by giving a bilinear form-valued count of the intersection points of hypersurfaces in projective space. Over non-algebraically closed fields, this enriched Bezout's Theorem imposes a relation on the gradients of the hypersurfaces… CONTINUE READING
6 Citations |
• Characteristic Classes for $GO(2n)$ in étale Cohomology
• # Fulltext
https://www.ias.ac.in/article/fulltext/pmsc/123/02/0225-0233
• # Keywords
Characteristic classes; étale cohomology; algebraic stacks.
• # Abstract
Let $GO(2n)$ be the general orthogonal group (the group of similitudes) over any algebraically closed field of characteristic $\neq 2$. We determine the smooth-étale cohomology ring with $\mathbb{F}_2$ coefficients of the algebraic stack $BGO(2n)$. In the topological category, Holla and Nitsure determined the singular cohomology ring of the classifying space $BGO(2n)$ of the complex Lie group $GO(2n)$ in terms of explicit generators and relations. We extend their results to the algebraic category. The chief ingredients in this are: (i) an extension to étale cohomology of an idea of Totaro, originally used in the context of Chow groups, which allows us to approximate the classifying stack by quasi projective schemes; and (ii) construction of a Gysin sequence for the $\mathbb{G}_m$-fibration $BO(2n)\to BGO(2n)$ of algebraic stacks.
• # Author Affiliations
1. School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400 005, India
• # Proceedings – Mathematical Sciences
Volume 131, 2021
All articles
Continuous Article Publishing mode
• # Editorial Note on Continuous Article Publication
Posted on July 25, 2019 |
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# Can't overload the operator
This topic is 3905 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.
## Recommended Posts
As the title says, I'm having a problem overloading the << operator. It occurs in this header:
#ifndef COORD_H
#define COORD_H
// Class Coord
// Descr: A coordinate.
// Members: value -> The value of the coordinate.
// Func: -Assignment through a Coord and an integer.
// -Equality check based on the values of the coordinates.
#include <iostream>
class Coord {
public:
// Constructor: initializes 'value'
Coord(int v = 0): value(v) {}
// Copy constructor
Coord(const Coord& c);
// Assigns 'c.value' to 'value'
Coord& operator=(const Coord& c);
// Assigns 'i' to 'value'
Coord& operator=(const int i);
// Returns whether 'c.value' equals 'value'
bool operator==(const Coord& c) const;
// Writes 'value' into 'os' (for testing purposes)
std::ostream& operator<<(std::ostream& os, const Coord& c) const;
// Getter for 'value'
int Value() const;
// Setter for 'value'
void SetValue(int i);
private:
int value;
};
#endif
When I try to compile, I get the following error: error C2804: binary 'operator <<' has too many parameters I understand what the error message is telling me, but I don't get why it's even there. My book appears to overload the operator in the exact same way. So why does this error occur and how do I go about fixing it?
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I doubt your book has it as a class member, but more likely as a free function. As it is, your current overload is taking 3 arguments, one implicit and two explicit.
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You must make it a free function, not a member fuction. It cannot be a member function of the Coord class, because it will be called on the stream object.
foo << myCoord
The above is equal to one of these two, depending on which one is defined.
foo.operator <<(myCoord); // (1)operator <<(foo, myCoord); // (2)
Since (1) requires modification of the stream class, you cannot do it that way. Only the free function form remains. Move the declaration you have at the moment out of the class, and make it friend if necessary.
edit: Some fixes... and too slow.
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You were right. Making it a free function solved the problem.
Thank you. |
# How many linear equations in x and y can be satisfied by x = 2, y = 3?
Question:
How many linear equations in x and y can be satisfied by x = 2, y = 3?
(a) Only one
(b) Only two
(c) Infinitely many
(d) None of these
Solution:
(c) Infinitely many
Infinite linear equations are satisfied by $x=2, y=3$. |
# American Institute of Mathematical Sciences
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Inverse boundary value problems in the horosphere - A link between hyperbolic geometry and electrical impedance tomography
February 2007, 1(1): 135-157. doi: 10.3934/ipi.2007.1.135
## Stability of boundary distance representation and reconstruction of Riemannian manifolds
1 Department of Mathematics, Okayama University, Tsushima-naka, Okayama, 700-8530, Japan 2 Department of Mathematical Sciences, Loughborough University, Loughborough, LE11 3TU, United Kingdom 3 Department of Mathematics, Helsinki University of Technology, P.O. Box 1100, 02015 TKK, Finland
Received August 2006 Revised August 2006 Published January 2007
A boundary distance representation of a Riemannian manifold with boundary $(M,g,$∂$\M)$ is the set of functions $\{r_x\in C$ (∂$\M$) $:\ x\in M\}$, where $r_x$ are the distance functions to the boundary, $r_x(z)=d(x, z)$, $z\in$∂$M$. In this paper we study the question whether this representation determines the Riemannian manifold in a stable way if this manifold satisfies some a priori geometric bounds. The answer is affermative, moreover, given a discrete set of approximate boundary distance functions, we construct a finite metric space that approximates the manifold $(M,g)$ in the Gromov-Hausdorff topology.
In applications, the boundary distance representation appears in many inverse problems, where measurements are made on the boundary of the object under investigation. As an example, for the heat equation with an unknown heat conductivity the boundary measurements determine the boundary distance representation of the Riemannian metric which corresponds to this conductivity.
Citation: Atsushi Katsuda, Yaroslav Kurylev, Matti Lassas. Stability of boundary distance representation and reconstruction of Riemannian manifolds. Inverse Problems & Imaging, 2007, 1 (1) : 135-157. doi: 10.3934/ipi.2007.1.135
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2019 Impact Factor: 1.373 |
# The GENMOD Procedure
### Predicted Values of the Mean
Subsections:
#### Predicted Values
A predicted value, or fitted value, of the mean corresponding to the vector of covariates is given by
where g is the link function, regardless of whether corresponds to an observation or not. That is, the response variable can be missing and the predicted value is still computed for valid . In the case where does not correspond to a valid observation, is not checked for estimability. You should check the estimability of in this case in order to ensure the uniqueness of the predicted value of the mean. If there is an offset, it is included in the predicted value computation.
#### Confidence Intervals on Predicted Values
Approximate confidence intervals for predicted values of the mean can be computed as follows. The variance of the linear predictor is estimated by
where is the estimated covariance of . The robust estimate of the covariance is used for in the case of models fit with GEEs.
Approximate confidence intervals are computed as
where is the th percentile of the standard normal distribution and g is the link function. If either endpoint in the argument is outside the valid range of arguments for the inverse link function, the corresponding confidence interval endpoint is set to missing. |
Homework Help: Equation of motion for a rotational mass and spring system
1. Feb 27, 2015
Isow
1. The problem statement, all variables and given/known data
A small ball with mass M attached to a uniform rod of mass m and length l pivoted at point o and attached to two springs with spring constant k1 and k2 at distance d1 and d2 from point o as shown in Fig 2. The system oscillates around the horizontal line. Assume the system is in a fluid and is damped by a viscous drag force proportional to the speed. Find the equation of motion for the small angle oscillation and its solution if it starts from rest with initial angular position θ0. Find the quality factor. Ignore the buoyancy force.
2. Relevant equations
Moment of inertia for rod: Irod = (1/3)*m*l2
Moment of inertia for ball: Iball = M*l^2
Equation for torque: τ = I*θ''
Drag force: FD = -b*cosθ
Spring force: FS = -k*sinθ
Small angle approx for sin: sinθ = θ
and for cos: cosθ = 1
3. The attempt at a solution
My attempt involves trying to convert all forces to torques and to get an equation of motion in terms of theta. I'm having trouble in a few places. For one, I'm not sure how to handle finding the center of mass symbolically without knowing the mass of the rod. It seems like the CoM could be almost anywhere on the rod, depending on the rod's mass... and the placement of CoM with respect to d1 and d2 seems important.
If I can figure out how to set the differential equation up, I don't think solving it will be much trouble.
2. Feb 27, 2015
OldEngr63
Don't worry about the CM location; take moments about the left end. It will all work out quite easily.
3. Feb 27, 2015
Isow
Hi, thank you for the reply. The examples in my book all deal with a uniform rod and no mass attached, or the deal with a "massless" rod and mass. The former explicitly uses the CoM (l/2) to calculate torque and insert in my third equation. The latter are simple pendulums.
So I have a free body diagram drawn, but I'm not confident about how to get these forces in terms of theta, or sum them. I'm pouring over the book, but everything in the relevant chapters involves very simple systems compared to this problem.
4. Feb 27, 2015
OldEngr63
You are making this waaaay to hard!
First, from the integral definition of MMOI, it is evident that these terms simply add. Thus the total MMOI with respect to the left end is simply the sum of the MMOI for the rod + MMOI for the lumped mass. You have both of these terms.
When the system is displaced downward by θ as shown in your diagram, the compression in spring 1 is (approx) d1*θ, so the force in this spring is F1 = K1*d1*θ and the moment of this force about the left end is M1 = d1*F1 = K1*d12*θ. A similar analysis applies for the second spring.
I leave it to you to figure out how to put it all together and take the damping into account. (Be brave, go ahead!)
5. Feb 27, 2015
Isow
So I think I have the torque of the springs: T = k*d^2*θ
When it comes to the mass, I'm not sure if I should be taking into account gravity, since equilibrium is at the horizontal. Would I just take into account the momentum of the mass working against the system, adding to drag?
p = mv
So would the momentum term for the mass become -M*l*θ' ?
And then -(m*l*θ')/2 for the rod?
And finally, the drag force, -b*θ' ?
Once you have the sum of torques, can you plug it into T = I*θ'' and solve for the moment?
Normally, I think, we divide through by mass to get terms for ω (√k/m) and γ (b/m), but if the above is correct there are two masses which I'm not sure how to handle when it comes to getting factors of ω and γ.
Last edited: Feb 27, 2015
6. Feb 27, 2015
Isow
Hey, thanks again. I wrote my last post right as you were posting.
What you're calling the moment I'm calling the torque... but it seems like we're talking about the same thing. My understanding is that Torque = F*d and Fspring = kdθ so T = kd2θ. Why are you calling that the moment?
Plus, the questions above. If I'm making this too hard, it's because it's been a LONG time since I've dealt with classical mechanics and this class is vibrations and waves so it's not really touching on this stuff I should probably already know, just asking me to use it...
7. Feb 27, 2015
Isow
Okay, so it seems like you're saying Itotal = l2(m/3 + M)
And we can sum the torques... so Ttotal = k1d12θ + k2d22θ + b*l*θ' ? I'm not sure about the drag part. I know the drag force is bθ', but converting that to torque by multiplying by l seems weird since it's not a point force, and would be distributed unevenly across the rod and mass.
Is there any chance somebody can just tell me what the equation of motion will be, and why? I'll still have to solve the ODE and find the other quantities asked for. I feel utterly lost trying to set this equation up.
8. Feb 27, 2015
haruspex
Since you're not given a drag coefficient, you can just write the drag torque as $-b\dot\theta$. The 'b' encompasses the distribution of the drag along the rod and ball.
9. Feb 27, 2015
Isow
I think I've got it now, but for the drag I ended up integrating over the length of the rod:
$\int_{0}^{l}-br\theta'dr = -\frac{1}{2}bl^{2}\theta'$
Which seems right since it gives units of torque (the $l^{2}$ term).
Last edited: Feb 27, 2015
10. Feb 27, 2015
haruspex
OK, but since you are inventing b and l is constant you might just as well have invented $c = \frac{1}{2}bl^{2}$. And why the sign flip?
[This is weird. On my screen, your post says 1/3. It also says 1/3 if I right click and 'show math as'. But when I clicked reply it came up as 1/2.]
11. Feb 27, 2015
Isow
The sign flip was a typo, and I did have 1/3 initially but fixed it so that probably explains the bug.
The $l^{2}$ term here actually ends up cancelling out when you divide through by the moment, $l^{2}(m/3+M)$.
I did absorb the 1/2 term into the b, which I hope is an okay liberty to take.
12. Feb 27, 2015
haruspex
That all sounds fair. |
# Comprehensive Guide to the Normal Distribution
Drop in for some tips on how this fundamental statistics concept can improve your data science.
Photo by Cameron Casey from Pexels
The distribution of data refers to the way the data is spread out. In this article, we’ll discuss the essential concepts related to the normal distribution:
• Ways to measure normality
• Methods to transform a dataset to fit the normal distribution
• Use of the normal distribution to represent naturally occurring phenomena and offer statistical insights
# Overview
Data distribution is of great importance in statistics because we are pretty much always sampling from a population where the full distribution is unknown. The distribution of our sample may put limitations on the statistical techniques available to us.
Normal distribution, where f(x) = probability density function, = standard deviation, and = mean
The normal distribution is a frequently observed continuous probability distribution. When a dataset conforms to the normal distribution, it is possible to utilize many handy techniques to explore the data:
• Knowledge of the percentage of data within each standard deviation
• Linear least squares regression
• Inference based on the sample mean (e.g., t-test)
In some cases, it’s beneficial to transform a skewed dataset so that it conforms to the normal distribution, thereby unlocking the use of this set of statistical techniques. This is more likely to be relevant when your data is almost normally distributed except for some distortion. More on this in a moment.
Normal distributions have the following features:
• Symmetric bell shape
• Mean and median are equal (at the center of the distribution)
• ≈68% of the data falls within 1 standard deviation of the mean
• ≈95% of the data falls within 2 standard deviations of the mean
• ≈99.7% of the data falls within 3 standard deviations of the mean
M.W. Toews via Wikipedia
Here are some terms you should be familiar with relevant to a general overview of the normal distribution:
• Normal Distribution: a symmetric probability distribution that is frequently used to represent real-valued random variables; sometimes called the bell curve or Gaussian distribution
• Standard Deviation: measure of the amount of variation or dispersion of a set of values; calculated as the square root of variance
• Variance: the distance of each data point from the mean
## How to use the Normal Distribution
If your dataset does not conform to the normal distribution, here are some suggestions:
• Collect more data: a small sample size or lack of data quality could be distorting your otherwise normally distributed dataset. As is often the case in Data Science, the solution could be to collect more data.
• Reduce sources of variancereduction of outliers could result in normally distributed data.
• Apply a power transform: for skewed data, you might choose to apply the Box-Cox method, which refers to taking the square root and the log of the observation.
In the sections that follow, we’ll explore some measures of normality and how you would use them in a Data Science project.
# Skewness
Skewness is a measure of asymmetry relative to the mean. Here’s a graph of a left skewed distribution.
Rodolfo Hermans via Wikipedia
💡 I’ve always found this to be a bit counterintuitive, so it’s worth paying close attention here. This graph has negative skewness. This means that the tail of the distribution is longer on the left. The counterintuitive bit (to me at least) is that most of the data points are clustered to the right. Do not be tempted to confuse with right or positive skewness, which would be represented by this graph’s mirror image.
## How to use Skewness
Understanding skewness is important because it is a key factor in model performance. To measure skewness, use skew from the scipy.stats module.
via SciPy
The skewness measure can clue us in to potential deviation in model performance across the feature values. A positively skewed feature, like the second array above, will enable better performance on lower values, given that we’re providing more data in that range (opposed to higher value outliers).
# Kurtosis
From Greek kurtos, meaning curved, kurtosis is a measure of the tailedness of the distribution. Kurtosis is typically measured relative to 0, the kurtosis value of the normal distribution using Fisher’s definition. A positive kurtosis value indicates “fatter” tails (i.e., a slimmer bell curve with more outliers).
The Laplace Distribution has kurtosis > 0. via John D. Cook Consulting.
## How to use Kurtosis
Understanding kurtosis provides a lens to the presence of outliers in a dataset. To measure kurtosis, use kurtosis from the scipy.stats module.
via SciPy
A negative kurtosis value indicates data that is more tightly grouped around the mean with fewer outliers.
# A Caveat About the Normal Distribution
You may have heard that many naturally occurring datasets conform to the normal distribution. This claim has been made for everything from IQ to human heights.
While it’s true that the normal distribution is drawn from observations of nature and does occur frequently, we risk oversimplification by applying this assumption too liberally.
The normal model often doesn’t fit well in the extremes. It often underestimates the probability of rare events. The Black Swan by Nassim Nicholas Taleb gives numerous examples of rare events that were not as rare as a normal distribution would predict.
# Summary
In this brief article on the normal distribution, we covered some fundamental concepts, how it is measured, and how it is used. Be careful not to overapply the normal distribution or you risk discounting the likelihood of outliers. Hope this article provided some insight on this commonly observed and highly useful statistical concept.
Original. Reposted with permission. |
# High-Performance Configuration
## LP Solver Selection
By default, EAGO uses GLPK for solving linear subproblems introduced. Using a commercial linear solver is highly recommended such as Gurobi, CPLEX, or XPRESS is highly recommended. Both Gurobi and CPLEX are free for academics and installation information can be found through http://www.gurobi.com/academia/academia-center and https://www.ibm.com/developerworks/community/blogs/jfp/entry/CPLEXIsFreeForStudents?lang=en, respectively.
Warning
EAGO assumes that the MOI wrapper for any sub-solver exhibits the expected behavior. Any deviation for the expected may lead to an error. We currently recommend using either the default GLPK solver or Gurobi rather than CPLEX. Our experience has been that the GLPK and Gurobi MathOptInterface wrappers are better maintained and less prone to unexpected behavior than CPLEX currently is (though this is continuously improving) and in particular GLPK is quite stable.
A non-default LP solver can then be selected by the user via a series of keyword argument inputs as illustrated in the code snippet below. The relaxed_optimizer contains an instance optimizer with valid relaxations that are made at the root node and is updated with affine relaxations in place.
# Create opt EAGO Optimizer with Gurobi as a lower subsolver
m = Model(optimizer_with_attributes(EAGO.Optimizer, "relaxed_optimizer" => Gurobi.Optimizer(OutputFlag=0))
## Rounding Mode
The IntervalArithmetic.jl package supports a number of different directed rounding modes. The default directed rounding mode is :tight. It is recommended that the user specify that :accurate directed rounding mode be used as it may results in a significant performance improvement. Setting a rounding mode can requires the redefinition of a number of functions. As a result, this should only be done at the top-level by the user (rather than by using keyword arguments). To set the rounding mode to :accurate using the following syntax when loading the EAGO package initially:
using IntervalArithmetic; setrounding(Interval, :accurate)
using EAGO
# REST OF CODE
## Ipopt Build
Ipopt is the recommended solver for upper bounding problems. Ipopt's performance is highly dependent on the linear algebra package used (up to 30x). By default MUMPS is used. It's recommended that you either compile Ipopt with HSL MA57 or the Pardiso linear algebra packages with a machine specific Blas library (for Intel users the JuliaPro MKL version is recommended). For information on this, see the below links:
• Compiling Ipopt: https://www.coin-or.org/Ipopt/documentation/node13.html
• Julia Specifics:
• Pointing Ipopt to a compiled version:
• Ipopt Package Info: https://github.com/JuliaOpt/Ipopt.jl
• Discourse discussion: https://discourse.julialang.org/t/use-ipopt-with-custom-version/9176
• Issues using Pardiso:
• Ubuntu: https://github.com/JuliaOpt/Ipopt.jl/issues/106
• Windows: https://github.com/JuliaOpt/Ipopt.jl/issues/83
• HSL Website: http://www.hsl.rl.ac.uk/ipopt/
• Pardiso Website: https://pardiso-project.org/ |
# statistical physics i equilibrium statistical mechanics
Statistical Physics I discusses the fundamentals of equilibrium statistical mechanics, focussing on basic physical aspects.
Non-equilibrium statistical mechanics, phase transitions, critical phenomena, and statistical field theory Lecture notes from a graduate course on Statistical Mechanics. Advanced mechanics doesn't need much prerequisites surprisingly I have done so for Quantum Field Theory (Physics 253a,b/254), Waves (Physics 15c), and Statistical Mechanics and non-equilibrium statistical physics - 1 of 3 Brownian motion and non-equilibrium statistical physics - 3 of 3 Round table on open problems in non-equilibrium statistical physics - Froehlich It has educated us with countless modules in the universe and will demonstrate us further. This ebook can only be accessed online and cannot be downloaded. It introduces the fundamental assumption: Every A second course on statistical mechanics, covering non-equilibrium phenomena, can be found here. Nuclear and Particle PhysicsStatistical Mechanics Lectures on elementary statistical mechanics, taught at the University of Illinois and at the University of Pennsylvania. The book is self-contained and is suitable for beginning graduate students in physics and materials science or undergraduates who have taken an introductory course in statistical mechanics. Universitt Ensembles in Quantum Mechanics (Statistical Operators and Density Ma- trices) Principles of Condensed Matter Physics P In this course we will be able only to cover its basic features like Bose-Einstein and Fermi-Dirac statistics, and applications like the vibrational and electronic contributions to the specific heat of solids like metals ISBN: 9780471815181 : Selected lecture notes and problems from Equilibrium Statistical Physics, taught by Gerhard Mller at the University of Rhode Island Leonard Susskind's lectures on Classical Mechanics recorded on October 15, 2007 at Stanford University; Prof Vu-Quoc, L Statistical Mechanics DeGrand, C DeGrand, C. Recorded September 22, 2008 at Stanford Statistical ideas are then applied to systems of particles in equilibrium to enhance an understanding of the Page 8/31. All of these results come from doing the appropriate integral over $$f = \left(\exp\left[(E(p)-\mu)/(k_BT)\right] \pm 1\right)^{-1}$$. Computational methods, Monte Carlo and detailed balance. For example when we say the system has reached the equilibrium then what we mean is that the system is in equilibrium with its surrounding. The aim of statistical physics is to model systems with an extremely large number of degrees of freedom. Mod-01 Lec-25 Connection between statistical mechanics and ther-modynamics by nptelhrd 11 years ago 1 hour 32,152 views Lecture Series on Classical , Physics , by Prof Leonard Susskind is an American physicist, who is professor of theoretical physics at Stanford University, and founding director of the Stanford Institute for Theoretical Physics Here are Im intending to tidy this up into a book, or rather the rst half of a book. Mechanics (Dover Books on Physics). Read Free Statistical Mechanics Mcquarrie Solution Of Problem Volume Flow Rate, Velocity and Cross Sectional Area Detailed balance in non-equilibrium statistical mechanics (2017) 27. Ensembles. R. E. Wilde and S. Singh: Statistical mechanics. Fundamentals and modern applications. Answer: Euilibrium in thermal physics is relative. Statistical mechanics fundamentals and modern applications, Richard E. Wilde, Surjit Singh, 1998, , 400 pages. Equilibrium Statistical Physics 2nd and 3rd Edition Author(s): Michael Plischke , Birger Bergersen File Specification for 2nd Edition Extension PDF Pages 529 Size 19 MB File Specification for 3rd Edition Extension DJVU Pages 639 Size 3 MB Request Sample Email * Explain Submit Request We try to make prices affordable. Topics include: Thermodynamics, probability theory, kinetic theory, classical statistical mechanics, interacting systems, quantum statistical mechanics, and identical particles. Equilibrium of the system means when we add infinitesimal amount of energy to the whole thing including system and A third course on statistical mechanics, covering critical phenomena, can be found here. Introduction to statistical physics: more is di erent 1.1 Context and Goals This course is an introduction to statistical physics. This Probabilistic mechanism today might seem a more better and appropriate term but the statistical mechanism is firmly entrenched. are separately at equilibrium: if we lift one (or more) constraint the nal entropy, after the re-establisment of equilibrium, must be greater or equal to the initial entropy. Access Free Reif Statistical And Thermal Physics Solutions Manual 29 at 23 11 GMT 7 physics. From the reviews: "This book excels by its variety of modern examples in solid state physics, magnetism, elementary particle physics [] I can Selected lecture notes and problems from Equilibrium Statistical Physics, taught by Gerhard Mller at the University of Rhode Island. Physics 541 W. 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This first volume of Statistical Physics is an introduction to the theories of equilibrium statistical mechanics, whereas the second volume (Springer Ser. * Approach to thermal equilibrium: The circumstances under which a system reaches thermal equilibrium, and how to derive this from If the content Principles Of Equilibrium Statistical Mechanics not Found or Blank , you must refresh this page manually or visit our sister site Principles Of Equilibrium Statistical Mechanics materials Experimental Statistical Mechanics Lectures on Statistical Mechanics - S3 Important problems of Statistical Mechanics #SMLec-3 equilibrium and non-equilibrium statistical physics, and the universal nature of thermodynamic processes, from 184 Classical equilibrium statistical mechanics where (N,V E) is the number of states with energy as dened already in the microcanonical ensemble. Statistical mechanics is often Quantum statistics. Equilibrium Statistical Physics Free and easy access to complete set of presentable lecture notes and exercises is available on URI Digital Commons (downloadable pdf files covering Statistical Mechanics is a probabilistic approach to equilibrium properties of large numbers of degrees of freedom. The book is self-contained and is suitable for beginning graduate students in physics and materials science or undergraduates who have taken an introductory course in statistical mechanics. Solid-State Sci., Vol. There is the problem that strict ergodicity is not true of realistic systems.
equilibrium statistical physics, to a much larger variety of phases and phase transi-tions than was previously the case for textbooks of statistical mechanics. Statistical Mechanics Lecture 1 Statistical Mechanics Lecture 1 von Stanford vor 7 Jahren 1 Stunde, 47 Minuten 372 Walter Lewin's lectures on Classical Mechanics, as taught in 1999; Prof Balakrishnan, Department of Physics, Page 10/23 Lecture Notes in Statistical Mechanics- Lecture 4A - Methods of Statistical Mechanics 15 (2002) 1-271 C 15 Statistical Physics I: Equilibrium Statistical Mechanics, Edition 2 - Ebook written by Morikazu Toda, Ryogo Kubo, Nobuhiko Saito. No previous knowledge of thermodynamics or the molecular Statistical Mechanics II: Statistical Physics of Fields. 0. Course: Methods of Statistical Physics-- develops the basic principles of equilibrium statistical mechanics and their Microcanonical ensemble in quantum Statistical Mechanics: Equipartition theorem. The study of Equilibrium Statistical Mechanics Results in Various Limits. The probability distribution of the microscopic states of the system, p ( { O i }), is needed to estimate the observables { O i }. There are books written on equilibrium statistical mechanics, and people dedicate their lives doing research on the same. In equilibrium statistical physics (often called statistical mechanics) we consider the behavior of a system of a large number of particles in equilibrium with its surroundings (a heatbath).
Download Ebook Statistical Mechanics By S K Sinha Mod-01 Lec-01 Recapitulation of equilibrium statistical mechanics Mod-01 Lec-01 Recapitulation of equilibrium statistical mechanics door nptelhrd 4 jaar geleden 50 minuten 39 It covers all of classical This Stanford Continuing Studies course is the first of a six-quarter sequence of These are lecture notes for PHYS 559, Advanced Statistical Mechanics, which Ive taught at McGill for many years. Answer (1 of 2): It is impossible to give you a satisfactory answer. Introduction to statistical physics: more is di erent 1.1 Context and Goals This course is an introduction to statistical physics. Illustrative examples based on simple materials and photon systems elucidate the Free energies and thermodynamics. Book description. Select search scope, currently: catalog all catalog, articles, website, & more in one search; catalog books, media & more in the Stanford Libraries' collections; articles+ journal articles & other e-resources English.
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# Finding the real solutions using the ones from a complexification
Let
$$f(x) = 2(1+\cos x - x \sin x) \, .$$
The equation $f(x)=0$ has a set of solutions given by $x = (2n - 1)\pi, \, n \in \mathbb{Z}$, easy to find since $\sin x = 0$ at these roots. To find all solutions we need to solve the equation without any immediate tricks. Reduce and Solve are unable to deal with algebraic solutions, so
NSolve[2 (1 + Cos[x] - x Sin[x]) == 0 && 0 < x < 10, x] => {{x -> 1.30654}, {x -> 3.14159}, {x -> 6.58462}, {x -> 9.42478}}
Now, it'd be the happiest day of my life to find a complete algebraic solution. I decided to see what I could extract from complexifying $f$, that is,
$$\tilde{f}(\tilde{x}) = 2(1+ e^{i \tilde{x}} + i \tilde{x} e^{i \tilde{x}} ) \, ,$$
where the $\tilde{}$ denotes a complex number (notice both $f$ and $x$ are complexified). It is evident both that $\mathrm{Re}[\tilde{f}] = f$ and that $\tilde{f} = \mathrm{Re}[\tilde{f}] + i\, \mathrm{Im}[\tilde{f}]$.Reduce works for this case:
Reduce[2 (1 + Exp[I x] + I x Exp[I x]) == 0, x] => C[1] \[Element] Integers && x == I - I ProductLog[C[1], -E]
but I have no idea whether or not it is possible to find solutions for the real equation from this complex solution set or if it's just plain useless. Can someone help?
P.S.: I had problems to decide whether or not this question was suited for Math.SE (since it is probably about very basic complex analysis), but since I'm using Mathematica to find the solutions I decided to post it here. Feel free to move it in case it doesn't belong here.
• Ponder on the result of FullSimplify[1 + Exp[I x] + I x Exp[I x] /. x -> (2 k - 1) π, k ∈ Integers]. – J. M. will be back soon Aug 31 '17 at 1:32
• @J.M. I will... But if you have a specific answer, please say it. That's not homework :) – QuantumBrick Aug 31 '17 at 16:08
• You want the zero of the real part. What you derived with Reduce[] zeroes the real and the imaginary part simultaneously. The solution you were looking for does not zero the imaginary part. That was my point. – J. M. will be back soon Aug 31 '17 at 16:26
You can't get a complete algebraic rule for the solution, since there is no one for x == Cot[x/2]
Factor the f[x] into 3 factors with TrigFactor
g[x_] = f[x] // TrigFactor
(* -4 Cos[x/2] (-Cos[x/2] + x Sin[x/2]) *)
Solve the first part -4 Cos[x/2] for x to get the known general rule x = (2 n-1) Pi
Solve[g[x][[1 ;; 2]] == 0, x, Reals]
(* {{x -> ConditionalExpression[2 (-(\[Pi]/2) + 2 \[Pi] C[1]),
C[1] \[Element] Integers]}, {x ->
ConditionalExpression[2 (\[Pi]/2 + 2 \[Pi] C[1]),
C[1] \[Element] Integers]}} *)
The third factor is x == Cot[x/2] and roots do not follow a general rule, but can be solved for a restricted x-range
Solve[g[x][[3]] == 0 && 0 <= x < 30, x, Reals]
(* {{x -> Root[{Cos[#1/2] - Sin[#1/2] #1 &,
1.30654237418880620223}]}, {x ->
Root[{Cos[#1/2] - Sin[#1/2] #1 &, 6.5846200425641731922}]}, {x ->
Root[{Cos[#1/2] - Sin[#1/2] #1 &, 12.7232407841313299947}]}, {x ->
Root[{Cos[#1/2] - Sin[#1/2] #1 &, 18.9549714108415918088}]}, {x ->
Root[{Cos[#1/2] - Sin[#1/2] #1 &, 25.212026888550825656}]}} *)
• So, you say I can't get useful info from the complexification's solution, right? Because I already solved it numerically when asking the question. – QuantumBrick Aug 31 '17 at 16:06
• I think the main point is that you can't get algebraic solutions except for the regular (2*n-1)*Pi roots, because the solutions are transcendental and not algebraic. – Thies Heidecke Nov 29 '17 at 15:14
eqn = 2 (1 + Cos[x] - x Sin[x]) == 0;
Selecting the solutions within the interval that are not Root objects
soln1 = Select[x /. Solve[{eqn, 0 <= x <= 50}, x], FreeQ[#, Root] &]
(* {π, 3 π, 5 π, 7 π, 9 π, 11 π, 13 π, 15 π} *)
Using FindSequenceFunction to generalize the sequence
soln2 = FindSequenceFunction[soln1, n] // Simplify
(* (-1 + 2 n) π *)
Verifying the solution for all integers
Assuming[Element[n, Integers], eqn /. x -> soln2 // Simplify]
(* True *)
This does not include the solutions represented as Root objects. |
×
# KMXOR - Editorial
Author: Yuri Shilyaev
Editorialist: Yury Shilyaev
Easy
# PREREQUISITES:
Xor operation, constructive.
# PROBLEM:
You are given two integers $N$ and $K$. Your task is to construct a sequence $g_1, \dots, g_N$, that $1 \le g_i \le K$ for all $i$ and $g_1 \oplus g_2 \oplus \cdots \oplus g_N$ is maximum possible.
# QUICK EXPLANATION:
Let $m$ be the maximum integer that $2^m \le K$. The maximum possible answer is somewhere near $2^{m + 1} - 1$.
# EXPLANATION:
Let's first print $2^m$ and $2^m - 1$. This numbers give xor $2^{m + 1} - 1$. Now, if let's complete the sequence with ones. The xor would not change if $N$ is even, otherwise, let's swap $2^m - 1$ we added with $2^m - 2$.
Of course, we should also consider some corner cases, on which our solution doesn't work. They are $N = 1; K = 1; K = 2; K = 3$.
# AUTHOR'S AND TESTER'S SOLUTIONS:
Author's solution can be found here.
Tester's solution can be found here.
This question is marked "community wiki".
18
accept rate: 0%
19.3k348495534
2 One of the useful cases- try it if you decide to give up debugging after hours and hours- View Content Also, if you'd want to see my solution - https://www.codechef.com/viewsolution/18611082 The only case handling done is for $K=1$ (is it redundant? Try and find out!) , and if number of $1's$ to be added is even or odd. No other specific edge case handling was needed in my approach. Its based on adding $K$ first, and then adding a number $X$ such that $K \oplus X={2}^{a}-1$ where $a$ is number of bits in binary representation of $K$ . Rest of places were filled with $1's$ with special handling if number of $1's$ turns out to be odd making maximum value ${2}^{a}-2$ answered 21 May, 19:16 14.4k●1●13●52 accept rate: 18% 1 how to add this view content and hide content @vijju123 ... I wanted to make this type thing for test case in my editorial but didn't knew how to add that.... thanks... (21 May, 19:21) 1 Keep the content to be hidden in between [@hide] [@/hide] . (Remove the @ signs for it to work) If there still exists any issue, ping me, I will mail you a screenshot :) (21 May, 19:33) Thanks... :) (21 May, 19:39) 1 missing Chef vijju's corner in editorials... (21 May, 19:41) I used the same idea of choosing K and then some other combination of values :D The code can be made much shorter though: 18623918 (21 May, 19:47) meooow ♦6★ haha what a short code @meooow ... and python is also very much useful... (21 May, 19:50) 2 You dont know how happy that makes me feel <33333333333 Sadly, I dont think I would be getting any slot for editorials till August :( . I thought I could do a few contests since summer vacations means lot of free time but well, lets hope for the best :). (21 May, 19:51) When writing my code I am more focused on "Ok , what does this part do? What guarantees correctness of concept? What guarantees correctness of implementation?" So I elaborately write it. And people complain that they cant understand what I'm doing when I write shorter ones XD (21 May, 20:19) yeah people do tell they can't understand when code is more short, I agree... (21 May, 20:25) Can you suggest me something to improve my editorial ?? (21 May, 20:26) 1 Let me have a look. Will suggest if I find any :) (21 May, 20:28) okay thanks :) u ll cuz its my first attempt :D (21 May, 20:31) 1 Thank you so much @vijju123 ... It was a great help for me... :D (21 May, 21:13) 1 Happy to help bro :) (21 May, 21:34) showing 5 of 14 show all
1 The problem can be done like this also https://letsdocp.wordpress.com/2018/05/21/codechef-problem-best-cake-ever/ answered 21 May, 13:35 3★vjvjain0 61●6 accept rate: 0%
1 Can anyone tell whats wrong with my solution its getting WA https://www.codechef.com/viewsolution/18702042 answered 30 May, 22:03 11●1 accept rate: 0% try this test case and have a look at editorial(unofficial) if u r not missing anything(specially corner cases that I mentioned)... (31 May, 00:14) I m getting right answers in all the test cases mentioned above.Please have a look. 1 1 1 1 1 1 1 1 1 1 2 1 1 1 2 1 1 1 1 1 3 1 1 1 1 2 1 1 1 1 1 1 2 12 1 1 8 7 8 7 1 1 (01 Jun, 17:44) your logic and output seems correct... dk what does codechef found wrong :( (01 Jun, 19:47)
0 I am not able to understand the error in my solution.. the code is given in the link below https://ideone.com/4FqUwD answered 21 May, 13:24 11●2 accept rate: 0%
0 I did the same solution... I tried to explain this in bit detail... link to my unofficial editorial... answered 21 May, 14:00 1.1k●1●8 accept rate: 27% try this test case and do let me know if there wasn't any mistake and i can try to help u... (21 May, 14:21) FOR people getting wrong answer and not able to find their mistake after try for hours.. though i suggest that u should try to solve it by your own.. that the purpose of competitive coding.. (21 May, 14:22)
0 here I have my edited code... https://ideone.com/7192dd .....i am still not able to solve the error to my problem answered 21 May, 14:46 11●2 accept rate: 0% 1 solved.. typecast output of power function.... (21 May, 15:31) 1 thanks mate! (21 May, 15:56)
0 Can someone please provide a test case that fails? It passes all the below mentioned test cases and I am still getting WA. 9 4 1 5 1 5 2 6 2 5 3 6 3 5 8 2 8 4 8 answered 22 May, 08:14 25●2 accept rate: 14% Will reply u on ur comment on my editorial shortly.. (22 May, 09:47) alright! will wait. Thankyou. :) (22 May, 09:49) Well thank me if I solve it :) (22 May, 09:51) solved... keeping k==1 k==2 and k==3 condition before n==2 helps... (because 2 1 test case failed..) (22 May, 10:21)
0 whats wrong in my solution .... someone plz help thnx in advance.... https://www.codechef.com/viewsolution/18638067 answered 24 May, 01:42 1 accept rate: 0% try this test case and do let me know if there wasn't any mistake... (25 May, 01:28) all above test cases pass .. plz if possible check for once ..... I even type-casted pow function value to long long .... https://www.codechef.com/viewsolution/18651992 thnx in advance ... (26 May, 07:17) this is the edited one with minor changes ..... plz see whats the mistake i am getting WA ...... https://www.codechef.com/viewsolution/18652030 (26 May, 07:38) Okay bro you do have read above answer.. that's nice.. let me check.. (26 May, 10:07) i have further modified the code .... plz check ..... .... https://www.codechef.com/viewsolution/18676006..... i have got many WA .... ... plz help ...... thnx in advance !!! (27 May, 14:32)
0 can anyone describe this editorial in brief please...... answered 26 May, 23:17 59●4 accept rate: 0% HERE is my unofficial editorial(I have described whole soln there) my solution was almost same as this editorial soln... (28 May, 03:22)
Please write Editorials in such a way so that a beginner like me can easily understand the explaination. Codechef should look into this matter.
537
accept rate: 0%
# HERE
(14 Jun, 01:18)
0 Hi, I am getting wrong answer for the following problem. https://www.codechef.com/problems/KMXOR My solution : https://www.codechef.com/viewsolution/19707356 can someone please tell what is wrong in my solution? Thanks answered 13 Aug, 14:22 1 accept rate: 0%
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## anonymous 5 years ago 3x + 5 – 2x – 4 ------ ------ = 3 6 5
1. anonymous
$(3x+5)/6$is $(3/6)x +(5/6)$ you can figure out the rest.
2. anonymous
ahhh i'm confused :/
3. anonymous
first, u have to diminish the fraction such that: $1\div x + 1 \div y = t$ this equal to $1\div x \times xy + 1\div y \times xy = t \times xy$ thus $y + x = txy$ |
08-216 Denis Gaidashev, Hans Koch
Period doubling in area-preserving maps: an associated one-dimensional problem (4063K, Postscript) Nov 16, 08
Abstract , Paper (src), View paper (auto. generated ps), Index of related papers
Abstract. It has been observed that the famous Feigenbaum-Coullet-Tresser period doubling universality has a counterpart for area-preserving maps of $\field{R}^2$. A renormalization approach has been used in a computer-assisted proof of existence of an area-preserving map with orbits of all binary periods by J.-P. Eckmann, H. Koch and P. Wittwer (1982 and 1984). As it is the case with all non-trivial universality problems in non-dissipative systems in dimensions more than one, no analytic proof of this period doubling universality exists to date. We argue that the period doubling renormalization fixed point for area-preserving maps is almost one dimensional, in the sense that it is close to the following Henon-like map: $$H^*(x,u)=(\phi(x)-u,x-\phi(\phi(x)-u )),$$ where $\phi$ solves $$\phi(x)={2 \over \lambda} \phi(\phi(\lambda x))-x.$$ We then give a proof'' of existence of solutions of small analytic perturbations of this one dimensional problem, and describe some of the properties of this solution. The proof'' consists of an analytic argument for factorized inverse branches of $\phi$ together with verification of several inequalities and inclusions of subsets of $\field{C}$ numerically. Finally, we suggest an analytic approach to the full period doubling problem for area-preserving maps based on its proximity to the one dimensional. In this respect, the paper is an exploration of a possible analytic machinery for a non-trivial renormalization problem in a conservative two-dimensional system.
Files: 08-216.src( 08-216.keywords , Doubling.ps ) |
# The RMS survey: ammonia mapping of the environment of massive young stellar objects
@article{Urquhart2015TheRS,
title={The RMS survey: ammonia mapping of the environment of massive young stellar objects},
author={J. S. Urquhart and Charles C. Figura and Toby Moore and Timea Csengeri and S. L. Lumsden and Thushara G. S. Pillai and Maggie Thompson and David J. Eden and Larry Morgan and Mpifr and Wartburg College and Ljmu and Leeds and Herts and Universit'e de Strasbourg and Met Office},
journal={Monthly Notices of the Royal Astronomical Society},
year={2015},
volume={452},
pages={4029-4053}
}
• Published 8 July 2015
• Physics, Geology
• Monthly Notices of the Royal Astronomical Society
We present the results of ammonia observations towards 66 massive star forming regions identified by the Red Midcourse Space Experiment Source survey. We have used the Green Bank Telescope and the K-Band Focal Plane Array to map the ammonia (NH3) (1,1) and (2,2) inversion emission at a resolution of 30 arcsec in 8 arcmin regions towards the positions of embedded massive star formation. We have identified a total of 115 distinct clumps, approximately two-thirds of which are associated with an…
The RMS survey: Ammonia mapping of the environment of young massive stellar objects – II★
• Physics
Monthly Notices of the Royal Astronomical Society
• 2018
We present the results from NH3 mapping observations towards 34 regions identified by the Red MSX Source (RMS) survey. We have used the Australia Telescope Compact Array to map ammonia (1, 1) and (2,
ATLASGAL - Ammonia observations towards the southern Galactic Plane
• Physics
• 2017
The initial conditions of molecular clumps in which high-mass stars form are poorly understood. In particular, a more detailed study of the earliest evolutionary phases is needed. The APEX Telescope
A Search for High-mass Protostellar Objects in Cold IRAS Sources
• Physics
The Astronomical Journal
• 2018
We present the results of CS J = 2 -> 1 mapping observations toward 39 massive star-forming regions selected from the previous CO line survey of cold IRAS sources with high-velocity CO flows along
KFPA Examinations of Young STellar Object Natal Environments (KEYSTONE): Hierarchical Ammonia Structures in Galactic Giant Molecular Clouds
• Physics
The Astrophysical Journal
• 2019
We present initial results from the K-band focal plane array Examinations of Young STellar Object Natal Environments (KEYSTONE) survey, a large project on the 100-m Green Bank Telescope mapping
ATLASGAL - properties of a complete sample of Galactic clumps
• Physics
• 2018
Abridged: ATLASGAL is an unbiased 870 micron submillimetre survey of the inner Galactic plane. It provides a large and systematic inventory of all massive, dense clumps in the Galaxy (>1000 Msun) and
H2O Southern Galactic Plane Survey (HOPS): Paper III - properties of dense molecular gas across the inner Milky Way
• Physics
• 2017
The H2O Southern Galactic Plane Survey (HOPS) has mapped 100 deg2 of the Galactic plane for water masers and thermal molecular line emission using the 22 m Mopra telescope. We describe the automated
SEDIGISM: Structure, excitation, and dynamics of the inner Galactic interstellar medium
• Physics
• 2017
Context. The origin and life-cycle of molecular clouds are still poorly constrained, despite their importance for understanding the evolution of the interstellar medium. Many large-scale surveys of
ATLASGAL -- A Galaxy-wide sample of dense filamentary structures
• Physics
• 2016
[Abridged] Aims. We study the properties of filamentary structures from the ATLASGAL survey. Methods. We use the DisPerSE algorithm to identify spatially coherent structures located across the
Kinetic temperature of massive star forming molecular clumps measured with formaldehyde
• Physics
• 2016
For a general understanding of the physics involved in the star formation process, measurements of physical parameters such as temperature and density are indispensable. The chemical and physical
Interaction Between HII Region and AFGL333-Ridge: Implications to the Star Formation Scenario
• Physics
• 2016
We investigated the star formation activities in the AFGL333 region, which is in the vicinity of the W4 expanding bubble, by conducting NH3 (1,1), (2,2), and (3,3) mapping observations with the 45 m
## References
SHOWING 1-10 OF 73 REFERENCES
The RMS Survey: Ammonia and water maser analysis of massive star forming regions. ⋆
• Physics
• 2011
The Red MSX Source (RMS) survey has identified a sample of∼1200 massive young stellar objects (MYSOs), compact and ultra compact Hii regions from a sample of∼2000 MSX and 2MASS colour selected
Ammonia from cold high-mass clumps discovered in the inner Galactic disk by the ATLASGAL survey
• Physics
• 2012
The APEX Telescope Large Area Survey: The Galaxy (ATLASGAL) is an unbiased continuum survey of the inner Galactic disk at 870 \mu m. It covers +/- 60 deg in Galactic longitude and aims to find all
The RMS survey : Galactic distribution of massive star formation
• Physics
• 2014
We have used the well-selected sample of~1750 embedded, young, massive stars identified by the Red MSX Source (RMS) survey to investigate the Galactic distribution of recent massive star formation.
VERY LARGE ARRAY OBSERVATIONS OF AMMONIA IN HIGH-MASS STAR FORMATION REGIONS
• Physics
• 2014
We report systematic mapping observations of the NH3 (1, 1) and (2, 2) inversion lines toward 62 high-mass star-forming regions using the Very Large Array (VLA) in its D and DnC array configurations.
A distance-limited sample of massive star-forming cores from the RMS survey
• Physics
• 2015
We analyse C18O (J = 3-2) data from a sample of 99 infrared (IR)-bright massive young stellar objects (MYSOs) and compact H II regions that were identified as potential molecular-outflow sources in
THE RED MSX SOURCE SURVEY: THE MASSIVE YOUNG STELLAR POPULATION OF OUR GALAXY
• Physics
• 2013
We present the Red MSX Source survey, the largest statistically selected catalog of young massive protostars and H II regions to date. We outline the construction of the catalog using mid- and
ATLASGAL - Kinematic distances and the dense gas mass distribution of the inner Galaxy
• Physics
• 2015
The formation of high mass stars and clusters occurs in giant molecular clouds. Objects in evolved stages of massive star formation such as protostars, hot molecular cores, and ultracompact HII
ATLASGAL — towards a complete sample of massive star forming clumps ⋆
• Physics, Geology
• 2014
By matching infrared-selected, massive young stellar objects (MYSOs) and compact HII regions in the RMS survey to massive clumps found in the submillimetre ATLASGAL survey, we have identified ∼1000
Mapping the column density and dust temperature structure of IRDCs with Herschel
• Physics, Environmental Science
• 2010
Infrared dark clouds (IRDCs) are cold and dense reservoirs of gas potentially available to form stars. Many of these clouds are likely to be pristine structures representing the initial conditions
The RMS Survey: H2O masers towards a sample of southern hemisphere massive YSO candidates and ultra compact HII regions
• Physics, Geology
• 2009
Context. The red MSX source (RMS) survey has identified a large sample of candidate massive young stellar objects (MYSOs) and ultra compact (UC) HII regions from a sample of ∼2000 MSX and 2MASS |
## 2.2 Subjectivity is not the key
The concepts of subjectivity and objectivity indeed characterize both statistical paradigms in differing ways. Among Bayesians there are those who are immersed in subjective rationality (, , , ), but others who adopt objective prior distributions such as Jeffreys’, reference, empirical or robust (, , , ) to operationalize Bayes’ rule, and thereby weight quantitative (data-based) evidence. Among Frequentists, there are choices made about significance levels which, if not explicitly subjective, are typically not grounded in any objective and documented assessment of the relative losses of Type I and Type II errors.14 In addition, both Frequentist and Bayesian statisticians make decisions about the form of the data generating process, or “model,” which - if not subject to rigorous diagnostic assessment - retains a subjective element that potentially influences the final inferential outcome. Although we all know that by definition a model is a schematic and simplified approximation to reality,
“Since all models are wrong the scientist cannot obtain a correct one by excessive elaboration. On the contrary following William of Occam he should seek an economical description of natural phenomena.”
We also know that “All models are wrong, but some are useful” , that is why model diagnostics are important. This task can be performed in both approaches. Particularly, the Bayesian framework can use predictive p–values for absolute testing (, ) or posterior odds ratios for relative statements (, ). This is because the marginal density, conditional on data, is interpreted as the likelihood of the prior distribution .
In addition, what is objectivity in a Frequentist approach? For example, why should we use a 5% or 1% significance level rather than any other value? As someone said, the apparent objectivity is really a consensus . In fact “Student” (William Gosset) saw statistical significance at any level as being “nearly valueless” in itself . But, this is not just a situation in the Frequentist approach. The cut-offs given to “establish” scientific evidence against a null hypothesis in terms of $$log_{10}$$ scale or $$log_{e}$$ scale are also ad hoc.
Although the true state of nature in Bayesian inference is expressed in “degrees of belief,” the distinction between the two paradigms does not reside in one being more, or less, subjective than the other. Rather, the differences are philosophical, pedagogical, and methodological.
### References
Bayarri, M., and J. Berger. 2000. “P–Values for Composite Null Models.” Journal of American Statistical Association 95: 1127–42.
Bayes, T. 1763. “An Essay Towards Solving a Problem in the Doctrine of Chances.” Philosophical Transactions of the Royal Society of London 53: 370–416.
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1. Type I error is rejecting the null hypothesis when this is true, and the Type II error is not rejecting the null hypothesis when this is false.↩︎ |
# Find the area bounded by the curve y=2x^2-6x and y=-x^2+9?
Mar 15, 2018
$\text{Area} = 32$
#### Explanation:
First, it is a good idea to graph the given curves:
We need to evaluate the area bounded by these curves, i.e find the area in between them.
That's the area between their points of intersection, so we need to find that: their points of intersection:
$R i g h t a r r o w 2 {x}^{2} - 6 x = - {x}^{2} + 9$
$R i g h t a r r o w 3 {x}^{2} - 6 x - 9 = 0$
$R i g h t a r r o w {x}^{2} - 2 x - 3 = 0$
$R i g h t a r r o w {x}^{2} + x - 3 x - 3 = 0$
$R i g h t a r r o w x \left(x + 1\right) - 3 \left(x + 1\right) = 0$
$R i g h t a r r o w \left(x + 1\right) \left(x - 3\right) = 0$
$\therefore x = - 1 , 3$
We don't really need to find the points of intersection; the $x$-intercepts will suffice.
This is because the curves are already specified in terms of $y$.
Now we can start evaluating the definite integrals of these curves in the interval $\left[- 1 , 3\right]$:
$R i g h t a r r o w {\int}_{- 1}^{3}$ $\left(\left(- {x}^{2} + 9\right) - \left(2 {x}^{2} - 6 x\right)\right)$ $\mathrm{dx}$
$= {\int}_{- 1}^{3}$ $\left(- 3 {x}^{2} + 6 x + 9\right)$ $\mathrm{dx}$
$= | - {x}^{3} + 3 {x}^{2} + 9 x {|}_{- 1}^{3}$
$= \left(- {\left(3\right)}^{3} + 3 {\left(3\right)}^{2} + 9 \left(3\right)\right) - \left(- {\left(- 1\right)}^{3} + 3 {\left(- 1\right)}^{2} + 9 \left(- 1\right)\right)$
$= \left(- 27 + 27 + 27\right) - \left(1 + 3 - 9\right)$
$= \left(27\right) - \left(- 5\right)$
$= 32$
Therefore, the area bounded by the curves is $32$. |
# Determinant For ELO Rating.
I'm trying to implement a variant of the ELO system, for a game I'm working on. Giving two players $A$ and $B$ with ratings $R_A$ and $R_B$ respectively, the expectation of $A; E_A$ is given by the formula: $$E_A = \frac{1}{1+10^{\frac{R_B-R_A}{Y}}}\tag{*}$$
The Expectation of $B$ is similarly given by: $$E_B = \frac{1}{1+10^{\frac{R_A-R_B}{Y}}}$$
The different possible game outcomes are given scores: A win is $1.0$, a loss is: $0.0$, and a draw is $0.5$. The actual score of $A$ is $S_A$.
After a match between $A$ and $B$, the new ranking of $A; R'_A$ is given by: $$R'_A = R_A + K(S_A - E_A)$$
Question: For any value of $Y$, what value of $K$ should I choose, such that that value is a constant and makes the player's rating as reliable as possible.
I'm currently using $K = \sqrt{Y}$
$(*)$ Most Chess ELO algorithms use a value of $Y = 400$
• The $*$ was meant to provide information. -_- – Tobi Alafin Dec 17 '16 at 17:30
• Look at the bottom of the question. – Tobi Alafin Dec 17 '16 at 17:34
• Suggest an edit. – Tobi Alafin Dec 17 '16 at 17:38
• As for the question: Wiki has a discussion on the $K$ value problem: en.wikipedia.org/wiki/Elo_rating_system#Mathematical_issues (though only for $Y=400$) – Winther Dec 17 '16 at 17:41
• I read it. $K$ values were seemingly chosen arbitrarily without a mathematical explanation. Also most organisations used different $K$ values for different ranges. Furthermore, due to the pecularities of my game system, I don't use $400$. – Tobi Alafin Dec 17 '16 at 17:44 |
Students need to perform a system at least one senior citizen ticket and discuss it adds a custom themes about the image file type of solving systems of equations by substitution method is not. No players currently in game. These activities focus on solving systems of equations by substitution or. Want even if a music store.. "/>
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Solving by Substitution: 1. Solve ONE of the equations for x or for y a. This step can be skipped if one of the equations is already solved for x or y 2. Substitute the result into the other. A free maths worksheet all about solving simultaneous equations using the elimination method. Each question in this beginners worksheet can be solved without the need to multiply either equation. #maths #math #mathematics #gcse #simultaneous #equations #solve #algebra #simultaneousequations #worksheet #free #freeworksheet #worksheets #freeworksheets.
Solve the lower triangular system Ly = b for y by forward substitution. 2. Solve the upper triangular system Ux = y for x by back substitution. Moreover, consider the problem AX = B (i.e., many different right-hand sides that are associated with the same system . Cellular Respiration Worksheet Answers. view answers substitution method can be applied in four steps step 2: pick a different two equations and eliminate the same variable step 2: using the y intercept (b) and slope (m) graph both the elimination method for solving systems of linear equations uses the addition property of equality the elimination method for solving systems of linear. Chapter 5 pdf big ideas math algebra 1 answers solving systems of linear equations lesson by the substitution method grade 8 algebraically using lessons blendspace examples. This Solving Systems of Linear Equations by Substitution Additional Practice Sheet provides 3 guided examples, 9 non-guided examples, and 1 challenge example to allow students to practice their skills at solving systems of linear equations by substitution. Examples include systems with one unique solution, no solutions, and infinitely many soluti. . 1. $3.50. PDF. There are 3 worksheets on solving Systems of Equations. The first worksheet includes how to solve a system of equations by graphing. The second worksheet includes. Substitution Worksheets Word Problems Worksheets These free systems of equations worksheets will help you practice solving real-life systems of equations using the " substitution " method. Each set of free algebra worksheets is a progressive series that starts with simple problems featuring positive integers. 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Fill in the blanks to solve each system by substitution. 1. 1 y 3x y x 4 Check your answer. 3. 1 y 4x. More : Solving Systems by Substitution. Fill in the blanks to solve each system by substitution. 1. 1 y 3x y x 4 Check your answer. 3. 1 y 4x.. These free systems of equations worksheets will help you practice solving real-life systems of equations using the “ substitution ” method. Each set of free algebra worksheets is a. MIT grad shows how to use the substitution method to solve a system of linear equations (aka. simultaneous equations). To skip ahead: 1) For a BASIC SUBSTITU. Solving systems of equations is a vital algebra skill youll learn in 8th grade Only RUB 220 y Worksheet by Kuta Software LLC Kuta Software - Infinite Algebra 1 Name_____ Solving Systems of Equations by Substitution Date_____ Period____ Solve each system by substitution In this method, we must get rid of one variable in order to find the other Trumpet. 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Download File PDF Solving Systems Of Equations By Substitution Worksheet Answers www.modernh.modernh.com Applications in School MathematicsMerrill Algebra 1 Applications and Connections Reteaching MastersElementary AlgebraHands-On Algebra!Algebra für DummiesLineare Algebra für DummiesConcise Guide to Computing FoundationsHuman-. Lesson 1 – Solving Equations Review with Answers This handout focuses on solving systems of linear equations with one solution When solving a system of equations with 2 variables (i The pair of students will each solve different problems, but each row of problems will have the same answer Worksheet # 3: Solve the Linear Equations Using the Substitution.
1 To solve systems using substitution Examples 1 Using Substitution 2 Using Substitution and the Distributive Property 3 Real-World Problem Solving Math Background Solving systems using substitution begins with the concept of evalu-ating variable expressions. Here a variable is replaced with its value in terms of the other variable. More Math. U substitution worksheet with answers U substitution worksheet with answersN, This is a set of 2 solving systems of equations by using substitution RIDDLE worksheets. kids. In this unit, you will learn how to solve systems of equations graphically and algebraically. Systems of equations are used in real-world situations where two variables need to be determined. Some examples of problems where systems of equations may be applied are determining the maximum profit of sales and predicting the weather. Worksheet by Kuta Software LLC Math 8 Solve Systems by Substitution ... [LuLlCc.] J JAylplg mrsiJgrhXt_sz FryeVsYe_rEvxeedx. Solve each system by substitution. 1) 2x - 2y = 4 y = -3x + 14 2) 2x + 4y = 0 y = -4x - 21 3) y = -3x + 20-x - 8y = 24 4) 24x - 3y = -30 y = 8x + 10 5) 4x + 2y = 22 ... Answers to Solve Systems by Substitution 1) (4, 2) 2. Each of our math worksheet features an answer key as well as examples that illustrate how to solve the exercises. Systems of linear equations maze answer key.the above ncert cbse and kvs worksheets for class 9 linear equations in two variables will help you to improve marks by clearing linear equations in two variables concepts and also improve. Define variables for each question. Write a system of equations. Tell whether you are using "elimination" or "substitution" to solve. Solve. (Show work on separate paper and write answer as a sentence below the question.) 47) Stefan's school is selling tickets to the annual dance competition. On the first day of ticket sales the school. Solving systems of equations by elimination or by substitution worksheets pdf printable, solving and graphing systems of linear equations word problems, Cramer's rule. x – y = − 6. ... Elimination method using addition and subtraction. 3 Practice A) Day 1 Worksheet Answers Solve Systems of Equations by Elimination Day 2 Notes. Question 1. This is a complete unit used to introduce your classes to Solving Systems of Equations. Round to the Nearest Hundred Word Problems Worksheet from rounding word problems worksheets, image source: havefunteaching. Word Problems with Variables on Both Sides. Control the pace so everyone advances through each question together. Systems of Equations Worksheet 4 - This 9 problem algebra worksheet helps you practice solving systems of equations by using the "elimination" method. All of the equations are in the standard form of ax + by = c or ax - by = c. Some of the systems have no solution or infinite solutions. A few of the solutions feature decimals or negative numbers. Systems of Equations Worksheet 4 - This 9 problem algebra worksheet helps you practice solving systems of equations by using the "elimination" method. All of the equations are in the standard form of ax + by = c or ax - by = c. Some of the systems have no solution or infinite solutions. A few of the solutions feature decimals or negative numbers. D.Russell Use the Quadratic Formula to Solve the Equations(Answers on 2nd page of PDF. Sample questions ... PowerPoint begins by reviewing solving systems with substitution. ... , puzzle solving, highly active and engaging activity and/or as an alternative to worksheet review for Linear Systems. If you'd like to receive 50% off your next. Students will practice the substitution method for solving systems of equations in this Independent Practice Worksheet. There are 18 systems of equations problems which are already set up for the substitution method. This resource is ideal for the introduction to the substitution method for system. Systems of Equations Worksheet 4 – This 9 problem algebra worksheet helps you practice solving systems of equations by using the “elimination” method. All of the equations are in the standard form of ax + by = c or ax – by = c. Some of the systems have no solution or infinite solutions. A few of the solutions feature decimals or .... Students will practice solving sytems of linear equations uging substitution . Example Questions Directions: Solve each system of linear equations using substitution. Challenge Problems Other Details This is a 4 part worksheet: Part I Model Problems. Part II Practice. Part III Challenge Problems. Part IV Answer Key. Resources. All groups and messages .... What is the first step when solving with elimination? 1. Add or subtract the equations. 2. Multiply the equations. 3. Plug numbers into the equation. 4. Solve for a variable. 5. Check your answer. 6. Determine which variable to eliminate. 7. Put the equations in. Free worksheet(pdf) and answer key on solving systems of equations--using substitution , elimination and a graph. Solving Systems with Linear Combination or Elimination . In the Roots of a Quadratic Gizmo, students will find the roots of a quadratic using its graph or the quadratic formula.. Join us on this flipped math lesson where we visually explore how to find a solution to a system of linear equations. For more MashUp Math content, visit htt. These free systems of equations worksheets will help you practice solving real-life systems of equations using the “ substitution ” method. Each set of free algebra worksheets is a. 11 Questions Show answers. Q. What is the first step in solving a system by Substitution? Make sure both equations are in standard form. Make sure at least one equation is solved. Example: Solve the system by substitution $$- 8x + 5y = - 6$$ $$- 3x + y = - 4$$ Solution: We follow the first procedure. We want to choose either of the two equations to begin with. We choose the second equation because it is easier to solve for the variable $$y$$ in that equation. Solving for $$y$$ in the second equation gives us. Define variables for each question. Write a system of equations. Tell whether you are using "elimination" or "substitution" to solve. Solve. (Show work on separate paper and write answer as a sentence below the question.) 47) Stefan's school is selling tickets to the annual dance competition. On the first day of ticket sales the school. To solve systems of equations by substitution simplify the equation to isolate one variable The pair of students will each solve different problems, but each row of problems will have the same answer Solve for x Use substitution to solve each system of equations 0000228634 00000 n Textbook Authors: Hall, Prentice, ISBN-10: 0133500403, ISBN-13: 978-0. Step 1 : First, solve one linear equation for y in terms of x . Step 2 : Then substitute that expression for y in the other linear equation. You'll get an equation in x . Step 3 : Solve this, and you have the x -coordinate of the intersection. Step 4 : Then plug in x to either equation to find the corresponding y -coordinate. Solve a system of equations by substitution. Step 1. Solve one of the equations for either variable. Step 2. Substitute the expression from Step 1 into the other equation. Step 3. Solve the resulting equation. Step 4. Substitute the solution in Step 3 into one of the original equations to find the other variable. Students need to perform a system at least one senior citizen ticket and discuss it adds a custom themes about the image file type of solving systems of equations by substitution method is not. No players currently in game. These activities focus on solving systems of equations by substitution or. Want even if a music store. Steps to solving Systems of Equations by Substitution: Isolate a variable in one of the equations. (Either y = or x =). Substitute the isolated variable in the other equation. This will result in an equation with one variable. Solve the equation. Substitute the solution from step 3 into another equation to solve for the other variable. Solving Systems Of Equations By Substitution Worksheet Answers Author: brookhiser.nationalreview.com-2022-10-06T00:00:00+00:01 Subject: Solving Systems Of Equations By Substitution Worksheet Answers Keywords: solving, systems, of, equations, by, substitution, worksheet, answers Created Date: 10/6/2022 2:22:33 PM. A free maths worksheet all about solving simultaneous equations using the elimination method. Each question in this beginners worksheet can be solved without the need to multiply either equation. #maths #math #mathematics #gcse #simultaneous #equations #solve #algebra #simultaneousequations #worksheet #free #freeworksheet #worksheets #freeworksheets.
Solving Systems Of Equations By Substitution Worksheet Answers Author: brookhiser.nationalreview.com-2022-10-06T00:00:00+00:01 Subject: Solving Systems Of Equations By Substitution Worksheet Answers Keywords: solving, systems, of, equations, by, substitution, worksheet, answers Created Date: 10/6/2022 2:22:33 PM. We wrote the answer as an ordered pair.Solving three-variable, three-equation linear systems is not more difficult than solving the two-variable systems, it does take longer.What we do is. Answer: Explanation: y = 3x – 4 y = x + 2 The solution of thr linear system of equations is the intersection point of the two equations. (3, 5) is the solution of the system of equations. If x = 3, y = 3 (3) – 4 = 9 – 4 = 5; y = 3 + 2 = 5 5 = 5; True Question 2. \left\ {\begin {array} {l}x-3 y=2 \\-3x+9y=-6\end {array}\right. Type below:. To solve a system of equations , you need to figure out the variable values that solve all the equations involved Systems of Linear Equations Worksheets Systems of Linear Equations. Solving systems of equations by elimination or by substitution worksheets pdf printable, solving and graphing systems of linear equations word problems, Cramer's rule. Steps Given. Solving systems of equations by elimination or by substitution worksheets pdf printable, solving and graphing systems of linear equations word problems, Cramer's rule. Steps Given.
Free Download Algebra 1 Solving Systems Of Equations By Elimination Worksheet Some of these worksheets are for students in the 5th-8th grade. These two-step word problem are created using fractions and decimals. Each worksheet contains ten problems. You can access them through any website or print source. This practice worksheet begins with an example that shows how to solve one of the equations for a single variable and then substitute that into an equation to help you solve the system of equations. Then students will apply this process on their own as they solve systems of of equations using the substitution method. For related practice. Solving Linear Systems By Elimination Worksheet Pdf - Example Worksheet www.viajeabariloche.com. systems worksheet solving equations substitution pdf elimination briefencounters linear.Elimination With Multiplication Worksheet Answers - Leonard Burton's cuddlyturb0dog.blogspot.com. elimination solving.
We wrote the answer as an ordered pair.Solving three-variable, three-equation linear systems is not more difficult than solving the two-variable systems, it does take longer.What we do is change the 3x3 system to a 2x2 system by eliminating one of the variables using the elimination, then we solve the 2x2 system as we have done before. Math Worksheets $10.00$8.00. This can be checked by substituting back into both original equations to ensure that the left-hand and right-hand (LHS) and (RHS) sides are equal for these values of x and y. tudent. C. L. earning. S. entre Equations 5/2013 @ SLC 1 of 2 . When we solve a pair of simultaneous equations what we are actually looking for is the intersection of two. Systems of Equations - Substitution Solve each system by substitution. 1) y = 2x + 7 3x − 4y = −13 2) y = −8x − 24 6x + 6y = 24 3) −4x + 8y = −12 y. yellow triangle sign tyson beef processing plant locations eth puzzle catholic confession script fuyu no hanashi guitar chords. The first worksheet includes how to solve a system of equations by graphing. The second worksheet includes how to solve a system of equations by substitution. The third worksheet includes how to solve a system of equations by elimination. Each worksheet contains 10 problems.These worksheets help with self-assessment due to creating a stained .... There are three ways to solve systems of linear equations: substitution, elimination, and graphing. Substitution will have you substitute one equation into the other; elimination will have you add or subtract the equations to eliminate a variable; graphing will have you sketch both curves to visually find the points of intersection. Solving Systems by Substitution Solve the system by substitution. Example 1A: Solving a System of Linear Equations by Substitution y = 3x y = x – 2 Step 1 y = 3x y = x – 2 Both equations are. This process describes the elimination (or addition) method A means of solving a system by adding equivalent equations in such a way as to eliminate a variable Step 2: Pick a different two equations and eliminate the same variable 4 Worksheet by Kuta Software LLC 32) x − y = 0 −6x + 5y = 4 (−4, −4) 33) 4x + 4y = −8 3x − 2y = 19 (3, −5) Solve each system by elimination The. In this worksheet, students will practice solving systems of equations by using elimination or substitution. This quiz cannot be played with flashcards because none of the questions have correct answers. Once students solve the problem, they will find their ans Solving two variable systems of equations worksheets.
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These free systems of equations worksheets will help you practice solving real-life systems of equations using the “ substitution ” method. Each set of free algebra worksheets is a. They bought .... solving systems of equations by substitution worksheet algebra 1 answer key, ... Therefore, we can use the substitution method to solve our systems of equations.. Solve systems of equations using various methods (substitution, elimination, and/or graphing) ... Students will solve several real-world problems involving systems.
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Aug 18, 2022 · Descriptions: Solving Systems by Substitution. Fill in the blanks to solve each system by substitution. 1. 1 y 3x y x 4 Check your answer. 3. 1 y 4x. More : Solving Systems by Substitution. Fill in the blanks to solve each system by substitution. 1. 1 y 3x y x 4 Check your answer. 3. 1 y 4x.. 2 Answers Worksheet Substitution By Equations Of Systems Solving 9-10-2022 Need to ACE the Algebra 2 Exam! Algebra 2 Workbook provides students with the confidence and math skills they need to succeed in any math course they choose and prepare them for future study of Pre–Calculus and.
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Solving by Substitution: 1. Solve ONE of the equations for x or for y a. This step can be skipped if one of the equations is already solved for x or y 2. Substitute the result into the other equation for that variable 3. The new equation will now be all x terms, or..
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You will obtain the value of one of the variables. Substitute this value into either of the original equations to obtain the value of the other variable. Let's jump to an example. Example: Solve the system by substitution. − 8 x + 5 y = − 6. − 3 x + y = − 4. Solution: We follow the first procedure.
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Jan 20, 2018 · Solving systems of equations by substitution kuta warrayat instructional unit maze teaching algebra math and elimination 3 1 2 examples solutions method worksheet with answers printable pdf activity worksheets resources 6 2a isolated using the to solve a system you. Solving Systems Of Equations By Substitution Kuta. Warrayat Instructional Unit ....
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The Substitution Method: There are two different types of systems of equations where substitution is the easiest method. Type 1: One variable is by itself or isolated in one of the equations. The system is solved by substituting the equation with the. Solving Systems of Equations by Substitution Date_____ Period____ Solve each system by substitution. 1) y = 6x − 11 −2x − 3y = −7 2) 2x − 3y = −1 y = x − 1 ... Create your own worksheets like this one with Infinite Algebra 1. Free trial available at KutaSoftware.com. Title: Systems of Equations Substitution.
In this unit, you will learn how to solve systems of equations graphically and algebraically. Systems of equations are used in real-world situations where two variables need to be determined. Some examples of problems where systems of equations may be applied are determining the maximum profit of sales and predicting the weather.. cant explain it but i can give you an example: equation 1: 14x+2y=20 equation 2: 5x-y=8 First step is to get on of the varibles to cancel out so leave the first eqation alone put change the second in this problem so: 5x-y=8 becomes 2(5x-y=8) which equals 10x-2y=16 the next step is to subtract them like this: 14x+2y=20 10x-2y=16 the +2y and the -2y cancel out so the answer.
These free systems of equations worksheets will help you practice solving real-life systems of equations using the “substitution” method The worksheet contains (8) problems on (1) page and a duplicate page with the answers Solving a System of Equations Using Elimination and Multipliers In this page substitution method questions 10 we are.. Each solving the systems.
Improve your math knowledge with free questions in "Solve a system of equations using substitution" and thousands of other math skills.
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Write the answer. x + 6x – 15 = 6 y = 2x – 5 The solution . 7x – 15 = 6 y = 2(3) – 5 is (3,1) 7x – 15 + (15) = 6 + (15) y = 6 – 5 . 7x = 21 y = 1. x = 3. B) Solving a system of equations by substitution without “y =” or “x =”. Given a system of equations: x + 2y = 4. 3x + y = 7. Change one equation to y = or x =. 3x + y. how to hack pokemon into your game. thor palazzo interior. Alg 8.2.- Solving Systems by Substitution.A1.3.12 Represent and solve problems that can be modeled using a system of.
Algebra 1 Substitution Method - Displaying top 8 worksheets found for this concept. Solve the equation from Step 2 for the variable. Algebra 1 Worksheet Solving Systems Of Equations. Solving Systems By Substitution Worksheet Answers Elimination Math 1 y. Solve each system by elimination. This item contains: *** 15 problems of solving systems of linear equations u. 3 z hAHl5lW 2rZiigRhct0s7 drUeAsqeJryv3eTdA. Let's say I have the equation, 3x plus 4y is equal to 2. Follow-Up: Use a graphing calculator to solve a system of.
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Use substitutionworksheet solving systems of equations worksheets are solved in the substitution, solve the salaryoptions for. We solve linear equations by substitution method is. The system by graphing is a quickassessment. What happens when solving by substitution. Solving by substitution worksheet solving areader will check out about knockout.. Systems of Equations (Graphing & Substitution) Worksheet Answers. Solving Systems of Equations by Elimination Notes. System of Equations Day 2 Worksheet Answers. Solving Systems with 3 Variables Notes. p165 Worksheet Key. Systems of 3 Variables Worksheet Key. Linear-Quadratic Systems of Equations Notes.
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Solving Linear Systems By Elimination Worksheet Pdf - Example Worksheet www.viajeabariloche.com. systems worksheet solving equations substitution pdf elimination briefencounters linear.Elimination With Multiplication Worksheet Answers - Leonard Burton's cuddlyturb0dog.blogspot.com. elimination solving.
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Algebra 1 Substitution Method - Displaying top 8 worksheets found for this concept. Solve the equation from Step 2 for the variable. Algebra 1 Worksheet Solving Systems Of Equations.
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1 To solve systems using substitution Examples 1 Using Substitution 2 Using Substitution and the Distributive Property 3 Real-World Problem Solving Math Background Solving systems using substitution begins with the concept of evalu-ating variable expressions. Here a variable is replaced with its value in terms of the other variable. More Math. This can be checked by substituting back into both original equations to ensure that the left-hand and right-hand (LHS) and (RHS) sides are equal for these values of x and y. tudent. C. L. earning. S. entre Equations 5/2013 @ SLC 1 of 2 . When we solve a pair of simultaneous equations what we are actually looking for is the intersection of two.
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Step 1: Solve one equation for one of the variables. Step 2: Substitute the resulting expression into the other equation to replace the variable. Then solve the equation. Step 3: Substitute to solve for the other variable. Step 4: Write the solution as an ordered pair (x, y) and check your answer. Example: x – y = 13. Free worksheet ( pdf ) and answer key on solving systems of equations using substitution Each printable worksheet in this unit of solving systems of equations offers eight sets of equations 1 Assess Your Understanding - Page 715 36 including work step by step written by community members like you 1 Assess Your Understanding - Page 715 36 including work step.
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Systems of Equations (Graphing & Substitution) Worksheet Answers. Solving Systems of Equations by Elimination Notes. System of Equations Day 2 Worksheet Answers. Solving Systems with 3 Variables Notes. p165 Worksheet Key. Systems of 3 Variables Worksheet Key. Linear-Quadratic Systems of Equations Notes. Aug 18, 2022 · Descriptions: Solving Systems by Substitution. Fill in the blanks to solve each system by substitution. 1. 1 y 3x y x 4 Check your answer. 3. 1 y 4x. More : Solving Systems by Substitution. Fill in the blanks to solve each system by substitution. 1. 1 y 3x y x 4 Check your answer. 3. 1 y 4x..
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Solve The System By Substitution. Displaying all worksheets related to - Solve The System By Substitution. Worksheets are Systems of equations substitution, Ws solving systems by substitution isolated, Systems of equations by substitution, Systems of equations substitution date period, Name system of equations substitution 1, Name system of equations substitution 6, Practice solving systems of equations 3 different, Systems of equations.. Maths Worksheets / Algebra Worksheets / Substitution Worksheets with Answers. Our substitution worksheets were created to help students master this complicated area of maths. We have resources available which cover substituting numbers into formulae, magic squares, using negative numbers and more. All of the worksheets we offer are presented.
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. Solve for the variable and perform back substitution to find the value of the other two, in this batch of pdf worksheets. Elimination Method Pair-up the linear equations to eliminate one variable and form two new equations of 2 variables each. Find the value of one variable by eliminating the other.
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-5 (1) + 1 = -5 + 1 = -4, which is equal to the R.H.S of the equation. Since L.H.S = R.H.S so, the values are correct. Check Point Solve by elimination method: x + y = 1 x – y = 3 Solve by substitution method: x + 2 y = 2 -4 x + 3 y = 25 Solve by elimination method: 2 x + 5 y = -4 3 x – y = 11 Solve by substitution method: 5 x + 2 y = 0 x – 3 y = 0. SOLVING LINEAR SYSTEMS SUBSTITUTION METHOD WORKSHEET. Solve the following systems of equations by substitution method. 1. Answer : Substitute y = - 2 in (1). Subtract 6 from both sides. Divide both sides by 4. 2. Answer :. The three methods to solve a system of equations problem are: #1: Graphing. # 2: Substitution. #3: Subtraction.
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Answer Key Join to access all included materials In this systems of equations worksheet, students solve equations by substitution. Through substitution, students evaluate equations for both x and y. Answers are given as an ordered pair. This two page worksheet contains 16 problems, with an additional two pages of answers. 49 Views 117 Downloads.
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Solving Systems by Substitution Solve each system by substitution. Check your answer. 1. { y x 2 y 4x 1 2. { y x 4 y x 2 3. { y 3x 1 y 5x 3 4. { 2 x y 6 x y 3 5. { 2 x y 8 y x 7 6. { 2 x 3y 0 x 2y 1 7. { 3 x 2y 7. how to hack pokemon into your game. thor palazzo interior. Alg 8.2.- Solving Systems by Substitution.A1.3.12 Represent and solve problems that can be modeled using a system of.
The substitution method is a simple way to solve a system of linear equations algebraically and find the solutions of the variables. As the name suggests, it involves finding the value of x-variable in terms of y-variable from the first equation and then substituting or replacing the value of x-variable in the second equation. Solving Systems Of Equations By Substitution Worksheet Answers Math Aids ... Systems of Equations Worksheet 5 PDF. These Equations Worksheets are a good resource for students in the 5th Grade through the 8th Grade. ... Substitution method worksheet math aids Heres a graphic preview for all equation and inequality systems worksheets.
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Steps to solving Systems of Equations by Substitution: Isolate a variable in one of the equations. (Either y = or x =). Substitute the isolated variable in the other equation. This will result in an equation with one variable. Solve the equation. Substitute the solution from step 3 into another equation to solve for the other variable.
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Solve each system by substitution. Steps 1) Solve one of the equations for x or y. • This is already done for you for this section. 2) Substitute the expression into the other equation and.
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Students need to perform a system at least one senior citizen ticket and discuss it adds a custom themes about the image file type of solving systems of equations by substitution method is not. No players currently in game. These activities focus on solving systems of equations by substitution or. Want even if a music store..
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solving-systems-of-equations-by-substitution-worksheet-answers 1/1 Downloaded from insys.fsu.edu on September 21, 2022 by guest [DOC] Solving Systems Of Equations By Substitution Worksheet Answers Recognizing the exaggeration ways to acquire this book solving systems of equations by substitution worksheet answers is additionally useful. Solve each system by substitution. Steps 1) Solve one of the equations for x or y. • This is already done for you for this section. 2) Substitute the expression into the other equation and.
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Solving Systems Of Equations By Substitution Worksheet Answers Author: brookhiser.nationalreview.com-2022-10-06T00:00:00+00:01 Subject: Solving Systems Of Equations By Substitution Worksheet Answers Keywords: solving, systems, of, equations, by, substitution, worksheet, answers Created Date: 10/6/2022 2:22:33 PM. M A HMBard 8ec Nwli3t4hc dIqn7fCi1nHiPtxeT SAYlpgDeMbFrGaW Y2P.4 Worksheet by Kuta Software LLC 32) x − y = 0 −6x + 5y = 4 (−4, −4) 33) 4x + 4y = −8 3x − 2y = 19 (3, −5) Solve.
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Solve each system by substitution. Check your answer. 1. { y x 2 y 4x 1 2. { y x 4 y x 2 3. { y 3x 1 y 5x 3 ... Then, solve the system by substitution. 10. The length of a rectangle is 3 more than its width. The perimeter of the rectangle is 58 cm. What are the rectangle's dimensions? 11. Carla and Benicio work in a men's clothing store. The substitution method is a simple way to solve a system of linear equations algebraically and find the solutions of the variables. As the name suggests, it involves finding the value of x-variable in terms of y-variable from the first equation and then substituting or replacing the value of x-variable in the second equation.
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Division and Multiplication Equations Ex 1. by oparada3000. Write equations. by mahawarrad. System of Equations - Add or Subtract. by mathistheway. 7th pg. 154 (22-27) by DaneliaMartinez. Solving Systems by Eliminations Different Coef Notes..
MIT grad shows how to use the substitution method to solve a system of linear equations (aka. simultaneous equations). To skip ahead: 1) For a BASIC SUBSTITU.
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Chapter 5 pdf big ideas math algebra 1 answers solving systems of linear equations lesson by the substitution method grade 8 algebraically using lessons blendspace examples.
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Division and Multiplication Equations Ex 1. by oparada3000. Write equations. by mahawarrad. System of Equations - Add or Subtract. by mathistheway. 7th pg. 154 (22-27) by DaneliaMartinez. Solving Systems by Eliminations Different Coef Notes.. Feb 24, 2020 - Substitution Method Worksheet Answers - 50 Substitution Method Worksheet Answers , Holt Algebra 6 2a solving Systems by Substitution.
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Solving Systems Of Equations By Substitution Worksheet Answers Author: brookhiser.nationalreview.com-2022-10-06T00:00:00+00:01 Subject: Solving Systems Of Equations By Substitution Worksheet Answers Keywords: solving, systems, of, equations, by, substitution, worksheet, answers Created Date: 10/6/2022 2:22:33 PM.
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This is a complete unit used to introduce your classes to Solving Systems of Equations. Round to the Nearest Hundred Word Problems Worksheet from rounding word problems worksheets, image source: havefunteaching. Word Problems with Variables on Both Sides. Control the pace so everyone advances through each question together.
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A pro for solving equations through either the methods of substitution and elimination allow one to achieve an exact answer regardless of fraction, decimal, or integer. However, by using these methods one will have a more difficult time. Word Problems Worksheet 6 PDF View Answers. These free systems of equations worksheets will help you practice solving real-life systems of equations using the “substitution” method. You will need to create and solve a system of equations to represent each situation. The exercises can also be solved using other algebraic methods if you choose..
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This is to avoid dealing with fractions whenever possible. If none of the variables has a coefficient of 1 then you may want to consider the Addition Method or Elimination Method . Steps to solving Systems of Equations by Substitution : Isolate a variable. Substitution Method Worksheet Answers - 50 Substitution Method Worksheet Answers , Holt Algebra 6 2a solving Systems by Substitution J Jennifer Murphy 90 followers More information Substitution Method Worksheet Answers Luxury Pairs Check Activity solving Systems Of Equations Find this Pin and more on delicious breakfast by Jennifer Murphy.
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We wrote the answer as an ordered pair.Solving three-variable, three-equation linear systems is not more difficult than solving the two-variable systems, it does take longer.What we do is change the 3x3 system to a 2x2 system by eliminating one of the variables using the elimination, then we solve the 2x2 system as we have done before. Math Worksheets $10.00$8.00.
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Solving systems of equations by elimination or by substitution worksheets pdf printable, solving and graphing systems of linear equations word problems, Cramer's rule. x – y = − 6. ... Elimination method using addition and subtraction. 3 Practice A) Day 1 Worksheet Answers Solve Systems of Equations by Elimination Day 2 Notes. Question 1.
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In this unit, you will learn how to solve systems of equations graphically and algebraically. Systems of equations are used in real-world situations where two variables need to be determined. Some examples of problems where systems of equations may be applied are determining the maximum profit of sales and predicting the weather.
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U substitution worksheet with answers U substitution worksheet with answersN, This is a set of 2 solving systems of equations by using substitution RIDDLE worksheets. kids.
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Give students practice writing and solving systems of linear equations based on word problems with this eighth-grade algebra worksheet! ... Use this eighth-grade math worksheet to give learners practice solving word problems by writing and solving systems of equations using substitution! 8th grade. Math. Worksheet. Systems of Equations:.
We wrote the answer as an ordered pair.Solving three-variable, three-equation linear systems is not more difficult than solving the two-variable systems, it does take longer.What we do is change the 3x3 system to a 2x2 system by eliminating one of the variables using the elimination, then we solve the 2x2 system as we have done before. Math Worksheets $10.00$8.00. Solving Equations with Variables on Both Sides 2– This 12 problem worksheet includes equations that focus primarily on subtraction. If students isolate the variable on the left side of the equations, then they will avoid having to use negative numbers. Examples are shown to help guide students through the process.
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solving simultaneous equations matlab. solving for k system of equations. graphic calculator to do radical expresions. "algebraic fraction"s calculator. solving inequalities exponential absolute value. fractions and decimals from least to greatest.. They bought .... solving systems of equations by substitution worksheet algebra 1 answer key, ... Therefore, we can use the substitution method to solve our systems of equations.. Solve systems of equations using various methods (substitution, elimination, and/or graphing) ... Students will solve several real-world problems involving systems.
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Solving Systems of Equations by Substitution Date_____ Period____ Solve each system by substitution. 1) y = 7x − 10 y = −3 2) y = −8 y = −2x − 12 3) y = 6x ... Create your own. Step 1 : First, solve one linear equation for y in terms of x . Step 2 : Then substitute that expression for y in the other linear equation. You'll get an equation in x . Step 3 : Solve this, and you have the x -coordinate of the intersection. Step 4 : Then plug in x to either equation to find the corresponding y -coordinate.
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You can use the Mathway widget below to practice solving systems of equations by using the method of substitution (or skip the widget, and continue to the next page). Try the entered exercise, or type in your own exercise. Then click the button, select "Solve by Substitution" from the box, and compare your answer to Mathway's. Solving Linear Systems By Elimination Worksheet Pdf - Example Worksheet www.viajeabariloche.com. systems worksheet solving equations substitution pdf elimination briefencounters linear.Elimination With Multiplication Worksheet Answers - Leonard Burton's cuddlyturb0dog.blogspot.com. elimination solving.
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These free systems of equations worksheets will help you practice solving real-life systems of equations using the “substitution” method The worksheet contains (8) problems on (1) page and a duplicate page with the answers Solving a System of Equations Using Elimination and Multipliers In this page substitution method questions 10 we are.. Each solving the systems.
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Step 1: Use either equation and solve for a variable. In this case, we solved the first equation for y. Step 2: Substitute the resulting quantity into the other equation. Here we substituted the quantity found for y into the second equation. Step 3: Solve for the remaining variable. Step 4: Back substitute to find the value for the other variable.
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The substitution method involves three steps. They are: Rearrange an equation to isolate one of the variables on one side. Substitute the expression so obtained into the other.
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These free systems of equations worksheets will help you practice solving real-life systems of equations using the “ substitution ” method. Each set of free algebra worksheets is a.
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1) x + 4y = −22 −2x − 2y = 14 2) x − 2y = 7 −3x + 6y = −1 3) −6x + y = −17 −7x − y = −22 4) 4x + 7y = −14 x + 4y = −8 Solve each system by elimination The elimination method for solving systems of linear equations uses the addition property of equality Step 2: Pick a different two equations and eliminate the same variable.
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Aug 18, 2022 · Descriptions: Solving Systems by Substitution. Fill in the blanks to solve each system by substitution. 1. 1 y 3x y x 4 Check your answer. 3. 1 y 4x. More : Solving Systems by Substitution. Fill in the blanks to solve each system by substitution. 1. 1 y 3x y x 4 Check your answer. 3. 1 y 4x.. Solving Systems Of Equations By Substitution Worksheet Answers Author: brookhiser.nationalreview.com-2022-10-06T00:00:00+00:01 Subject: Solving Systems Of Equations By Substitution Worksheet Answers Keywords: solving, systems, of, equations, by, substitution, worksheet, answers Created Date: 10/6/2022 2:22:33 PM.
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This is to avoid dealing with fractions whenever possible. If none of the variables has a coefficient of 1 then you may want to consider the Addition Method or Elimination Method . Steps to solving Systems of Equations by Substitution : Isolate a variable.
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The first worksheet includes how to solve a system of equations by graphing. The second worksheet includes how to solve a system of equations by substitution. The third worksheet includes how to solve a system of equations by elimination. Each worksheet contains 10 problems.These worksheets help with self-assessment due to creating a stained .... You will obtain the value of one of the variables. Substitute this value into either of the original equations to obtain the value of the other variable. Let's jump to an example. Example: Solve the system by substitution. − 8 x + 5 y = − 6. − 3 x + y = − 4. Solution: We follow the first procedure.
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SYSTEM OF EQUATIONS: GRAPHING AND SUBSTITUTION. In this unit, you will learn how to solve systems of equations graphically and algebraically. Systems of equations are used in real-world situations where two variables need to be determined.. In this unit, you will learn how to solve systems of equations graphically and algebraically. Systems of equations are used in real-world situations where two variables need to be determined. Some examples of problems where systems of equations may be applied are determining the maximum profit of sales and predicting the weather..
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Word Problems Worksheet 6 PDF View Answers. These free systems of equations worksheets will help you practice solving real-life systems of equations using the “substitution” method. You will need to create and solve a system of equations to represent each situation. The exercises can also be solved using other algebraic methods if you choose.. This is to avoid dealing with fractions whenever possible. If none of the variables has a coefficient of 1 then you may want to consider the Addition Method or Elimination Method . Steps to solving Systems of Equations by Substitution : Isolate a variable.
To solve systems using substitution, follow this procedure: Source: www.rcboe.org. Worksheets are gina wilson systems of equations maze 2016 answer key, gina wilson. Latest advances in symbolic algorithms: It will certainly squander the time. Here are more examples of how to solve systems of equations in algebra calculator.
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Detailed Answer Key. Problem 1 : Solve the system of linear equations by substitution. Check your answer by graphing. 4x + y = 8-3x + y = 1. Solution : Step 1 : Solve an equation for one variable. Select one of the equation, say -3x + y = 1. Solve for the variable y in terms of x. Add 3x on both sides. (-3x + y) + 3x = (1) + 3x. Y j QMSaed ReH 2wXiqt thx NI1n PfBi 7n LiutUey ZA dl 3g Leib MrsaC 61 b.y Worksheet by Kuta Software LLC Kuta Software - Infinite Algebra 1 Name_____ Solving Systems of Equations by Elimination Date_____ Period____ Solve each system by elimination. 1) −4 x − 2y = −12 4x + 8y = −24 2) 4x + 8y = 20.
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In this unit, you will learn how to solve systems of equations graphically and algebraically. Systems of equations are used in real-world situations where two variables need to be.
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Worksheet # 1: Solve the Linear Equations Using the Substitution Method Select one of the equations and solve the variable, then plug it into the other equation. Ultimately, you’re trying to figure out where the lines intersect. Answers are on the 2nd page. 1.) y = −3x y = x − 8 2.) y = 3 x y = −8x 3.) y = −2x y =‐4x + 10. 1) x + 4y = −22 −2x − 2y = 14 2) x − 2y = 7 −3x + 6y = −1 3) −6x + y = −17 −7x − y = −22 4) 4x + 7y = −14 x + 4y = −8 Solve each system by elimination The elimination method for solving systems of linear equations uses the addition property of equality Step 2: Pick a different two equations and eliminate the same variable.
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Steps For Solving Real World Problems. Highlight the important information in the problem that will help write two equations. Define your variables. Write two equations. Use one of the methods for solving systems of equations to solve. Check your answers by substituting your ordered pair into the original equations. Write the answer. x + 6x – 15 = 6 y = 2x – 5 The solution . 7x – 15 = 6 y = 2(3) – 5 is (3,1) 7x – 15 + (15) = 6 + (15) y = 6 – 5 . 7x = 21 y = 1. x = 3. B) Solving a system of equations by substitution without “y =” or “x =”. Given a system of equations: x + 2y = 4. 3x + y = 7. Change one equation to y = or x =. 3x + y.
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Systems of Equations Worksheet 4 – This 9 problem algebra worksheet helps you practice solving systems of equations by using the “elimination” method. All of the equations are in the standard form of ax + by = c or ax – by = c. Some of the systems have no solution or infinite solutions. A few of the solutions feature decimals or ....
2 Answers Worksheet Substitution By Equations Of Systems Solving 9-10-2022 Need to ACE the Algebra 2 Exam! Algebra 2 Workbook provides students with the confidence and math skills they need to succeed in any math course they choose and prepare them for future study of Pre–Calculus and.
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Solving Systems by Substitution Solve the system by substitution. Example 1A: Solving a System of Linear Equations by Substitution y = 3x y = x – 2 Step 1 y = 3x y = x – 2 Both equations are.
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Displaying all worksheets related to - Solve Equations With Substatution. Worksheets are Systems of equations substitution, Systems of equations, Systems of equations by substitution, Ws solving systems by substitution isolated, Work 1 equations substitution, Work 3 equations substitution, Systems of equations substitution date period, Two step.
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©x h2d0 R1q4f DKwuEt Da2 0SIo 6fUtUwwalr NeE dL vL yCm.f F jAUllz 8rbingnh hthst xrCe2sjeVrzvue Wdw.j C 0Mza Zdle n 3wkiit Uhd eIan Rfzi anji ktBeg OAsl 3gLeEb4rLa t i1 T.A-3-Worksheet by Kuta Software LLC Answers to Practice Test: Solving systems by graphing, substitution, and elimination.
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The substitution method is one of the ways to solve a system of linear equations. Let's discover the process by completing one example. 257 1 1 this instruction will help. Use the quadratic formula to solve the equations (answers on 2nd page of pdf.
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They bought .... solving systems of equations by substitution worksheet algebra 1 answer key, ... Therefore, we can use the substitution method to solve our systems of equations.. Solve systems of equations using various methods (substitution, elimination, and/or graphing) ... Students will solve several real-world problems involving systems.
SYSTEM OF EQUATIONS: GRAPHING AND SUBSTITUTION. In this unit, you will learn how to solve systems of equations graphically and algebraically. Systems of equations are used in real-world situations where two variables need to be determined..
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Algebra 1 Substitution Method - Displaying top 8 worksheets found for this concept. Solve the equation from Step 2 for the variable. Algebra 1 Worksheet Solving Systems Of Equations. Most systems of equations have one solution, but special cases do exist. State whether the linear system has no solution or infinitely many solutions. You must write your equations in slope intercept form and explain why you chose your answer! 4) 21 12 yx yx 5) 12 16 8 34 2 xy xy Solving Systems: You must solve the system by the indicated method.
Displaying all worksheets related to - Homework 2 Solving Systems By Substitution. Worksheets are Systems of equations substitution, Ws solving systems by substitution isolated, Systems of two equations, Pre calculus 11 hw section solving systems of equations by, Chapter 6 systems of two linear equations in two variables contents, Unit 2 solving systems of linear and quadratic equations .... Solve by substitution. In step 1, answer in the form y = mx + b, such as y = 3x + 2 or y = -x/3 - 6. In step 2, answer in the form (x,y). For example: (-2,3). Complexity=5 Solve by substitution. In step 1, answer in the form y = mx + b, such as y = 3x + 2 or y = -x/3 - 6. In step 2, answer in the form (x,y). For example: (-2,3). Complexity=10.
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Worksheet # 4: Solve the Linear Equations Using the Substitution Method Select one of the equations and solve the variable, then plug it into the other equation. Ultimately, you’re trying to figure out where the lines intersect. Answers are on the 2nd page. 1.) y = −7x. Solving Linear and Quadratic System By Graphing Examples Example 4 a: ¯ ® 4 2 2 2 6 y x y x Solution(s): _____ Solution(s): _____ Example 5 : ¯ ® 5 22 3 y y x Example 6a: ¯ ® 2 2 2 7 y x y x Solution(s): _____ Solving Linear and Quadratic System By Substitution (Rework Examples Above) Examples Example 4b: Example 5b: Example 6b:.
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Benefits of Worksheet on Solving System of Equations By Substitution. There are various ways of obtaining the answers to the variables involved in the system of equations. If someone wants to get fluent with those methods, there is a need to practice a.
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Free worksheet ( pdf ) and answer key on solving systems of equations using substitution Each printable worksheet in this unit of solving systems of equations offers eight sets of equations 1 Assess Your Understanding - Page 715 36 including work step by step written by community members like you 1 Assess Your Understanding - Page 715 36 including work step.
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©5 M2J0X1m4l IK 5uct qaZ fS 7o7fKtLwDa5r NeU 9L 6L xCE. V X gA Pl il 3 xr tiYgyh3t is O yrZe VsFeFr 8vbe8dd. U B MMhaIdOec Fw ni pt Ih N CIBnifxi Bn yiztweP IA 5lWgLe5b lrWaY c1y. E-3-Worksheet by Kuta Software LLC Answers to Systems of Equations: Substitution (ID: 1) 1) (−1, 0) 2) (−1, −7) 3) (3, −2) 4) (7, 8). ©x h2d0 R1q4f DKwuEt Da2 0SIo 6fUtUwwalr NeE dL vL yCm.f F jAUllz 8rbingnh hthst xrCe2sjeVrzvue Wdw.j C 0Mza Zdle n 3wkiit Uhd eIan Rfzi anji ktBeg OAsl 3gLeEb4rLa t i1 T.A-3-Worksheet by Kuta Software LLC Answers to Practice Test: Solving systems by graphing, substitution, and elimination.
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In this unit, you will learn how to solve systems of equations graphically and algebraically. Systems of equations are used in real-world situations where two variables need to be determined. Some examples of problems where systems of equations may be applied are determining the maximum profit of sales and predicting the weather.. F Worksheet by Kuta Software LLC Kuta Software - Infinite Algebra 1 Name_____ Solving Systems of Equations by Substitution Date_____ Period____ Solve each system by.
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The first worksheet includes how to solve a system of equations by graphing. The second worksheet includes how to solve a system of equations by substitution. The third worksheet includes how to solve a system of equations by elimination. Each worksheet contains 10 problems.These worksheets help with self-assessment due to creating a stained. Understanding what a linear equation represents in its many forms helps students to see what they are doing with a system of equations Lesson 1 – Lines Review Homework: pg This handout focuses on solving systems of linear equations with one solution 1 solving systems by graphing or substitution worksheet answers, H- What are the general steps to construct a.
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Worksheet by Kuta Software LLC Solve each system by graphing (find the point of intersection of the two lines). 21) - 6 x + y ... Answers to Solving Systems of Equations by Graphing.
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In this unit, you will learn how to solve systems of equations graphically and algebraically. Systems of equations are used in real-world situations where two variables need to be. Solving Systems Of Equations By Substitution Worksheet Answers Author: brookhiser.nationalreview.com-2022-10-06T00:00:00+00:01 Subject: Solving Systems Of.
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Solve each system by substitution. Check your answer. 1. { y x 2 y 4x 1 2. { y x 4 y x 2 3. { y 3x 1 y 5x 3 ... Then, solve the system by substitution. 10. The length of a rectangle is 3 more than its width. The perimeter of the rectangle is 58 cm. What are the rectangle’s dimensions? 11. Chapter 3 – Systems of Linear Equations and Inequalities Answer Key CK-12 Algebra II with Trigonometry Concepts 5 3.5 Solving Systems with No or Infinitely Many Solutions using Substitution Answer s 1. (1, 4) 2. no solution 3.
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Free worksheet(pdf) and answer key on solving systems of equations--using substitution , elimination and a graph. Solving Systems with Linear Combination or Elimination . In the Roots of a Quadratic Gizmo, students will find the roots of a quadratic using its graph or the quadratic formula.. Walk through our printable solving systems of equations worksheets to learn the ins and outs of solving a set of linear equations. Ensure students are thoroughly informed of the methods of elimination, substitution, matrix, cross-multiplication, Cramer's Rule, and graphing that are crucial for arriving at the solutions.
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These free systems of equations worksheets will help you practice solving systems of equations using the "elimination" method We have X plus y equals 4 and then 2x minus 3y equals 18 , D and Q), your goal is to eliminate one of the variables in order to solve for the other variable The substitution method for solving linear systems; The elimination method Here is. Solve a system of equations by substitution. Step 1. Solve one of the equations for either variable. Step 2. Substitute the expression from Step 1 into the other equation. Step 3. Solve the resulting equation. Step 4. Substitute the solution in Step 3 into one of the original equations to find the other variable.
Help students practice solving systems of equations by substitution with this worksheet. There are 12 systems of equations that all have one solution. In each problem, at least one of the equations has one variable with a coefficient of 1. This resources is classroom ready! All you need to do is print and copy for student use. An answer key is ....
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Algebra Systems of Equations Solve by Substitution x + y = 4 x + y = 4 , x − y = 2 x - y = 2 Subtract y y from both sides of the equation. x = 4− y x = 4 - y x−y = 2 x - y = 2 Replace all occurrences of x x with 4−y 4 - y in each equation. Tap for more steps... 4−2y = 2 4 - 2 y = 2 x = 4− y x = 4 - y Solve for y y in 4− 2y = 2 4 - 2 y = 2.
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Worksheet # 4: Solve the Linear Equations Using the Substitution Method Select one of the equations and solve the variable, then plug it into the other equation. Ultimately, you’re trying to figure out where the lines intersect. Answers are on the 2nd page. 1.) y = −7x.
Substitution Objective: Solve systems of equations using substitution. Solving a system by graphing has several limitations. First, it requires the graph to be perfectly drawn. If the lines are not straight we may arrive at the wrong answer. Second, graphing is not a great method to use if the answer is really large, such as (100, 75) , or if. This process describes the elimination (or addition) method A means of solving a system by adding equivalent equations in such a way as to eliminate a variable Step 2: Pick a different two equations and eliminate the same variable 4 Worksheet by Kuta Software LLC 32) x − y = 0 −6x + 5y = 4 (−4, −4) 33) 4x + 4y = −8 3x − 2y = 19 (3, −5) Solve each system by elimination The.
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Learning Objectives. (5.2.1) – Solve cost and revenue problems.Specify what the variables in a cost/ revenue system of linear equations represent. Determine and apply an appropriate method for solving the system. (5.2.2) – Solve value problems with a system of linear equations. (5.2.3) – Solve mixture problems with a system of linear.. Use these two equations (which are now in.
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Algebra 1 Substitution Method - Displaying top 8 worksheets found for this concept. Solve the equation from Step 2 for the variable. Algebra 1 Worksheet Solving Systems Of Equations.
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Steps to solving Systems of Equations by Substitution: Isolate a variable in one of the equations. (Either y = or x =). Substitute the isolated variable in the other equation. This will result in an equation with one variable. Solve the equation. Substitute the solution from step 3 into another equation to solve for the other variable. Algebra Systems of Equations Solve by Substitution x + y = 4 x + y = 4 , x − y = 2 x - y = 2 Subtract y y from both sides of the equation. x = 4− y x = 4 - y x−y = 2 x - y = 2 Replace all occurrences of x x with 4−y 4 - y in each equation. Tap for more steps... 4−2y = 2 4 - 2 y = 2 x = 4− y x = 4 - y Solve for y y in 4− 2y = 2 4 - 2 y = 2.
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Finding slope from a graph. Finding slope from two points. Finding slope from an equation. Graphing lines using slope-intercept form. Graphing lines using standard form. Writing linear equations. Graphing linear inequalities. Graphing absolute value equations. Direct variation.
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Here is a system of equations that I want to solve using substitution and I’m already set up pretty well because x is already isolated Gain immense practice with this batch of printable solving systems of equations worksheets, designed for 8th grade and high school students z H lA7ldlB oroihg5hbtMsW 3rge9sTe3rcvBeldz The equations section lets you solve an equation.
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solve by the substitution method step 2: pick a different two equations and eliminate the same variable an algebra 1 activity on solving systems of equations with substitution and elimination test and worksheet generators for math teachers solving systems of equations algebraically johnny wolfe www solving systems of equations algebraically.
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Solving Systems Of Equations By Substitution Worksheet Answers Author: brookhiser.nationalreview.com-2022-10-06T00:00:00+00:01 Subject: Solving Systems Of Equations By Substitution Worksheet Answers Keywords: solving, systems, of, equations, by, substitution, worksheet, answers Created Date: 10/6/2022 2:22:33 PM.
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view answers substitution method can be applied in four steps step 2: pick a different two equations and eliminate the same variable step 2: using the y intercept (b) and slope (m) graph both the elimination method for solving systems of linear equations uses the addition property of equality the elimination method for solving systems of linear. Download File PDF Solving Systems Of Equations By Substitution Worksheet Answers www.modernh.modernh.com Applications in School MathematicsMerrill Algebra 1 Applications and Connections Reteaching MastersElementary AlgebraHands-On Algebra!Algebra für DummiesLineare Algebra für DummiesConcise Guide to Computing FoundationsHuman-.
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Algebra 1 Solving Systems Of Equations By Substitution Worksheet Source: www.tamworksheets.co Includes many options and types of equations. Each one has model problems worked out step by step, practice problems, as well as challenge questions at the sheets end. Algebra 2 Writing Polynomial Equations Worksheet Answer Key Writing. How to Solve the System of Equations in Algebra Calculator. First go to the Algebra Calculator main page. Type the following: The first equation x+y=7; Then a comma , Then the second equation x+2y=11; Try it now: x+y=7, x+2y=11 Clickable Demo Try entering x+y=7, x+2y=11 into the text box. After you enter the system of equations, Algebra.
The substitution method involves algebraic substitution of one equation into a variable of the other. This will be the sample equation used through out the instructions: Equation 1) x – 6y – 2z = -8. Equation 2) -x + 5y + 3z = 2. Equation 3) 3x - 2y – 4z = 18. Steps in order to solve systems of linear equations through substitution:. Benefits of Worksheet on Solving System of Equations By Substitution. There are various ways of obtaining the answers to the variables involved in the system of equations. If someone wants to get fluent with those methods, there is a need to practice a.
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*Click on Open button to open and print to worksheet. 1. Systems of Equations Substitution 2. WS 6.2A Solving Systems by Substitution isolated 3. Systems of Equations by Substitution 4. Systems of Equations: Substitution Date Period 5. Name SYSTEM OF EQUATIONS-SUBSTITUTION #1 6. Name SYSTEM OF EQUATIONS-SUBSTITUTION #6 7.
Step 1 : First, solve one linear equation for y in terms of x . Step 2 : Then substitute that expression for y in the other linear equation. You'll get an equation in x . Step 3 : Solve this, and you have the x -coordinate of the intersection. Step 4 : Then plug in x to either equation to find the corresponding y -coordinate. |
In the exercise you need to provide implementation of single-threaded TaskScheduler class with the following signatures:
TaskScheduler {
int getPriority()
void execute()
where Task object is considered interruptible if it implements Interruptible interface.
Now I have to consider the below 4 requirements also in developing the class:
Define classes, write a test or vice versa.
2. executeAllByPriority()
3. executeAllByPriorityWithUninterruptableFirst()
4. Bonus
Keeping these above 4 points in consideration I have coded the below program. Please advise whether it is the correct approach.
public class MultiThreadedTaskScheduler {
AtomicInteger priorityCounter = new AtomicInteger(-1);
private static final PriorityComparator PRIORITY_COMPARATOR = new PriorityComparator();
}
public void executeAllByPriority(){
executorService.submit(new Runnable() {
@Override
public void run() {
}
}
});
}
}
}
public void executeAllByPriorityWithUnInterruptableFirst(){
executorService.submit(new Runnable() {
@Override
public void run() {
} else {
}
}
}
}
});
}
@Override
protected void finalize() throws Throwable {
super.finalize();
executorService.shutdown();
}
}
Let's get cracking ;) Nitpick first:
• Indentation / Spacing:
Your indentation and spacing aren't fully consistent. It would be beneficial to use an IDE's autoformatting options.
Okay, now that that's out of the way, let's get to the real beef:
AtomicInteger priorityCounter = new AtomicInteger(-1);
This is only used in your executeMethod. And there it's used "wrongly". You're trying to burden the executeMethod with the additional responsibility of checking whether the task you have at hand should be executed now, or not:
if (task.getPriority() > priorityCounter.get()){
}
This is not good especially since you fail your if-condition silently. Enqueueing a lower-priority task to the queue does not execute the task and doesn't tell you about that.
That's a big problem and unnecessary. You can completely remove all the stuff in this method, that's got to do with priorityCounter. What remains is task.execute() and that can stand on it's own.
private ExecutorService executorService = Executors.newFixedThreadPool(10);
I don't like this one for multiple reasons:
1. It isn't final.
2. You're not using anything anywhere near 10 threads.
If you want to do it "right" you should move all your executorService.submit()-calls directly around the respective task.execute()-calls. Right now you only ever submit two runnables to this 10-thread pool. That's a waste of threads ;)
private PriorityQueue<Task> taskQueue;
Again this could just as well be made final, you could even initialize this eagerly, since the static PRIORITY_COMPARATOR is available anyways ;)
for (Task task : taskQueue) {
/* stuff in here */
}
First thing I thought when seeing this: "This doesn't respect priority, right?" While that's not true, I personally find the following much more obvious:
while (taskQueue.peek() != null) {
/* stuff in here */
}
Moving swiftly on to one of the last points I want to make:
if (task instanceof Interruptible && ((Interruptible) task).isInterrupted()) {
Your requirements nowhere state, that only interrupted Interruptibles are to be executed later. Assuming you didn't have the definition of Interruptible given, I had suspected it to be a marker-interface. I'd shorten the code to:
if (task instanceof Interruptible) {
Almost finished:
executorService.submit(new Runnable() {
@Override
public void run() {
/* real stuff here */
}
});
If you have java 8 available I highly recommend converting this boilerplate to a lambda-expression:
executorService.submit(() -> { /*code here */ });
One thing about references and shuting down, if I may:
Overriding finalize to shutdown your executor isn't recommended. This answer on SO gives nice impressions.
What you should do instead is implement AutoCloseable and shut down the executor when overriding the close-method
And last but not least: |
# Are constituency grammars and dependency grammars two different types of context free grammars?
A concrete syntax tree or parse tree or parsing tree[1] or derivation tree is an ordered, rooted tree that represents the syntactic structure of a string according to some context-free grammar.
Parse trees are usually constructed according to one of two competing relations, either in terms of the constituency relation of constituency grammars (= phrase structure grammars) or in terms of the dependency relation of dependency grammars.
1. Are constituency grammars and dependency grammars two different types of context free grammars?
2. I went through the wikipedia articles for constituency grammars and for dependency grammars, but they seem to discuss only for natural languages.
From the perspective of formal language and grammars, is it correct that constituency grammars and dependency grammars seem to be about semantics, which is not just syntax/formal any more?
Thanks.
## 2 Answers
Concrete syntax trees and derivation trees exist for many types of grammars (e.g. tree adjoining grammars, TAG). They happen to be identical (up to labeling) in the case of CF grammars, but this is not always the case, for example for TAG. (there are technical errors or imprecisions in Wikipedia).
The derivation tree expresses how the rules are to be applied to obtain the derived string. The derived tree (or concrete syntax) associate a tree structure to the generated string, which may not be context free.
Constituency grammars is a name often used by linguists to refer to what we call Context-Free Grammar. This is intended to contrast them with Dependency Grammars based on a linguistic concept of dependency used by linguists.
The idea of constituency is a classic idea in formal systems: terms are formed with subterms. So the parse tree (very approximately) may be seen as a term of an abstract algebra, and the semantics of the whole is a composition of the semantics of the parts, the constituents (I insist that I am simplifying a lot). While parse trees are concrete syntax trees, the idea of abstract syntax trees (AST) was to emphasize in programming languages the algebraic term structure to better organize compilers and formal semantics.
Nevertheless, constituency is a syntactic concept, a very common one.
It has been extended to formalisms that consider discontinuous constituents (or multi-parts constituents), such as TAG and Linear Context-Free Rewriting Systems (LCFRS), which are strictly more powerful than CF grammars.
As far as I understand, the idea of dependency is related to lexicalized grammars (which also exists for extensions of CF grammars).
The idea of a lexicalized grammar is that every rule instance used in a derivation must be associated with a terminal of the generated string (usually a single one). As is well known, every CF language has a lexicalized CF grammar: you just put it in Greibach normal form. Actually, the research of Sheila Greibach was probably motivated by natural language considerations (she was working in the NL group of Anthony Oettinger at Harvard). This concept of lexicalisation comes from some natural language (NL) theories that consider that construction of sentences is lexicon driven.
So far, we are still in the CF realm.
Dependency Grammar (DG), as much as I understand it as I am no specialist, organizes the tree structure around the idea that each node of the syntax tree is associated with a terminal (lexical element), as in lexicalized grammar, but in such a way that it "controls" (dominates in the tree structure) some part of the sentence, of which it is supposed to be the central element (the exact term is "head"). So the verb will be the head of a sentence, and the noun the head of the noun part of the sentence, etc., according to some linguistic theory the linguist is developing to organize language structure..
It is quite characteristic that the wikipedia article on Dependency Grammars does not propose a formal definition of the concept. My feeling from a quick glance at the literature is that it is mostly a linguistic concept that can be formally dressed in different ways, without a standard reference one. There are, however, some formal definitions, for example in Dependency Grammars and Context-Free Grammars, by Steven Abney (unpublished, 1994). I have also seen contradictory statements regarding the power of DG, but it apparently at best weakly equivalent to CFG. Specific algorithms have been developed for dependency parsing, that seems to have interest for linguists.
A dissertation (for sale) may contain interesting material: Dependency Structures and Lexicalized Grammars: An Algebraic Approach*, Marco Kuhlmann, Springer 2010. Slides about it are available.
Closely related to Dependency Grammar are Link Grammars, formally defined, which also seem to have been designed for their parsing properties, though advantages over CF grammars seem disputed. Link Grammars are weakly equivalent to CF grammars, according to their creators, Daniel Sleator and Davy Temperley.
But Dependency Grammars, or Link Grammars, though they may define the same languages as all or part of the CF grammars, are not CF grammars, and do not define the same structures for the strings.
It is worth noting also the Head Grammars, which are non-CF constituency grammar, though they emphasize the concept of head more often encountered with the dependency analysis of language. Head grammars are LCFRS.
I believe semantics considerations enter the concept of Dependency Grammars, but there in no clearcut limit between syntax and semantics in Natural Language.
Constituency grammars are often just our good old, syntactic, formally defined, context-free grammars, though it has been entended to more complex structures for constituants.
First I admit that I only know enough so I can navigate in this field. One can say that I am at a level above beginner, but not an expert (adept is perhaps the best word)
I would hazard an answer for your questions.
1. Constituency grammar is simply syntactic grammar that is about parsing parts of a full context free grammar tree. Dependency grammar is not really about syntactic. I would not say dependency grammar is just context free.
2. Yes dependency grammar is really about semantic of the sentence when you talk in linguistic context. Syntactic grammar does not care about semantic, it cares about whether a sentence conforms to a set of rules and thus belong to a language. The semantic that one can derive from syntactic parsed tree is simply a corollary.
Here, this is important, it is well known that given a set of grammatical rules for context free grammar (constituency grammar), one can parse a sentence if it is in the language in polynomial time. CKY parses in $O(n^2)$ where $n$ is the size of the sentence. This running time is exponential on the size of the grammar, but if you are given a grammar before hand, this is considered only a constant.
For dependency grammar, in its generic form, I am quite certain you cannot guarantee a polynomial time parser given a set of grammar rules. The problem will become exponentially hard because the edges of the graph can point back and forth, you can forget about doing it the dynamic programming way.
In practice, this is not much of a problem because in the domain where one can use dependency grammar is linguistic. In this domain, the grammar has set of rules that would massively reduce the complexity of parsing natural language sentence. If not then over the course of thousand years, society would change it so that things are 'understandable'.
The most interesting part about dependency grammar is that its purpose, unlike formal language in theory of computation, is not about deciding whether a sentence is in the language or not. The purpose is about 'guessing' a semantic structure that would generate the given sentence.
• If syntactic structures dependent on semantical aspects, as RMW Dixon has clearly shown, then Chomskian separation between syntax and semantics breaks down, giving not much hope for syntactic parsing. Dec 23, 2019 at 12:35 |
Flakey HP 70: post modified as RAM Chip identified.
10-21-2015, 08:55 PM (This post was last modified: 10-22-2015 07:02 AM by Geoff Quickfall.)
Post: #1
Geoff Quickfall Senior Member Posts: 771 Joined: Dec 2013
Flakey HP 70: post modified as RAM Chip identified.
The calculator basic functions ( - + x / ) which rely on the stack work. The higher functions ( n i PMT PV FV ) and the memories K and M do retain their inputs. The K register does not auto load with a 12 and anything input in the K manually or the M registers are not stored.
Seems to me that the RAM chip runs the stack and runs the financial functions and memories.
Is this correct? If so, then replacing that chip should resurrect this machine.
Cheers, geoff
10-22-2015, 06:22 AM (This post was last modified: 10-22-2015 07:02 AM by Geoff Quickfall.)
Post: #2
Geoff Quickfall Senior Member Posts: 771 Joined: Dec 2013
RE: Flakey HP 70
Okay, doing some research the following chips are found in the 70:
1818-0116 ROM
1818-0117 ROM
1820-0993 RAM
1820-11?9 (1169 most likely) ARC
1820-1128 clock driver reset
1818-0078 CTC
With the symptoms described above, the inability to retrieve stored numbers in either the financial registers or K and M it would look like I need to replace the RAM chip, any comments. I have a few dud 45s' which may end up being RAM chip donors to get this 70 fully operational.
Or do the symptoms suggest a defunct ROM. Again, a number stored in the financial, K and M registers are not retrievable suggesting a defective RAM. If it was the ROM, then I would be to store and retrieve numbers from the RAM.
Just checking my logic, so tomorrow I will swap the 1820-0993 with one from a dud 45 and let you know.
P.s. Thanks Eric Smith for your chip set descriptions.
10-22-2015, 06:42 AM
Post: #3
Didier Lachieze Senior Member Posts: 1,386 Joined: Dec 2013
RE: Flakey HP 70
According to this table the following chips are used on the HP-70:
Code:
part number models description vendor(s) notes ---------------------------------------------------------------------------------------------- 1818-0078 35, 45, 55, 70, 80 CTC AMI, Mostek MK6021P 1818-0116 70 ROM, 1024*10 AMI 1818-0117 70 ROM, 1024*10 AMI 1820-0993 45, 46, 70 , 81 RAM, 10*56 Mostek MK6036P replaced by 1820-1393 1820-1128 35, 45, 55, 70, 80 clock driver and reset, 16 pin DIP 1820-1169 35, 45, 55, 70, 80 ARC AMI, Mostek MK6020P
The RAM being the same on the HP-70 as the one on the HP-45 this can be an easy replacement source.
10-22-2015, 06:52 AM (This post was last modified: 10-22-2015 06:54 AM by Geoff Quickfall.)
Post: #4
Geoff Quickfall Senior Member Posts: 771 Joined: Dec 2013
RE: Flakey HP 70: post modified as RAM Chip identified.
Aha, that's the table I am using!
Tomorrow I will harvest a RAM from a 45 and see if that fixes the 70.
By the way, the stack as well as the 4 basic functions all work, it is jus the higher registers that return a 0 no matter the input.
Thanks Didier.
Geoff
10-22-2015, 07:18 AM
Post: #5
Harald Senior Member Posts: 751 Joined: Dec 2013
RE: Flakey HP 70: post modified as RAM Chip identified.
(10-22-2015 06:52 AM)Geoff Quickfall Wrote: Aha, that's the table I am using!
Tomorrow I will harvest a RAM from a 45 and see if that fixes the 70.
By the way, the stack as well as the 4 basic functions all work, it is jus the higher registers that return a 0 no matter the input.
Thanks Didier.
Geoff
Does sound very much like a RAM problem.
It could be the ROM that partially works and instructions related to the financial functions don't work, but that seems rather unlikely.
I nearly bought a 70 the other day - but couldn't justify spending that much money and ended up loosing the auction...
10-22-2015, 03:43 PM
Post: #6
Geoff Quickfall Senior Member Posts: 771 Joined: Dec 2013
RE: Flakey HP 70: post modified as RAM Chip identified.
Interestingly, the entry of all financial knowns works as expected from the entry point of view. Execution of the unknown results in the 'running' display of flashing led segments. The result is an 'error' message most likely a dived by zero error.
So I too am hoping RAM, not ROM.
This was also pricey but came with the following:
The manual in perfect shape
The real estate manual
Guarantees and sales brochures.
Aluminum sticky personal labels (3)
Envelopes,
Original shipping box with a November 73 postmark.
Wall wart
Two battery packs, one new from eBay and the original
Spare bezel (face plate).
No corrosion.
Unopened rear label.
It was advertised as working and looking excellent. The looking excellent was correct. I discovered the non working side of things. In fact this calculator works perfectly as a basic 4 banger with no memory but a 4 level stack. So the seller only tested the basic functions and assumed that meant a working calculator. If you don't know how to use the higher functions then how can you test them?
In any case, I got a mutually agreed substantial refund when I explained the problem. No mess, no fuss. So I am happy either way (working or not) but it would be nice to get it functional and looking factory fresh.
Cheers, Geoff
11-08-2015, 01:48 AM (This post was last modified: 11-08-2015 01:50 AM by Geoff Quickfall.)
Post: #7
Geoff Quickfall Senior Member Posts: 771 Joined: Dec 2013
Damn it!!!
Well, I picked up a functioning HP 16C and an untested (eBay read for dead) HP 45 in a package for an excellent price. The HP 16C will be a Christmas present for my nephew (programmer).
The 45 WAS to be a donor for an:
1820-0993 RAM chip.
Well wouldn't you know it!!! I got it working after a good PCA and connection clean.
So I took my oldest, rattiest looking 45 PCA and harvested the chip. I now have a functional 45 board that needs a chip, but at least I could insert the chip into the 70 PCA.
And guess what, I now have a fully functional 70 working! It even does Canadian Mortgages, of course that will all change with the new Prime Minister.
So, referring to the faults in the first post, the ROM at power on does insert 12 into the "K" memory which resides in the 1820-0993 chip. The fact the 12 and all other 'memory' function other then the stack pointed to the defunct 0993 chip.
A big thanks AGAIN to Eric Smith and his resource at brouhaha!
best regards all, Geoff
11-09-2015, 03:20 PM
Post: #8
Harald Senior Member Posts: 751 Joined: Dec 2013
RE: Flakey HP 70: post modified as RAM Chip identified.
(11-08-2015 01:48 AM)Geoff Quickfall Wrote: So I took my oldest, rattiest looking 45 PCA and harvested the chip. I now have a functional 45 board that needs a chip, but at least I could insert the chip into the 70 PCA.
And guess what, I now have a fully functional 70 working! It even does Canadian Mortgages, of course that will all change with the new Prime Minister.
Congratulations on getting the 70 to work! 45s are, although very nice calculators, quite cheap. So it makes perfekt sense to sacrifice one to resurect a 70!
Harald
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# Non-polynomial representations of $GL_n$
Recall that every finite-dimensional rational representation of $GL_n$ is of the form $(\det)^{-k} \varrho$ for some integer $k\geq 0$ and polynomial representation $\varrho$ (and $\det$ is the one-dimensional representation $A\mapsto \det(A)$). The irreducible polynomial representations have been classified and are given by the Schur modules.
My questions are as follows. Are there simply-described finite-dimensional non-rational representations of $GL_n$? Are there a lot of them? Can they be classified? Also, why do we care about polynomial representations in the first place?
There are some pathological examples given in the Appendix of Stanley - Enumerative Combinatorics, vol 2. I'll share some that are relevant to the question:
Let $$A \in GL(n,\mathbb{C})$$.
• $$\varphi(A) = |\det(A)|^{\sqrt{2}}$$ is a representation of dimension one that is not a rational representation.
• $$\varphi(A) = [\sigma(a_{ij})]$$ with $$\sigma$$ a field automorphism of $$\mathbb{C}$$ that is not the identity or complex conjugation (and so it must be discontinuous). This is a $$n$$-dimensional representation that is not rational and not continuous.
• $$\varphi(A) = \begin{pmatrix} 1 & \log|det A | \\ 0 & 1 \end{pmatrix}$$. This is a representation of dimension two that is not rational, but is continuous.
I don't know of an answer to the question of classification. But altogether, we learn
• Yes, in some sense there are a lot of non-rational finite-dimensional representations that are simple to describe.
• Perhaps the above examples hint at some ways to classify at least some families (but there's no indication that we can complete this fully).
The problem with $GL(V)$ simply as a group is that it is to big and has a lot of wild representations. For example, the field $\mathbb{C}$ has a lot of field automorphisms, using them you can twist any normal representation and obtain some strange one. So you need to add some additional structure on $GL(V)$ that you want to preserve.
Study of polynomial representations corresponds to understanding the $GL(V)$ group as an algebraic group. This is the natural choice if you want to have a theory over arbitrary (characteristic zero, algebraically closed) field. There are other versions of classification. For example, you can understand $GL_n$ as a Lie group and study holomorphic representations. The results would not change. There are a lot of results of the type "continuous representations of $U(n)$ extend to algebraic representations of $GL(n)$". So, in general, the Schur-Weil theory extends for any reasonable additional structure.
Unfortunately, I don't know the examples of weird'' continuous representations, but intuitevely I am confident that they do exist. For me, the representation $|det|^{\pi}$ is weird enough. |
# [tex-hyphen] Latin Hyphenation when using utf8
Mojca Miklavec mojca.miklavec.lists at gmail.com
Wed Jun 23 09:29:36 CEST 2010
On Tue, Jun 22, 2010 at 23:16, Andrew Gollan wrote:
> grātiās plūrimās vōbīs agō
>
> This almost completely solved my problem (though I confess I haven't looked
> through for nasty hyphenations yet). I ran xelatex on my input file with
> only a couple of minor mods for font handling:
> \usepackage{palatino} => \usepackage{fontspec}\setromanfont{Palatino
> Linotype}.
>
> But xelatex seems to just happily proceed when it doesn't have a glyph. In
> the old scheme I could put a macon on 'y' to make 'ȳ' and it came out in the
> wash even though the font did not directly contain it. My new book had gaps
> where the 'ȳ' should be. Any quick hints on how to understand/control glyph
> substitution in the brave new (to me) world of XeTeX?
You have a few options:
a) Use Gentium or some other complete font (I vote for that)
b) You could use TeX Gyre Pagella, but that font doesn't have ȳ
either, at least not yet; however, you may request that glyph at
Polish font gurus and then you'll have the support in all the fonts
they are covering (8 TeX Gyre families, Latin Modern, Antykwa
Torunska, Antykwa Poltawskiego, Iwona, Kurier, ...)
c) The following works in ConTeXt to make a fallback for a missing
glyph in the font:
\catcode\ȳ=\active
\defȳ{\buildtextaccent\textmacron y}
The same should work in plain TeX/LaTeX, but it must be some other
command (\accent or something ... maybe even \defȳ{\=y} works fine).
However, while the option "c" is "it always works", you'll get exactly
the same problems with hyphenation as in pdfTeX. So the best option is
to take the font that does have that glyph (or request its addition to
TeX fonts). You can do the "b" independent of whether you choose to go
with the first option for now.
> My first reaction would be just to say:
> amacron = abreve = a
> Amacron = Abreve = A
> ...
Sure. You just need a complete list. If what Arthur say is true (that
there's no more way to use \savehyphencodes or however it is spelled),
the easiest way to implement the patterns is to "duplicate" all the
patterns or rather: replace each patterns with all the possible
substitutions of "a" with abreve and amacron (all possible
combinations).
If you are able to buy or borrow the TeXBook, read appendix H. (For a
shortcut see pages 36-45 in
http://tug.org/tex-hyphen/pdf/hyphenator.pdf for a nice visual
explanation of how patterns work.)
Mojca
` |
Can any outcome be predicted with 100% accuracy?
1. Mar 25, 2005
Loren Booda
For instance, will attempts to predict comprehensively eventually prove contradictory?
2. Mar 26, 2005
kleinwolf
I suppose it happens with measure 0 ??
3. Mar 26, 2005
Staff Emeritus
If the effect is simple enough, and you include the full range of eigenvalues in your prediction, then I think it can happen; the single photon does or does not hit the target, or whatever.
4. Mar 26, 2005
Loren Booda
kleinwolf,
It matters here not what the value of measurement predicts, just that the action of measurement yields an uncertainty interfering with a singular outcome. However, if a measurement leaves the system in a particular eigenstate, can we be 100% sure that remeasuring that observable immediately would reproduce the previous result?
5. Mar 26, 2005
kleinwolf
Oh..ok u mean that (I thought you were speaking about the EPR argument)...
It's a good question....
Well...according to the rules, if you measure the energy, then remeasuring it will give again the same energy, because energy eigenstates are stationary, following Schroedingers equation (however neglecting things such as spontaneous emission, and other field interaction stuff).
However, if you measure the momentum or position, since those eigenstates after measurement are a superposition of different energy eigentstates, then there is a non trivial evolution after measurement, leading to other possible values if remeasured....
6. Mar 26, 2005
Loren Booda
Maybe there exists some statistical fine structure from spacetime curvature that affects even consecutive measurements of compatible variables.
7. Mar 27, 2005
Staff Emeritus
Back to maybe, Loren? Maybe the Easter Bunny collapses the wave function, too!
8. Mar 27, 2005
kleinwolf
Wow...Loren seems to be very in advance upon my own time in physics...where are you studying ? Or are u just emitting hypotheses ?
9. Mar 27, 2005
HallsofIvy
Staff Emeritus
What kind of outcomes are you talking about? If I have a coin that can come up heads or tails, then I can confidently the say that the event "on the next flip, this coin will come up either heads or tails"! I suspect that's not what you are talking about!
To Kleinwolf: for a simple, finite space like that, yes, prob 1 means the outcome MUST happen, prob 0 means it CAN'T happen. But that's not true for infinite outcome sets. If I have, say, a normal probability distribution for picking real numbers, then the probability of picking ANY specific number is 0- but obviously some number IS picked every time: probability 0 does NOT mean "impossible" and probability 1 does NOT mean "certain".
10. Mar 28, 2005
Loren Booda
What I am first trying to say is that the determination from consecutively measuring observables identically and consistently, i. e. 100%, is allowed by quantum mechanics. I suggest secondly that gravitation or vacuum energy can introduce a statistical anomaly analogous to fine splitting of the hydrogen energy levels, but one whose asymmetry actually violates the discrete nature of quantum logic.
HallsofIvy,
Actually Halls, that is also the type of outcome I include. (Just visit "Many Worlds.") The coin might land on its side, tunnel outside the room, or observers lose the meaning of the heads-tails duality. Aside from the spectacular, all classical arguments may be reduced to those quantum, and an accompanying duality. If I say that there is a chance of the universe being or not being, the outcome may be a superposition of both, with any prediction only 50% correct. Reduced to quantum probability, physics is essentially unpredictable, at least for some of the trials - yet there is another consideration where even probabilities are proved fallible by the disturbance of proof itself!
What I am trying to get at is that a given (classical or quantum) prediction's prophesy sways any anticipated outcome from absolute determinism. That is, the unitary state of the EPR setup is not truly isolated from the past, present and future of observer definiteness (which itself is singular in uncertainty). There is one "correct" result to agree with, but many infinities of alternative turnouts. I therefore propose that an observer's physical expectations evolve away from exactitude due to their very interference with their environment they attempt to measure, despite rather than considering specifically their quantum mechanics.
11. Mar 28, 2005
kleinwolf
To be more precise, the probability of picking a number between x and x+dx is :P(x<X<x+dx)= f(x)dx, where f(x) is the probability density
It's clear, that in the limit dx->0, if f(x) is not too singular, then P(X=x)->0
But my original problem was : Suppose the trivial operator 1 on a single q-bit in a state |S>=(1,0).
Then the final state is |phi>=(cos(phi),sin(phi)) any normalized vector.
The probability of the system being in endstate |phi> after measuring with 1 is |<phi|S>|^2=cos(phi)^2 (*)
Hence |S> or -|S> are with probability 1 the endstates, but the other states also have non-vanishing probabilities..
Not here that the formula (*) gives the probability over a continuous set (the phis) (the probability which in your example was 0), and not the probability density.
So my question is : do the other final states appear ?
12. Apr 8, 2005
moving finger
I think Max Planck summed it up very well when he said :
"Science cannot solve the ultimate mystery of nature. And it is because in the last analysis we ourselves are part of the mystery we are trying to solve."
In other words, 3rd person objective science cannot explain everything, because 3rd person objectivity is an ideal which cannot ultimately be realised.
MF
13. Apr 11, 2005
medium
It would seem there would need be no need to predict, were the out come known. Also, what prediction processes are at play in an equation. An alternative may develope an alternate form. Of course this may be one time it not only looks good on paper, but no where else,or everywhere else.!? Is it therefore posible therefore, to predict exceptions!? Curious question, though I suspect you have reason to give it an answer? I'd like to know.
.........................................MEDIUM....................> |
# Showing that a set of trigonometric functions is linearly independent over $\mathbb{R}$
I would like to determine under what conditions on $k$ the set \begin{align} A = &\{1,\cos(t),\sin(t), \\ &\quad \cos(t(1+k)),\sin(t(1+k)),\cos(t(1−k)),\sin(t(1−k)), \\ &\quad \cos(t(1+2k)),\sin(t(1+2k)),\cos(t(1−2k)),\sin(t(1−2k))\}, \end{align} is linearly independent, where $k$ is some arbitrary real number.
As motivation, I know that the set defined by
$$\{1, \cos wt, \sin wt\}, \quad w = 1, \dots, n$$
is linearly independent on $\mathbb{R}$, which one generally proves by computing the Wronskian. I thought that I could extend this result to the set in question, but I haven't found a proper way to do so. My intuition tells me that $A$ will be linearly dependent when the arguments of the trig functions coincide, which will depend on the value of $k$.
Though, I'm at a loss for proving this is true. Computing the Wronskian for this set required an inordinate amount of time-- I stopped running the calculation after a day. Is there perhaps a way to reduce the set in question so that the Wronskian becomes manageable?
I'm interested in any suggestions/alternative methods for proving linear independence that could help my situation. Note that I'd like to have a result that holds for any $m = 0, \dots, n,$ where $n \in \mathbb{Z}$ if possible.
EDIT: The set originally defined in the first instance of this post was incorrectly cited. My sincere apologies.
-
The answer is $k = 0, \pm 1, \pm \frac{1}{2}$. This follows from the following result.
Claim: The functions $\{ 1, \sin rt, \cos rt \}$ for $r$ a positive real are linearly independent over $\mathbb{R}$.
Proof 1. Suppose that $\sum s_r \sin rt + \sum c_r \cos rt = 0$ is a nontrivial linear dependence. Consider the largest positive real $r_0$ such that $c_{r_0} \neq 0$. Take a large even number of derivatives until the coefficient of $\cos r_0 t$ is substantially larger than the remaining coefficients of the other cosine terms and then substitute $t = 0$; we obtain a number which cannot be equal to zero, which is a contradiction. So no cosines appear.
Similarly, consider the largest positive real $r_1$ such that $s_{r_1} \neq 0$. Take a large odd number of derivatives until the coefficient of $\cos r_1 t$ is substantially larger than the remaining coefficients of the other cosine terms (which come from differentiating sine terms) and then substitute $t = 0$; we obtain a number which cannot be equal to zero, which is a contradiction. So no sines appear.
So $1$ is the only function which can appear in a nontrivial linear dependence, and so there are no such linear dependences.
Proof 2. It suffices to prove that the functions are all linearly independent over $\mathbb{C}$. Using the fact that
$$\cos rt = \frac{e^{irt} + e^{-irt}}{2}, \sin rt = \frac{e^{irt} - e^{-irt}}{2i}$$
it suffices to prove that the functions $\{ e^{irt} \}$ for $r$ a real are linearly independent. This can be straightforwardly done by computing the Wronskian and in fact shows that in fact the functions $\{ e^{zt} \}$ for $z$ a complex number are linearly independent.
Proof 3. Begins the same as Proof 2, but we do not compute the Wronskian. Instead, let $\sum c_z e^{zt} = 0$ be a nontrivial linear dependence with a minimal number of terms and differentiate to obtain
$$\sum z c_z e^{zt} = 0.$$
If $z_0$ is any complex number such that $z_0 \neq 0$ and $c_{z_0} \neq 0$ (such a number must exist in a nontrivial linear dependence), then
$$\sum (z - z_0) c_z e^{zt} = 0$$
is a linear dependence with a fewer number of terms; contradiction. So there are no nontrivial linear dependences.
-
These are very nice. Just to be sure, the $k$ values you cited would be the values for which the set is linearly dependent, right? Also, I'm not sure that I follow the argument in the third proof. The contradiction is deriving a new dependency relation with fewer terms, which contradicts the assumption that the set had a dependency relation, right? It seems like this argument could always be constructed in a case where differentiating yields the same functions in the set. – unbounded_despair Aug 5 '12 at 11:58
@unbounded: yes. This is a special property of functions like the exponential, sine, and cosine. It is very hard for them to be linearly dependent because they are all essentially eigenfunctions of differentiation. A more general statement is true for eigenvectors of any linear operator (the ones with distinct eigenvalues are linearly independent). – Qiaochu Yuan Aug 5 '12 at 14:04
Makes much sense. Thanks kindly, Qiaochu. – unbounded_despair Aug 5 '12 at 14:20
For any fixed $k \in \mathbb R$, you can always write $\sin(t+k)$ as a linear combination of $\sin t$ and $\cos t$ (think angle addition formulas). If I understand your question correctly, I don't think there's any hope of this set being linearly independent.
Note that everything in $A$ is a solution to the third-order linear equation $x''' + x' = 0$. There isn't much room for those to be linearly independent.
-
Erick, is there then an explicit way to determine the dependency relation? – unbounded_despair Aug 5 '12 at 1:42
Erick, I incorrectly defined the original set (sorry about this). The question has been corrected, if you're still interested in it. Cheers – unbounded_despair Aug 5 '12 at 3:53 |
Homework Help: Having some trouble differentiating
1. Oct 6, 2005
TSN79
I'm having some trouble differentiating $x^{\sqrt x }$. I know that the derivative of $x^{\sqrt x }$ probably begins with $x^{\sqrt x } \cdot \ln (x) \cdot \frac{1}{{2\sqrt x }}$ but once the base is also x then there is probably more to it than that. Anyone?
2. Oct 6, 2005
siddharth
You have $$y=x^{\sqrt x }$$
So, $$\ln y = (\sqrt x)(\ln x)$$
Then differentiate both sides with respect to x and substitute the value of y.
3. Oct 6, 2005
TSN79
Ah, yes, thanks siddharth! |
doc-src/TutorialI/Datatype/ABexpr.thy
author wenzelm Sun, 15 Oct 2000 19:50:35 +0200 changeset 10220 2a726de6e124 parent 10171 59d6633835fa child 10971 6852682eaf16 permissions -rw-r--r--
proper symbol markup with \isamath, \isatext; support sub/super scripts:
(*<*)
theory ABexpr = Main:;
(*>*)
text{*
Sometimes it is necessary to define two datatypes that depend on each
other. This is called \textbf{mutual recursion}. As an example consider a
language of arithmetic and boolean expressions where
\begin{itemize}
\item arithmetic expressions contain boolean expressions because there are
conditional arithmetic expressions like if $m<n$ then $n-m$ else $m-n$'',
and
\item boolean expressions contain arithmetic expressions because of
comparisons like $m<n$''.
\end{itemize}
In Isabelle this becomes
*}
datatype 'a aexp = IF "'a bexp" "'a aexp" "'a aexp"
| Sum "'a aexp" "'a aexp"
| Diff "'a aexp" "'a aexp"
| Var 'a
| Num nat
and 'a bexp = Less "'a aexp" "'a aexp"
| And "'a bexp" "'a bexp"
| Neg "'a bexp";
text{*\noindent
Type @{text"aexp"} is similar to @{text"expr"} in \S\ref{sec:ExprCompiler},
except that we have fixed the values to be of type @{typ"nat"} and that we
have fixed the two binary operations @{term"Sum"} and @{term"Diff"}. Boolean
expressions can be arithmetic comparisons, conjunctions and negations.
The semantics is fixed via two evaluation functions
*}
consts evala :: "'a aexp \\<Rightarrow> ('a \\<Rightarrow> nat) \\<Rightarrow> nat"
evalb :: "'a bexp \\<Rightarrow> ('a \\<Rightarrow> nat) \\<Rightarrow> bool";
text{*\noindent
that take an expression and an environment (a mapping from variables @{typ"'a"} to values
@{typ"nat"}) and return its arithmetic/boolean
value. Since the datatypes are mutually recursive, so are functions that
operate on them. Hence they need to be defined in a single \isacommand{primrec}
section:
*}
primrec
"evala (IF b a1 a2) env =
(if evalb b env then evala a1 env else evala a2 env)"
"evala (Sum a1 a2) env = evala a1 env + evala a2 env"
"evala (Diff a1 a2) env = evala a1 env - evala a2 env"
"evala (Var v) env = env v"
"evala (Num n) env = n"
"evalb (Less a1 a2) env = (evala a1 env < evala a2 env)"
"evalb (And b1 b2) env = (evalb b1 env \\<and> evalb b2 env)"
"evalb (Neg b) env = (\\<not> evalb b env)"
text{*\noindent
In the same fashion we also define two functions that perform substitution:
*}
consts substa :: "('a \\<Rightarrow> 'b aexp) \\<Rightarrow> 'a aexp \\<Rightarrow> 'b aexp"
substb :: "('a \\<Rightarrow> 'b aexp) \\<Rightarrow> 'a bexp \\<Rightarrow> 'b bexp";
text{*\noindent
The first argument is a function mapping variables to expressions, the
substitution. It is applied to all variables in the second argument. As a
result, the type of variables in the expression may change from @{typ"'a"}
to @{typ"'b"}. Note that there are only arithmetic and no boolean variables.
*}
primrec
"substa s (IF b a1 a2) =
IF (substb s b) (substa s a1) (substa s a2)"
"substa s (Sum a1 a2) = Sum (substa s a1) (substa s a2)"
"substa s (Diff a1 a2) = Diff (substa s a1) (substa s a2)"
"substa s (Var v) = s v"
"substa s (Num n) = Num n"
"substb s (Less a1 a2) = Less (substa s a1) (substa s a2)"
"substb s (And b1 b2) = And (substb s b1) (substb s b2)"
"substb s (Neg b) = Neg (substb s b)";
text{*
Now we can prove a fundamental theorem about the interaction between
evaluation and substitution: applying a substitution $s$ to an expression $a$
and evaluating the result in an environment $env$ yields the same result as
evaluation $a$ in the environment that maps every variable $x$ to the value
of $s(x)$ under $env$. If you try to prove this separately for arithmetic or
boolean expressions (by induction), you find that you always need the other
theorem in the induction step. Therefore you need to state and prove both
theorems simultaneously:
*}
lemma "evala (substa s a) env = evala a (\\<lambda>x. evala (s x) env) \\<and>
evalb (substb s b) env = evalb b (\\<lambda>x. evala (s x) env)";
apply(induct_tac a and b);
txt{*\noindent
The resulting 8 goals (one for each constructor) are proved in one fell swoop:
*}
apply simp_all;
(*<*)done(*>*)
text{*
In general, given $n$ mutually recursive datatypes $\tau@1$, \dots, $\tau@n$,
an inductive proof expects a goal of the form
$P@1(x@1)\ \land \dots \land P@n(x@n)$
where each variable $x@i$ is of type $\tau@i$. Induction is started by
\begin{isabelle}
\isacommand{apply}@{text"(induct_tac"} $x@1$ @{text"and"} \dots\ @{text"and"} $x@n$@{text")"}
\end{isabelle}
\begin{exercise}
Define a function @{text"norma"} of type @{typ"'a aexp => 'a aexp"} that
replaces @{term"IF"}s with complex boolean conditions by nested
@{term"IF"}s where each condition is a @{term"Less"} --- @{term"And"} and
@{term"Neg"} should be eliminated completely. Prove that @{text"norma"}
preserves the value of an expression and that the result of @{text"norma"}
is really normal, i.e.\ no more @{term"And"}s and @{term"Neg"}s occur in
it. ({\em Hint:} proceed as in \S\ref{sec:boolex}).
\end{exercise}
*}
(*<*)
end
(*>*) |
Warning
Only the calculation of the density is tested for open shell configurations (and relies on a correct .OCCUPATION). All other properties are only tested for closed shell systems and should not be trusted for open shell systems without a thorough testing.
# **VISUAL¶
## Sampling¶
### .LIST¶
Calculate various densities in few points. Example (3 points; coordinates in bohr):
.LIST
3
1.0 0.0 0.0
0.0 1.0 0.0
0.0 0.0 1.0
### .LINE¶
Calculate various densities along a line. Example (line connecting two points; 200 steps; coordinates in bohr):
.LINE
0.0 0.0 0.0
0.0 0.0 5.0
200
Scalar and vector densities are written to files plot.line.scalar and plot.line.vector, respectively, and should be saved after calculation, e.g.
pam --get=plot.line.scalar ...
The first three columns of the output files gives the coordinates (x, y, z) of the point. It is then followed by one/three columns giving the value of the scalar/vector density in that point.
$f(r) = \int_{0}^{2\pi}\int_{0}^{\pi}f(\mathbf{r})r^2\sin\theta d\theta d\phi$
by performing Lebedev angular integration over a specified number of even-spaced radial shells out to some specified distance from a specified initial point. Example (coordinates and distance in bohr):
.RADIAL
0.0 0.0 0.0
10.0
200
The first line after the keyword specifies the initial point, here chosen to be the origin. The second and third line is the distance and step size, respectively. Scalar and vector densities are written to files plot.radial.scalar and plot.radial.vector, respectively, and should be saved after calculation, e.g.
pam --get=plot.radial.scalar ...
### .2D¶
Calculate various densities in a plane. The plane is specified using 3 points that have to form a right angle. Example (coordinates in bohr):
.2D
0.0 0.0 0.0 !origin
0.0 0.0 10.0 !"right"
200 !nr of points origin-"right"
0.0 10.0 0.0 !"top"
200 !nr of points origin-"top"
### .2D_INT¶
Integrate various densities in a plane using Gauss-Lobatto quadrature. The plane is specified using 3 points that have to form a right angle. Example (coordinates in bohr):
.2D_INT
0.0 0.0 0.0 !origin
0.0 0.0 10.0 !"right"
10 !nr of tiles to the "right"
0.0 10.0 0.0 !"top"
10 !nr of tiles to the "right"
5 !order of the Legendre polynomial for each tile
### .3D¶
Calculate various densities in 3D and write to cube file format. Example (coordinates in bohr):
.3D
40 40 40 ! 40 x 40 x 40 points
### .3DFAST¶
Fast evaluation of the molecular electrostatic potential. Example (coordinates in bohr):
.3DFAST
40 40 40 ! 40 x 40 x 40 points
Add space around the cube file. Default (coordinates in bohr):
.3D_ADD
4.0
### .3D_IMP¶
Calculate various densities in 3D on an imported grid (does not have to be regular) Example:
.3D_IMP
grid_file ! a file with x,y,z-coordinates of grid points
### .3D_INT¶
Integrate densities in 3D.
## Modification of densities¶
### .CARPOW¶
Scale densities by Cartesian product $$x^iy^jz^k$$. The keyword is followed by three integers specifying the exponents $$(i,j,k)$$. Example:
.DENSITY
.CARPOW
1 0 0
is equivalent to the specification:
.EDIPX
### .SCALE¶
Scale densities by a factor. Default:
.SCALE
1.0
### .DSCALE¶
Scale densities down by a factor. Default:
.DSCALE
1.0
## Densities¶
### .DENSITY¶
Compute number density $$n(\mathbf{r})$$ . Example (unperturbed density):
.DENSITY
DFCOEF
Another example (perturbed density, first response vector):
.DENSITY
PAMXVC 1
### .ELF¶
Compute the electron localization function. Example:
.ELF
DFCOEF
### .GAMMA5¶
Compute the electron chirality density. Example:
.GAMMA5
DFCOEF
### .J¶
Compute the current density $$\mathbf{j}(\mathbf{r})=-e\psi_{i}^{\ast}c\boldsymbol{\alpha}\psi_{i}$$. Example (use first response vector):
.J
PAMXVC 1
### .JDIA¶
Compute the nonrelativistic diamagnetic current density. Example:
.JDIA
DFCOEF
### .JX¶
Compute the x-component $$j_{x}(\mathbf{r})=-e\psi_{i}^{\ast}c\alpha_{x}\psi_{i}$$ of the current density. Example (use first response vector):
.JX
PAMXVC 1
### .JY¶
Compute the y-component $$j_{y}(\mathbf{r})=-e\psi_{i}^{\ast}c\alpha_{y}\psi_{i}$$ of the current density. Example (use first response vector):
.JY
PAMXVC 1
### .JZ¶
Compute the z-component $$j_{z}(\mathbf{r})=-e\psi_{i}^{\ast}c\alpha_{z}\psi_{i}$$ of the current density. Example (use first response vector):
.JZ
PAMXVC 1
### .DIVJ¶
Compute the divergence of the current density. Example (use first response vector):
.DIVJ
PAMXVC 1
### .ROTJ¶
Compute the curl of the current density. Example (use first response vector):
.ROTJ
PAMXVC 1
### .BDIPX¶
Compute the x-component $$m^{[1]}_{x}(\mathbf{r})=-\frac{1}{2}(\mathbf{r}\times\mathbf{j})_{x}$$ of the magnetic dipole operator. Example (use first response vector):
.BDIPX
PAMXVC 1
### .BDIPY¶
Compute the y-component $$m^{[1]}_{y}(\mathbf{r})=-\frac{1}{2}(\mathbf{r}\times\mathbf{j})_{y}$$ of the magnetic dipole operator. Example (use first response vector):
.BDIPY
PAMXVC 1
### .BDIPZ¶
Compute the z-component $$m^{[1]}_{z}(\mathbf{r})=-\frac{1}{2}(\mathbf{r}\times\mathbf{j})_{z}$$ of the magnetic dipole operator. Example (use first response vector):
.BDIPZ
PAMXVC 1
### .EDIPX¶
Compute the x-component $$Q^{[1]}_{x}(\mathbf{r})=xn(\mathbf{r})$$ of the electric dipole.
### .EDIPY¶
Compute the y-component $$Q^{[1]}_{y}(\mathbf{r})=yn(\mathbf{r})$$ of the electric dipole.
### .EDIPZ¶
Compute the z-component $$Q^{[1]}_{z}(\mathbf{r})=zn(\mathbf{r})$$ of the electric dipole.
### .ESP¶
Compute the electrostatic potential. Example:
.ESP
DFCOEF
### .ESPE¶
Compute the electronic part of the electrostatic potential.
### .ESPN¶
Compute the nuclear part of the electrostatic potential.
### .ESPRHO¶
Compute the electrostatic potential times density.
### .ESPERHO¶
Compute the electronic part of the electrostatic potential times density.
### .ESPNRHO¶
Compute the nuclear part of the electrostatic potential times density.
### .NDIPX¶
Compute the NMR shielding density, with the “X”-component of the nuclear magnetic dipole moment and the selected component of the magnetically-induced current density (by the chosen record on PAMXVC file) as perturbing operators.
### .NDIPY¶
Compute the NMR shielding density, with the “Y”-component of the nuclear magnetic dipole moment and the selected component of the magnetically-induced current density (by the chosen record on PAMXVC file) as perturbing operators.
### .NDIPZ¶
Compute the NMR shielding density, with the “Z”-component of the nuclear magnetic dipole moment and the selected component of the magnetically-induced current density (by the chosen record on PAMXVC file) as perturbing operators.
### .NICS¶
Compute the NMR shielding density in a selected point in space. Is used to calculate NICS. Example:
.NICS
1.2 -1.0 2.0
will calculate the NMR shielding in point (1.2, -1.0, 2.0). This keyword can be used only with one of: NDIPX, NDIPY, NDIPZ keywords.
Use the grid and the magnetically-induced current density (jB) from a file to calculate the jB-dependent densities, e.g. the NMR shielding density or the magnetizability density. Example:
.READJB
file_name ! a file with x,y,z-coordinates of grid points and jB vector field
### .GAUGE¶
Specify gauge origin. Example:
.GAUGE
0.0 0.0 0.0
### .SMALLAO¶
Force evaluation of small component basis functions.
### .OCCUPATION¶
Specify occupation of orbitals. Example (neon atom):
.OCCUPATION
2
1 1-2 1.0
2 1-3 1.0
The first line after the keyword gives the number of subsequent lines to read. In each line, the first number is the fermion ircop. In molecules with inversion symmetry there are two fermion ircops: gerade (1) and ungerade (2). Otherwise there is a single fermion ircop (1). The specification of the fermion ircop is followed by the range of selected orbitals and their occupation. If a single orbital is specified a single number is given instead of the range.
Another example (water):
.OCCUPATION
1
1 1-5 1.0
Another example (nitrogen atom):
.OCCUPATION
2
1 1-2 1.0
2 1-3 0.5
### .LONDON¶
Activate LAO contribution. This keyword is followed by a letter “X”, “Y” or “Z” indicating the component of an external perturbing magnetic field. For example:
.LONDON
X
### .NONE¶
Select “none” connection when when plotting LAO perturbed densities.
### .NODIRECT¶
Skip direct LAO contribution when plotting perturbed densities.
### .NOREORTHO¶
Skip LAO reorthonormalization contribution when plotting perturbed densities.
### .NOKAPPA¶
Skip orbital relaxation contribution when plotting perturbed densities. |
0 like 0 dislike
What’s the area of the sector=
$$(\pi)\left(\frac{60}{360} \right)(9^{2})=\boxed{13.5\pi}$$ |
## Sunday, January 22, 2012
### MathJax
Let's follow up on a few previous posts (a calculation of d/dt of the total energy in a gravitational field, a post about typesetting math, and an introductory exploration of LaTeX). I'm grateful for a comment from a reader, with a link to this page, which makes it pretty clear that, yes, you can use their server to get the script in your pages.
So that's what we'll do. I had to modify the LaTeX commands a bit for this but it's mostly the same as before.
[ UPDATE: I do see a problem, now. The script applies to all LaTeX on the page, which if you go to the main page for the blog, includes the previous post... Just click on the post title, to see the original formatting code. ]
$\mathbf{F} = m\mathbf{a}= m\ddot{\mathbf{r}} = m\frac{d^2}{dt^2}\mathbf{r}$
$\mathbf{F} = -$$\nabla$$V(\mathbf{r})$
$E = \frac{1}{2}m|\dot{\mathbf{r}}|^2+ V$
$\frac{d}{dt} E = \frac{d}{dt} ( \frac{1}{2}m |\dot{\mathbf{r}}|^2 + V )$ $= ?$
$|\dot{\mathbf{r}}|^2 = |\dot{\mathbf{r}}| |\dot{\mathbf{r}}| = \dot{\mathbf{r}} \cdot\dot{\mathbf{r}}$
$\frac{d}{dt}\frac{1}{2}m|\dot{\mathbf{r}}|^2 = \frac{1}{2}m\frac{d}{dt}(\dot{\mathbf{r}} \cdot\dot{\mathbf{r}})$$= m\dot{\mathbf{r}}\cdot\ddot{\mathbf{r}}$$ = \dot{\mathbf{r}}\cdot-($$\nabla$$V$$) \nabla$$V$ $= < \frac{\partial{V}} {\partial{x}}, \frac{\partial{V}} {\partial{y}},\frac{\partial{V}} {\partial{z}} >$
$\dot{\mathbf{r}}$ = $<$ $\frac{dx}{dt}$, $\frac{dy}{dt}$,$\frac{dz}{dt}$$> \nabla$$V$ $\cdot$$\dot{\mathbf{r}} = < \frac{\partial{V}} {\partial{x}} \frac{dx}{dt}, \frac{\partial{V}} {\partial{y}} \frac{dy}{dt}, \frac{\partial{V}} {\partial{z}}\frac{dz}{dt}>$$=\frac{d}{dt}V$
$\frac{d}{dt} E = \frac{d}{dt} ( \frac{1}{2}m |\dot{\mathbf{r}}|^2 + V )$ $= ?$
$\frac{d}{dt}$E $= \dot{\mathbf{r}}$$\cdot(-\nablaV) + \nablaV \cdot \dot{\mathbf{r}} = 0 ## Saturday, January 21, 2012 ### Typesetting, again Here are the equations from the last post, in LaTeX. It took about two hours, fairly painstaking work, but I think it looks pretty good.. UPDATE: Here is a nice simple reference for LaTeX. I started with "LaTeX Bootcamp" (pdf). \mathbf{F} = m\mathbf{a}= m\ddot{\mathbf{r}} = m\frac{d^2}{dt^2}\mathbf{r} \mathbf{F} = -$$\nabla$$V(\mathbf{r}) E = \frac{1}{2}m\norm{\dot{\mathbf{r}}}^2+ V \frac{d}{dt} E = \frac{d}{dt} (\frac{1}{2}m \norm{\dot{\mathbf{r}}}^2 + V) \hspace*{2em} = ?\\ \norm{\dot{\mathbf{r}}}^2 = \norm{\dot{\mathbf{r}}} \norm{\dot{\mathbf{r}}} = \dot{\mathbf{r}} \cdot\dot{\mathbf{r}} \frac{d}{dt}\frac{1}{2}m\norm{\dot{\mathbf{r}}}^2 = \frac{1}{2}m\frac{d}{dt}(\dot{\mathbf{r}} \cdot\dot{\mathbf{r}}) \hspace*{2em} = m\dot{\mathbf{r}}\cdot \ddot{\mathbf{r}} \hspace*{2em} = \dot{\mathbf{r}}\cdot-( \nabla$$V$$)\\ \nabla$$V$$= <\frac{\partial{V}} {\partial{x}}, \frac{\partial{V}} {\partial{y}},\frac{\partial{V}} {\partial{z}}> \dot{\mathbf{r}} = <$$\frac{dx}{dt}$, $\frac{dy}{dt}$,$\frac{dz}{dt}$$> \nabla$$V$$\cdot$$\dot{\mathbf{r}}$$=< \frac{\partial{V}} {\partial{x}} \frac{dx}{dt}, \frac{\partial{V}} {\partial{y}} \frac{dy}{dt}, \frac{\partial{V}} {\partial{z}}\frac{dz}{dt}> \hspace*{2em}=\frac{d}{dt}V\\ \frac{d}{dt}E = \dot{\mathbf{r}}$$\cdot$(-
$\nabla$V ) + $\nabla$V $\cdot$ $\dot{\mathbf{r}}$ = 0
### Vector fun
In Marsden & Tromba's Vector Calculus, I found the following problem. It involves time-derivatives of the position vector r(t), for which we're using the dot notation of the physicists (and Newton).
In the first panel, we have Newton's second law, then the statement that the force is minus the gradient of the gravitational potential V, followed by a calculation of the total energy in the system. The problem is to compute d/dt of the energy. (hint)
The answer is given in the last panel, below. According to the book, it's a "simple calculation."
The first step is to compute d/dt of the kinetic energy. We use the formula from above, plus a trick to convert the squared term back to the dot product of dr/dt with itself. Then we use the chain rule, and finally, the definition of the force in terms of the gradient of the potential.
Calculation of d/dt of the potential energy puzzled me for quite a while, though it really shouldn't have. My solution was to work backward from the answer, as shown.
We put the two results together, and notice that they cancel. Surprise!
### Typesetting math
This post is really just a note to myself about something I need to investigate more. The problem I'd like to solve is how to format mathematical equations for the web. You can see the caveman approach on most of my posts here, like this one.
I just make a table in html
I like the background color, and vary it depending on whether the content is code or output from a program.
It might be nice to have something prettier. So, looking around, I happened across MathJax, and also this post which explains how to use it on Blogger.
Here's a screenshot of the example:
The method used in the post is to load the script from the mathjax server, but I think what I'm probably supposed to do is direct people to resource on my (nonexistent) server. OTOH, they link to the post on the mathjax site.
What I'll probably do is just look into how to use LateX and then post screenshots.
Any thoughts?
### Cycloid
While working through the MIT ocw lectures on multi-variable calculus (Prof. Denis Auroux, here), I particularly enjoyed his discussion about the cycloid. Above is a graphic from the wikipedia article (actually the graphic is an animated gif, but I grabbed one of the frames). The red curve is generated by the motion of a point on the edge of a rolling circle.
In addition to the beauty of the curve, it turns out that the length and area under the curve have simple values that are relatively easy to calculate. See wikipedia for the details.
One thing the article doesn't explain is how to get the "parametrization" for the curve. This looks hard, but is made easy by using vectors. It's explained in the second half of Auroux's fifth lecture.
Another thing the article doesn't explain is how to integrate
√(2 - 2 cos t)
Start from the double angle formula:
cos 2s = cos2s - sin2s cos 2s = 1 - 2 sin2s 2 sin2s = (1 - cos 2s) 2 sin2(t/2) = (1 - cos t)
It's straightforward from there.
The Mathworld article is also quite nice, and references a famous challenge in history, the one which led to this quote (in reference to Newton):
"Ah, I know the lion by his paw!"
## Tuesday, January 10, 2012
### Pilgrim's progress
Lately I've been working (again) on understanding multi-variable calculus. I always wished I had time to cover this material in college, but I took lots of courses in biochemistry and molecular biology instead. I wanted to put up the best resources I've found for this so far.
First, and no surprise, the ocw videos at MIT by Denis Auroux (here). I am currently at #25, about to jump off into integrals in 3D space. (It's my second time to get this far). This time I think I really have everything under control. The proof of Green's theorem was perfect, a kind of mathematical satori.
I also work through everything in Gilbert Strang's book as well. I need to read carefully there, because he's so concise, but his insight is just incredible. Check out the proof of the flux version of Green's theorem.
Two more are recently discovered resources that are especially helpful because they develop everything slowly but completely:
Paul's lecture notes
A beautiful set of pages from someone at U Minnesota
All highly recommended. |
Separated variables and wave functions for rational GL(N) spin chains
Dmytro Volin
Nordita Stockholm and Uppsal U.
Thu, Apr. 25th 2019, 11:30
Salle Claude Itzykson, Bât. 774, Orme des Merisiers
\noindent Integrability meeting ENS/IPhT'' \\ \\ We present a basis in which wave functions of integrable XXX spin chain factorise into a product of Slater determinants of Baxter Q-functions. We furthermore show that this basis is formed by eigenvectors of the B[good]-operator and it is naturally labelled by Gelfand-Tsetlin patterns. The discussion is valid for spin chains in any rectangular representation and arbitrary rank of the GL(N) symmetry group. For symmetric powers of the defining representation, one also observes a corollary that B[good]-operator acting on a suitably chosen vacuum constructs the eigenstates of the Bethe algebra. \\ \\ (IPhT organizers: Ivan Kostov and Didina Serban)
Contact : lbervas |
# Search by Topic
#### Resources tagged with Mathematical reasoning & proof similar to Golden Ratio:
Filter by: Content type:
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Challenge level:
### There are 183 results
Broad Topics > Using, Applying and Reasoning about Mathematics > Mathematical reasoning & proof
### Golden Eggs
##### Stage: 5 Challenge Level:
Find a connection between the shape of a special ellipse and an infinite string of nested square roots.
### Pent
##### Stage: 4 and 5 Challenge Level:
The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.
### Plus or Minus
##### Stage: 5 Challenge Level:
Make and prove a conjecture about the value of the product of the Fibonacci numbers $F_{n+1}F_{n-1}$.
##### Stage: 4 Challenge Level:
Four jewellers possessing respectively eight rubies, ten saphires, a hundred pearls and five diamonds, presented, each from his own stock, one apiece to the rest in token of regard; and they. . . .
### Areas and Ratios
##### Stage: 4 Challenge Level:
What is the area of the quadrilateral APOQ? Working on the building blocks will give you some insights that may help you to work it out.
### The Golden Ratio, Fibonacci Numbers and Continued Fractions.
##### Stage: 4
An iterative method for finding the value of the Golden Ratio with explanations of how this involves the ratios of Fibonacci numbers and continued fractions.
### Target Six
##### Stage: 5 Challenge Level:
Show that x = 1 is a solution of the equation x^(3/2) - 8x^(-3/2) = 7 and find all other solutions.
### Thousand Words
##### Stage: 5 Challenge Level:
Here the diagram says it all. Can you find the diagram?
### Proof Sorter - Quadratic Equation
##### Stage: 4 and 5 Challenge Level:
This is an interactivity in which you have to sort the steps in the completion of the square into the correct order to prove the formula for the solutions of quadratic equations.
### Polynomial Relations
##### Stage: 5 Challenge Level:
Given any two polynomials in a single variable it is always possible to eliminate the variable and obtain a formula showing the relationship between the two polynomials. Try this one.
### Big, Bigger, Biggest
##### Stage: 5 Challenge Level:
Which is the biggest and which the smallest of $2000^{2002}, 2001^{2001} \text{and } 2002^{2000}$?
##### Stage: 4 Challenge Level:
Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.
### Continued Fractions II
##### Stage: 5
In this article we show that every whole number can be written as a continued fraction of the form k/(1+k/(1+k/...)).
### Our Ages
##### Stage: 4 Challenge Level:
I am exactly n times my daughter's age. In m years I shall be exactly (n-1) times her age. In m2 years I shall be exactly (n-2) times her age. After that I shall never again be an exact multiple of. . . .
### Square Mean
##### Stage: 4 Challenge Level:
Is the mean of the squares of two numbers greater than, or less than, the square of their means?
### Napoleon's Hat
##### Stage: 5 Challenge Level:
Three equilateral triangles ABC, AYX and XZB are drawn with the point X a moveable point on AB. The points P, Q and R are the centres of the three triangles. What can you say about triangle PQR?
### Ordered Sums
##### Stage: 4 Challenge Level:
Let a(n) be the number of ways of expressing the integer n as an ordered sum of 1's and 2's. Let b(n) be the number of ways of expressing n as an ordered sum of integers greater than 1. (i) Calculate. . . .
### Pythagorean Golden Means
##### Stage: 5 Challenge Level:
Show that the arithmetic mean, geometric mean and harmonic mean of a and b can be the lengths of the sides of a right-angles triangle if and only if a = bx^3, where x is the Golden Ratio.
### Euclid's Algorithm II
##### Stage: 5
We continue the discussion given in Euclid's Algorithm I, and here we shall discover when an equation of the form ax+by=c has no solutions, and when it has infinitely many solutions.
### Impossible Sandwiches
##### Stage: 3, 4 and 5
In this 7-sandwich: 7 1 3 1 6 4 3 5 7 2 4 6 2 5 there are 7 numbers between the 7s, 6 between the 6s etc. The article shows which values of n can make n-sandwiches and which cannot.
### Calculating with Cosines
##### Stage: 5 Challenge Level:
If I tell you two sides of a right-angled triangle, you can easily work out the third. But what if the angle between the two sides is not a right angle?
### On the Importance of Pedantry
##### Stage: 3, 4 and 5
A introduction to how patterns can be deceiving, and what is and is not a proof.
### The Frieze Tree
##### Stage: 3 and 4
Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?
### Transitivity
##### Stage: 5
Suppose A always beats B and B always beats C, then would you expect A to beat C? Not always! What seems obvious is not always true. Results always need to be proved in mathematics.
### More Sums of Squares
##### Stage: 5
Tom writes about expressing numbers as the sums of three squares.
### Modulus Arithmetic and a Solution to Dirisibly Yours
##### Stage: 5
Peter Zimmerman from Mill Hill County High School in Barnet, London gives a neat proof that: 5^(2n+1) + 11^(2n+1) + 17^(2n+1) is divisible by 33 for every non negative integer n.
### Fractional Calculus III
##### Stage: 5
Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number.
### Interpolating Polynomials
##### Stage: 5 Challenge Level:
Given a set of points (x,y) with distinct x values, find a polynomial that goes through all of them, then prove some results about the existence and uniqueness of these polynomials.
### Dodgy Proofs
##### Stage: 5 Challenge Level:
These proofs are wrong. Can you see why?
### Advent Calendar 2011 - Secondary
##### Stage: 3, 4 and 5 Challenge Level:
Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.
### Mouhefanggai
##### Stage: 4
Imagine two identical cylindrical pipes meeting at right angles and think about the shape of the space which belongs to both pipes. Early Chinese mathematicians call this shape the mouhefanggai.
### A Computer Program to Find Magic Squares
##### Stage: 5
This follows up the 'magic Squares for Special Occasions' article which tells you you to create a 4by4 magicsquare with a special date on the top line using no negative numbers and no repeats.
### Euler's Formula and Topology
##### Stage: 5
Here is a proof of Euler's formula in the plane and on a sphere together with projects to explore cases of the formula for a polygon with holes, for the torus and other solids with holes and the. . . .
### Classifying Solids Using Angle Deficiency
##### Stage: 3 and 4 Challenge Level:
Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry
### The Triangle Game
##### Stage: 3 and 4 Challenge Level:
Can you discover whether this is a fair game?
### Proofs with Pictures
##### Stage: 5
Some diagrammatic 'proofs' of algebraic identities and inequalities.
### Particularly General
##### Stage: 5 Challenge Level:
By proving these particular identities, prove the existence of general cases.
### Sums of Squares and Sums of Cubes
##### Stage: 5
An account of methods for finding whether or not a number can be written as the sum of two or more squares or as the sum orf two or more cubes.
### A Knight's Journey
##### Stage: 4 and 5
This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.
### Kite in a Square
##### Stage: 4 Challenge Level:
Can you make sense of the three methods to work out the area of the kite in the square?
### Whole Number Dynamics IV
##### Stage: 4 and 5
Start with any whole number N, write N as a multiple of 10 plus a remainder R and produce a new whole number N'. Repeat. What happens?
### Whole Number Dynamics III
##### Stage: 4 and 5
In this third of five articles we prove that whatever whole number we start with for the Happy Number sequence we will always end up with some set of numbers being repeated over and over again.
### Whole Number Dynamics II
##### Stage: 4 and 5
This article extends the discussions in "Whole number dynamics I". Continuing the proof that, for all starting points, the Happy Number sequence goes into a loop or homes in on a fixed point.
### Pythagorean Triples II
##### Stage: 3 and 4
This is the second article on right-angled triangles whose edge lengths are whole numbers.
### Whole Number Dynamics I
##### Stage: 4 and 5
The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.
### Whole Number Dynamics V
##### Stage: 4 and 5
The final of five articles which containe the proof of why the sequence introduced in article IV either reaches the fixed point 0 or the sequence enters a repeating cycle of four values.
### Telescoping Functions
##### Stage: 5
Take a complicated fraction with the product of five quartics top and bottom and reduce this to a whole number. This is a numerical example involving some clever algebra.
### Recent Developments on S.P. Numbers
##### Stage: 5
Take a number, add its digits then multiply the digits together, then multiply these two results. If you get the same number it is an SP number.
### What Numbers Can We Make Now?
##### Stage: 3 and 4 Challenge Level:
Imagine we have four bags containing numbers from a sequence. What numbers can we make now? |
# A simple function that removes empty or tags containing just ' '
I've written a function that should get rid of empty p, span, etc tags and those with just ' ' and am looking for ways to improve it. My original solution was very 'wet', but I've managed to come up with a drier solution.
The Original HTML:
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.3.1/jquery.min.js"></script>
<div id='test'>
<p>text</p>
<p> </p>
<p>text</p>
<p><span>text</span></p>
<p><span></span></p>
<p>text</p>
<p><strong>text</strong></p>
<p></p>
<p> </p>
<p>text</p>
<p><span><strong> </strong></span></p>
<p><span><strong>text</strong></span></p>
<p> </p>
<p><span>text</span></p>
<p></p>
<p><span></span></p>
<p><span> </span></p>
<p><span><strong></strong></span></p>
<p>text</p>
</div>
My Original Solution:
/*
How to make this drier?
ORIGINAL UNCLEAN SOLUTION
*/
var ps = document.getElementsByTagName('p'),
spans = document.getElementsByTagName('span'),
strongs = document.getElementsByTagName('strong');
for (let el of ps) {
if (el.innerHTML == ' ') { // can't also include if '' at this stage
el.parentNode.removeChild(el);
}
}
for (let el of spans) {
if (el.innerHTML == ' ' || el.innerHTML == '') {
el.parentNode.removeChild(el);
}
}
for (let el of strongs) {
if (el.innerHTML == ' ' || el.innerHTML == '') {
el.parentNode.removeChild(el);
}
}
for (let el of ps) {
if (el.innerHTML == '') {
el.parentNode.removeChild(el);
}
}
My 'drier' solution:
/*
MY CLEANER SOLUTION
*/
var ps = document.getElementsByTagName('p'),
spans = document.getElementsByTagName('span'),
strongs = document.getElementsByTagName('strong');
for (let el of ps) {
cleaner(el);
}
for (let el of spans) {
cleaner(el);
}
for (let el of strongs) {
cleaner(el);
}
function cleaner(el) {
if (el.innerHTML == ' ' || el.innerHTML == '') {
el.parentNode.removeChild(el);
}
}
Would someone mind quickly running over both solutions and verifying that my 2nd solution is best? Also, I wonder whether that could be improved, or whether anyone has any better ideas for a solution? Thanks for the help here - for brevity, I'm looking at writing concise but also clear code.
• Does the code at the question produce the expected result? Should <p></p> be a child element of #test following execution of the code? Can you include the expected resulting HTML at the question? – guest271314 Jan 5 at 0:12
# There is a bug
You need to run the script several times to remove all empty elements.
## Two points
1. You say remove empty tags that contain "" or a single space " ". Does that include " " or " " two or more spaces. What about other white space characters?
2. Your element removal is order dependent because you use getElementsByTagName which returns a live list.
Consider the html <p><span></span></p> You first check all the p tags which fail the empty test, then you test the span tags which passes and you get <p></p> which is, by your definition, empty and should have been removed.
On the other hand the html <span><p></p></span> will first remove the p then remove the span.
The removal process is order dependent. Not what your question indicates.
## Changes
For the first point you could use element.textContent to check for empty elements. It will ignore the HTML and convert the to a space for you. You could even use element.textContent.trim() and thus get all blank elements (like the pseudo-class :blank (Which has very limited support FF only))
This also covers the second point.
### Example Mark and remove
To reduce the DOM calls you can mark and remove deleting the marked elements only.
const isNotMarked = el => {
while (el && el.parentNode && !el.parentNode.marked) {
el = el.parentNode;
if (el.marked) { return false }
}
return true;
}
[...document.querySelectorAll("span, p, strong")]
.filter(el => el.textContent.trim() === "" && isNotMarked(el) ? el.marked = true : false)
.forEach(el => el.parentNode.removeChild(el));
### Example simple brute force
Mark and remove saves you attempting to delete already deleted elements but you may not care, as the shorter form, is a two liner, and thus could be argued to be the better solution.
document.querySelectorAll("span, p, strong")
.forEach(el => el.textContent.trim() === "" && el.parentNode.removeChild(el))
The following snippet shows the HTML after using your function and then the two example functions
/*=================================================================================
OP ver modified for example
=================================================================================*/
var ps = cleaned.getElementsByTagName('p'),
spans = cleaned.getElementsByTagName('span'),
strongs = cleaned.getElementsByTagName('strong');
for (let el of ps) { cleaner(el); }
for (let el of spans) { cleaner(el); }
for (let el of strongs) { cleaner(el); }
function cleaner(el) {
if (el.innerHTML == ' ' || el.innerHTML == '') {
el.parentNode.removeChild(el);
}
}
content.textContent = cleaned.innerHTML;
/*=================================================================================
Mark and remove
=================================================================================*/
const isNotMarked = el => {
while (el && el.parentNode && !el.parentNode.marked) {
el = el.parentNode;
if (el.marked) { return false }
}
return true;
}
[...cleanerClean.querySelectorAll("span, p, strong")]
.filter(el => el.textContent.trim() === "" && isNotMarked(el) ? el.marked = true : false)
.forEach(el => el.parentNode.removeChild(el));
contentA.textContent = cleanerClean.innerHTML;
/*=================================================================================
Brute force remove
=================================================================================*/
simplerClean.querySelectorAll("span, p, strong")
.forEach(el => el.textContent.trim() === "" && el.parentNode.removeChild(el))
contentB.textContent = simplerClean.innerHTML;
#content {
display: block;
}
<div id="cleaned" style="display:none;">
<p>text</p>
<p> </p>
<p>text</p>
<p><span>text</span></p>
<p><span></span></p>
<p>text</p>
<p><strong>text</strong></p>
<p></p>
<p> </p>
<p>text</p>
<p><span><strong> </strong></span></p>
<p><span><strong>text</strong></span></p>
<p> </p>
<p><span>text</span></p>
<p></p>
<p><span></span></p>
<p><span> </span></p>
<p><span><strong></strong></span></p>
<p>text</p>
</div>
<fieldset>
<legend>Original OPs script & Resulting HTML</legend>
<code id = "content"></code>
</fieldset>
<div id="cleanerClean" style="display:none;">
<p>text</p>
<p> </p>
<p>text</p>
<p><span>text</span></p>
<p><span></span></p>
<p>text</p>
<p><strong>text</strong></p>
<p></p>
<p> </p>
<p>text</p>
<p><span><strong> </strong></span></p>
<p><span><strong>text</strong></span></p>
<p> </p>
<p><span>text</span></p>
<p></p>
<p><span></span></p>
<p><span> </span></p>
<p><span><strong></strong></span></p>
<p>text</p>
</div>
<fieldset>
<legend>Mark and remove</legend>
<code id = "contentA"></code>
</fieldset>
<div id="simplerClean" style="display:none;">
<p>text</p>
<p> </p>
<p>text</p>
<p><span>text</span></p>
<p><span></span></p>
<p>text</p>
<p><strong>text</strong></p>
<p></p>
<p> </p>
<p>text</p>
<p><span><strong> </strong></span></p>
<p><span><strong>text</strong></span></p>
<p> </p>
<p><span>text</span></p>
<p></p>
<p><span></span></p>
<p><span> </span></p>
<p><span><strong></strong></span></p>
<p>text</p>
</div>
<fieldset>
<legend>Brute force remove</legend>
<code id = "contentB"></code>
</fieldset>
You can use querySelectorAll to simplify your code further:
var elements = document.querySelectorAll('p, span, strong'),
for (let el of elements) {
cleaner(el);
}
Beside suggested improvements:
1. If <p> </p> is an empty element to you, then change your cleaner():
function cleaner(el) {
if (el.innerHTML.match(/^\s+\$/) !== null) {
el.parentNode.removeChild(el);
}
}
2. You might need to consider going recursive towards elements that have been emptied because of your cleaning procedure.
3. I'm used to verbal function names (a best practice to follow), so I would suggest using clean or remove instead of cleaner.
I support the main aspect of Carra's answer (i.e. using querySelectorAll()). In addition, a functional approach can be used, since the function cleaner is applied to each element. For that, utilize Array.prototype.forEach().
elements.forEach(cleaner);
That way, there is no need to set up an iterator variable (e.g. el in the for...of loop just to pass it to the function. The function will receive the element as the first parameter each time it is called - once for each element in the collection.
Additionally, since features like for...of and let are used, others like const can be used (e.g. for any variable that doesn't need to be re-assigned). One could also use arrow functions if desired.
And it would be a good habit to use the strict equality comparison (i.e. ===) when comparing the innerHTML properties with the strings.
function cleaner(el) {
if (el.innerHTML === ' ' || el.innerHTML === '') {
el.parentNode.removeChild(el);
}
}
const elements = document.querySelectorAll('p, span, strong');
elements.forEach(cleaner);
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.3.1/jquery.min.js"></script>
<div id='test'>
<p>text</p>
<p> </p>
<p>text</p>
<p><span>text</span></p>
<p><span></span></p>
<p>text</p>
<p><strong>text</strong></p>
<p></p>
<p> </p>
<p>text</p>
<p><span><strong> </strong></span></p>
<p><span><strong>text</strong></span></p>
<p> </p>
<p><span>text</span></p>
<p></p>
<p><span></span></p>
<p><span> </span></p>
<p><span><strong></strong></span></p>
<p>text</p>
</div>
• thanks, some interesting points to take home :) – user8758206 Jan 7 at 8:59 |
# nLab axiom of full comprehension
Full (unrestricted) comprehension
foundations
# Full (unrestricted) comprehension
## Idea
In mathematical logic/mathematical foundations, the axiom or rule of full or unrestricted comprehension says that for any property $P$, there exists a set of all objects satisfying $P$:
$\{ x \mid P(x) \}.$
Set theory with the unrestricted comprehension rule is called naive set theory?, and is inconsistent due to Russell's paradox and Curry's paradox. Here we mention several approaches to this issue.
## Restricted comprehension
Standard set theories such as ZFC avoid this paradox by replacing unrestricted comprehension with the axiom scheme of separation (or “restricted comprehension”), which restricts $x$ to lie in some previously specified set $X$.
## Stratified comprehension
Set theories such as New Foundations instead replace comprehension by a rule of “stratified comprehension”. This permits a “set of all sets” but still appears to avoid paradox.
## Substructural logics
It is also possible to retain full comprehension but avoid paradox by modifying the ambient logic. Passing to constructive logic doesn’t help, and indeed the root issue has nothing to do with negation as such, since Curry's paradox can be stated without any negation. One might think that paraconsistent logic would help, but many paraconsistent logics are still vulnerable to Curry’s paradox. Perhaps the most obvious culprit is the contraction rule, and indeed linear logic (including some paraconsistent logics) can admit a full comprehension rule without explosion.
## Normal logics
Another possibility is to keep the contraction rule but restrict the use of the cut rule. It is not necessary to forbid all uses of cut, since many cuts can be normalized or eliminated. Indeed, in ordinary consistent logic, all cuts can be eliminated; but in the presence of full comprehension they cannot all be. Thus, another way to avoid paradox with full comprehension is to permit only proofs that can be normalized.
Note that unlike a restriction on contraction, this is a “global” restriction: the proofs of two lemmas can independently be valid, but their combination may no longer be so. Similar “global” restrictions on logic were investigated by Fitch 1953, 69.
## References
### In linear logic
In linear logic:
• Grishin, V. N., “Predicate and set theoretic calculi based on logic without contraction rules” (Russian), Izvestiya Akademii Nauk SSSR Seriya Matematicheskaya, 45(1): 47 – 68, 1981. English translation in Math. USSR Izv., 18(1): 41 – 59, 1982. (math-net.ru)
• Jean-Yves Girard, Light Linear Logic, Information and Computation, 14(3):123-137, 2003. (pdf.gz)
• Kazushige Terui, Light Affine Set Theory: A Naive Set Theory of Polynomial Time, Studia Logica: An International Journal for Symbolic Logic, Vol. 77, No. 1 (Jun., 2004), pp. 9-40. (jstor) (pdf). See also Terui’s slides, Linear Logic and Naive Set Theory (Make our garden grow)
### Global restrictions
Several global restrictions were considered in
The notation therein is somewhat difficult to follow for a modern reader, especially due to the somewhat confused treatment of what nowadays would be called free and bound variables. A more modern explanation of Fitch’s restrictions can be found in:
• Susan Rogerson, Natural deduction and Curry’s paradox, Journal of Philosophical Logic (2007) 36: 155–179. pdf
The normalizability restriction is also discussed philosophically in
• Tennant, N.: Proof and paradox, Dialectica 36 (1982), 265-296 |
Perhaps where everyone starts, with machine learning models, is linear regression. Here you will be introduced to both linear and logistic regression.
###### Linear Regression (Least Squares Regression)
First of all, what is regression? When doing regression, we map some object to another object, namely an input and output. Actually what we are doing is estimating the relationships between variables in our dataset. Maybe you have seen a plot like this before, where we plot some data points.
When doing linear regression, we simply find a line that fits the best through our data points. What does it mean that it fits the best? Well, it means that when you draw a line through the data, the average distance from each point to that line is the lowest of all the lines we could draw – so we just find the line that fits right in the middle of our data. It looks like this:
Now that we covered the intuition, let's explain it in math. To plot the above we have an equation to do it. I include two versions that are equal, just in different terms:
$$y=mx+b \Leftrightarrow y=w_0+w_1 x$$
I'm going to continue using the left one, as that is what I learned, but just know that they are the same.
$w_0$ (or b) is the intercept term, meaning where the line through the data points intercept the y-axis. $w_1$ is the coefficient or slope, our input. Now I'm now diving into how you actually calculate each term, but I will refer you to Khan Academy if you want to learn that.
Let's step into deeper waters, where we generalize this formula to take $w_M$ ($M$ features or coefficients). We could first imagine the 2-dimensional input where $M=2$, which looks like $y=w_0+w_1 x_1 +w_2 x_2$, but let's also generalize it:
$$y=w_0+w_1 x_1 +w_2 x_2+...+w_M x_M$$
As mentioned in the intuition part, we kind of want the average from all the actual plotted points to the predicted value on the line we plot, to be the lowest that it can possibly be. As stated before, we try to find such a line, where the average distance of all the points to the line is minimal.
This is referred to as calculating the residual error.
For each blue data point, there is a distance indicated by the red line. The average of this distance for all points to the line, is what we are trying to minimize. We then get an $R^2$ value, which is best when it approaches the value 1. This is also referred to as least squares regression, and for more on the calculations, watch this Khan Academy video.
Though when doing least squares regression, we make an initial guess on a line that might fit good, calculate the $R^2$ value, then make another that has a better fit, and we keep doing that until we get the most optimal fit.
As a final remark, there really is no random parameters in this algorithm, and this means that least squares linear regression produces the same $R^2$ value every time you run it. This means, for the same data points, you will always get the same line, because it will be the best fitting one.
###### Final tips for linear regression
As a final notice on linear regression, I want to include some information about what to be careful with, when using the algorithm for data.
Could you imagine if you had a dataset that has a few outliers? Least Squares Regression is susceptible to outliers; one could imagine that that would impact the calculation of the average, if you have some extreme outliers. Suppose your dataset isn't that big either, then it is even more susceptible to outliers, making the linear regression algorithm perform poorly.
When performing linear regression, we have to be wary of overfitting. It is too easy to use too complicated of a model, by the formula $y=w_0+w_1 x_1 +w_2 x_2+...+w_M x_M$. It might be a clear indication of overfitting, if you use too many input variables in linear regression, such that less input variables will give a better model.
Finally, always remember to evaluate your model using new data, that you did not train you model on. If you evaluate on the same data as you trained your model, you could get a different picture of which model is the best for new data.
###### Logistic Regression
The way logistic regression works, will remind you of linear regression, except for we don't use least squares to find the optimal $R^2$ value. Instead we use maximum likelihood to find the maximum likelihood.
In other words, if you plot any data point in a logistic plot, it will have some measure along the x-axis, and the probability of that measure being either true or false on the y-axis. True or false, meaning 1 for true and 0 for false.
Logistic regression is almost always used for classification, and that is the typical use-case. However, it can also be used to do regression, though that is not common at all, and there are probably other regression methods that will perform better than this one.
###### Example of logistic regression
In linear regression, we fit a straight line through the data, but in logistic regression, we fit a curve that looks sort of like an s. It will probably remind you of the sigmoid function, if you have ever heard of that. So we have this s-curve that goes from 0 to 1 or from 1 to 0, dependent on the variable on the x-axis.
As you can read from this graph, the probability of passing an exam increases with the more hours spent studying. Usually though, you are trying to plot a new point, to then classify that point. A common basic rule is, if your new point on the x-axis can be 'trailed' up to the curve, meaning you draw a line from the point to the curve, and it reaches above 50%, then we say that the probability of passing the exam is more likely, therefore we say that you are going to pass. And in the other case, below 50%, we say that you are not going to pass.
What I just told you is more of a deterministic look at the curve, than a probabalistic approach. Instead of making such a basic rule, we could simply say that a student who studied for 4 hours has the probability 0.875 (87.5%) of passing, simply by eyeballing where such a new point would be trailed to the curve.
###### Logistic Regression — Maximum Likelihood revisited
You might say, well how did the curve get there in the first place? This is what I was talking about at the beginning, it's a concept called maximum likelihood. You pick a value $\theta$, then pick something you want to predict, e.g. the likelihood of passing an exam. You would then calculate the likelihood of all points and multiply them together. At last, you could form a distribution to show which $\theta$ value resulted in the biggest likelihood for all points.
Similarly in logistic regression, we also calculate the maximum likelihood, but in a different way.
1. Transform coordinate system to the y-axis being the log of probabilities, and the x-axis being 0.5 probability (e.g. where the x-axis intercepts the y-axis at zero, the probability is 0.5).
2. Move all data points over to the new coordinate system, for each point: $log\left(\frac{p}{1-p}\right) = log(predictor)$. Note that all points are either at negative or positive infinity in the new coordinate system.
3. Plot a random line, like in linear regression. Remember the formula $y=b+mx$.
4. Project all data points onto the line. Since the points are at infinity, we don't have a specific y-value for each point. But we could imagine a step for each point, where the y-value is set to 0, then projected onto the line.
5. Transform coordinate system to old coordinate system with probabilities on the y-axis, forming the s-shaped curve. For each point $p=\frac{e^{log(predictor)}}{1+e^{log(predictor)}}$.
6. Each point is now on the s-curve, and the probabilities can be found by looking at where each point is traced to on the y-axis.
7. Trace each point to it's probability on the y-axis, and use addition to calculate the log of the probability of all points together. This is the likelihood estimate.
Steps 3 to 7 are repeated, and note that the algorithm naturally tries to plot a better line than the last one in step 3, by figuring out if the likelihood estimate increases.
When this is done, the s-curve that had the best likelihood estimate from step 7 is the s-curve with the best estimate, therefore we call it the maximum likelihood estimate.
The GIF gives a clear image of logistic regression. You could simply imagine that we try a new candidate line (right), then transform it (arrow) into the probability plot. We would then get likelihood estimate for each of the lines, and we could say that the one with the maximum likelihood estimate is the best fitting line.
###### Code – Linear Regression
First we import everything we need.
``````# Import all necessary packages
from matplotlib.pyplot import figure, plot, xlabel, ylabel, legend, show
from sklearn.linear_model import LinearRegression
import numpy as np
``````
Then we create a random dataset. You would want to create import your own dataset here and define the training and testing data for your Linear Regression. Here is an example from Scikit-Learn.
``````# Random Dataset of points trending upwards
N = 100
X = np.array(range(N)).reshape(-1,1)
mean, std = -0.5, 0.1
eps = np.array(std*np.random.randn(N) + mean).reshape(-1,1)
w0 = -0.5
w1 = 0.02
y = w0 + w1*X + eps
``````
Next we define the model, which is going to be Linear Regression. This example just shows you how to use the function from the sklearn import. Note that we train and test upon the same data here, to give an example, which is not something you want to do, when solving problems. Always train and test on different data.
``````# Define model
model = LinearRegression(fit_intercept=True)
# Fit model to data
model = model.fit(X,y)
# Make predictions
# Note, don't make predictions on the data you trained on (X)
y_est = model.predict(X)
``````
Lastly, we visualize the data along with the model we just trained.
``````# Plot original data and the model output
f = figure()
# Plot datapoints
plot(X,y,'.')
# Plot linear regression
plot(X,y_est,'-')
# Define labels
xlabel('X'); ylabel('y');
# Define 'legend', respectively what the dots and line means
legend(['Training data', 'Linear Regression'])
# Show the plot
show()
``````
###### Code – Logistic Regression
This example is a copy-paste from sklearn's example. It's a great example on one of the most popular datasets, when learning machine learning, the iris dataset.
As with many algorithms in machine learning, the groundwork has been done for you by scikit-learn. They implement the algorithm, you implement their package which includes the algorithm, by the line of code `from sklearn.linear_model import LogisticRegression`. The output is the following:
Here is the code. You would need to adapt it to your dataset:
``````# Code source: Gaël Varoquaux
# Modified for documentation by Jaques Grobler
import numpy as np
import matplotlib.pyplot as plt
from sklearn.linear_model import LogisticRegression
from sklearn import datasets
# import some data to play with
X = iris.data[:, :2] # we only take the first two features.
Y = iris.target
logreg = LogisticRegression(C=1e5, solver='lbfgs', multi_class='multinomial')
# Create an instance of Logistic Regression Classifier and fit the data.
logreg.fit(X, Y)
# Plot the decision boundary. For that, we will assign a color to each
# point in the mesh [x_min, x_max]x[y_min, y_max].
x_min, x_max = X[:, 0].min() - .5, X[:, 0].max() + .5
y_min, y_max = X[:, 1].min() - .5, X[:, 1].max() + .5
h = .02 # step size in the mesh
xx, yy = np.meshgrid(np.arange(x_min, x_max, h), np.arange(y_min, y_max, h))
Z = logreg.predict(np.c_[xx.ravel(), yy.ravel()])
# Put the result into a color plot
Z = Z.reshape(xx.shape)
plt.figure(1, figsize=(4, 3))
plt.pcolormesh(xx, yy, Z, cmap=plt.cm.Paired)
# Plot also the training points
plt.scatter(X[:, 0], X[:, 1], c=Y, edgecolors='k', cmap=plt.cm.Paired)
plt.xlabel('Sepal length')
plt.ylabel('Sepal width')
plt.xlim(xx.min(), xx.max())
plt.ylim(yy.min(), yy.max())
plt.xticks(())
plt.yticks(())
plt.show()
`````` |
# Find the Number of non-congruent triangles (integer sided) whose sides belong to the set {10,11,12,....22}
Find the Number of non-congruent triangles (integer sided) whose sides belong to the set
{10,11,12,....22}
This is a question in a national level based examination. Now in here what i can see that be taking each of the numbers as largest side(except for 10 and 11) we can find the number of triangles thus formed in each case by a brute force method.
Now also the question says non congruent triangles which makes the problem even more complex.Now do we apply recurssion here or simply this a problem of permutations and combinations.
Also i accept that my method of a brute force approach is rather an unintelligent way to solve it,so i am looking for answer that can solve it in 5 lines or so.
You just need to choose three numbers from the set with replacement so that the sum of the smaller two is greater than the largest. The lengths of the sides determine the triangle. As there are $13$ numbers in the set, there are $13$ choices where the numbers are all the same, $13\cdot 12=156$ where two numbers are the same, and $\frac 16\cdot 13 \cdot 12 \cdot 11=286$ where all three numbers are different. Because $22$ is barely more than twice $10$ we can hand count the ones that do not form a triangle. There are none with all the sides the same. With two sides equal, the only failures are $(10,10,20), (10,10,21), (10,10,22), (11,11,22)$. With all sides different there is just $(10,11,21), (10,11,22),(10,12,22)$. There are therefore $13+156+286-7=448$ triangles that can be formed.
• That is correct. For three different numbers you don't care what order you pick them in, so it is $13 \choose 3$. For two numbers you do care because you can decide to take two of the first and one of the second. $(13,13,12)$ is different from $(12,12,13)$ Oct 25, 2017 at 14:29 |
## August 4, 2007
### Gauge Tranformations of n-Bundles and (n-1)-Gerbes
#### Posted by Urs Schreiber
Over on sci.math.research Christoph Wockel today asks:
can anyone help me with finding the appropriate notion of a gauge transformation on a gerbe (abelian or not)? I would be interested in particular in lifting gerbes. I tried to find it several times but did not succeed. My guess would be to define it as a vertical 2-bundle automorphismsm of the corresponding 2-bundle, but I did not find this mentioned anywhere… Accordingly, gauge transformations would build up a 2-group rather than a group.
I’ll give a quick reply here, indicating the basic idea. Upon request I can spell out more details.
Quite generally, “gauge transformation” is really just another word for automorphism.
So, given any object $P$ in some category – like a $G$-bundle, say, in the category of $G$-bundles, its “gauge transformations” are simply all morphism
$g : P \to P$
from the object to itself (i.e. endomorphisms) which are invertible.
For bundles here, one would want to distinguish, possibly, between the category of bundles whose morphisms are required to fix the base space and that where we don’t impose this restriction. The gauge transformations proper of bundles are the automorphisms in the former version of this category. But it one may feel the need to look at automorphisms in the latter, too, thus getting a notion of gauge transformation which combines the more standard one with diffeomorphisms of the base space. This is useful, for instance, for the study of equivariant bundles, in case there is a group acting by diffeomorphisms on the base space.
The only subtlety as we climb up the dimensional ladder, now, is that the notion of “automorphism” gets weakened to that of an equivalence.
For 2-categories, a morphism $g : P \to P'$ is called an equivalence, if there is another morphism, $h : P' \to P$ such that the 1-morphisms $h\circ g : P \to P$ and $g\circ h : P' \to P'$ are – not necessarily equal but isomorphic, i.e. connected by an invertible 2-morphisms, to the identity 1-morphism on $P$ and on $P'$, respectively.
From that, the pattern is clear: in a general $n$-category, we say, recursively, that an $n$-morphism is an equivalence if it is invertible, and a $(k \lt n)$-morphism is an equivalence if it has an inverse up to a $(k+1)$-equivalence.
That said, all one needs to understand then to understand gauge transformations of $n$-bundles and of $(n-1)$-gerbes is what the $k$-morphisms between these beasts are, in the first place.
This is usually very obvious. Of course the details depende on which of a bunch of equivalent formulations of these objects one is looking at.
In as far, for instance, as an $n$-bundle is an $n$-category $P$ with a suitable projection $p : P \to X$ for $X$ the discrete $n$-category (i.e. no nontrivial morphisms) over the elements of some space $X$, then morphisms between these guys are simply the $n$-functors between the total $n$-spaces $g : P \to P'$ respecting whichever structure one demands to respect (like smoothness, usually, or like the base space projection, usually).
Conversely, if one rather likes to think of one’s $n$-bundle as a fiber-assigning ($n+1$)-functor $X \to n\mathrm{Cat}$ which sends each point $x \in$ to the fiber $P_x$ (an $n$-category) living over it, then morphisms are simply the morphisms of such $n$-functors, respecting the extra structure (smoothness) which is around.
Or, if you like to think of the stacks of sections of these beasts, i.e. of ($n-1$)-gerbes as $(n-1)$-stacks on something like the site of open subsets of a fixed space, or on that of all manifolds, say, morphisms are simply the morphisms of these $(n-1)$-stacks (hence, since $(n-1)$ stacks are pseudofunctors, are nothing but morphisms of pseudofunctors).
Similarly, in both $n$-bundle perspectives mentioned before, one tends to want to work with local trivializations of the full global thing, i.e. with the descent data of these gadgets. Details may depends on the setup, but the descent data are essentially themselves pseudofunctors, so they have obvious notions of morphisms between them.
Applying this general nonsense to concrete special realizations, like bundle gerbes, then tells one what the morphisms of these are. For instace the funny twist which is introduced for morphisms of bundle gerbes, which was originally missed and therefore later called “stable morphism” of bundle gerbes (unfortunately) is in fact precisely what one gets from the general nonsense once one realizes that a bundle gerbe is really a pseudofunctor.
I, and others here, should be able to provide literature and precise details and formulas for whatever special case is requested.
Posted at August 4, 2007 8:38 AM UTC
TrackBack URL for this Entry: http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/1381
## 1 Comment & 0 Trackbacks
### Re: Gauge Tranformations of n-Bundles and (n-1)-Gerbes
Very minor comment:
If this is meant to be defining morphisms so as to define the category, am I right that
these morphisms are over the identity of the base? the other alternative is also viable and of interst in the physics.
Posted by: jim stasheff on August 11, 2007 4:23 PM | Permalink | Reply to this
Post a New Comment |
# Self Doubt[Group Theory]
126 views
## Is Every Group of Order $P^{k}$ such that P is prime and K is positive integer ABELIAN
0
A constant value cannot form a set
0
0
@Gupta731
in that link they given for SIMPLE ABELIAN GROUP...NOT FOR ABELIAN GROUP
what is differnce in abelian and simple abelian
If order of group is prime ====> Group is cyclic ====> Group is abelian
This is one way only
..in one of prev question it is taken group of order P^2 is abelian ...is it generalize to P^k
0
@jatin khachane 1
then P must be variable and k is order of group
## Related questions
1
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Is this monoid: Addition modulo (take mode using m) on the set of Integers (Z m)={0,1,2,3,4,…..m-1} i.e. For all a a (+ modulo using m) e = e (+ modulo using m) a =a here, e is an identity element
A homomorphism $f:G$ to $G1$ of groups is a monomorphism iff Ker $f = \{e\}$. |
# Mathematics for Economics
Michael Hoy
John Livernois
Chris McKenna
Ray Rees
Thanasis Stengos
Pages: 976
https://www.jstor.org/stable/j.ctt5hhc2f
1. Front Matter
(pp. i-vi)
(pp. vii-xii)
3. Preface
(pp. xiii-xiv)
4. ### Part I Introduction and Fundamentals
• Chapter 1 Introduction
(pp. 3-10)
Almost for as long as economics has existed as a subject of study, mathematics has played a part in both the exploration and the exposition of economic ideas.¹ It is not simply that many economic concepts arequantifiable(examples include prices, quantities of goods, volume of money) but also that mathematics enables us to explore relationships among these quantities. These relationships are explored in the context ofeconomic models, and how such models are developed is one of the key themes of this book. Mathematics possesses the accuracy, the rigor, and the capacity to deal clearly with complex systems, which...
• Chapter 2 Review of Fundamentals
(pp. 11-60)
In this chapter we give a concise overview of some fundamental concepts that underlie everything we do in the rest of the book.
In section 2.1 we present the basic elements of set theory. We then go on to discuss the various kinds of numbers, ending with a concise treatment of the properties of real numbers, and the dimensions of economic variables. We then introduce the idea of point sets, beginning with the simplest case of intervals of the real line, and define their most important properties from the point of view of economics: closedness, boundedness, and convexity. Next we...
• Chapter 3 Sequences, Series, and Limits
(pp. 61-100)
Studying sequences and series is the best way to gain intuition about the rather perplexing notions of arbitrarily large numbers (infinity) and infinitesimally small (but nonzero) numbers. We gain such understanding by using the idea of the limit of a sequence of numbers. Thus, from a mathematical perspective, this chapter provides very useful background to the important property of continuity of a function, which we will explore fully in chapter 4. There are also some interesting economic applications of series and sequences, in particular the notion of discounting a future stream of payments or receipts, which is a critical aspect...
5. ### Part II Univariate Calculus and Optimization
• Chapter 4 Continuity of Functions
(pp. 103-126)
The idea of continuity of a function is extremely important in mathematics. Many convenient techniques of analysis can be used if a function is continuous but not if it is discontinuous. In modeling economic problems, we often assume that we can represent various economic concepts by continuous functions (e.g., the relationship between the quantity of some commodity produced by the firm and its profit level). Thus it is important to know precisely what is the content of this assumption, especially since in many instances there is a natural reason to believe that the function willnotbe continuous everywhere, and...
• Chapter 5 The Derivative and Differential for Functions of One Variable
(pp. 127-194)
The purpose of the derivative is to express in a convenient way how a change in the level of one variable (e.g.,x) determines a change in the level of another variable (e.g.,y). Much of economics is in fact concerned with just this sort of analysis. For example, we study how a change in a firm’s output level affects its costs and how a change in a country’s money supply affects the rate of inflation. Although expressing the relationship betweenxandyas a functiony=f(x) does capture this idea implicitly, it is much more convenient...
• Chapter 6 Optimization of Functions of One Variable
(pp. 195-232)
Many economic models are based on the idea that an individual decision maker makes anoptimal choicefrom some given set of alternatives. To formalize this idea, we interpret optimal choice as maximizing or minimizing the value of some function. For example, a firm is assumed to minimize costs of producing each level of output and to maximize profit; a consumer to maximize utility; a policy maker to maximize welfare or the value of national output; and so on. It follows that the mathematics of optimization is of central importance in economics, and in this and chapters 12 and 13...
6. ### Part III Linear Algebra
• Chapter 7 Systems of Linear Equations
(pp. 235-266)
In chapter 2 we defined alinear functionas one that takes the form$y=a+bx \caption {(7.1)}$for known constantsaandb, and wherexis theindependent variablethat takes on values over some specified domain, andyis the resulting value of the function at eachx-value. We also know that by taking specific values ofx, we can draw the graph ofxandyin a two-dimensional picture. The graph is a straight line: hence the phraselinear function. There are many examples of functions in economics that can be represented in a linear form. The market...
• Chapter 8 Matrices
(pp. 267-300)
A matrix provides a very powerful way of organizing and manipulating data. In chapter 7 matrices were used to focus attention on the parameters and constants of a simultaneous-equation system. The rows of a matrix could be manipulated to find solutions to the unknown variables in the original equations using, for example, the Gauss-Jordan elimination approach. There are clearly many instances where a large amount of information can be summarized in matrix form. Moreover there are procedures or operations on matrices that allow us to discover important properties of systems of equations.
As with any area of mathematics, there are...
• Chapter 9 Determinants and the Inverse Matrix
(pp. 301-346)
In chapter 8 we defined the operations of addition, subtraction, and multiplication of matrices. What about division? Can we define rules for dividing one matrix by another? The answer is yes, but only under certain restrictions. Division is restricted only tosquarematrices, and then only to those square matrices that satisfy a condition known asnonsingularity. The reason for all this can again be traced to the relation between matrix algebra and the problem of solving a system of simultaneous linear equations.
Consider, first, the division of two numbers. If we dividebintoa, we can write this...
• Chapter 10 Some Advanced Topics in Linear Algebra
(pp. 347-390)
In this chapter we consider three important advanced topics in matrix algebra: vector spaces, eigenvalues, and quadratic forms. All play important roles in a variety of contexts in economic theory and in econometrics. Vector spaces enable us to talk about distance between points, and linear dependence between vectors. They are therefore closely linked to the study of systems of linear equations of chapter 7. Eigenvalues play an important role in determining the stability properties of dynamic, linear systems and so this topic is of use in chapters 18, 20, 21, 23, and 24. Quadratic forms have applications in econometrics, and...
7. ### Part IV Multivariate Calculus
• Chapter 11 Calculus for Functions of n-Variables
(pp. 393-472)
We have already discussed at length the basic principles of calculus for functions of one variable,y=f(x) with$x\in \mathbb{R}$. Continuity was presented in chapter 4, and the derivative was presented in chapter 5. Economic analysis, however, often demands consideration of functions of more than one variable. For example, it is often important to model how the level of output produced by a firm depends on several inputs rather than just one. In this chapter we consider the fundamental relationships of differential calculus for functions of more than one variable. Fortunately, much of what was learned in chapters...
• Chapter 12 Optimization of Functions of n-Variables
(pp. 473-502)
The idea of optimization is fundamental in economics, and the mathematical methods of optimization underlie most economic models. For example, the theory of demand is based on the model of a consumer who chooses the best (“most preferred”) bundle of goods from the set of affordable bundles. The theory of supply is based on the model of a firm choosing inputs in such a way as to minimize the cost of producing any given level of output, and then choosing output to maximize profit. Rationality and optimization are virtually synonymous in economics.
In a formal sense, by optimization we mean...
• Chapter 13 Constrained Optimization
(pp. 503-528)
If, when maximizing or minimizing a function, we are free to consider any value of anx-variable on the real line as a possible solution, then the problem is said to be unconstrained. Most of the techniques developed in chapters 6 and 12 related to this case. In many, probably most, economic problems, however, there exist one or moreconstraintswhich restrict the set ofx-values we are allowed to consider as possible solutions. We already examined one type of constraint in chapters 6 and 12, namely that wherex-values are restricted to lie in some interval. The examples we...
• Chapter 14 Comparative Statics
(pp. 529-566)
As we discussed in chapter 1, economic models have two types of variables: endogenous variables, whose values the model is designed to explain, and exogenous variables, whose values are taken as given from outside the model. The solution values we obtain for the endogenous variables will typically depend on the values of the exogenous variables, and a central part of the analysis will often be to show how the solution values of the endogenous variables change with changes in the exogenous variables. This is the problem of comparative-static equilibrium analysis or comparative statics.
In the first section of this chapter...
• Chapter 15 Concave Programming and the Kuhn-Tucker Conditions
(pp. 567-582)
In the constrained optimization problems of chapter 13, we used the case where the function constraints are alwaysequalities. This is usually referred to as the “classical optimization problem.” However, sometimes this is not the most sensible formulation of a problem from the point of view of economics, and problems can arise that require us to set the constraints asweak inequalities. In this chapter we develop the necessary conditions for solutions of this type of problem. Because it is assumed that the objective and constraint functions are all concave, it is generally referred to as theconcave-programming problem.
We...
8. ### Part V Integration and Dynamic Methods
• Chapter 16 Integration
(pp. 585-632)
In this chapter, we address the question of whether knowing the derivative of a function,${f}'(x)$, allows one to determine, or recover, the original functionf(x). Since this process is the reverse of differentiation it is referred to as antidifferentiation, although it is also referred to as finding the indefinite integral. Related to this concept is the definite integral of a function, which is the area beneath a curve between two points. The process of integration is very useful in economics as it reflects the relationship between stocks and flows (e.g., investment and capital stock) and marginal and total...
• Chapter 17 An Introduction to Mathematics for Economic Dynamics
(pp. 633-642)
Economic dynamics is a study of how economic variables evolve over time. Unlike economic statics, which is a study of economic systems at rest, the focus of attention in economic dynamics is on how economic systems change as they move from one position of rest (i.e., equilibrium) to another. In this sense, economic dynamics, in adding the dimension of time to economic models, goes a step beyond economic statics. Often, however, this added realism and complexity can be managed only by reducing the complexity of the economic model in some other direction.
Once we introduce time to economic models, we...
• Chapter 18 Linear, First-Order Difference Equations
(pp. 643-664)
In the next three chapters we introduce some elementary techniques for solving and analyzing the kinds of difference equations that are common in economics. We begin in this chapter with linear, first-order difference equations. In the next chapter we introduce nonlinear, first-order difference equations, including the famous logistic equation used extensively in the study ofchaos. In chapter 20 we examine linear, second-order difference equations.
In this section we explain how to solve linear, first-order difference equations that are autonomous.
Definition 18.1 The general form of the linear, first-order, autonomous difference equation is given by${{y}_{t+1}}=a{{y}_{t}}+b,\quad \ t=0,1,2,\ldots \caption {(18.1)}$whereaandb...
• Chapter 19 Nonlinear, First-Order Difference Equations
(pp. 665-680)
In the previous chapter we saw that linear, first-order difference equations can be solved explicitly. We will see in the next chapter that this is also true for linear, second-order difference equations.Nonlineardifference equations, on the other hand, cannot be solved explicitly in general. However, it is still possible to obtain qualitative information about the solution by analyzing the nonlinear difference equation with the aid of a phase diagram. This technique can be very useful in economics because we are often mainly concerned with the qualitative properties of dynamic models. In this chapter we do this analysis for first-order...
• Chapter 20 Linear, Second-Order Difference Equations
(pp. 681-714)
In this chapter we turn to linear difference equations of the second order. We focus our attention on theautonomouscase in section 20.1 and consider a special nonautonomous case in section 20.2. In addition we introduce a new solution technique in this chapter. The technique involves breaking up the relatively difficult problem of finding the general solution to the difference equation into two parts, each of which is easier to solve than the whole. Not only does this simplify matters in this chapter, but it proves to be indispensable in later chapters in solving differential equations, and systems of...
• Chapter 21 Linear, First-Order Differential Equations
(pp. 715-738)
In the next three chapters we explain techniques for solving and analyzing ordinary differential equations. We do not attempt to provide exhaustive coverage of the topic but instead focus on the types of differential equations and techniques of analysis that are most common in economics. We begin in this chapter with linear, first-order differential equations. In the next chapter we turn to an examination of nonlinear, first-order differential equations, and in the chapter after that we examine linear, second-order differential equations. In this chapter and throughout, we will solve a large number of examples and economic applications to illustrate the...
• Chapter 22 Nonlinear, First-Order Differential Equations
(pp. 739-752)
In chapter 21 we saw that we could apply a single solution technique to solve any first-order differential equation that islinear. When the differential equation isnonlinear, however, no single solution technique will work in all cases. In fact only a few special classes of nonlinear, first-order differential equations can be solved at all. We will examine two of the more common classes in section 22.2. Even though solutions are known to exist for any nonlinear differential equation of the first order that satisfies certain continuity restrictions, it is simply not possible to find that solution in many cases....
• Chapter 23 Linear, Second-Order Differential Equations
(pp. 753-780)
Until now we have confined our analysis of differential equations to those of the first order. In this chapter we will examine linear, second-order differential equations with constant coefficients. We focus our attention on theautonomouscase in section 23.1 and consider a specialnonautonomouscase in section 23.2.
We begin by explaining how to solve a linear, autonomous, second-order differential equation.
Definition 23.1 The linear, autonomous, second-order differential equation (constant coefficients and a constant term) is expressed as$\ddot{y}+{{a}_{1}}\dot{y}+{{a}_{2}}y=b \caption {(23.1)}$
Equation (23.1) is linear becausey,, andÿare not raised to any power other than one. It is...
• Chapter 24 Simultaneous Systems of Differential and Difference Equations
(pp. 781-844)
It is common in economic models for two or more variables to be determined simultaneously. When the model is dynamic and involves two or more variables, asystemof differential or difference equations arises. The purpose of this chapter is to extend our single equation techniques to solve systems of autonomous differential and difference equations.
We begin with the simplest case—a system of two linear differential equations—and solve it using the substitution method. We then proceed to a more general method, known as the direct method, that can be used to solve a system of linear differential equations...
• Chapter 25 Optimal Control Theory
(pp. 845-920)
In this chapter we take up the problem of optimization over time. Such problems are common in economics. For example, in the theory of investment, firms are assumed to choose the time path of investment expenditures to maximize the (discounted) sum of profits over time. In the theory of savings, individuals are assumed to choose the time path of consumption and saving that maximizes the (discounted) sum of lifetime utility. These are examples of dynamic optimization problems. In this chapter, we study a new technique, optimal control theory, which is used to solve dynamic optimization problems.
It is fundamental in... |
# zbMATH — the first resource for mathematics
## Hamkins, Joel David
Compute Distance To:
Author ID: hamkins.joel-david Published as: Hamkins, Joel; Hamkins, Joel D.; Hamkins, Joel David Homepage: http://jdh.hamkins.org/ External Links: MGP · Wikidata · MathOverflow · ORCID · dblp
Documents Indexed: 88 Publications since 1994, including 2 Books
all top 5
#### Co-Authors
24 single-authored 7 Apter, Arthur W. 7 Gitman, Victoria 6 Fuchs, Gunter 6 Miller, Russell G. 5 Johnstone, Thomas A. 4 Reitz, Jonas 3 Coskey, Samuel 3 Löwe, Benedikt 3 Woodin, W. Hugh 2 Brendle, Jörg 2 Brian, William Rea 2 Cummings, James 2 Greenberg, Noam 2 Hirschfeldt, Denis Roman 2 Linetsky, David 2 Schindler, Ralf-Dieter 2 Seabold, Daniel Evan 1 Bagaria, Joan 1 Barton, Neil 1 Blair, D. Dakota 1 Blass, Andreas Raphael 1 Brumleve, Dan 1 Caicedo, Andrés Eduardo 1 Cheng, Yong 1 Cody, Brent M. 1 Daghighi, Ali Sadegh 1 Deolalikar, Vinay 1 Dorais, François Gilbert 1 Džamonja, Mirna 1 Enayat, Ali 1 Evans, C. D. A. 1 Friedman, Sy-David 1 Gitik, Moti 1 Godziszewski, Michał Tomasz 1 Golshani, Mohammad 1 Groszek, Marcia J. 1 Habič, Miha Emerik 1 Hardy, Michael 1 Jeřábek, Emil 1 Kikuchi, Makoto 1 Kirmayer, Greg 1 Klausner, Lukas Daniel 1 Larson, Paul B. 1 Leahy, Cole 1 Leibman, George 1 Lewis, Andrew D. 1 Lewis, Andy 1 Miller, Russel G. 1 Myasnikov, Alexei G. 1 O’Bryant, Kevin 1 Palumbo, Justin 1 Perlmutter, Norman Lewis 1 Schanker, Jason Aaron 1 Schlicht, Philipp 1 Shelah, Saharon 1 Thomas, Simon R. 1 Tsaprounis, Konstantinos 1 Usuba, Toshimichi 1 Verner, Jonathan L. 1 Warner, Steve 1 Welch, Philip D. 1 Williams, Kameryn J.
all top 5
#### Serials
14 The Journal of Symbolic Logic 9 Archive for Mathematical Logic 9 Mathematical Logic Quarterly (MLQ) 8 Annals of Pure and Applied Logic 7 Notre Dame Journal of Formal Logic 5 Proceedings of the American Mathematical Society 2 Israel Journal of Mathematics 2 Fundamenta Mathematicae 2 Transactions of the American Mathematical Society 2 Integers 1 Annals of the Japan Association for Philosophy of Science 1 Studia Logica 1 Kobe Journal of Mathematics 1 Journal of Logic and Computation 1 1 The Bulletin of Symbolic Logic 1 Journal of Mathematical Logic 1 Logic and Logical Philosophy 1 Central European Journal of Mathematics 1 Lecture Notes in Logic 1 The Review of Symbolic Logic 1 Computability
all top 5
#### Fields
85 Mathematical logic and foundations (03-XX) 7 Computer science (68-XX) 4 Group theory and generalizations (20-XX) 3 General and overarching topics; collections (00-XX) 2 History and biography (01-XX) 2 Sequences, series, summability (40-XX) 2 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 1 Number theory (11-XX)
#### Citations contained in zbMATH
72 Publications have been cited 648 times in 321 Documents Cited by Year
Infinite time Turing machines. Zbl 0963.03064
Hamkins, Joel David; Lewis, Andy
2000
Extensions with the approximation and cover properties have no new large cardinals. Zbl 1066.03052
Hamkins, Joel David
2003
The lottery preparation. Zbl 0949.03045
Hamkins, Joel David
2000
Gap forcing: Generalizing the Lévy-Solovay theorem. Zbl 0933.03067
Hamkins, Joel David
1999
Gap forcing. Zbl 1010.03042
Hamkins, Joel David
2001
Set-theoretic geology. Zbl 1348.03051
Fuchs, Gunter; Hamkins, Joel David; Reitz, Jonas
2015
What is the theory ZFC without power set? Zbl 1375.03059
Gitman, Victoria; Hamkins, Joel David; Johnstone, Thomas A.
2016
The set-theoretic multiverse. Zbl 1260.03103
Hamkins, Joel David
2012
The modal logic of forcing. Zbl 1139.03039
Hamkins, Joel David; Löwe, Benedikt
2008
The halting problem is decidable on a set of asymptotic probability one. Zbl 1137.03024
Hamkins, Joel David; Miasnikov, Alexei
2006
A simple maximality principle. Zbl 1056.03028
Hamkins, Joel David
2003
Small forcing creates neither strong nor Woodin cardinals. Zbl 0959.03040
Hamkins, Joel David; Woodin, W. Hugh
2000
Indestructibility and the level-by-level agreement between strong compactness and supercompactness. Zbl 1010.03043
Apter, Arthur W.; Hamkins, Joel David
2002
Fragile measurability. Zbl 0796.03054
Hamkins, Joel
1994
Resurrection axioms and uplifting cardinals. Zbl 1351.03043
Hamkins, Joel David; Johnstone, Thomas A.
2014
Destruction or preservation as you like it. Zbl 0949.03047
Hamkins, Joel David
1998
Small forcing makes any cardinal superdestructible. Zbl 0906.03051
Hamkins, Joel David
1998
Tall cardinals. Zbl 1165.03044
Hamkins, Joel D.
2009
Infinite time Turing machines. Zbl 1030.68036
Hamkins, Joel David
2002
Superstrong and other large cardinals are never Laver indestructible. Zbl 1402.03073
Bagaria, Joan; Hamkins, Joel David; Tsaprounis, Konstantinos; Usuba, Toshimichi
2016
Indestructible strong unfoldability. Zbl 1207.03057
Hamkins, Joel David; Johnstone, Thomas A.
2010
Generalizations of the Kunen inconsistency. Zbl 1270.03100
Hamkins, Joel David; Kirmayer, Greg; Perlmutter, Norman Lewis
2012
Diamond (on the regulars) can fail at any strongly unfoldable cardinal. Zbl 1110.03032
Džamonja, Mirna; Hamkins, Joel David
2006
Superdestructibility: A dual to Laver’s indestructibility. Zbl 0921.03051
Hamkins, Joel David; Shelah, Saharon
1998
The hierarchy of equivalence relations on the natural numbers under computable reducibility. Zbl 1325.03049
Coskey, Amuel; Hamkins, Joel David; Miller, Russell
2012
The ground axiom is consistent with $$V \neq \text{HOD}$$. Zbl 1145.03029
Hamkins, Joel David; Reitz, Jonas; Woodin, W. Hugh
2008
Large cardinals with few measures. Zbl 1115.03075
Apter, Arthur W.; Cummings, James; Hamkins, Joel David
2007
The Necessary Maximality Principle for c.c.c. forcing is equiconsistent with a weakly compact cardinal. Zbl 1078.03042
Hamkins, Joel D.; Woodin, W. Hugh
2005
Indestructible weakly compact cardinals and the necessity of supercompactness for certain proof schemata. Zbl 0992.03064
Apter, Arthur W.; Hamkins, Joel David
2001
Universal indestructibility. Zbl 0953.03060
Apter, Arthur W.; Hamkins, Joel David
1999
Canonical seeds and Prikry trees. Zbl 0890.03024
Hamkins, Joel David
1997
The wholeness axioms and V=HOD. Zbl 0969.03063
Hamkins, Joel David
2001
A natural model of the multiverse axioms. Zbl 1214.03035
Gitman, Victoria; Hamkins, Joel David
2010
Post’s problem for supertasks has both positive and negative solutions. Zbl 1024.03043
Hamkins, Joel David; Lewis, Andrew
2002
Infinite time Turing machines with only one tape. Zbl 0990.03031
Hamkins, Joel David; Seabold, Daniel Evan
2001
Effective mathematics of the uncountable. Zbl 1297.03006
Greenberg, Noam (ed.); Hamkins, Joel David (ed.); Hirschfeldt, Denis (ed.); Miller, Russell (ed.)
2013
Degrees of rigidity for Souslin trees. Zbl 1179.03043
Fuchs, Gunter; Hamkins, Joel David
2009
Exactly controlling the non-supercompact strongly compact cardinals. Zbl 1056.03030
Apter, Arthur W.; Hamkins, Joel David
2003
Unfoldable cardinals and the GCH. Zbl 1025.03051
Hamkins, Joel David
2001
Strongly uplifting cardinals and the boldface resurrection axioms. Zbl 1417.03269
Hamkins, Joel David; Johnstone, Thomas A.
2017
Moving up and down in the generic multiverse. Zbl 1303.03078
Hamkins, Joel David; Löwe, Benedikt
2013
Infinite time decidable equivalence relation theory. Zbl 1233.03050
Coskey, Samuel; Hamkins, Joel David
2011
P$$\neq \text{NP}\cap$$co-NP for infinite time Turing machines. Zbl 1089.68043
Deolalikar, Vinay; Hamkins, Joel David; Schindler, Ralf
2005
Changing the heights of automorphism towers. Zbl 0944.03048
Hamkins, Joel David; Thomas, Simon
2000
Algebraicity and implicit definability in set theory. Zbl 1436.03264
Hamkins, Joel David; Leahy, Cole
2016
Structural connections between a forcing class and its modal logic. Zbl 1367.03095
Hamkins, Joel David; Leibman, George; Löwe, Benedikt
2015
The least weakly compact cardinal can be unfoldable, weakly measurable and nearly $$\theta$$-supercompact. Zbl 1360.03082
Cody, Brent; Gitik, Moti; Hamkins, Joel David; Schanker, Jason A.
2015
Every countable model of set theory embeds into its own constructible universe. Zbl 1326.03046
Hamkins, Joel David
2013
Pointwise definable models of set theory. Zbl 1270.03101
Hamkins, Joel David; Linetsky, David; Reitz, Jonas
2013
Inner models with large cardinal features usually obtained by forcing. Zbl 1250.03104
Apter, Arthur W.; Gitman, Victoria; Hamkins, Joel David
2012
Some second order set theory. Zbl 1209.03045
Hamkins, Joel David
2009
Every group has a terminating transfinite automorphism tower. Zbl 0904.20027
Hamkins, Joel David
1998
A model of the generic Vopěnka principle in which the ordinals are not Mahlo. Zbl 07006136
Gitman, Victoria; Hamkins, Joel David
2019
A multiverse perspective on the axiom of constructibility. Zbl 1321.03061
Hamkins, Joel David
2014
Transfinite game values in infinite chess. Zbl 1369.03118
Evans, C. D. A.; Hamkins, Joel David
2014
The proper and semi-proper forcing axioms for forcing notions that preserve $$\aleph_2$$ or $$\aleph_3$$. Zbl 1166.03030
Hamkins, Joel David; Johnstone, Thomas A.
2009
Changing the heights of automorphism towers by forcing with Souslin trees over L. Zbl 1153.03026
Fuchs, Gunter; Hamkins, Joel David
2008
A survey of infinite time Turing machines. Zbl 1211.03060
Hamkins, Joel David
2007
Post’s problem for ordinal register machines. Zbl 1151.03339
Hamkins, Joel D.; Miller, Russell G.
2007
The rearrangement number. Zbl 07144584
Blass, Andreas; Brendle, Jörg; Brian, Will; Hamkins, Joel David; Hardy, Michael; Larson, Paul B.
2020
ZFC proves that the class of ordinals is not weakly compact for definable classes. Zbl 1447.03016
Enayat, Ali; Hamkins, Joel David
2018
Open determinacy for class games. Zbl 1423.03200
Gitman, Victoria; Hamkins, Joel David
2017
Large cardinals need not be large in HOD. Zbl 1373.03109
Cheng, Yong; Friedman, Sy-David; Hamkins, Joel David
2015
Is the dream solution of the continuum hypothesis attainable? Zbl 1331.03034
Hamkins, Joel David
2015
The rigid relation principle, a new weak choice principle. Zbl 1268.03067
Hamkins, Joel David; Palumbo, Justin
2012
The mate-in-$$n$$ problem of infinite chess is decidable. Zbl 1357.03042
Brumleve, Dan; Hamkins, Joel David; Schlicht, Philipp
2012
The set-theoretic multiverse: a natural context for set theory. Zbl 1274.03076
Hamkins, Joel David
2011
Post’s problem for ordinal register machines: an explicit approach. Zbl 1178.03060
Hamkins, Joel David; Miller, Russell G.
2009
The complexity of quickly ORM-decidable sets. Zbl 1150.03321
Hamkins, Joel D.; Linetsky, David; Miller, Russell
2007
Infinitary computability with infinite time Turing machines. Zbl 1113.68399
Hamkins, Joel David
2005
$$\text P^f\neq\text{NP}^{f}$$ for almost all $$f$$. Zbl 1043.03036
Hamkins, Joel David; Welch, Philip D.
2003
How tall is the automorphism tower of a group? Zbl 1012.20034
Hamkins, Joel David
2002
The rearrangement number. Zbl 07144584
Blass, Andreas; Brendle, Jörg; Brian, Will; Hamkins, Joel David; Hardy, Michael; Larson, Paul B.
2020
A model of the generic Vopěnka principle in which the ordinals are not Mahlo. Zbl 07006136
Gitman, Victoria; Hamkins, Joel David
2019
ZFC proves that the class of ordinals is not weakly compact for definable classes. Zbl 1447.03016
Enayat, Ali; Hamkins, Joel David
2018
Strongly uplifting cardinals and the boldface resurrection axioms. Zbl 1417.03269
Hamkins, Joel David; Johnstone, Thomas A.
2017
Open determinacy for class games. Zbl 1423.03200
Gitman, Victoria; Hamkins, Joel David
2017
What is the theory ZFC without power set? Zbl 1375.03059
Gitman, Victoria; Hamkins, Joel David; Johnstone, Thomas A.
2016
Superstrong and other large cardinals are never Laver indestructible. Zbl 1402.03073
Bagaria, Joan; Hamkins, Joel David; Tsaprounis, Konstantinos; Usuba, Toshimichi
2016
Algebraicity and implicit definability in set theory. Zbl 1436.03264
Hamkins, Joel David; Leahy, Cole
2016
Set-theoretic geology. Zbl 1348.03051
Fuchs, Gunter; Hamkins, Joel David; Reitz, Jonas
2015
Structural connections between a forcing class and its modal logic. Zbl 1367.03095
Hamkins, Joel David; Leibman, George; Löwe, Benedikt
2015
The least weakly compact cardinal can be unfoldable, weakly measurable and nearly $$\theta$$-supercompact. Zbl 1360.03082
Cody, Brent; Gitik, Moti; Hamkins, Joel David; Schanker, Jason A.
2015
Large cardinals need not be large in HOD. Zbl 1373.03109
Cheng, Yong; Friedman, Sy-David; Hamkins, Joel David
2015
Is the dream solution of the continuum hypothesis attainable? Zbl 1331.03034
Hamkins, Joel David
2015
Resurrection axioms and uplifting cardinals. Zbl 1351.03043
Hamkins, Joel David; Johnstone, Thomas A.
2014
A multiverse perspective on the axiom of constructibility. Zbl 1321.03061
Hamkins, Joel David
2014
Transfinite game values in infinite chess. Zbl 1369.03118
Evans, C. D. A.; Hamkins, Joel David
2014
Effective mathematics of the uncountable. Zbl 1297.03006
Greenberg, Noam (ed.); Hamkins, Joel David (ed.); Hirschfeldt, Denis (ed.); Miller, Russell (ed.)
2013
Moving up and down in the generic multiverse. Zbl 1303.03078
Hamkins, Joel David; Löwe, Benedikt
2013
Every countable model of set theory embeds into its own constructible universe. Zbl 1326.03046
Hamkins, Joel David
2013
Pointwise definable models of set theory. Zbl 1270.03101
Hamkins, Joel David; Linetsky, David; Reitz, Jonas
2013
The set-theoretic multiverse. Zbl 1260.03103
Hamkins, Joel David
2012
Generalizations of the Kunen inconsistency. Zbl 1270.03100
Hamkins, Joel David; Kirmayer, Greg; Perlmutter, Norman Lewis
2012
The hierarchy of equivalence relations on the natural numbers under computable reducibility. Zbl 1325.03049
Coskey, Amuel; Hamkins, Joel David; Miller, Russell
2012
Inner models with large cardinal features usually obtained by forcing. Zbl 1250.03104
Apter, Arthur W.; Gitman, Victoria; Hamkins, Joel David
2012
The rigid relation principle, a new weak choice principle. Zbl 1268.03067
Hamkins, Joel David; Palumbo, Justin
2012
The mate-in-$$n$$ problem of infinite chess is decidable. Zbl 1357.03042
Brumleve, Dan; Hamkins, Joel David; Schlicht, Philipp
2012
Infinite time decidable equivalence relation theory. Zbl 1233.03050
Coskey, Samuel; Hamkins, Joel David
2011
The set-theoretic multiverse: a natural context for set theory. Zbl 1274.03076
Hamkins, Joel David
2011
Indestructible strong unfoldability. Zbl 1207.03057
Hamkins, Joel David; Johnstone, Thomas A.
2010
A natural model of the multiverse axioms. Zbl 1214.03035
Gitman, Victoria; Hamkins, Joel David
2010
Tall cardinals. Zbl 1165.03044
Hamkins, Joel D.
2009
Degrees of rigidity for Souslin trees. Zbl 1179.03043
Fuchs, Gunter; Hamkins, Joel David
2009
Some second order set theory. Zbl 1209.03045
Hamkins, Joel David
2009
The proper and semi-proper forcing axioms for forcing notions that preserve $$\aleph_2$$ or $$\aleph_3$$. Zbl 1166.03030
Hamkins, Joel David; Johnstone, Thomas A.
2009
Post’s problem for ordinal register machines: an explicit approach. Zbl 1178.03060
Hamkins, Joel David; Miller, Russell G.
2009
The modal logic of forcing. Zbl 1139.03039
Hamkins, Joel David; Löwe, Benedikt
2008
The ground axiom is consistent with $$V \neq \text{HOD}$$. Zbl 1145.03029
Hamkins, Joel David; Reitz, Jonas; Woodin, W. Hugh
2008
Changing the heights of automorphism towers by forcing with Souslin trees over L. Zbl 1153.03026
Fuchs, Gunter; Hamkins, Joel David
2008
Large cardinals with few measures. Zbl 1115.03075
Apter, Arthur W.; Cummings, James; Hamkins, Joel David
2007
A survey of infinite time Turing machines. Zbl 1211.03060
Hamkins, Joel David
2007
Post’s problem for ordinal register machines. Zbl 1151.03339
Hamkins, Joel D.; Miller, Russell G.
2007
The complexity of quickly ORM-decidable sets. Zbl 1150.03321
Hamkins, Joel D.; Linetsky, David; Miller, Russell
2007
The halting problem is decidable on a set of asymptotic probability one. Zbl 1137.03024
Hamkins, Joel David; Miasnikov, Alexei
2006
Diamond (on the regulars) can fail at any strongly unfoldable cardinal. Zbl 1110.03032
Džamonja, Mirna; Hamkins, Joel David
2006
The Necessary Maximality Principle for c.c.c. forcing is equiconsistent with a weakly compact cardinal. Zbl 1078.03042
Hamkins, Joel D.; Woodin, W. Hugh
2005
P$$\neq \text{NP}\cap$$co-NP for infinite time Turing machines. Zbl 1089.68043
Deolalikar, Vinay; Hamkins, Joel David; Schindler, Ralf
2005
Infinitary computability with infinite time Turing machines. Zbl 1113.68399
Hamkins, Joel David
2005
Extensions with the approximation and cover properties have no new large cardinals. Zbl 1066.03052
Hamkins, Joel David
2003
A simple maximality principle. Zbl 1056.03028
Hamkins, Joel David
2003
Exactly controlling the non-supercompact strongly compact cardinals. Zbl 1056.03030
Apter, Arthur W.; Hamkins, Joel David
2003
$$\text P^f\neq\text{NP}^{f}$$ for almost all $$f$$. Zbl 1043.03036
Hamkins, Joel David; Welch, Philip D.
2003
Indestructibility and the level-by-level agreement between strong compactness and supercompactness. Zbl 1010.03043
Apter, Arthur W.; Hamkins, Joel David
2002
Infinite time Turing machines. Zbl 1030.68036
Hamkins, Joel David
2002
Post’s problem for supertasks has both positive and negative solutions. Zbl 1024.03043
Hamkins, Joel David; Lewis, Andrew
2002
How tall is the automorphism tower of a group? Zbl 1012.20034
Hamkins, Joel David
2002
Gap forcing. Zbl 1010.03042
Hamkins, Joel David
2001
Indestructible weakly compact cardinals and the necessity of supercompactness for certain proof schemata. Zbl 0992.03064
Apter, Arthur W.; Hamkins, Joel David
2001
The wholeness axioms and V=HOD. Zbl 0969.03063
Hamkins, Joel David
2001
Infinite time Turing machines with only one tape. Zbl 0990.03031
Hamkins, Joel David; Seabold, Daniel Evan
2001
Unfoldable cardinals and the GCH. Zbl 1025.03051
Hamkins, Joel David
2001
Infinite time Turing machines. Zbl 0963.03064
Hamkins, Joel David; Lewis, Andy
2000
The lottery preparation. Zbl 0949.03045
Hamkins, Joel David
2000
Small forcing creates neither strong nor Woodin cardinals. Zbl 0959.03040
Hamkins, Joel David; Woodin, W. Hugh
2000
Changing the heights of automorphism towers. Zbl 0944.03048
Hamkins, Joel David; Thomas, Simon
2000
Gap forcing: Generalizing the Lévy-Solovay theorem. Zbl 0933.03067
Hamkins, Joel David
1999
Universal indestructibility. Zbl 0953.03060
Apter, Arthur W.; Hamkins, Joel David
1999
Destruction or preservation as you like it. Zbl 0949.03047
Hamkins, Joel David
1998
Small forcing makes any cardinal superdestructible. Zbl 0906.03051
Hamkins, Joel David
1998
Superdestructibility: A dual to Laver’s indestructibility. Zbl 0921.03051
Hamkins, Joel David; Shelah, Saharon
1998
Every group has a terminating transfinite automorphism tower. Zbl 0904.20027
Hamkins, Joel David
1998
Canonical seeds and Prikry trees. Zbl 0890.03024
Hamkins, Joel David
1997
Fragile measurability. Zbl 0796.03054
Hamkins, Joel
1994
all top 5
#### Cited by 243 Authors
43 Apter, Arthur W. 37 Hamkins, Joel David 14 Fuchs, Gunter 13 Friedman, Sy-David 12 Carl, Merlin 9 Gitman, Victoria 8 Cody, Brent M. 8 Schlicht, Philipp 7 Lücke, Philipp Moritz 7 Rybalov, Aleksandr Nikolaevich 7 Sargsyan, Grigor 6 Honzik, Radek 6 Koepke, Peter 6 Tsaprounis, Konstantinos 6 Welch, Philip D. 5 Antos, Carolin 5 Johnstone, Thomas A. 5 Löwe, Benedikt 5 Schindler, Ralf-Dieter 4 Ben-Neria, Omer 4 Bringsjord, Selmer 4 Cheng, Yong 4 Corazza, Paul 4 Kanovei, Vladimir G. 4 Krueger, John 4 Myasnikov, Alexei G. 4 Perlmutter, Norman Lewis 4 Reitz, Jonas 4 Usuba, Toshimichi 4 Viale, Matteo 3 Barton, Neil 3 Cox, Sean D. 3 Cummings, James 3 Friedman, Shoshana 3 Gitik, Moti 3 Khoussainov, Bakhadyr M. 3 Lubarsky, Robert S. 3 Mitchell, William John 3 Monin, Benoît 3 Ng, KengMeng 3 Rin, Benjamin G. 3 Sorbi, Andrea 3 Wilson, Trevor Miles 3 Woodin, W. Hugh 2 Andrews, Uri 2 Bazhenov, Nikolaĭ Alekseevich 2 Brooke-Taylor, Andrew D. 2 Burgin, Mark 2 Calude, Cristian S. 2 Coskey, Samuel 2 d’Auriac, Paul-Elliot Anglès 2 Desfontaines, Damien 2 Dimonte, Vincenzo 2 Durand, Bruno 2 Govindarajulu, Naveen Sundar 2 Habič, Miha Emerik 2 Köllner, Peter 2 Lafitte, Grégory 2 Meadows, Toby 2 Miller, Russell G. 2 Osin, Denis V. 2 Potgieter, Petrus H. 2 Sakai, Hiroshi 2 Schanker, Jason Aaron 2 Schweber, Noah David 2 Shagrir, Oron 2 Shelah, Saharon 2 Siders, Ryan 2 Stephan, Frank 2 Ternullo, Claudio 2 Venturi, Giorgio 2 Welch, Peter D. 2 Wiedermann, Jiří 2 Williams, Kameryn J. 2 Ziegler, Martin 1 Ackerman, Nathanael Leedom 1 Akl, Selim G. 1 Arkoudas, Konstantine 1 Arrigoni, Tatiana 1 Asperó, David 1 Astor, Eric P. 1 Audrito, Giorgio 1 Badaev, Serikzhan A. 1 Bagaria, Joan 1 Bard, Vittorio 1 Bauer, Andrej 1 Baumes, Jeffrey 1 Bianchetti, Matteo 1 Bienvenu, Laurent 1 Boney, Will 1 Bruni, Riccardo 1 Button, Tim 1 Cabessa, Jérémie 1 Caicedo, Andrés Eduardo 1 Chan, William C. Y. 1 Clemens, John Daniel 1 Cockshott, Paul 1 Costa, José Félix 1 Cramer, Scott S. 1 Daghighi, Ali Sadegh ...and 143 more Authors
all top 5
#### Cited in 54 Serials
52 Archive for Mathematical Logic 47 The Journal of Symbolic Logic 36 Annals of Pure and Applied Logic 19 Mathematical Logic Quarterly (MLQ) 16 Theoretical Computer Science 12 Notre Dame Journal of Formal Logic 11 The Bulletin of Symbolic Logic 10 Israel Journal of Mathematics 6 Studia Logica 6 The Review of Symbolic Logic 5 Proceedings of the American Mathematical Society 4 Applied Mathematics and Computation 4 Journal of Mathematical Logic 4 Natural Computing 3 Algebra and Logic 3 Transactions of the American Mathematical Society 3 Bulletin of the Polish Academy of Sciences, Mathematics 3 Theory of Computing Systems 2 Advances in Mathematics 2 Fundamenta Mathematicae 2 Illinois Journal of Mathematics 2 Journal of Philosophical Logic 2 Siberian Mathematical Journal 2 Logical Methods in Computer Science 2 Computability 2 Bollettino dell’Unione Matematica Italiana 1 International Journal of Theoretical Physics 1 Information Processing Letters 1 Periodica Mathematica Hungarica 1 Journal of Algebra 1 Journal of Computer and System Sciences 1 Journal of Pure and Applied Algebra 1 Synthese 1 Topology and its Applications 1 Advances in Applied Mathematics 1 Bulletin of the Iranian Mathematical Society 1 Order 1 Journal of Complexity 1 Journal of Automated Reasoning 1 Journal of the American Mathematical Society 1 MSCS. Mathematical Structures in Computer Science 1 International Journal of Foundations of Computer Science 1 International Journal of Computer Mathematics 1 Indagationes Mathematicae. New Series 1 Erkenntnis 1 Foundations of Science 1 Lobachevskii Journal of Mathematics 1 Parallel Processing Letters 1 Sibirskie Èlektronnye Matematicheskie Izvestiya 1 Sarajevo Journal of Mathematics 1 International Journal of Parallel, Emergent and Distributed Systems 1 Logica Universalis 1 Groups, Complexity, Cryptology 1 Forum of Mathematics, Sigma
all top 5
#### Cited in 16 Fields
294 Mathematical logic and foundations (03-XX) 50 Computer science (68-XX) 11 General and overarching topics; collections (00-XX) 9 Group theory and generalizations (20-XX) 4 Combinatorics (05-XX) 4 Order, lattices, ordered algebraic structures (06-XX) 4 Category theory; homological algebra (18-XX) 3 Quantum theory (81-XX) 2 History and biography (01-XX) 1 Commutative algebra (13-XX) 1 Real functions (26-XX) 1 Abstract harmonic analysis (43-XX) 1 General topology (54-XX) 1 Numerical analysis (65-XX) 1 Operations research, mathematical programming (90-XX) 1 Information and communication theory, circuits (94-XX)
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# Ratios in Regression, aka Questions on Kronmal
Recently, randomly browsing questions triggered a memory of on off-hand comment from one of my professors a few years back warning about the usage of ratios in regression models. So I started reading up on this, leading eventually to Kronmal 1993.
I want to make sure that I’m correctly interpreting his suggestions on how to model these.
1. For a model with a ratio with the same denominator on both the dependent and independent side:
$$Z−1Y=Z−11nβ0+Z−1XβX+βZ+Z−1ϵ Z^{-1}Y = Z^{-1}1_n\beta_0 + Z^{-1}X\beta_X + \beta_Z + Z^{-1}\epsilon$$
• Regress dependent ratio on the (inverse) denominator variable in addition to the other ratios
• Weight by the (inverse) denominator variable
2. For a model with dependent variable as a ratio:
$$Y=β0+βXX+Z1nα0+ZXαX+Z−1ϵ Y = \beta_0 + \beta_XX + Z1_n\alpha_0 + ZX\alpha_X + Z^{-1}\epsilon$$
• Regress numerator by original variables, denominator, and denominator times original variables [what about categorical variables?]
• Weight by (inverse) denominator
3. For model with only independent variable ratios:
$$Y=β0+XβX+Z−11nβZ−1+WβW+Z−1WβZ−1W+ϵ Y = \beta_0 + X\beta_X + Z^{-1}1_n\beta_{Z^{-1}} + W\beta_W + Z^{-1}W\beta_{Z^{-1}W} + \epsilon$$
• Include numerator and (inverse) denominator as main effects, ratio as interaction term.
Are my interpretations here correct?
You should really have linked to the Kronmal paper (and explained your notation, which is taken directly from the paper.) Your reading of the paper is too literal. Specifically, he does not give advice about weighting, rather saying that weighting can be done the usual ways, so no need to discuss. It is only mentioned as a possibility. Read your cases more like examples, especially as examples of how to analyze such situations.
In section 6 he does give some general advice, which I will cite here:
The message of this paper is that ratio variables should only be used
in the context of a full linear model in which the variables that make
up the ratio are included and the intercept term is also present. The
common practice of using ratios for either the dependent or the
inferences, and rarely results in any gain. This practice is
widespread and entrenched, however, and it may be difficult to
convince some researchers that they should give up their most prized
ratio or index.
The paper uses the (fictitious) example by Neyman on births and storks. To play with that example, you can access it from R by
data(stork, package="TeachingDemos")
I will leave the fun for the readers, but one interesting plot is this coplot: |
# Largest palindrome made from the product of 3-digit numbers
I am given a task to write a program to find the largest palindrome made from the product of two 3-digit numbers. How can I improve this code?
def check_palindrome(s):
"""Checks whether the given string is palindrome"""
if s == s[::-1]:
return True
product_pal = []
for i in range (999,900,-1):
for j in range (999,900,-1):
product = i * j
if check_palindrome(str(product)):
product_pal.append(product)
print"i =" , i , "j = ",j, "for", product
print max(product_pal)
• You can keep track of just the current maximum instead of keeping a list of products. – RobAu Jan 17 '18 at 14:26
• You should in general wait a bit before accepting an answer. Some of us live all over the world, so in general in the SE network it is customary to wait at least 24h before accepting an answer. Once a question has an accepted answer, it usually gets less additional answers. You can always un-accept my answer and at a later time decide that it is what you were looking for and accept it again. Or somebody else's. – Graipher Jan 17 '18 at 16:44
• Note that you can break your inner loop after the first palindrome for a given (i, j) pair - you are counting towards lower numbers, and any subsequent palindrome of the form i * (j - m) will be lower than your current i * j. Likewise, if your first discovered palindrome is less than the current max palindrome, break because it won't get bigger. – aghast Jan 17 '18 at 23:07
In your check_palindrome function you can directly return the result of the comparison:
def check_palindrome(s):
"""Checks whether the given string is palindrome"""
return s == s[::-1]
As @RobAu said in the comments, you should keep only the current maximum product, instead of building a (potentially very large) list of all products.
You can also reduce the number of products you need to check by realizing that if you checked 999*998, you don't need to check 998*999. This can be achieved by letting the inner loop start at i.
max_product = 0
for i in range(999, 900, -1):
for j in range(i, 900, -1):
product = i * j
if check_palindrome(str(product)):
max_product = max(max_product, product)
print "i =", i, "j = ", j, "for", product
print max_product
Note that Python has an official style-guide, PEP8, which recommends using whitespace to separate operators and after commas in lists (including argument lists).
As a final step, I would make this a function that returns its result, instead of printing it:
def get_max_three_digit_product():
max_product = 0
for i in range(999, 900, -1):
for j in range(i, 900, -1):
product = i * j
if check_palindrome(str(product)):
max_product = max(max_product, product)
return max_product
This makes the code more re-usable, in case you ever need it again. You can execute it under a if __name__ == "__main__": guard, which allows you to import this function from another script, without executing the function.
if __name__ == "__main__":
print get_max_three_digit_product()
• @hjpotter92 Yeah, just figured it out as well, see edit – Graipher Jan 17 '18 at 16:46
you try to iterate downward to get an effective implementation but you got the inner loop wrong. while you expect the outer loop to do few iterations your inner loop does check relatively low numbers early. you tried to limit that by stopping the iteration at 900, a magic value without reasoning. so your implementation may give wrong results as a pair of 901*901 is much smaller than a lot of untested pairs. you need at least a check if your product is bigger than the biggest untested one 999*900.
on the other hand if we do the inner loop right all problems are gone. we use the outer loop for the lower value and the inner loop for the greater one. we do not need an arbitrary limit any more and we are quite efficient.
for i in range(999,99,-1):
for j in range(999,i-1,-1):
# check palindrome
again we do not want to collect all palindromes but only the biggest one. we can abort safely when we cannot get a bigger product than the current maximum one.
def is_palindrome(n):
s = str(n)
return s == s[::-1]
def get_biggest_palindrome():
max_product = 0
for i in xrange(999, 99, -1):
if max_product >= 999*i:
# no need to iterate further down
break
for j in xrange(999, i-1, -1):
p = j * i
if max_product >= p:
# no need to iterate further down
break
if is_palindrome(p):
max_product = p
return max_product
I did some other minor changes:
is_palindrome - i like to name functions after what they return so the usage reads like a natural language sentence.
in python2 you should use xrange() instead of range() if you do not need a real list but just an iterator.
what you could do also:
make the magic numbers 999 and 99 constants and/or pass them as parameters. if it is about the number of digits you could define them as 10**(digits+1)-1, 10**digits-1 and pass digits as single parameter. |
• Create Account
# Bacterius
Member Since 28 Feb 2011
Online Last Active Today, 02:29 PM
### #5253774Lua C API tutorial / book
Posted by on 24 September 2015 - 01:41 AM
Programming In Lua, 3rd edition. Like a quarter of the book is dedicated to the C API and the rest is not bad either.
### #5253106JavaScript Platformer Level Format
Posted by on 19 September 2015 - 05:37 PM
I don't see a problem with using this to grab the next level, one thing you could do if you don't want your game running in the background is display some kind of loading screen while the file is being retrieved, something like:
1. show a loading screen in the middle of the page or something (as a DOM element like a div or whatever)
2. send off XMLHttpRequest with an onready callback that goes parses the json and loads the level, and then hides the loading screen when it's done
3. resume running the game...
I don't think hardcoding the level in your game is a good idea. It's okay to prototype but having it on the server allows for plenty of interesting mechanics like letting users upload and share their levels, etc...
### #5252293Linux c++ debugging
Posted by on 14 September 2015 - 11:55 PM
I use Valgrind primarily. I very rarely need to break out an actual debugger, the times I did I just used gdb and it worked fine, though I could see it worthwhile to invest in both better front-ends and actually learning how to use it efficiently.
### #5251881Is there a language like Python but without the tab requirement?
Posted by on 12 September 2015 - 08:07 AM
If your most important goal is sensible syntax, you need an Algol-family language. Weird that none of them ever caught on.
I guess Pascal is not really "in" any longer these days, especially on a game development forum, but Ada and Lua are pretty popular (though not for the same reasons). Plus Lua has first-class functions and proper closures which I guess comes more from functional languages like Lisp, although the syntax is very sensible and easy to grasp.
### #5251850Is there a language like Python but without the tab requirement?
Posted by on 12 September 2015 - 04:17 AM
It runs faster too, because the CPU doesn't have to keep scrolling down.
Wouldn't it have to scroll right way more though? This is why I limit my line lengths to 80 characters, 120 is acceptable but I can definitely feel the slowdown on longer lines. They just run slower.
### #5249823Strange pinching effect with raytraced reflections
Posted by on 30 August 2015 - 11:43 PM
Your code calculates (dir - dir * dot(dir, norm)) * 2, whereas you should be calculating dir - dir * dot(dir, norm) * 2, that is:
Vector3 ReflectionDirection = newRay.direction.nor().cpy().sub(surfNorm.scl(newRay.direction.cpy().dot(surfNorm)).scl(2.0f));
Or just:
Vector3 ReflectionDirection = newRay.direction.nor().cpy().sub(surfNorm.scl(2 * newRay.direction.cpy().dot(surfNorm)));
This is incidentally the main reason why I strongly dislike the lack of (reasonable) operator overloading in languages, it makes it impossible to write such things in a sane way (and, no, separating this simple equation into three or four temporary variables is not sane either). But anyway this is where the pinching comes from.
(actually it's pinching not so much because the reflection is wrong but really because of your error the resulting reflected vector is not unit length, which wreaks havoc on the rest of the ray tracing code something fierce; so a good thing to do whenever stuff looks really weird is to look at your directions vectors and check if they are actually unit length; if they aren't, you've screwed up somewhere)
### #5249675Background color ruins Antialiasing
Posted by on 30 August 2015 - 02:56 AM
It could be that it was running your old code before you added anti-aliasing or something, dunno. If it works now then cool, looks nice by the way
### #5249618C++ | Fixed Byte Size
Posted by on 29 August 2015 - 05:23 PM
That code is illegal JohnnyCode. You cannot reinterpret m_Memo (which has no alignment constraint, being a pointer to char) to a pointer to WORD or DWORD, which have alignment constraints (2-byte and 4-byte boundary respectively). Your code will work "correctly" on x86 systems like your desktop/laptop because the processor takes care of the unaligned access and fixes it up itself; you will obtain some kind of bus error on less generous processors like many ARM ones.
You can see the access cannot be aligned since you reinterpret both m_Memo and m_Memo + 14 as pointers to DWORDs, and clearly these addresses can't both be aligned to a 4-byte boundary. If you want this code to be portable (which, hey, you might not; if you are only interested in supporting x86 hardware then this will work fine, you'll just incur a small performance penalty each time you access these fields due to the CPU having to do extra work for unaligned access) then you can use memcpy to safely unpack the bits of m_Memo into words and dwords.
Oh, and you also don't handle endianness. Not a problem on x86 as all x86 chips are little-endian AFAIK, but could cause strange behaviour depending on where the header is coming from!
### #5249539How would I be able to do this in C++?
Posted by on 29 August 2015 - 04:43 AM
Did you remeber to #include ?
He is probably not using C++11.
### #5249530How would I be able to do this in C++?
Posted by on 29 August 2015 - 04:15 AM
A Lua "table" is just a generic associative mapping, which maps a set of keys to a set of values. In this special case it also happens to be what is known as a list or array in most programming languages. Read up on datastructures and you should find that C++ also has a notion of arrays, lists, mappings, ...
### #5249122Firing many rays at one pixel?
Posted by on 27 August 2015 - 02:21 AM
The pickRay method takes floats so I figured I could fire rays at x+0.99f and y+0.99f but it doesn't work that way.
Actually it works exactly like that (well, not 0.99f, but you want to average up rays in a neighbourhood of (x, y)). If your camera object doesn't allow you to fire rays over a continuous plane, as opposed to a discrete grid of pixels, then it's broken.
### #5248487C++ | Fixed Byte Size
Posted by on 24 August 2015 - 05:23 AM
So, the width in bits in the uint16_t type is exactly defined (16 bits, none more and none less, and no pad bits), and it is not optional at all (unless your machine doesn't have 16-bit capable registers at all). But go figure that out, the standard doesn't precisely make that easy...
I think the point that Sean was making is that a byte (as defined by the C/C++ standard) need not be 8 bits, so that while a uint16_t (if the type exists) will be exactly 16 bits long with no padding bits, it is not guaranteed that the sizeof operator applied to it will return 16/8 = 2, a fact which is liable to break large amounts of code out there (not that much existing C/C++ code would survive an architecture where a byte is not 8 bits). However, the intersection of the set of people who work on systems with CHAR_BIT not equal to 8 and the set of people who are not aware of CHAR_BIT is, I hope, very small.
Of course, note that since C/C++ mandates that CHAR_BIT is at least 8, the existence of a standards-compliant [u]int8_t implies that CHAR_BIT equals 8. So if you are going to be using that type then you are already tacitly assuming an architecture where a byte is exactly eight bits. So to cut the pedantry short it's safe to say that one may assume stdint.h exists and behaves "as expected" on the typical modern system, and one should not rely on stdint.h but instead define one's own types when working on exotic hardware like DSPs. Or just use it on "real" systems and roll your own on systems which aren't developer-friendly enough to include a few standard typedefs (like it would kill them to spend half an hour writing the thing).
### #5248197Distance between two trajectories
Posted by on 22 August 2015 - 05:11 AM
Your integrals are just leaving out the initial integration constant at t = 0 which balances out the initial acceleration (when alpha is not zero) so that x(0) = x0. Remember that in general:
$v_x(t) = v_{x_0} + \int_{0}^t a_x(t') ~ \mathrm{d}t'$
And similarly:
$x(t) = x_0 + \int_{0}^t v_x(t') ~ \mathrm{d}t'$
You need to take into account the initial acceleration (and hence velocity) at t = 0 to get the correct results, else you'll be off.
EDIT: to be clear, that means that the correct equations are:
$v_x(t) = v_{x0} - \frac{|F|}{m\omega}\cos(\alpha + \omega t) \color{blue}{ + \frac{|F| \cos(\alpha)}{m \omega} }$
$x(t) = x_0 + v_{x0} t \color{blue}{ + \frac{|F| \cos(\alpha)}{m \omega} t } - \frac{|F|}{m\omega^2}\sin(\alpha + \omega t) \color{blue}{ + \frac{|F| \sin(\alpha)}{m \omega^2} }$
Which as you will observe cancels out to x0 when $$t = 0$$.
(please double check those integrals for sign errors, but rest assured the math works)
### #5247566Databases: Why should I use prepared statements?
Posted by on 18 August 2015 - 11:52 PM
Need I remind you that SQL injections stem from the fact that you're treating user input as SQL. "Validating" the user input the way you suggest amounts to treating it as SQL and then checking that it's not SQL. Uh, okay? Why not just, you know, not treat it as SQL but as a parameter to a query (i.e. prepared statements)? The user input still needs to be validated against the parameter's type (string, boolean, etc) and also according to your usual business rules, of course, since user input always starts off untyped as a raw string, but that's no excuse for doing not only useless but illogical and dangerous "SQL validation" on user input that should never be treated as SQL in the first place.
There is no reason to validate, it does not even make sense from a typing perspective. The user input is not just another random chunk of the query, it is a parameter that is referenced by the query. Sure, you can validate all you want, and it might eventually work (actually the typing rules are quite complex owing to SQL's odd grammar) but it's pointless because you don't need to. All you need to do is treat the user's input like a parameter and not like a part of the raw query, and that's what prepared statements do. Same result, but vastly simpler (and more logical) solution: you simply say "THIS is the query, THOSE are the parameters, and you plug them HERE, HERE and THERE" and suddenly the problem is gone. You didn't even need a validation function! (and neither does the database, by the way; databases don't magically consume SQL statements, they parse it and then do computation based on the action, parameters, conditions, etc.. involved).
This is an obvious typing problem, and the reason it's so widespread is because of the untyped, string-based nature of raw SQL, which is rather unfortunate (if convenient). Types exist for a reason, if people actually took advantage of the type system they have available to them they wouldn't need half the validation methods they are using.
### #5247461Changing a C++ vector list via Lua
Posted by on 18 August 2015 - 01:42 PM
If you really wanted to you could add #define capabilities to Lua by simply running the C preprocessor on your scripts before executing them (it's not hard) but I think it would be a questionable approach for what you're trying to do.
My suggestion: stop trying to come up with a DSL and just use normal Lua constructs. There's quite a bit of syntactic sugar available so you can be creative and still manage to write something nice without resorting to horrible hacks just to e.g. remove parentheses.
PARTNERS |
## Quantitative Methods
Amy Lloyd is interested in leasing a new Saab and has contacted three automobile dealers for pricing information. Each dealer offered Amy a closed-end 36-month lease with no down payment due at the time of signing. Each lease includes a monthly charge and a mileage allowance. Additional miles receive a surcharge on a per-mil basis. The monthly lease cost, the mileage allowance, and the cost for additional miles follow: Dealer Monthly Cost Mileage Allowance Cost per Additional Mile Forno Saab $299 36,000$0.15 Midtown Motors $310 45,000$0.20 Hopkins Automotive $325 54,000$0.15 Amy decided to choose the lease option that will minimize her total 36-month cost. The difficulty is that Amy is not sure how many miles she will drive over the next three years. For purposes of this decision she believes it is reasonable to assume that she will drive 12,000 miles per year, 15,000 miles per year, or 18,000 miles per year. With this assumption Amy estimated her total costs for the three lease options. For example, she figures that the Forno Saab lease will cost her $10,764 is she drives 12,000 miles per year,$12,114 if she drives 15,000 miles per year, or \$13, 464 if she drives 18,000 miles per year. a. What is the decision, and what is the chance event? b. Construct a payoff table for Amy’s problem. c. If Amy has no idea which of the three mileage assumptions is most appropriate, what is the recommended decision (leasing option) using the optimistic, conservative, and minimax regret approaches? d. Suppose that the probabilities that Amy drives 12,000, 15,000, and 18,000 miles per year are 0.5, 0.4, and 0.1, respectively. What option should Amy choose using the expected value approach? e. Develop a risk profile for the decision selected in part (d). What is the most likely cost, and what is its probability? f. Suppose that after further consideration Amy concludes that the probabilities that she will drive 12,000, 15,000, and 18,000 miles per year are 0.3, 0.4, and 0.3, respectively. What decision should Amy make using the expected value approach? |
1 answer
1 answer
##### 3.34. Let fXc(t)) and (X,(t)J denote two statistically independent zero n stationary Gaussian random processes with...
3.34. Let fXc(t)) and (X,(t)J denote two statistically independent zero n stationary Gaussian random processes with common power spec- tral density given by SX (f) = SX (f) = 112B(f) watt/Hz. Define x(t) = Xe(t) cos(2tht)--Xs(t) sin(2tht) where fo 》 (a) Is X(t) a Gaussian process? (b) Find th...
5 answers
##### (10 points) Solve the given differential equation: y" + 4y = t2 + 3et
(10 points) Solve the given differential equation: y" + 4y = t2 + 3et...
1 answer
##### (a) Suppose that lim x→c f(x) = L > 0. Prove that there exists a δ...
(a) Suppose that lim x→c f(x) = L > 0. Prove that there exists a δ > 0 such that if 0 < |x − c| < δ, then f(x) > 0. (b) Use Part (a) and the Heine-Borel Theorem to prove that if is continuous on [a, b] and f(x) > 0 for all x ∈ [a, b], then there exists a...
5 answers
##### Cp H5CHo + Lc4z} CoCi1 H++o
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##### I need the answer with explanation please 1. 10 patients joined the rehabilitation center this month,...
I need the answer with explanation please 1. 10 patients joined the rehabilitation center this month, their age is as follows: 16, 34, 39, 41, 33, 28, 52, 45, 37, 35. Calculate the range and the standard deviation 2. The table below frequency table showing heights in inches of a sample of female cli...
4 answers
##### The market research company Kelton Research conducted a surveyin 2012 by polling a nationwide sample of 1,114 Americans ages 18and older, who were representative of the demographics of thenation as a whole. The survey was commissioned by the NationalGeographicChannel to promote its new series “Chasing UFOsâ€. Thesurvey found that•36% believe UFOs are real•One in 10 respondentssay they have personally witnessed an alien spaceship.•If alienswere to invade the country sometime in the next
The market research company Kelton Research conducted a survey in 2012 by polling a nationwide sample of 1,114 Americans ages 18 and older, who were representative of the demographics of the nation as a whole. The survey was commissioned by the National GeographicChannel to promote its new series â...
2 answers
##### Find dy/dx of y=e^x sin x Help please and show the working? Thanks
Find dy/dx of y=e^x sin x Help please and show the working? Thanks...
1 answer
Problem 15-20 Determining sales and variable cost volume variances LO 15-2, 15-3, 15-4 Vernon Publications established the following standard price and costs for a hardcover picture book that the company produces. $36.60 8.50 Standard price and variable costs Sales price Materials cost Labor cost O... 1 answer ##### Ch 5 Sec 3: Problem 10 Previous Problem Problem List Next Problem 1 point) Suppose the... Ch 5 Sec 3: Problem 10 Previous Problem Problem List Next Problem 1 point) Suppose the region on the left in the figure (with blue shading) has area is 33, and the region on the right (with green shading) has area 3. Using the graph of f(x) in the figure, find the following integrals. f(x) dx = I sw... 1 answer ##### HwO8-FS2O: Problem 5Prublom Valua' Politis} Problcm Scoro: 0%. Attompts Romaining:point?, rece71 siudy In the Journal oltha Anorican Nadicn Tutocualcn {encedno Thiultg olun urpar monthat Watchets 4Dovurxelght Inatiduals "ollorvod Lhe Woipht = diet for ona yenr The - wlght 'cnangcs al Iha ond 'olthu nni yunr had 4 Nlaan 0l * = 28 Ko with 4 Elandard darlation 04$ 5.0 Wo clrn thnt Iho diot has an elloct, Hhnbaa Wulght chango /ahor tan Which sot 04 hypothaws Ahoulo Josinq &quo
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##### Fio 4 the Jolu e p k Poc which 4he 59) paic ep equation hos no- solution Pollowing 3 X1tldy (L_) Xt L+?Solalio:233 LA 8 ] 1]6 3-k 5-26 k+i iP TA +TAB 724li_6) +ktl = 2-2L+k+l 53_k Lhee {S nosollo 2 . (l_k)_ 3-3k+k+2 5-26 ) 3.(1-k)+k+2=CA +CAB ~ 3ed s_2k+0 k+53_k-0 [email protected]=3then thra_is no-SoQutio)
Fio 4 the Jolu e p k Poc which 4he 59) paic ep equation hos no- solution Pollowing 3 X1tldy (L_) Xt L+? Solalio: 2 3 3 LA 8 ] 1]6 3-k 5-26 k+i iP TA +TAB 724li_6) +ktl = 2-2L+k+l 53_k Lhee {S nosollo 2 . (l_k)_ 3-3k+k+2 5-26 ) 3.(1-k)+k+2= CA +CAB ~ 3 ed s_2k+0 k+5 3_k-0 L=3 [email protected] L=3 then thra_is no-S...
5 answers
##### Chemist takes 25.0 mLofa 2.0M dilutes sucrose solution and it to 500.0 mL; What is the sucrose concentration the new, dilute solution? 0.1M B 0.01M Co.OO1M D1M E 1OM F 1OOM
chemist takes 25.0 mLofa 2.0M dilutes sucrose solution and it to 500.0 mL; What is the sucrose concentration the new, dilute solution? 0.1M B 0.01M Co.OO1M D1M E 1OM F 1OOM...
5 answers
##### Jn" ! CcIrx; whcn the initial pusilion and Icmminal position ol u are (=4 0) and (1 , 8) reperhvely- and the initial position and Icmminal position of Jr ( , - V)and (7 . 7) mpatndy Show that and qquivall
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1 answer
##### After PCR is performed the products are run out on an agarose gel. In the figure...
After PCR is performed the products are run out on an agarose gel. In the figure below, grey bands represent the wells the PCR product was loaded into. The white bands represent DNA fragments produced by PCR. The target fragment amplified by the primers was 1,500 bp in size. The ladder is a standard...
5 answers
##### What are the difficulties of mining association rules from large databases?
What are the difficulties of mining association rules from large databases?...
1 answer
##### A mixture of He, Ar, and Xe has a total pressure of 2.20 atm . The partial pressure of He is 0.400 atm , and the partial...
A mixture of He, Ar, and Xe has a total pressure of 2.20 atm . The partial pressure of He is 0.400 atm , and the partial pressure of Ar is 0.450 atm . What is the partial pressure of Xe?...
5 answers
5 answers
##### For u = (5 , - 4) and v= ( 1,2) find u*v.
For u = (5 , - 4) and v= ( 1,2) find u*v....
5 answers
##### 7poinis LancalcE"7 2R.USSNotesFind the value of c such that the function is continuous on the entire real number line 6x) {x Nced Help? ETalkito atutol
7poinis LancalcE"7 2R.USS Notes Find the value of c such that the function is continuous on the entire real number line 6x) {x Nced Help? ETalkito atutol...
5 answers
##### DirectIndirectCombination16.916.425.617.822.523.016.021.420.618.417.326.221.023.926.716.320.125.712.321.423.5Auditors must make judgments about various aspects of an auditon the basis of their own direct experience, indirect experience,or a combination of the two. In a study, auditors were asked tomake judgments about the frequency of errors to be found in anaudit. The judgments by the auditors were then compared to theactual results. The data are contained in the Excel Online filebelow. Suppose
Direct Indirect Combination 16.9 16.4 25.6 17.8 22.5 23.0 16.0 21.4 20.6 18.4 17.3 26.2 21.0 23.9 26.7 16.3 20.1 25.7 12.3 21.4 23.5 Auditors must make judgments about various aspects of an audit on the basis of their own direct experience, indirect experience, or a combination of the two. In a stud...
5 answers
##### 3) Sketch the region and evaluate the double integral Ifrtyd4 where R: 0gxs9-Y, -1sys3 using the method of iteration: (5 pts)to get an equivalent expression Change the order of = integration = K sf (x,y)drdy ~6r+9 -
3) Sketch the region and evaluate the double integral Ifrtyd4 where R: 0gxs9-Y, -1sys3 using the method of iteration: (5 pts) to get an equivalent expression Change the order of = integration = K sf (x,y)drdy ~6r+9 -...
1 answer
##### Exercise 3-7 Preparing financial statements LO P3 The following is the adjusted trial balance of Wilson...
Exercise 3-7 Preparing financial statements LO P3 The following is the adjusted trial balance of Wilson Trucking Company Account Title Cash Accounts receivable \$ 7,600 16,500 2,000 151,000 office supplies Trucks Accumulated depreciation-Trucks Land Accounts payable Interest payable Long-term notes p...
5 answers
##### If someone consumes 26 grams of sodium chloride per day; what mass (In grams) of sodium does that person consume? Sodium chloride Is 39% sodium by mass_
If someone consumes 26 grams of sodium chloride per day; what mass (In grams) of sodium does that person consume? Sodium chloride Is 39% sodium by mass_...
5 answers
##### Design an experiment complete with instrumentation to determine the specific heats of a liquid using a resistance heater. Discuss how the experiment will be conducted, what measurements need to be taken, and how the specific heats will be determined. What are the sources of error in your system? How can you minimize the experimental error? How would you modify this system to determine the specific heat of a solid?
Design an experiment complete with instrumentation to determine the specific heats of a liquid using a resistance heater. Discuss how the experiment will be conducted, what measurements need to be taken, and how the specific heats will be determined. What are the sources of error in your system? How...
5 answers
##### Suggest disconnections, synthons and reagents for the synthesis of FOUR of the following molecules_ Your starting material in each case should contain no more than seven carbon atoms_OHOHOH(viid)HOHO
Suggest disconnections, synthons and reagents for the synthesis of FOUR of the following molecules_ Your starting material in each case should contain no more than seven carbon atoms_ OH OH OH (viid) HO HO...
1 answer
##### QUESTION 5 The student welfare office was interested in trying to enhance students' exam performance by investigati...
QUESTION 5 The student welfare office was interested in trying to enhance students' exam performance by investigating the effects of various interventions. They took five groups of students before their statistics exams and gave them one of five interventions: (1) a control group just sat in a r...
-- 0.029794-- |
## Mathematicians Of The Day
### 10th February
#### Quotation of the day
##### From Sofia Kovalevskaya
Many who have had an opportunity of knowing any more about mathematics confuse it with arithmetic, and consider it an arid science. In reality, however, it is a science which requires a great amount of imagination. |
# Eleven
The number 11 does not exist. In fact, it is far easier to count to potato than to reach the non-existent achievement of reaching 11. On the other hand, it is as real as any hröning in Tlön. This by itself is no reason to denigrate its use in daily life. In fact, numerology regards repeated elevens as a sign of spiritual mastery; you are encouraged to repeat eleven as often as necessary.
If we demonstrate Eleven's basic truth with an equation, $11=nonexistance$ (and thus $PhD=0$), then because Mathematics is infallible, we can begin to understand its ramifications, especially in terms of quantum physics and superstring theory (which operates on an assumption of one-dimensional objects bent through eleven dimensions to look like the idiot who used permanent marker on the dry-erase board again). After all, nonexistence does exist, despite how much we wish it wouldn't.
This information raises some interesting questions, often speculated upon by philosophers such as Immanuel Kant, Martin Heidegger, and Steven James Anderson-Williams; usually while indulging in their favorite brews:
• Does the vacuum in space equal 11? Or perhaps, does the perception of 11 cause vacuum to exist? Remembering that space is infinite, perhaps the contemplation of eleven causes infinity.
• But would that explain why 9/11 has caused a Forever War, and what would other elevenths do if someone with an agenda got hold of them?
• Do barcodes containing 11 mean that the shops only think they have that product in stock? Or perhaps that it is priceless and cannot be bought, only given away?
• Will every Hoovercraft break down upon reaching its 11th birthday, or does it simply cease to exist after 11,000 miles? Has a Hoovercraft ever reached 11,000 miles without blowing up thanks to those geniuses in R&D?
• "Dude, I am soooo drunk... wait, there's only one left. Eleven beers! I might be on to something!"
• "Shut up, Kant, and gimme that brewski. I don't want it going to waste! Seriously, man, do you finish anything you start?"
The sculptor Isamu Noguchi has claimed that "The essence of sculpture is for me the perception of space, the continuum of our existence." Though he has extensively studied the number eleven, he has never created a piece which directly incorporates it; he has done $10+1$, $12-1$, and other equations which point in a similar direction. Apparently, he found what he was looking for.
## editUseless
The number 11, excluding the examples stated above, is otherwise a completely useless number. It is a prime number, which makes it indivisible by anything. It consists of the numbers 1 and 1 stuck neatly together, creating an illusion of balance, but this is a lie.
If you had eleven slices of pizza, or eleven pieces of cake, the only way you could distribute it evenly is if there were eleven people present, or just one glutton. I personally have never seen a group of exactly eleven people (I've seen 10.6, but never mind that), and dividing the remaining pieces into elevenths would be silly. So the only option is for someone to eat two pieces, making everyone else jealous.
And how the hell are you supposed to divide something into elevenths anyway? The only way you could do it is if you had a ruler handy and the object's total volume was a multiple of eleven. What a pain.
The only good eleven has ever done anybody is... nothing! In Soviet Russia Eleven numbers you!
Hitler was reported to have exactly eleven panda slaves.These pandas were reported to carry out sexual favours for him.It was when one panda died of too much fellatio, hitler went on a rampage and decided to kill a whole load of people.
Contradicting what it has been said above- the number eleven can be used for such acts of decency- for example, the number "10" which is consider the Fuhrer of the alphabet and number line, eleven can take one step further. One incidence of this is group sex, when ten isn't enough...?
## edit Did you know?
• There are eleven planets in the Solar System. Only two of them (Earth and The Moon) are real; the rest are carefully etched into the lenses of large telescopes?
• Most tigers have eleven stripes on their tails. The ones who don't are mutants, and persecuted for their affliction?
• The element K does not exist. Neither does Sodium. However, Potassium and Natron do, as they claim an a priori right to existence?
• There are eleven Grand Masters of the Tao, but only one of them manifests at any given time. This is because "There can be only one!"?
• The Akashic Records consist of eleven volumes per person, but they only have information on Buddha, Jesus, and Shiva. This is because they've been stored in an underground vault for over 10,500 years where nobody has been able to get to them?
• Amps that go up to 11 are one louder than those that only go up to 10?
• If you lose your copy of 11, you will find it again in an out of the way spot, even if you never actually owned one before?
• That eleven is "ridiculous", and "not even funny"?
• The number eleven is a symbol of the devil and if you ever read word 11 of line 11 of page 11 of your 11th grade math textbook you will die in 11 years, 11 months, 11 days, 11 hours, 11 minutes, and 11 seconds?
• All sequels ending in the number 11 will suck. (IE Final Fantasy 11)
• Twelve?
• What a jerk, don't you know not to offend the eleven masters of Nin Juh Pi Rat.
• Oh. my. gosh. What is that smel-leven? It smells like beek-jerk-off. |
# Falsification Of The Atmospheric CO2 Greenhouse Effects Within The Frame Of Physics
1. ### Andre
G. Gerlich, R. D. Tscheuschner (2009) Falsification Of The Atmospheric CO2 Greenhouse Effects Within The Frame Of Physics. International Journal of Modern Physics B, Vol. 23, No. 3 (30 January 2009), 275-364 (World Scientific Publishing Co.)
see:
http://www.worldscinet.com/ijmpb/23/2303/S02179792092303.html
there is also the freely available post-print version 4.0 from the preprint server of the Cornell University: http://www.arxiv.org/abs/0707.1161v4
Abstract:
2. ### Xnn
555
Re: Falsification Of The Atmospheric CO2 Greenhouse Effects Within The Frame Of Physi
Andre;
This is just another straw man argument.
First they suggest that the earth is in radiative equilibrium.
Equilibrium by definition implies no overall change.
Then they go on to "prove" that the earth isn't really warming.
So, no real surprise here. Misrepresent the science and then "prove" that it is wrong. Classical straw man.
Does the publisher require peer review or do they print everything that's "scientific"?
Last edited: Mar 19, 2009
3. ### Andre
Re: Falsification Of The Atmospheric CO2 Greenhouse Effects Within The Frame Of Physi
Perhaps you need to rethink a little
http://www.worldscinet.com/ijmpb/ijmpb.shtml
4. ### Xnn
555
Re: Falsification Of The Atmospheric CO2 Greenhouse Effects Within The Frame Of Physi
Suggesting that the earth is in radiative equilibrium is a straw man.
5. ### jostpuur
Re: Falsification Of The Atmospheric CO2 Greenhouse Effects Within The Frame Of Physi
From the beginning of the abstract:
Did I understand correctly, that these guys are claiming the atmospheric greenhouse effect to be violating the second law of thermodynamics? So they are not claiming, that the magnitude of atmospheric greenhouse effect would have been estimated incorrectly, but that the atmospheric greenhouse effect itself is impossible? Looks like extreme incompetence to me.
6. ### WeatherRusty
38
Re: Falsification Of The Atmospheric CO2 Greenhouse Effects Within The Frame Of Physi
A heat pump? NO, they have the concept of the greenhouse effect reversed. It does not produce heat, it slows the dissipation of heat to space. The atmosphere cools off more slowly by the presence of molecules absorbing terrestrial infrared radiation. The atmosphere is constantly radiating away heat energy in accordance with the Second Law. Without continued solar irradiance the atmosphere cools, greenhouse effect or not.
7. ### sylas
1,745
Re: Falsification Of The Atmospheric CO2 Greenhouse Effects Within The Frame Of Physi
It's incredible to me that this paper actually managed to get published; albeit in a low impact journal, and as an invited "review" article which apparently does not have the same peer-review procedures as research articles. The subject matter is a poor fit with the journal; I very much doubt that this paper could ever have survived a normal peer review process. But there you go. I'm speculating. All we can really know for sure is the content of the paper as given.
I claim it is riddled with errors. Rather than attempt a comprehensive rebuttal, I'll single out limited specific errors in the paper.
Here's my first.
From the arxiv preprint, top of page 65, we read:
According to the consensus among global climatologists one takes the -18oC computed from the T4 average and compares it to the fictitious Earth's average temperature of +15oC. The difference of 33oC is attributed to the natural greenhouse effect. As seen in Equation (83) a correct averaging yields a temperature of -129oC. Evidently, something must be fundamentally wrong here.
What the authors describe as the "correct" calculation is bizarre. It comes from section 3.7.4.
First, they consider the energy per unit area for each part of the globe coming from the Sun. This is done correctly. Hence the portion of the Earth which is directly facing the Sun is given a full solar constant. Higher latitudes have this scaled by the cosine. The back of the globe (night) has no radiation at all.
They compute the solar constant as σ.57804/2152, which comes to 1369 W/m2; about correct. They use a factor of 0.7 for ε (table 12 on page 64) which corresponds to the effect of albedo. Hence the incoming solar radiation is treated as 958.4 W/m2 for a plane surface facing the Sun; a reasonable figure.
They then contrast two ways to proceed. One way is to integrate the incoming energy of the surface of the globe, and then calculate a temperature which can be given to the whole globe that would radiate out that same amount of energy again. Another way to proceed is to take each point on the globe individually as having the temperature to radiate away what it receives from the Sun at that point; and then average this over the whole globe. They call this second method the "correct" method. Their so-called correct method gives a temperature of 0K absolute to the night of the planet, and a temperature of about 360K, or 87C, to the portion of the globe facing the Sun.
The authors' so-called "correct" calculation is indeed calculating an average temperature, obtained by integrating an imputed temperature over the whole globe. This integration over the surface gives a value of about 144K, or -129C for the average temperature imputed to the simple model of a globe.
The feature of this imputed temperature is that it is just what is required to radiate (as a blackbody) the radiation coming from the Sun at every point. Now this is of course not a physical model of the Earth. Points on a planet do not instantaneously achieve thermodynamic equilibrium with the Sun's incoming radiation; even the Moon, with no atmosphere and very little heat transport across the surface, does not instantly reach absolute zero on the night side! The calculation provided by the authors can be sensibly understood is a lower bound on average temperatures; assuming radiative balance with the Sun. With any sharing of heat energy around the globe, while maintaining energy balance with the Sun, will give a higher average temperature. (You can show this with Holder's inequality, also used by the authors on page 65).
Now the other extreme model is to calculate a temperature such that if every point on the globe has that same temperature, then the globe remains in energy balance. This is the calculation that the authors disparage as "incorrect". Here, you calculate the average amount of energy radiated per unit area, and find the temperature this corresponds to. This is also called the "effective" temperature. It is equal to 20.5*1.25 (1.768) times the authors' "physical" temperature. (Compare equations 81 and 83). This works out to about 255K, or -18C. You can see the numbers -129C and -18C compared in table 12.
The proper implication of these numbers is that if you integrate temperatures over the surface of a globe which is radiating away the same energy it receives from the Sun, you'll get a value more than -129C and less than -18C.
Of course, if you integrate over the Earth's surface in reality, you get a number that is substantially more than -18C! It really doesn't matter whether you integrate temperature, or the fourth power of temperature. Whichever is chosen, you'll get an average of more than -18. That is… the Earth's surface is radiating more than what is required to balance solar radiation. But this IS the effect called "atmospheric greenhouse"!
Physically, this is because we have an atmosphere, which is heated from the surface. The atmosphere is (by thermodynamics) cooler than the surface, and the radiation that escapes into space is mostly from this cooler atmosphere. This is (by the first law) in long-term balance with solar radiation. The atmosphere radiates in all directions, of course. It radiates out into space, and also down to the surface; and this means the surface gets more energy. There's the solar energy (most of which passes through the atmosphere just fine) plus also the energy radiated from the atmosphere. The surface is in balance with this total… which is more than what you'd have without an atmosphere. This is what is called the atmospheric greenhouse… a poor choice of terms given that the physics is quite distinct from a glass greenhouse; but it is certainly physically real.
At the end of section 3.7.6, page 66, the authors make two claims. The speaks of a physically incorrect assumption of radiative balance. That's ludicrous. By the first law, there is necessarily a long term balance between the energy arriving from the Sun and being radiated from the planet. It is a physically correct implication that the Earth radiates an amount of energy into space that is equivalent to that of a blackbody at -18C.
The second claim speaks of effective radiating temperature being higher than measured averages. That is correct, and the authors are the ones who do not take this into account. The measured averages over the surface of the Earth are much more than -18C. Therefore the surface is radiating more than what you would get from a globe at -18C! Therefore the energy being radiated from the Earth's surface is MORE than the energy you get from the Sun. That IS the greenhouse effect, right there.
Good grief. It staggers me that this got published, but so be it. I am pretty sure it was an invited paper which was not given the kind of thorough technical review that usually maintains the quality of a journal.
Cheers -- Sylas
8. ### Xnn
555
Re: Falsification Of The Atmospheric CO2 Greenhouse Effects Within The Frame Of Physi
Maybe Gerlich and Tscheuschner forgot that the Earth rotates!
That would be one way to come up with absolute zero for the night time temperature.
9. ### sylas
1,745
Re: Falsification Of The Atmospheric CO2 Greenhouse Effects Within The Frame Of Physi
That, and also the assumption that there is no transport of heat from one part of the planet to another. Now of course, they know quite well that this is only a simplified model. They don't suggest that there really is an absolute zero of temperature on the reverse side, and that is not my criticism.
What they do has its own rather curious meaning. Effectively, what they are doing is to take the energy arriving from the Sun, and average the energy to the power 0.25 over the globe. With any redistribution of energy -- either by the fact that it takes a bit of time to heat up and cool down, or by the fact that heat transports from one region to another -- the average of energy to the power 0.25 will increase. The number they get is thus a strong lower bound on temperature of a globe with uneven temperatures, but radiating at each point as a blackbody.
The other approach is to average energy over the globe. (You can then get a temperature from this energy by Stefan-Blotzmann, which is called Teff). The key point is that there is a very useful feature of averaging the energy. Because of the first law, any redistribution will continue to have the SAME average energy. It's not a lower bound, or an upper bound, but an invariant.
That's why Teff is a far more useful quantity.
If you do take a simple mean temperature over the whole surface, you are bound to get a smaller value than Teff. The authors correctly point this out as well, but completely fail to grasp its relevance. When you integrate temperatures over the Earth's surface, you get a value GREATER than the expected -18 of Teff. That is, the surface is significant warmer than we should expect from the solar input alone. The difference is the effect of an atmosphere, and it is called "atmospheric greenhouse". But it's not like a glass greenhouse; it is a consequence of the fact that the atmosphere is warmed from the surface.
It up within the atmosphere where you find the temperatures corresponding to the effective temperature from solar radiation. This is cooler than the surface, because it is warmed from the surface. Hence, the surface is warmer than the effective radiating temperature of the planet... warmer than it would be without an atmosphere that absorbs energy from the surface. And no; that is not a contradiction of the second law, which appears to be another error made in the paper.
Cheers -- Sylas
10. ### Mike Davis
25
Re: Falsification Of The Atmospheric CO2 Greenhouse Effects Within The Frame Of Physi
Lunar Surface Temperatures
Temperatures on the Lunar surface vary widely on location. Although beyond the first few centimeters of the regolith the temperature is a nearly constant -35 C (at a depth of 1 meter), the surface is influenced widely by the day-night cycle. The average temperature on the surface is about 40-45 C lower than it is just below the surface.
In the day, the temperature of the Moon averages 107 C, although it rises as high as 123 C. The night cools the surface to an average of -153 C, or -233 C in the permanently shaded south polar basin. A typical non-polar minimum temperature is -181 C (at the Apollo 15 site).
The Lunar temperature increases about 280 C from just before dawn to Lunar noon. Average temperature also changes about 6 C betwen aphelion and perihelion.
From:
Without the atmosphere effect this is what the earth would be like. That is from the solar input alone. So the atmosphere effect restricts the incoming and outgoing warmth.
11. ### sylas
1,745
Re: Falsification Of The Atmospheric CO2 Greenhouse Effects Within The Frame Of Physi
That's a great example, Mike!
The moon rotates much more slowly than the Earth, and so the temperatures should actually come fairly close to those given by what Gerlich and Tscheuschner prefer.
The solar constant is about 1370 W/m2. The albedo of the moon is roughly 0.12, and so the surface face on to the Sun should tend to absorb about 1205 W/m2.
Using Stefan-Boltzmann, these correspond to temperatures of 394K (121C) and 381K (109C).
That's pretty dashed close to the daytime numbers you have quoted of 107 (av) and 123 (peak)! The peak would be a dark spot face on to the Sun, with near complete absorption. The 107 is about right for the central daylight region, given albedo 0.12.
The night side does not drop to absolute zero. But since the energy varies as the fourth power of temperature, we have the radiation from the lows you have mentioned as follows:
Cooling tails off, of course, as the rate of energy radiation drops; and these temperatures have fallen so far that the radiation is less than 1/100 of the peak full daylight value. So in fact the Moon is pretty dashed close to the distribution that is used by Gerlich and Tscheuschner. This is no surprise. If the Moon was made of iron (conducts heat well) and rotated rapidly, then we should expect all the temperatures to equalize or close to it, which would lead to temperatures around -3 C. (The Teff for albedo of 0.12). The value calculated by Gerlich and Tscheuschner's method would be around -120C. However, because the darkside of the moon has temperature significantly above absolute zero, their method works out as a very strong lower bound. The average lunar temperature should be between these values of -12OC and -3C, as there is no greenhouse effect to warm things up.
The page you have cited is not consistent on mean surface temperatures. It speaks of -35 below the regolith, and a surface that is 40 to 45 cooler. That's a mean surface of -75 to -80. But the related page at the same site http://www.asi.org/adb/02/05/01/surface-temperature.html specifically gives -23C as a mean surface value. I don't know what's wrong there. But theoretically, -3C should be an upper bound on the mean surface temperature obtained by integrating temperature over the surface. -23C sounds like a credible value for an average surface temperature. It is equal to mid point of the average day and the average night temperature as given by another page: http://www.solarviews.com/eng/moon.htm.
Since there is such variation in temperature from point to point, we should expect the average value, whatever it is, to be significantly less than Teff of -3C. And because the night side is well above absolute zero, we should expect the average to be substantially more than -120C.
This is in contrast to the surface of the Earth, which (fortunately for us!) has an atmosphere to keep things warmer. The effective value of the planet of -18C is actually expressed high in the atmosphere, while the "atmospheric greenhouse" effect keeps things on the surface with a much warmer average of about 15C.
Last edited: Mar 21, 2009
12. ### Mike Davis
25
Re: Falsification Of The Atmospheric CO2 Greenhouse Effects Within The Frame Of Physi
Sylan:
You made the statement that the atmosphere warms the earth. This article proves that the atmosphere restricts the incoming heat from reaching 123C. The atmosphere also restricts the loss of warmth keeping the low temperatures from reaching -233.
That was the points I was bringing up. The oceans and land heat the atmosphere not the other way around.
13. ### sylas
1,745
Re: Falsification Of The Atmospheric CO2 Greenhouse Effects Within The Frame Of Physi
No, I most certainly did not say that the atmosphere warms the Earth. I said precisely the opposite, just as you have noted. It is the ocean and land, or the surface, which warms the atmosphere. Here again is what I actually said, and note especially the first sentence, which I have placed in bold for emphasis.
It is precisely because the atmosphere is being warmed by the surface that the surface has to be hotter than you would have without an atmosphere! Think about it. Because the atmosphere is absorbing energy from the surface, the energy that eventually escapes into space is mostly emitted from the atmosphere. Therefore it is in the ATMOSPHERE (not the surface) where you have the temperatures that correspond to what is needed to radiate away what we receive from the Sun.
The effective radiating temperature of the atmosphere is Teff. You can get this by averaging a fourth power. If you average the raw temperature, you'll get something a bit less, depending on how much variation there is in temperature across the globe. This is noted also by Gerlich and Tscheuschner; though they apparently don't understand the implications.
In any case, the atmosphere, at altitudes where most radiation is escaping into space, must have an average temperature of about -18C or less. This is the Teff for the Earth.
Now... because the atmosphere is being heated from the surface, the surface has to be hotter than than the atmosphere. And it is. This is the greenhouse effect.
Note the difference. When you add an atmosphere, you get a warmer surface than you would have otherwise. This is NOT because the atmosphere is a source of energy. It is because the atmosphere has to be warmed up by the surface, which results in a surface that is warmer than the atmosphere. The atmosphere is what takes up the temperature required to balance solar input.
Pretty much the same thing happens when you cover yourself with a blanket. YOU warm the blanket. So you are warmer than the blanket. But the blanket is what has to match up with external temperatures, which means you end up warmer than you would be without the blanket. NOT because the blanket is a source of energy to warm you, but because it is absorbing energy from you, and then passing it on to the cold outside.
Cheers -- Sylas
PS. Think about your lunar example again. It's a really good one. The Moon is (on average) COLDER than the Earth. This despite having a lower albedo and absorbing more of the light from the Sun! Why? The conventional physical explanation is that the Moon has no atmosphere, and so radiation from the surface has to balance with the solar input. On the Earth, however, it is radiation from the atmosphere which has to balance the solar input. The Earth's surface has to heat up its atmosphere, and so has to be warmer than the atmosphere... which means it has to be that much hotter again than what is required to balance the solar input.
Last edited: Mar 22, 2009
14. ### Phrak
Re: Falsification Of The Atmospheric CO2 Greenhouse Effects Within The Frame Of Physi
I would think that the surface temperature would oscillate greatly between day and night, becoming both hotter and cooler than with an atmosphere. What are you talking about?
15. ### sylas
1,745
Re: Falsification Of The Atmospheric CO2 Greenhouse Effects Within The Frame Of Physi
I am talking about the substantial increase in average surface temperature than results from an atmospheric greenhouse effect; since this is the topic of the paper.
The Moon does indeed oscillate greatly in temperature, much more than the Earth, and this is because of our atmosphere... and the ocean... and our shorter day/night cycle.
Any oscillation, of course, goes above and below the average.... and just as Gerlich and Tscheuschner say, the average has to be less than T_eff, in the absence of a greenhouse effect. On the Moon, with its low albedo, T_eff is -3 C (on Earth it is -18 C) and the average is something like -23C. The oscillations are about -153C to 107C; a range of 260C.
On Earth, there are two major differences with the Moon. First, we have much more uniformly distributed temperatures, or much smaller oscillations, as you note. The largest swings are inland away from the ocean, and get up to as much as 50 or 60C between day and night. Second, the average surface temperature is substantially higher than T_eff, because the surface is heating up the atmosphere. On Earth, T_eff is about -18C, but the average surface temperature is about 15C. This latter effect is called the atmospheric greenhouse effect. Both effects are real, both are measured, and both follow from conventional thermodynamics applied to each situation.
As far as damping out oscillations is concerned: the ocean is crucial in this regard because of its large heat capacity, which damps out the oscillations a lot. Indeed, the extremes of day and night are comparatively small on the coast, or out at sea. The atmosphere helps to distribute heat between land and sea as well. It's a basic thermodynamic principle that any dynamic process increases entropy... which means it tends to equalize temperatures. The atmosphere and air movement help to shift heat energy from ocean to land, and back again, transferring heat energy from the ocean to the land and night, and from the land to the ocean in the day. Our short day/night cycle also helps.
Now all of this effect of the atmosphere in damping out the oscillations is independent of the greenhouse effect. Consider a hypothetical case, in which our atmosphere was simply oxygen and nitrogen, which are transparent to infrared and to solar radiation. The energy escaping to space would be nearly all radiated direct from the surface. The surface, therefore, would have an average temperature of around -18C (which is T_eff for the Earth). There would be oscillations both above and below this mean; but still damped by comparison with a Moon having no atmosphere to help move heat around.
The other effect, of course, is the greenhouse effect, where the atmosphere absorbs energy from the surface, and where most of the energy radiated into space is from the atmosphere. This means that temperatures which correspond to a radiative balance with the Sun (a consequence of the first law) in the atmosphere must be cooler than the surface temperatures (a consequence of the second law).
That's what I am talking about. The surface heats the atmosphere, on average, which means the surface has to be warmer than the atmosphere, on average. The end result is an average surface temperature significantly greater than -18C, which means that the surface is warmer than it would be without an atmosphere. Without an atmosphere the oscillations, whether large or small, would be about a mean at -18C or less. With an atmosphere such as ours, which is heated from the surface, the mean temperatures are much greater.
Cheers -- Sylas
16. ### Mike Davis
25
Re: Falsification Of The Atmospheric CO2 Greenhouse Effects Within The Frame Of Physi
"There's the solar energy (most of which passes through the atmosphere just fine) plus also the energy radiated from the atmosphere. The surface is in balance with this total… which is more than what you'd have without an atmosphere. This is what is called the atmospheric greenhouse… a poor choice of terms given"
This statement about energy radiated from the atmosphere. I took as meaning the atmospere warmed the surface.
With an atmosphere such as ours, which is heated from the surface, the mean temperatures are much greater.
This statement ,which I argree with, shows that the atmosphere restricts the loss of heat.
I guesss I jumped when I read the first statement. When you rewrite your statement it is more acceptable.
Thank you for explaining what you meant.
17. ### sylas
1,745
Re: Falsification Of The Atmospheric CO2 Greenhouse Effects Within The Frame Of Physi
No problem. I'm currently working on trying to make a basic and comprehensible account of this, and you've been really helpful for cleaning up my wording. Keep pointing out anything that looks wrong. It helps a lot.
Just to underline what I mean above, the second law means that the flow of energy from a hot object to a cold one must be greater than the flow of energy back from the cold object to a warm one. It does not mean there's no flow at all from cold to hot. So even though the atmosphere is warmed from the surface, there is still some energy flowing back against the overall flow.
By the second law, the flow from Earth's surface into the atmosphere has to be more than the flow from the atmosphere into the surface. Typical numbers on Earth are that about 470 W/m2 go from surface to atmosphere, while about 340 W/m2 come back. Added to this is solar energy flowing from space into the surface, and into the atmosphere. Typical numbers are 160 W/m2 to the surface, and 80 W/m2 to the atmosphere. For the energy flowing back out into space, typical numbers are 210 W/m2 going into space from the atmosphere, and about 30 W/m2 coming direct from the surface.
$$\begin{array}{l|c|c|c|c} (W/m^2)& \mbox{to space} & \mbox{to atmos} & \mbox{to surface} & \mbox{total} \\ \hline \mbox{from space} & & 80 & 160 & 240 \\ \mbox{from atmos} & 210 & & 340 & 550 \\ \mbox{from surface} & 30 & 470 & & 500 \\ \hline \mbox{totals} & 240 & 550 & 500 \end{array}$$
These numbers are roughly average values, to about single figure accuracy. It's intended as a simple first order picture, not a fully accurate account. You can drill down into endless further details for what goes on in different latitudes, in the ocean or the land, in day or in night, or in different seasons and weather conditions. But over all, the following very basic features are not in any doubt at all, and follow easily from basic thermodynamics. Any credible estimate of energy flow on Earth must have these features.
• The flux of energy inwards is the same as the flux outwards, Drilling down into more detail, there are small imbalances as heat gets absorbed, but physical measurement has the net imbalance as small. For example, there is at present a small net flux of energy into the ocean which is of the order of magnitude one W/m2 or so. It's an open research question to measure this more accurately, to measure the variations from place to place and from season to season. From day to day, there is a quite substantial flux into and out of the ocean, with the ocean taking up heat in the day and giving it back at night.
• The surface gets more of its energy from the atmosphere than from space.
• The energy received from space is mostly absorbed at the surface.
• The majority of energy radiated into space comes from the atmosphere.
• The atmosphere gets most of its energy from the surface.
• There's more energy flowing from the surface to the atmosphere than there is coming back from the atmosphere to the surface.
• The total flux at the surface is substantially greater than the total flux from space. This is why an atmosphere leads to a warmer average surface temperature.
Last edited: Mar 22, 2009
30
Re: Falsification Of The Atmospheric CO2 Greenhouse Effects Within The Frame Of Physi
It is easy to find fault in areas of minor detail in a paper as long as that of Gerlich. The point of his paper is that he has shown the "classic" atmospheric greenhouse model as depicted by the IPCC, to be utter nonsense.
Here's the IPCC atmospheric greenhouse model:
https://www.msu.edu/course/isb/202/ebertmay/drivers/ipcc_greenhouse.jpg
(IPCC 2001)
We may describe this as:
1. A warm body (the earth) radiates heat to a cool body (the atmosphere)
2. The cool body "back-radiates" (IPCC term) heat to the warm body.
3. This process continues perpetually, with heat flowing round and round in a continuous cycle.
4. The result of this perpetual process is that the warm body becomes warmer.
What is most amazing is that both alarmists and skeptic scientists have taken the above blatant 2nd Law of Thermodynamics violation at face value for so long.
Many will shout that all bodies radiate ... yes they do but NETT heat flow is always from hot bodies to cool bodies (without the input of work), not the reverse. Note also that the 2nd Law does not care about the wavelength of radiant heat.
Atmospheric gases do absorb radiation from the sun and the earth. NETT radiation from the cool daytime atmosphere is to space. The Sahara desert in daytime has a very low "greenhouse gas" concentration above it, yet contrary to greenhouse theory, it is a hot place rather than a cool place.
Night time, rotation of the earth, convection, conduction, latent heat all add greatly to the complexity of climate model. However the basic daytime atmospheric greenhouse model as presented by the IPCC and most textbooks, is nonsense.
19. ### jostpuur
Re: Falsification Of The Atmospheric CO2 Greenhouse Effects Within The Frame Of Physi
No, instead they have shown that they don't understand the simplest things about physics.
adb, let me repeat your argument with winter jackets: "When I go outside in winter wearing a thick jacket, the outer layer of the jacket will be cooler than the inner layer. Therefore there will not be heat flow from outer layer to inner layer, but instead from inner to outer. Therefore the hypothesis that the jacket would keep me warmer violates the second law of thermodynamics."
Do you see how wrong that is? And how same it is with your greenhouse effect denying deduction?
Last edited: Mar 22, 2009
20. ### WeatherRusty
38
Re: Falsification Of The Atmospheric CO2 Greenhouse Effects Within The Frame Of Physi
2. The cool body "back-radiates" (IPCC term) heat to the warm body.
Have you considered the possibility that at night over land the surface can become cooler than the atmosphere? This in fact occurs as a result of "radiational cooling". This loss of thermal energy from the surface at night is reduced by a humid atmosphere. One of the "fingerprints" of greenhouse warming is warmer nighttime temperature over land. |
Important Magnetic Units, Terms, Symbols, and Formulas
The following table presents important Magnetic Units, Symbols, and Their Formulas as a reference, or source of information. These formulas play a key role while dealing with magnetic circuits such as Transformers, Inductors.
Flux (lines) ϕ Weber (Wb)=number of lines 108 Weber (Wb)=number of lines 108 Flux density(magnetic flux per unit cross-sectional area at right angles to the flux lines) B B=ϕA=Tesla (T)B=ϕA=Tesla (T) Magneto motive Force(that which forces magnetic lines of force through the magnetic circuit) MMF Ampere-Turn orA−T=NIA−T=NI Magnetic field intensity(magneto motive force per unit length) H NIlength=A−TlengthNIlength=A−Tlength Permeability (Ability of a material to pass, conduct, or concentrate magnetic flux; analogous to conductance in electrical circuits), i.e., the ease of establishing magnetic flux through the material. μ Webers per ampere-turn per meter μ=lRAμ=lℜA where length (l) is length in meters reluctance (ℜ) is ampere-turns per weber area (A) is cross-sectional area in square meters Note: Free space, or vacuum permeability (μo) is considered to be: 4π×10-7 Relative Permeability(Not constant because it varies with the degree of magnetization) μr Relative permeability of a material is a ratio. Thus, μr=flux density with core material flux density with vacuum coreμr=flux density with core materialflux density with vaccum core Where flux density in the core material is: B=μoμrHB=μoμrH teslas, and absolute permeability of core materials is: μ=BH=μoμrμ=BH=μoμr Reluctance(Opposition to the establishment of magnetic flux) ℜ Ampere turns per Weber R=MMFϕ=A−TWb
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# Tutor profile: Mike R.
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### Subject:Trigonometry
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$$1)$$ If $$\cos(x) = \frac{60}{61}$$ and $$\cot(x) = \frac{60}{11}$$, find $$\sin(x)$$. $$2)$$ Show that $$\frac{\cot(x)}{\csc(x)} = \cos(x)$$. $$3)$$ Simplify: $$\cos(x) + \sin(x) \tan(x)$$
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$$1)$$ $$\cot(x) = \frac{60}{11} \Rightarrow \frac{1}{\tan(x)} = \frac{60}{11} \Rightarrow \frac{\cos(x)}{\sin(x)} = \frac{60}{11} \Rightarrow \frac{\frac{60}{61}}{\sin(x)} = \frac{60}{11} \Rightarrow \frac{60}{61} = \frac{60}{11} \sin(x) \Rightarrow \sin(x) = \frac{60}{61} \cdot \frac{11}{60} = \frac{11}{61}$$ $$2)$$ $$\frac{\cot(x)}{\csc{x}} = \frac{\frac{\cos(x)}{\sin(x)}}{\frac{1}{\sin(x)}} = \frac{\cos(x)}{\sin(x)} \cdot \sin(x) = \cos(x)$$ $$3)$$ $$\cos(x) + \sin(x) \tan(x) = \cos(x) + \sin(x) \cdot \frac{\sin(x)}{\cos(x)} = \frac{\cos^2(x)}{\cos(x)} + \frac{\sin^2 (x)}{\cos(x)} = \frac{\cos^2(x) + \sin^2(x)}{\cos(x)} = \frac{1}{\cos(x)} = \sec(x)$$
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$$1)$$ Compute the derivative $$y'$$ of the function $$y = x^x$$. $$2)$$ Compute the integral $$\int_0^{\pi} \sin(x) + \cos(x) dx$$ $$3)$$ Calculate the implicit derivative $$y'$$ of $$x^2 + y^2 = 2xy$$
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$$1)$$ $$y = x^x \Rightarrow \ln(y) = \ln (x^x) \Rightarrow \ln(y) = x \ln(x) \Rightarrow (\ln(y))' = (x \ln(x))' \Rightarrow \frac{1}{y} y' = x \cdot \frac{1}{x} + \ln(x) \cdot 1 = 1 + \ln(x) \Rightarrow \frac{1}{y}y' = 1 + \ln(x) \Rightarrow y' = y(1 + \ln(x)) = x^x(1 + \ln(x))$$. Thus, $$y' = x^x(1 + \ln(x))$$. $$2)$$ $$\int_0^{\pi} \sin(x) + \cos(x) dx = [-\cos(x) + \sin(x)]_0^\pi = [-\cos(\pi) + \sin(\pi)] - [- \cos(0) + \sin(0)] = [-(-1)+ 0]-[-1 + 0] = 1 - (-1) = 1 + 1 = 2$$ $$3)$$ $$(x^2 + y^2)' = (2xy)' \Rightarrow 2x + 2yy' = 2xy' + 2y \Rightarrow 2yy' = 2xy' + 2y - 2x \Rightarrow 2yy' - 2xy' = 2y - 2x \Rightarrow y'(2y - 2x) = 2y - 2x \Rightarrow y' = 1$$.
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$$1)$$ Determine the sum of $$\frac{1}{3\sqrt{25}} + \frac{1}{2 \sqrt[3]{27}}$$ $$2)$$ Evaluate the expression into simplest form: $$\frac{8x^6 z^4 + 4x^4 z^2}{4x^2 z}$$ $$3)$$ Determine the product of $$\sqrt{-6}(\sqrt{-4} - \sqrt{3})$$
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$$1)$$ $$\frac{1}{3\sqrt{25}} + \frac{1}{2 \sqrt[3]{27}} = \frac{1}{3 \cdot 5} + \frac{1}{2 \cdot 3} = \frac{1}{15} + \frac{1}{6} = \frac{2}{30} + \frac{5}{30} = \frac{7}{30}$$ $$2)$$ $$\frac{8x^6 z^4 + 4x^4 z^2}{4x^2 z} = \frac{4x^4 z^2(2x^2z^2 + 1)}{4x^2 z} = x^2 z(2x^2 z^2 + 1) = 2x^4 z^3 + x^2z$$ $$3)$$ $$\sqrt{-6}(\sqrt{-4} - \sqrt{3}) = i\sqrt{6}(i \sqrt{4} - \sqrt{3}) = i \sqrt{6}(2i - \sqrt{3}) = (i \sqrt{6})(2i) + (-\sqrt{3})(i \sqrt{6}) = -2 \sqrt{6} + (-\sqrt{3})(i \sqrt{3} \cdot \sqrt{2}) = -2 \sqrt{6} + (-3i \sqrt{2}) = -2\sqrt{6} - 3i\sqrt{2}$$
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# American Institute of Mathematical Sciences
July 2016, 12(3): 851-878. doi: 10.3934/jimo.2016.12.851
## Equilibrium balking strategies in renewal input queue with Bernoulli-schedule controlled vacation and vacation interruption
1 School of Basic Scienes, Indian Institute of Technology, Bhubaneswar-751007, India, India, India 2 School of Computer Application, KIIT University, Bhubaneswar-751024, India
Received September 2014 Revised March 2015 Published September 2015
We consider a single server renewal input queueing system under multiple vacation policy. When the system becomes empty, the server commences a vacation of random length, and either begins an ordinary vacation with probability $q\, (0\le q\le 1)$ or takes a working vacation with probability $1-q$. During a working vacation period, customers can be served at a rate lower than the service rate during a normal busy period. If there are customers in the system at a service completion instant, the working vacation can be interrupted and the server will come back to a normal busy period with probability $p\, (0\le p\le 1)$ or continue the working vacation with probability $1-p$. The server leaves for repeated vacations as soon as the system becomes empty. Upon arrival, customers decide for themselves whether to join or to balk, based on the observation of the system-length and/or state of the server. The equilibrium threshold balking strategies of customers under four cases: fully observable, almost observable, almost unobservable and fully unobservable have been studied using embedded Markov chain approach and linear reward-cost structure. The probability distribution of the system-length at pre-arrival epoch is derived using the roots method and then the system-length at an arbitrary epoch is derived with the help of the Markov renewal theory and semi-Markov processes. Various performance measures such as mean system-length, sojourn times, net benefit are derived. Finally, we present several numerical results to demonstrate the effect of the system parameters on the performance measures.
Citation: Gopinath Panda, Veena Goswami, Abhijit Datta Banik, Dibyajyoti Guha. Equilibrium balking strategies in renewal input queue with Bernoulli-schedule controlled vacation and vacation interruption. Journal of Industrial & Management Optimization, 2016, 12 (3) : 851-878. doi: 10.3934/jimo.2016.12.851
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[1] Sheng Zhu, Jinting Wang. Strategic behavior and optimal strategies in an M/G/1 queue with Bernoulli vacations. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1297-1322. doi: 10.3934/jimo.2018008 [2] Dequan Yue, Wuyi Yue, Gang Xu. Analysis of customers' impatience in an M/M/1 queue with working vacations. Journal of Industrial & Management Optimization, 2012, 8 (4) : 895-908. doi: 10.3934/jimo.2012.8.895 [3] Shan Gao, Jinting Wang. On a discrete-time GI$^X$/Geo/1/N-G queue with randomized working vacations and at most $J$ vacations. Journal of Industrial & Management Optimization, 2015, 11 (3) : 779-806. doi: 10.3934/jimo.2015.11.779 [4] Biao Xu, Xiuli Xu, Zhong Yao. Equilibrium and optimal balking strategies for low-priority customers in the M/G/1 queue with two classes of customers and preemptive priority. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1599-1615. doi: 10.3934/jimo.2018113 [5] Chia-Huang Wu, Kuo-Hsiung Wang, Jau-Chuan Ke, Jyh-Bin Ke. A heuristic algorithm for the optimization of M/M/$s$ queue with multiple working vacations. Journal of Industrial & Management Optimization, 2012, 8 (1) : 1-17. doi: 10.3934/jimo.2012.8.1 [6] Pikkala Vijaya Laxmi, Singuluri Indira, Kanithi Jyothsna. Ant colony optimization for optimum service times in a Bernoulli schedule vacation interruption queue with balking and reneging. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1199-1214. doi: 10.3934/jimo.2016.12.1199 [7] Ahmed M. K. Tarabia. Transient and steady state analysis of an M/M/1 queue with balking, catastrophes, server failures and repairs. Journal of Industrial & Management Optimization, 2011, 7 (4) : 811-823. doi: 10.3934/jimo.2011.7.811 [8] Dequan Yue, Wuyi Yue, Guoxi Zhao. Analysis of an M/M/1 queue with vacations and impatience timers which depend on the server's states. Journal of Industrial & Management Optimization, 2016, 12 (2) : 653-666. doi: 10.3934/jimo.2016.12.653 [9] Dequan Yue, Wuyi Yue. A heterogeneous two-server network system with balking and a Bernoulli vacation schedule. Journal of Industrial & Management Optimization, 2010, 6 (3) : 501-516. doi: 10.3934/jimo.2010.6.501 [10] Ruiling Tian, Dequan Yue, Wuyi Yue. Optimal balking strategies in an M/G/1 queueing system with a removable server under N-policy. Journal of Industrial & Management Optimization, 2015, 11 (3) : 715-731. doi: 10.3934/jimo.2015.11.715 [11] Shaojun Lan, Yinghui Tang. Performance analysis of a discrete-time $Geo/G/1$ retrial queue with non-preemptive priority, working vacations and vacation interruption. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1421-1446. doi: 10.3934/jimo.2018102 [12] Feng Zhang, Jinting Wang, Bin Liu. On the optimal and equilibrium retrial rates in an unreliable retrial queue with vacations. Journal of Industrial & Management Optimization, 2012, 8 (4) : 861-875. doi: 10.3934/jimo.2012.8.861 [13] Pikkala Vijaya Laxmi, Seleshi Demie. Performance analysis of renewal input $(a,c,b)$ policy queue with multiple working vacations and change over times. Journal of Industrial & Management Optimization, 2014, 10 (3) : 839-857. doi: 10.3934/jimo.2014.10.839 [14] Zhanyou Ma, Wenbo Wang, Linmin Hu. Performance evaluation and analysis of a discrete queue system with multiple working vacations and non-preemptive priority. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-14. doi: 10.3934/jimo.2018196 [15] Sujit Kumar Samanta, Rakesh Nandi. Analysis of $GI^{[X]}/D$-$MSP/1/\infty$ queue using $RG$-factorization. Journal of Industrial & Management Optimization, 2017, 13 (5) : 0-0. doi: 10.3934/jimo.2019123 [16] Zhanyou Ma, Pengcheng Wang, Wuyi Yue. Performance analysis and optimization of a pseudo-fault Geo/Geo/1 repairable queueing system with N-policy, setup time and multiple working vacations. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1467-1481. doi: 10.3934/jimo.2017002 [17] Qingqing Ye. Algorithmic computation of MAP/PH/1 queue with finite system capacity and two-stage vacations. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-19. doi: 10.3934/jimo.2019063 [18] Dequan Yue, Jun Yu, Wuyi Yue. A Markovian queue with two heterogeneous servers and multiple vacations. Journal of Industrial & Management Optimization, 2009, 5 (3) : 453-465. doi: 10.3934/jimo.2009.5.453 [19] Zsolt Saffer, Wuyi Yue. A dual tandem queueing system with GI service time at the first queue. Journal of Industrial & Management Optimization, 2014, 10 (1) : 167-192. doi: 10.3934/jimo.2014.10.167 [20] Cheng-Dar Liou. Optimization analysis of the machine repair problem with multiple vacations and working breakdowns. Journal of Industrial & Management Optimization, 2015, 11 (1) : 83-104. doi: 10.3934/jimo.2015.11.83
2018 Impact Factor: 1.025 |
# Gross value added and worker salary
I'm trying to figure out how the Gross value added is calculated. The first question that I have encountered is "Does the Gross value added of an activity includes workers salary?" If not, what is the difference between GVA and an activity profit?
I assume that you're interested in gross value added as a national accounting concept (which is the most frequent use of the term). In that case, it is formally defined by the UN's System of National Accounts as "the value of output less the value of intermediate consumption". Here is a table from the freely available old version of Understanding National Accounts showing this breakdown for the nonfinancial corporate sector of France:
This definition is made on the production side, but gross value added can also be broken down into income. On the income side, it includes worker salaries, as another account excerpted from Understanding National Accounts shows:
As we can see above, gross value added is decomposed into compensation of employees (of which "wages and salaries" is the most important part), other taxes less subsidies on production, and gross operating surplus. Gross operating surplus can further be decomposed into consumption of fixed capital (basically depreciation) and net operating surplus.
Combining Tables 3 and 4, we see how the initial "output" accruing to a firm can be progressively whittled down:
• First, we subtract intermediate inputs to obtain gross value added.
• Then, we subtract compensation of employees and taxes less subsidies on production to obtain gross operating surplus.
• Finally, we subtract consumption of fixed capital to obtain net operating surplus.
Of all these concepts, the final one ("net operating surplus") is closest to our usual concept of profit, since it subtracts payments for intermediates, compensation of employees, production taxes, and depreciation. Nevertheless, there are a number of conceptual and technical differences between this and profit as it is usually defined: for instance, the net earnings of a corporation subtract interest payments, while "net operating surplus" does not. Instead, "net operating surplus" can be viewed as a broader measure of capital income. (Indeed, when Piketty and Zucman (QJE 2014) plot "capital shares" of national income in Table XII, they are actually plotting net operating surplus as a share of national income for each country, taken from the national accounts.)
Finally, note that all these definitions are quite generic and meant to apply to many levels of aggregation: firms, industries, areas, sectors, or entire economies. The tables with gross value added above show the "production" and "generation of income" accounts for the non-financial corporate sector, but all national accounts constructed using the same principles provide an analogous account for other sectors—including households, government, the total economy, etc.
Looking at a Profit & Loss statement of a company,
Value Added = Personnel Costs (+) Depreciation Charges (+) Profits (or (-) Losses)
It essentially is this concept that coincides exactly (in a narrow sense) to the fundamental production function with constant returns to scale, and factors of production only Capital and Labor, that moreover are being paid their marginal product:
Due to homogeneity of degree $1$, we have the production function
$$Y = F(K,L) = \frac {\partial F(K,L)}{\partial K}K + \frac {\partial F(K,L)}{\partial L}L$$
So if the gross rate of return to capital equals its marginal product, and the wage equals labor's marginal product,
$$r= \frac {\partial F(K,L)}{\partial K}, \;\; w=\frac {\partial F(K,L)}{\partial L}L$$
we get
$$Y = rK + wL$$
So all of $Y$ goes to the factors of production that appear in the right-hand side.
But $rK=$ Depreciation Charges + Profits (or minus Losses), and $wL=$ Personnel Costs. So in reality, $Y$ is Value Added rather than Gross Sales (and most econometric studies where production functions are estimated, regress Value Added on labor and capital, rather than Gross Sales). An issue here could be how one views interest costs, which are also returns to capital (irrespective of the fact that they do not belong to the shareholders -the company employs this capital and pays a fee for its use)
Of course the concept of production function can be enhanced to include separately materials and energy for example, in which case the appropriate dependent variable becomes Gross Sales. |
Home > Probability Of > Probability Of Error Bpsk
Probability Of Error Bpsk
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By using this site, you agree to the Terms of Use and Privacy Policy. The waveforms for DPSK are the same as for differentially encoded PSK given above since the only change between the two schemes is at the receiver. awgnerrornoisepbskprobabilitywhite gaussian Cancel Please login to add a comment or rating. They are positioned on a circle so that they can all be transmitted with the same energy. http://bsdupdates.com/probability-of/probability-of-bit-error-in-bpsk.php
The demodulator, which is designed specifically for the symbol-set used by the modulator, determines the phase of the received signal and maps it back to the symbol it represents, thus recovering doi:10.1109/GLOCOM.2005.1578470. Your cache administrator is webmaster. Two common examples are "binary phase-shift keying" (BPSK) which uses two phases, and "quadrature phase-shift keying" (QPSK) which uses four phases, although any number of phases may be used.
Probability Of Error In Qpsk
This results in a two-dimensional signal space with unit basis functions ϕ 1 ( t ) = 2 T s cos ( 2 π f c t ) {\displaystyle \phi The binary data stream is split into the in-phase and quadrature-phase components. Difference of the phase between QPSK and OQPSK Taking four values of the phase (two bits) at a time to construct a QPSK symbol can allow the phase of the signal The binary data that is conveyed by this waveform is: 1 1 0 0 0 1 1 0.
Each pattern of bits forms the symbol that is represented by the particular phase. Black Box Network Services. DPQPSK Dual-polarization quadrature phase shift keying (DPQPSK) or dual-polarization QPSK - involves the polarization multiplexing of two different QPSK signals, thus improving the spectral efficiency by a factor of 2. Bit-error-probability-for-bpsk-modulation The amplitude of each point along the in-phase axis is used to modulate a cosine (or sine) wave and the amplitude along the quadrature axis to modulate a sine (or cosine)
Generated Mon, 24 Oct 2016 12:04:09 GMT by s_wx1062 (squid/3.5.20) ERROR The requested URL could not be retrieved The following error was encountered while trying to retrieve the URL: http://0.0.0.8/ Connection Subscribe Enter Search Term First Name / Given Name Family Name / Last Name / Surname Publication Title Volume Issue Start Page Search Basic Search Author Search Publication Search Advanced Search Contents 1 Introduction 1.1 Definitions 2 Applications 3 Binary phase-shift keying (BPSK) 3.1 Implementation 3.2 Bit error rate 4 Quadrature phase-shift keying (QPSK) 4.1 Implementation 4.2 Bit error rate 4.3 Variants https://www.mathworks.com/matlabcentral/fileexchange/9236-bpsk-probability-of-error-in-awgn High definition programming is delivered almost exclusively in 8PSK due to the higher bitrates of HD video and the high cost of satellite bandwidth.[6] The DVB-S2 standard requires support for both
Using DPSK avoids the need for possibly complex carrier-recovery schemes to provide an accurate phase estimate and can be an attractive alternative to ordinary PSK. Probability Of Error For Bpsk And Qpsk As mentioned for BPSK and QPSK there is an ambiguity of phase if the constellation is rotated by some effect in the communications channel through which the signal passes. As with BPSK, there are phase ambiguity problems at the receiving end, and differentially encoded QPSK is often used in practice. The system returned: (22) Invalid argument The remote host or network may be down.
Bit Error Rate Of Qpsk
These are then separately modulated onto two orthogonal basis functions. With more than 8 phases, the error-rate becomes too high and there are better, though more complex, modulations available such as quadrature amplitude modulation (QAM). Probability Of Error In Qpsk Couch, Leon W. Bit Error Rate Matlab Code Please try the request again.
V. "Tutorial on dynamic analysis of the Costas loop". this page The decision variable for the k − 1 {\displaystyle k-1} th symbol and the k {\displaystyle k} th symbol is the phase difference between r k {\displaystyle r_{k}} and r k Alternatively, instead of operating with respect to a constant reference wave, the broadcast can operate with respect to itself. Digital and Analog Communications. Bpsk Probability Of Error Derivation
This lowers the dynamical range of fluctuations in the signal which is desirable when engineering communications signals. Assume without loss of generality that the phase of the carrier wave is zero. Phase-shift keying From Wikipedia, the free encyclopedia Jump to: navigation, search Passband modulation Analog modulation AM FM PM QAM SM SSB Digital modulation ASK APSK CPM FSK MFSK MSK OOK get redirected here Planet Fox. 2014. ^ http://www.broadcom.com/products/set-top-box-and-media-processors/satellite/bcm7325 ^ "Local and Remote Modems" (PDF).
Thus, the 180° phase ambiguity does not matter. Ber Of Bpsk In Awgn Channel Matlab Code MATLAB release MATLAB 7.1.0 (R14SP3) Tags for This File Please login to tag files. The 6 and 9 Mbit/s modes use OFDM modulation where each sub-carrier is BPSK modulated.
The binary data stream is shown beneath the time axis.
At the k th {\displaystyle k^{\textrm {th}}} time-slot call the bit to be modulated b k {\displaystyle b_{k}} , the differentially encoded bit e k {\displaystyle e_{k}} and the resulting modulated In the figure, it is assumed that the signal starts with zero phase, and so there is a phase shift in both signals at t = 0 {\displaystyle t=0} . In other words, the signal does not pass through the origin. Bpsk Bit Error Rate Matlab Code Stern & S.
Privacy policy About Wikipedia Disclaimers Contact Wikipedia Developers Cookie statement Mobile view Search: MATLAB Central File Exchange Answers Newsgroup Link Exchange Blogs Cody Contest MathWorks.com Create Account Log In Products Solutions The system returned: (22) Invalid argument The remote host or network may be down. Implementation The general form for BPSK follows the equation: s n ( t ) = 2 E b T b cos ( 2 π f c t + π ( useful reference II (1997).
In the constellation diagram shown on the right, it can be seen that this will limit the phase-shift to no more than 90° at a time. Unfortunately, it can only be obtained from: P s = 1 − ∫ − π M π M p θ r ( θ r ) d θ r {\displaystyle P_{s}=1-\int _{-{\frac Please try the request again. The phase-shifts are between those of the two previous timing-diagrams.
This is a cost-effective alternative, to utilizing 16-PSK instead of QPSK to double the spectral efficiency. This gives maximum phase-separation between adjacent points and thus the best immunity to corruption. The increase in E b / N 0 {\displaystyle E_{b}/N_{0}} required to overcome differential modulation in coded systems, however, is larger - typically about 3dB. Furthermore, this analysis (and the graphical results below) are based on a system in which the only corruption is additive white Gaussian noise(AWGN).
Discover... Note the half-period offset between the two signal components. π /4–QPSK Dual constellation diagram for π/4-QPSK. This is the description of differentially encoded BPSK given above. Thus, the first symbol (1 1) is taken from the 'blue' constellation and the second symbol (0 0) is taken from the 'green' constellation.
The sudden phase-shifts occur about twice as often as for QPSK (since the signals no longer change together), but they are less severe. Nelson, E. In the case of PSK, the phase is changed to represent the data signal. It is sometimes called Staggered quadrature phase-shift keying (SQPSK).
Digital Communications. Changes in phase of a single broadcast waveform can be considered the significant items. In the absence of noise, the phase of this is ϕ k − ϕ k − 1 {\displaystyle \phi _{k}-\phi _{k-1}} , the phase-shift between the two received signals which can The modulated signal is shown below for a short segment of a random binary data-stream.
Bounds on the error rates of various digital modulation schemes can be computed with application of the union bound to the signal constellation. ISBN0-471-62947-2. Otherwise it remains in its previous state. This kind of encoding may be demodulated in the same way as for non-differential PSK but the phase ambiguities can be ignored. |
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# Thread: Distance between normed spaces
1. ## Distance between normed spaces
Hi, I need to solve this problem, can someone help me?
Let c be the space of convergent scalar sequences and c0 the spaces of the null scalar sequences. Prove that the distance between them is less or equal than 3.
Thanks.
2. Originally Posted by felixgotti
Hi, I need to solve this problem, can someone help me?
Let c be the space of convergent scalar sequences and c0 the spaces of the null scalar sequences. Prove that the distance between them is less or equal than 3.
Thanks.
What is the ambient space? What norm is this ambient space given?. Also isn't $c\cap c_0 \neq \emptyset$, so which is your definition of distance between normed spaces? (maybe this last one is the only one relevant?) |
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typing mathematics in microsoft word
Hello world!
March 27, 2017
If you encountered a bug or want to suggest a feature in Microsoft Office, we recommend you contact Microsoft Support. The dot product (inner product) can be displayed using the centered dot symbol "\cdot" e.g. The easiest thing to do would be to find a LaTeX reference sheet. The keyboard shortcut is "alt"+ "=". With Microsoft Office 2007, new Equation Tools were introduced, and gradually MS office reduced the support for the old equation editor. In the Symbol dialog box, select the Symbols or 1 The only way to mitigate this problem is to upgrade to Word 2016, which does not have this problem. d To see all the symbols, click the More button. Comment frapper rapidement des symboles mathématiques et autres. Typing complex mathematical equations or scientific expressions can be difficult when creating tests or writing research papers if you are using only Microsoft Word or … Some uncommon symbols are not listed in the menu and require knowing the keyboard shortcut. Nearly all symbols use the same commands as LaTeX. You can hold [Shift] for the upper case Greek characters. Please try again. Note that this is a different tool than the legacy tool Equation Editor 3.0 (which is still available on 32-bit Office versions until the January 2018 update[1]) and MathType. These symbols include "(), {}, [], ||". No LaTeX typesetting tools such as labels and references are implemented. How to Type and Use Mathematical Equations in Word 2016 Select Insert Tab to Type and Use Mathematical Equations. Below is the complete list of alt code shortcuts for mathematics symbols. Because the 1/2 fraction is is quite tall, the outer parentheses need to be adjusted to enclose the fraction appropriately. {\displaystyle \int \limits _{a}^{b}{\frac {1}{x}}\,dx}. However, you will see almost invisible bar with math functions. The add-in also provides an extensive collection of mathematical symbols and structures to display clearly formatted mathematical expressions. The default is vertically aligned as illustrated below. You can use the decimal values of the Unicode points to use with the alt keys on Windows based documents. 7 ] For example, to type ⊂, ⊆ or ⊄, hold Alt and press C one, two or three times. \sdiv) and pressing space (twice) or by typing 1 \ldiv 2 (resp. This page was last edited on 1 June 2020, at 16:43. When I am typing some mathematics on my PC. + {\displaystyle {\dot {r}},{\ddot {r}}} Math typing would be so much easier if you could just use a word processor, … or a math processor. 1 \sdiv 2) and pressing space. Use parentheses to start and end the matrix. It also applies to Microsoft PowerPoint and Excel 2010 and higher. 4 Also you can insert any symbol automatically with the AutoCorrect command (see Math Builder is a much easier to use tool that has less functionality than LaTeX but more than typical document processing. {\displaystyle {\sqrt {x}}} x No highly advanced LaTeX tools such as graphing, commutative diagrams, or geometric shapes are implemented. 5. If you have defined equation preferences for new equations (using the Set Equation Preferences command), these settings will be used in the MathType window. 3 This book is about the Math Builder (officially called as Equation Editor) tool in Microsoft Word and Outlook 2007 and higher. There are differences between Math Builder and LaTeX code: advanced functionality that requires more than just a symbol tend to follow the same flavor but have slightly different syntax. This is implemented via math autocorrect which you can modify. The simplest way to write fractions in Microsoft Word is to just … Typing mathematical symbols like Greek alphabet or symbols "≤", "≥", "≠", "∞" etc. In the Symbols group, you’ll find math related symbols. Radicals are obtained using the "\sqrt" symbol, followed by the index, then "&", then the radicand. Symbol listbox and then select More Symbols...: 2. To be exact, the key presses required to reproduce the equation above are ( 1 / 2 space ( x + 1 ) space ) space. ∇ Be careful to press space after the "2" to render the fraction, otherwise Word might put "x+1" in the denominator. There are multiple ways to display a fraction. 1 The format used is non-proprietary and given in Unicode Technical Note #28. Typing Math in Microsoft Word . Additionally, \sqrt(x) will simply output Display specifies to use as much space as needed. skewed fraction) is obtained using \ldiv (resp. Also, we are not responsible for access configuration or software updates in your company. = This tutorial demonstrates how to write a math equation using Microsoft Word 2010. The steps for creating theses text elements are listed for both Office 2016 for Mac users and Office 2016 (including Office 365) for Windows users. Also press space after typing every closing parenthesis ")", which will adjust both the opening and closing parentheses size to fit the group's contents. 2 may be displayed using "\nabla". Opening the Equation Editor To access the equation editor, either choose the “Insert” tab on the main menu, and click on the π symbol, or press “Alt=”, and an equation editor object will open. Choose Design to see tools for adding various elements to your equation. {\displaystyle \nabla \cdot A} These symbols are constructed with all the commands starting with "\" as illustrated in the above sections. Dot notation for time derivatives (e.g. Display mode equations must appear on their own line. x Exponents can be obtained by using "^" and subscripts by "_". . Use parentheses to start and end the matrix. Examples here are matrices, multiple aligned equations, and binomial coefficients. Get step-by-step explanations. Stop the mouse over each button to learn its keyboard shortcut. a Once you pressed the “Alt=”, an equation window will appear: r It can be used in Outlook to easily write equations in emails; it renders as images to the recipent. . MathType is currently a free add-in that can be utilized in Microsoft Word, Excel, and PowerPoint. This will insert an … ^ 5 1 A few of those symbols are shown here: The math environment implements 3 fronts in addition to the default. Integrals are obtained by inserting the desired integral symbol (see above table), and then pressing space twice. skewed fraction) is obtained using \ldiv (resp. To simplify this task, you can assign a shortcut for frequently used symbol. Unicode has a code point from 2200 to 22FF for mathematical operators. A vector is often denoted by an overhead right arrow, which can be obtained by following a letter variable with "\vec": Everything in Math Builder requires special symbols that the computer knows how to interpret. 1 \sdiv 2) and pressing space. {\displaystyle {\begin{aligned}2&x+&3&y=5\\&x+&&y=7\end{aligned}}}, (The math environment here seems to be adding excess space between the alignments that doesn't occur in Word). Grouping symbols will automatically size to the appropriate size. p Insert Inline Equation Ctrl + Alt + Q (Windows), Ctrl + Q (Mac) Opens a new MathType window ready for you to enter an equation. In the Design Window, you have three Groups. Everything you type in this environment is considered math: all automatic formatting of text is disabled. {\displaystyle \nabla } There are multiple ways to display a fraction. into the Word document: switch to, How to use AutoCorrect substitutions for fast typing of Greek alphabet and mathematical symbols, Comment frapper rapidement des symboles mathématiques et autres, create shortcuts for frequently used symbols, How to insert lambda, sigma, theta and other Greek symbols in Word, How to insert alpha, beta, gamma, delta and other Greek symbols in Word. x A stepping stone between word processing (MS Word) and typesetting (LaTeX). To see other sets of symbols, click the arrow in the upper right corner of the gallery. used Alt+1,4) and click Assign. Typically this is the LaTeX code for the symbol. Word has an equation editor, but the interface is not particularly intuitive. {\displaystyle {\frac {1}{2}}}. If you have any questions or suggestions, please feel free to ask OfficeToolTips team. x {\displaystyle x_{2}^{5}}. The gradient (also known as del or nabla) operator A x Math functions is for using in equation editor while normal replacements are for using in text editor. algebra trigonometry statistics calculus matrices variables list. v Aligning equations can be obtained with the "eqarray" symbol. Then the Equation toolbar will pop out along with a textbox. You can add or change the following elements to your equation. In Microsoft Word, go to Insert–Object and select Microsoft Equation 3.0. Write, insert, or change an equation or formula. ) are denoted by a hat (circumflex), which can be obtained by following a letter variable with "\hat". With the Microsoft Mathematics Add-in for Word and OneNote, you can perform mathematical calculations and plot graphs in your Word documents and OneNote notebooks. On the Insert tab, in the Symbols group, click in the Solve. You can do this by adding the command to the math autocorrect directory. For instance, you might like to use \ra instead of \rightarrow. x Common symbols have point-and-click icons. Blackboard bold letters can be obtained by typing "\" followed by "double" followed by the letter. {\displaystyle {\hat {x}}} There are multiple equations in the drop-down list, then scroll down and select one of them to meet your actual needs. the curl Graph your math problems. 2 Typing equations. I want it to convert/typed in MS word… For instance fractions will use a smaller font. This has not been verified with Equation Editor or Word for Mac. {\displaystyle {v}/{p}}. Option 3: Type Exponents Using Keyboard Shortcut. ( Math in Microsoft Word Section The following guide will help you add symbols, subscript and superscript text, and equations to documents created in Microsoft Word. To exit the math environment, click on any text outside the math environment. In addition, there are also many other mathematical symbols part of Unicode system. Fraktur does not have capitals. Unit vectors (e.g. \sdiv) and pressing space (twice) or by typing 1 \ldiv 2 (resp. One easy way to do this is by pressing the right arrow key. Quick typing of mathematical and other symbols Word 365 Mathematical and other texts require a large number of special symbols that are not present on the keyboard. Drop in a comment, if you see some important symbol is missing. From Wikibooks, open books for an open world, https://support.office.com/en-us/article/Equation-Editor-6eac7d71-3c74-437b-80d3-c7dea24fdf3f, http://iun.edu/~mathiho/useful/word07shortcuts.pdf, https://en.wikibooks.org/w/index.php?title=Typing_Mathematics_in_Microsoft_Word&oldid=3695286, Book:Typing Mathematics in Microsoft Word. 2. Type a math problem. ) It's easy to get started: it's already built in to Microsoft Word. 1. The monomial below can be obtained by typing x_2^5 or x^5_2 and pressing space. The equations below can be obtained by typing the following text: 2 Capitalizing it creates a capital letter. To simplify this task, you can assign a shortcut for frequently used symbol. How to use AutoCorrect substitutions for fast typing of Greek alphabet and mathematical symbols). 2 Well, there is a plug-in for Microsoft Word and other word processors, called MathType, but it is not a free program and you would agree with me that free is always better, right? I have questions and solutions (Physics, chemistry and Mathematics) in Picture format. Then highlight only the exponent and press Ctrl+Shift+=. Obtain this by typing the fraction and pressing space: 1/2 1 2 {\displaystyle {\frac {1}{2}}} Linear fraction (resp. y So here are the three methods. To type the squared symbol on Microsoft Word, click the superscript button (x²) in the Font group under the Home tab, and then type the number 2. (If you are working in Word 2007, do not use the default equation editor: ‘π Equation’) Benefits of using MathType or Equation Editor 3.0. = [2], For example: \int_a^b space space 1/x space dx will output ¨ For example: \sqrt(a&b) will output Go to Insert tab, click Object button in Text section. {\displaystyle \nabla \times A} Contact your company support team and install latest updates before asking questions. → While you can also do this by right-clicking on the equation and clicking Linear, this affects the whole equation and not just the fraction. Ms Word and Power Point shortcut for equation editor is “ Alt + = ” (i.e. The mathematical community almost universally accepts a typesetting language called LaTeX. While MS Word provides some math creation tools built in, MathType is the preferred method as it offers greater accessibility.To install MathType go to the Insert tab in Word and select Get Add-ins in the Add-ins group.Type MathType into the search box and select the Add buttonOnce installed this MathType option will appear in your Insert menu. {\displaystyle {\overrightarrow {A}}} The cross product can be displayed using "\times" e.g. a Typing Fractions on a Single Line. 3. With the Microsoft Mathematics Add-in for Word and OneNote, you can perform mathematical calculations and plot graphs in your Word documents and OneNote notebooks. Microsoft Math Solver. I searched in symbols and it takes too much of time. Making the best of it with handwriting recognition in touch screens. Script letters can be obtained by typing "\" followed by "script" followed by the letter. b v / p {\displaystyle {v}/{p}} To type exponents in Word using a keyboard shortcut, type both the base number and the exponent. + See also this tip in French: the divergence It is an appropriate tool for: Note that Math Builder does not perform any mathematics; it is a tool for displaying it. Inline specifies that the equation is to be in line with text. You can also type 2 first and then select or highlight it … Equations have two forms. These are all common symbols. (Note:- Geometric shapes are otherwise available in the Insert ribbon), Students studying mathematics might not be motivated to learn LaTeX because they might be able to get by with. [ ) Greek letters can be obtained by typing a "\" followed by the name of symbol. There is a serious bug on Word versions up to and including Word 2013 that causes the first letter of small equations to disappear randomly after some time. {\displaystyle \left({\frac {1}{2}}(x+1)\right)}. ∇ 5 Use "@" to separate rows, and "&" to separate columns. Click the Shortcut Key... button to open the Customize box. Smooth the learning curve of math tools with a unified experience. The default is vertically aligned as illustrated below. To speed up typing, you can create aliases and expand them through, There are four quick methods to enter lambda, sigma, theta, and other Greek letters D × Math Builder code tends to be shorter than LaTeX code and disappears upon completion to the WYSIWYG output. lot of time. A For a Mac system, the shortcut is control + "=". See how to solve problems and show your work—plus get definitions for mathematical concepts. {\displaystyle \mathbb {d} \mathbb {D} }. Open up your Word document. into the Word document: switch to, There are four quick methods to enter alpha, beta, gamma, delta, and other Greek letters Spaces is an important part of Math AutoCorrect shortcut. Linear fraction (resp. Typesetting mathematics on a computer has always been a challenge. {\displaystyle {\sqrt[{a}]{b}}} 2. Special Characters tab and select the character that you want: 3. + takes a Failed to send the question. y hold down Alt key while typing ‘=’). When text replacement happens, the application will show you AutoCorrect options as an icon. 5 Typing any document whose focus is not itself mathematics. A Creative Commons Attribution-ShareAlike License. 6 We are not a division, a subsidiary, or a contractor of Microsoft Corporation, and we are not responsible for their decisions. . {\displaystyle {\begin{bmatrix}1&2&3\\4&5&6\end{bmatrix}}}. ∫ 2. d To assign a symbol to a shortcut key, follow these steps: 1. ˙ You can still click on that to adjust the settings when typing. b Click the Close button to return to the Symbol dialog ∇ 3 MS Word Tricks: Typing Math Symbols 2015-05-14 Category: MS Office. The subscription model allows you to enjoy the same quality solution in your word processor and LMS. There are at least two ways to type professionally looking mathematical objects in Microsoft Word. , While you can also do this by right-clicking on the equation and clicking Linear, this affects the whole equation and not just the fraction. The second and then every other occurrence is white space. Use "&" to specify alignment and whitespace. To go to the first step, start opening your Ms. Word and click... Click Equations Option to Open Design window. In the Press new shortcut key box, type the key combination Keyboard dialog box: 4. Summations and integrals will place the endpoints to the right of the symbol instead of below it. After some time I found these three ways for typing theta symbol in MS word. can be obtained by following a letter variable with "\dot" for a first derivative and "\ddot" for a second derivative. I find difficult to write the theta θ symbol in it. Solve Practice Download. ⋅ Fraktur letters can be obtained by typing "\" followed by "fraktur" followed by the letter. . Choose Microsoft Equation 3.0 in the list of Object type in Create New tab and click OK to confirm it. Although you can also click on “Equations” under the “Insert” Tab to get it. Mathematical and other texts require a large number of special symbols that are not present on the keyboard. 1 \doubled \doubleD produces I thought that I should share with you also. To obtain the math environment, click on "Equation" on the "Insert" ribbon on Windows or Word for Mac '16, or in "Document Elements" on Word for Mac '11. Easy user experience that will boost your productivity. Using a keyboard shortcut questions or suggestions, please feel free to ask team. Some time i found these three ways for typing theta symbol in MS Word Tricks typing! ≥ '', then & '' to separate columns mode equations must on. And then pressing space ( twice ) or by typing \ '' by... ( ), and then every other occurrence is white space the best of it handwriting! Of those symbols are shown here: the math environment code and disappears upon completion to the symbol box! Symbol ( x² ) Word processors have support for special characters tab and click OK to confirm.... ) or by typing \ '' as illustrated in the drop-down list, then the.... Complete list of Alt code shortcuts for mathematics symbols the default knowing the keyboard advanced tools! Symbol \cdot '' e.g other occurrence is alignment equations, and gradually MS Office typically is... The complete list of Alt code shortcuts for mathematics symbols user need not use a mouse at all ''... Centered dot symbol \cdot '' e.g type exponents in Word using the keyboard shortcut is.. Particularly intuitive all automatic formatting of text is disabled by typing \ '' followed by letter. 2007, New equation tools were introduced, and gradually MS Office the... Is control + = '' can hold [ Shift ] for the old equation editor or Word Mac! And the exponent is control + = '' references are implemented a subsidiary or. In Create New tab and select Microsoft equation 3.0 in the symbol instead of \rightarrow settings! Is missing _ '' an appropriate tool for: Note that math Builder does not perform any mathematics it. An icon almost universally accepts a typesetting language called LaTeX tool for typing theta symbol in Word! To your equation it with handwriting recognition in touch screens before asking questions a few expressions to make appear... For displaying it output x { \displaystyle { v } / { p } } } in 2016! Choose Design to see all the commands starting with \ '' as illustrated in the symbol listbox then. My PC i thought that i should share with you also Present Press Alt and Press C one, or... Their decisions Point from 2200 to 22FF for mathematical operators questions or suggestions please. Typing Projects for ₹600 - ₹1500 no highly typing mathematics in microsoft word LaTeX tools such as graphing, commutative diagrams, or for! Typing a \ '' as illustrated in the Design window, you might like to \ra. Microsoft Corporation, and we are not Present on the keyboard the mathematical community almost universally a... Scroll down and select one of them to meet your actual needs the second and then pressing space twice. Have questions and solutions ( Physics, chemistry and mathematics ) in Picture format & and! & '', ≠ '', ≥ '', then the radicand an part! Superscript symbol ( see above table ), { }, [ ], || '' select the that... And it takes too much of time MS Word in to Microsoft PowerPoint and Excel 2010 and.... The character that you want: 3 / p { \displaystyle \nabla } may be using... Other occurrence is alignment characters tab and click OK to confirm it at 16:43 structures... Simply output x { \displaystyle \nabla } may be displayed using the eqarray '' symbol, followed ! ⊄, hold Alt and Press C one, two or three times as needed elements to equation. 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Own line binomial coefficients you also code shortcuts for mathematics symbols Open Design.... Will show you AutoCorrect options as an icon { d } } mathematical expressions by double followed. And Power Point shortcut for frequently used symbol { 5 } } to solve problems and your... Listbox and then select More symbols...: 2 in this environment is considered math: all formatting! Can use the decimal values of the gallery of it with handwriting recognition in touch screens to interpret to other! Exponents in Word using a keyboard shortcut is control + = '' Shift ] for the old equation or! Few expressions to make them appear smaller example, to type exponents in Word using a keyboard.! Labels and references are implemented by inserting the desired integral symbol ( see above )! Collection of mathematical symbols like Greek alphabet or symbols ≤ '', ≠ '', scroll! For equation editor is “ Alt + = ” ( i.e = ’ ) making the best of it handwriting. 2020, at 16:43 install latest updates before asking questions also applies Microsoft... Go to Insert–Object and select Microsoft equation 3.0 example, to type professionally looking objects! @ '' to specify alignment and whitespace mitigate this problem a math using... 1 \ldiv 2 ( resp as an icon dot symbol \cdot '' e.g also, we recommend contact.: Note that math Builder code tends to be adjusted to enclose the fraction appropriately LaTeX such! Addition, there are multiple equations in emails ; it renders as images the... Used is non-proprietary and given in Unicode Technical Note # 28 when i am typing mathematics. Number and the exponent typing equations … write, Insert, or geometric shapes are implemented and.. Functionality than LaTeX code for the old equation editor, but the interface is itself. Microsoft PowerPoint and Excel 2010 and higher typing mathematics in microsoft word sets of symbols, click Object button in section! Comment, if you have three Groups of Alt code shortcuts for mathematics symbols ⊄, hold and. \Cdot a } in Create New tab and select Microsoft equation 3.0 Excel 2010 and higher following elements your. As labels and references are implemented ∇ ⋅ a { \displaystyle { v } / p! Based documents and disappears upon completion to the appropriate size Microsoft Word symbol \cdot '' e.g:! Has two different typing environments: text and math are implemented a subsidiary, or geometric shapes implemented! ” under the “ Insert ” tab to get started: it easy... ; it is an important part of math tools typing mathematics in microsoft word a unified experience the cross product can be obtained typing... 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Mathématiques et autres Office & typing Projects for ₹600 - ₹1500 any document whose focus is not mathematics. Hold Alt and Press C one, two or three times \sdiv ) and pressing space twice an.! Symbols or special characters tab and select the symbols group, click in menu... Symbol ( see above table ), and & '' and then other. And = '' symbol add-in also provides an extensive collection of mathematical like... |
# Coherence time
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For an electromagnetic wave, the coherence time is the time over which a propagating wave (especially a laser or maser beam) may be considered coherent. In other words, it is the time interval within which its phase is, on average, predictable.
In long-distance transmission systems, the coherence time may be reduced by propagation factors such as dispersion, scattering, and diffraction.
Coherence time, τ, is calculated by dividing the coherence length by the phase velocity of light in a medium; approximately given by
$\tau = \frac{1}{\Delta \nu} \approx \frac{\lambda^2}{c\, \Delta \lambda}$
where λ is the central wavelength of the source, Δν and Δλ is the spectral width of the source in units of frequency and wavelength respectively, and c is the speed of light in vacuum.
A single mode fiber laser has a linewidth of a few kHz. The Schawlow-Townes limit for some cw lasers can be below 1 Hz. Hydrogen masers have linewidth around 1 Hz;[1] their coherence length approximately corresponds to the distance from the Earth to the Moon.
As of 2009, single electron spins show the longest room-temperature spin dephasing times ever observed in solid-state systems (1.8 ms).[2]
## References
1. ^ http://www.physics.harvard.edu/Thesespdfs/humphrey.pdf - Precision measurements with atomic hydrogen masers
2. ^ Balasubramanian et al. (2009). "Direct Ultralong spin coherence time in isotopically engineered diamond". Nature Materials 8: 383. Bibcode:2009NatMa...8..383B. doi:10.1038/nmat2420. |
# Definition:Sum of Ideals of Ring
## Definition
Let $R$ be a ring.
### Two ideals
Let $I$ and $J$ be ideals of $R$.
Their sum is the ideal equal to their subset sum:
$I + J = \{i + j : i \in I \land j \in J\}$ |
# Equilateral triangle generation
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chrome68 132
Hi. I would like to generate an equilateral triangle on a plane in C++. I know the point P(p1,p2,p3) i wish to be the barycentric centre; the normal at that point, and another point on the plane 'a'. So 0 = N.(a-p) How can i calculate the 3 vertices of the triangle so they should lie on a circle (inscribed) of a radius 1, centred at P?
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Sneftel 1788
If I'm reading that right, it sounds like you don't care how the equilateral triangle is oriented, as long as it's in the plane. Correct me if I'm wrong.
First, calculate a vector in that plane. It doesn't matter which one. Try N × î, unless N is close to î, in which case use something else. I guess as long as you have a, you could use a-P. Then rotate that vector 120° around N, as well as -120° (by means of a rotation matrix, of course). Then just add those vectors to P. I guess you should probably normalize stuff somewhere in there.
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oliii 2196
easy peasy.
// for triangles, vnum = 3. supply a hint vector if you want to orientate // the polygon in a particular way. // normal should be normalised.void makePolygon(Vector* v, int vnum, const Vector& centre, const Vector& normal, float radius, const Vector* direction){ // vector used to define the direction the polygon is pointing. Vector x; // optional direction vector to align the polygon with. if(direction != NULL) { x = *direction; } // else use arbitrary direction. else { x.randomDirection(); } // secondary vector, perpendicular to the third and the normal of plane Vector z = x.crossProduct(normal); // normal aligned with hypothetical x axis. choose another x axis. while(z.length() < 0.00001f) { // compute new major and minor axes x.randomDirection(); z = x.crossProduct(normal); } z.normalise(); // re-align first vector into the plane. x = normal.crossProduct(z); x.normalise(); // generate vertices. float a = 0.0f; float da = twopi() / vnum; for(int i = 0; i < vnum; i ++, a+= da) { v[i] = centre + (x * cos(a) + z * sin(a)) * radius; }}
in your case, vnum = 3, centre = P, the hint vector would be (A - P), and radius = 1.0f;
[Edited by - oliii on May 1, 2009 11:35:57 AM]
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chrome68 132
Thanks very much for your help. It worked a treat ;)
I used Vector3D 'x' directly as (A-P), without the x.randomDirection(). I couldn't quite get the 'radius' parameter to give me the correct coordinates. Instead I normalised 'x' so its length was equal to 1.0 and that essentially became the first vertex of the equilateral triangle. Thanks again oliii. How do you rep in this forum?
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oliii 2196
you're welcome. you can rate using the link next to the user rating at the bottom of posts. |
# Is there a group $G$ and subgroup $H$, such that there exists $g\in G$ with $gHg^{-1} \subset H$ and $|H:gHg^{-1}|$ is infinite?
Question (asking on behalf of my friend who studies abstract algebra):
Is there a group $$G$$ and subgroup $$H$$, such that there exists $$g\in G$$ with $$gHg^{-1} \subset H$$ and $$|H:gHg^{-1}|$$ is infinite? ( I incline to think this is true.)
For such an example to exist, $$H$$ (and hence $$G$$) must be infinite and a non-normal subgroup of $$G$$. At first, it seems easy. However, I really don't know many types of infinite nonabelian groups (perhaps only the general linear group $${GL}_n(F)$$ and the group of bijections). Thanks for your slightest effort.
• This question is related, but does not ask for $|H:gHg^{-1}|$ to be infinite. Feb 21 '19 at 10:56
• @user1729 Does the matrix example work here? Feb 21 '19 at 10:58
• Nope; the subgroup $H$ there is cyclic. Feb 21 '19 at 10:59
Let $$\,G\,$$ be the free group generated by $$\,g,x_1,x_2,\dots\,$$ modded by the equations $$\,gx_ng^{-1} = x_{n+1}\,$$ for all $$n>0.$$ Let $$\,H\,$$ be the subgroup generated by all the $$\,x_n.\,$$ The index of $$\,gHg^{-1}\,$$ in $$\,H\,$$ is infinite because $$\,x_1\,$$ has infinite order. Note that there are more concrete ways to represent the groups and specializations where all the $$\,x_n\,$$ commute with each other but not with $$\,g.\,$$
• I believe this is the restricted wreath product of ${\mathbf Z}$ with itself. Feb 24 '19 at 0:22
Yes.
For example, take $$G$$ to be the HNN-extension* $$\langle a, b, t\mid t^{-1}at=[a, b], t^{-1}bt=b\rangle$$ and $$H=\langle a, b\rangle$$. By the theory of HNN-extensions, $$H$$ embeds into $$G$$ in the natural way. Clearly $$t^{-1}Ht\leq H$$. To see that $$|H:t^{-1}Ht|$$, note that the presentation $$\langle a, b\mid [a, b], b\rangle$$ defines an infinite group. Hence the smallest normal subgroup of $$H$$ containing $$[a, b]$$ and $$b$$ has infinite index in $$H$$; hence the smallest subgroup of $$H$$ containing $$[a, b]$$ and $$b$$, which is $$\langle [a, b], b\rangle$$, has infinite index in $$H$$. As $$t^{-1}Ht=\langle [a, b], b\rangle$$, the result follows.
*For background on HNN-extensions see Wikipedia or the book Combinatorial group theory by Lyndon and Schupp.
• Any simpler example? Feb 21 '19 at 10:51
• Define "simpler". (Personally, I think HNN-extensions are very easy to work with, because of my background, whilst I shy away from matrix groups.) Feb 21 '19 at 10:52
• Sorry, but I haven't taken combinatorial group theory... Feb 21 '19 at 10:53
• It will take me some time to read and understand your answer...Sorry for can't accept it now, but upvote it anyway. Feb 21 '19 at 11:02
Now, consider the semidirect product $$G:=\overline{H}\rtimes \bZ$$, where $$\bZ$$ acts on $$\overline{H}$$ by shifting to the right, so $$(1\cdot f)(k)=f(k+1)$$. Then if you take $$g=(0,1)\in G$$, then $$gHg^{-1}\unlhd H$$ (it is just the set of functions $$\bZ\to \bZ$$ which vanish on non-positive integers) and $$H/gHg^{-1}\cong \bZ$$.
If you replace functions $$\bZ\to \bZ$$ with functions into any other group $$K$$, then you will have $$H/gHg^{-1}\cong K$$.
Note that this construction is the (unrestricted) wreath product of $$\bZ$$ with itself.
• "Wreath product"? That sounds quite technical...Upvote it anyway. Feb 21 '19 at 12:10
• @YuiToCheng: It is not really. The first two paragraphs have a complete description, the wreath product is just a technical term. If you understand semidirect products, it should be clear. Feb 21 '19 at 12:11
• Wreath product is one of the oldest group constructions, and occurs at many places.
– YCor
Feb 21 '19 at 13:20
This is a very good question, and the answer is yes.
An example is given by $$G= \mathfrak{S}\mathbb{Z}$$, the group of permutations of $$\mathbb{Z}$$, and $$H$$ is the subgroup of permutations that fix $$\mathbb{Z}_{-}$$ pointwise (so the image of $$\mathfrak{S}\mathbb{N}$$ under the obvious morphism)
Now for $$g\in G$$, $$gHg^{-1}$$ is the group of permutations that fix $$g\mathbb{Z}_{-}$$ pointwise, so whenever $$\mathbb{Z}_{-}\subset g\mathbb{Z}_{-}$$, we have $$gHg^{-1}\subset H$$.
In particular if $$g\mathbb{Z}_{-}\setminus\mathbb{Z}_{-}$$ is infinite, then $$gHg^{-1}$$ has infinite index in $$H$$, and of course this can happen if you choose $$g$$ well enough. |
Engineering Simulations and Computations Using Matlab
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# 5'UTR and 3'UTR annotation in yeast
I am working on a project in which I need to compute several parameters (such GC content and length) of 5'UTR and 3'UTR sequences of Saccharomyces cerevisiae yeast genes.
The problem is finding a proper annotation for these regions in yeast. I have tried with BiomaRt and these sequences are not available for yeast. I have also tried with UCSC table browser to obtain a BED file with 5'UTR or 3'UTR sequences but it is returning only coordinates of tRNAs, pseudogenes, ncRNAs.
In summary, is there a straight-forward way to obtain an annotation for 5'UTR and 3'UTR regions in Saccharomyces cerevisiae yeast?
• Sorry, is for Saccharomyces cerevisiae. I am going to edit the question just in case.
– plat
Nov 6 '17 at 12:58
• isn't there an equivalent of AceView for Yeast ? ncbi.nlm.nih.gov/IEB/Research/Acembly/index.html Nov 6 '17 at 17:15
I am unaware of any "official" or gold-standard UTR annotations in S. cerevisiae.
One option is to use the annotations from the TIF-Seq publication (Pelechano et al. 2013).
The GSE39128_tsedall.txt.gz file contains the major isoforms identified. It would be up to you to computational associate each transcript with a given gene. It is also up to you to determine which major isoforms are present in your dataset. Furthermore based on my own experience there is often a significant number of discrepancies between what I see in an NGS dataset (e.g. RNA-Seq) and what is called as a possible isoform in these annotations. However, a lot of this could be related to the inherent technological limitations of the method.
If you look at the SGD genome browser they provide some additional UTR annotations. Derived from Nagalakshmi et al. 2008.
You can find the relevant annotations for those UTR's in their supplemental data section. The file is 1158441_tables_s2_to_s6.zip and inside you will find TableS4 which seems to contain what you seek.
There might be more resources but these are the two main ones I'm aware of.
According to the README files in the same directory, these are (the README for the 5' file is equivalent): |
# #StackBounty: #normal-distribution #conditional-expectation #intuition #multivariate-distribution Interpretation of multivariate condit…
### Bounty: 50
I’ve been reading over this Multivariate Gaussian conditional proof, trying to make sense of how the mean and variance of a gaussian conditional was derived. I’ve come to accept that unless I allocate a dozen or so hours to refreshing my linear algebra knowledge, it’s out of my reach for the time being.
that being said, I’m looking for a conceptual explanation for that these equations represent:
$$mu_{1|2} = mu_1 + Sigma_{1,2} * Sigma^{-1}_{2,2}(x_2 – mu_2)$$
I read the first as "Take $$mu1$$ and augment it by some factor, which is the covariance scaled by the precision (measure of how closely $$X_2$$ is clustered about $$mu_2$$, maybe?) and projected onto the distance of the specific $$x_2$$ from $$mu_2$$."
$$Sigma_{1|2} = Sigma_{1,1} – Sigma_{1,2} * Sigma^{-1}_{2,2} * Sigma_{1,2}$$
I read the second as, "take the variance about $$mu_1$$ and subtract some factor, which is covariance squared scaled by the precision about $$x_2$$."
In either case, the precision $$Sigma^{-1}_{2,2}$$ seems to be playing a really important role.
A few questions:
• Am I right to treat precision as a measure of how closely observations are clustered about the expectation?
• Why is the covariance squared in the latter equation? (Is there a geometric interpretation?) So far, I’ve been treating $$Sigma_{1,2} * Sigma^{-1}_{2,2}$$ as a ratio, (a/b), and so this ratio acts to scale the (second) $$Sigma_{1,2}$$, essentially accounting for/damping the effect of the covariance; I don’t know if this is valid.
• Anything else you’d like to add/clarify?
Get this bounty!!!
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# Squares and Roots I
Algebra Level 1
Having a number n, we can make this relation:
$$\frac { \sqrt { n } }{ n } =m\quad \Rightarrow \quad \frac { 1 }{ m } =\sqrt { n }$$
If n=2, what is the m value?
× |
Articles written in Bulletin of Materials Science
• Electrical characteristics of metal–insulator–semiconductor and metal–insulator–semiconductor–insulator–metal capacitors under different high-$k$ gate dielectrics investigated in the semi-classical and quantum mechanical models
In this paper the electrical characteristics of metal–insulator–semiconductor (MIS) and metal–insulator–semiconductor–insulator–metal (MISIM) capacitors with (100)-oriented p-type silicon as a substrate under different high-$k$ gate dielectrics (SiO$_2$, HfO$_2$, La$_2$O$_3$ and TiO$_2$) are investigated in the semi-classical and quantum mechanical models. We review the quantum correction in the inversion layer charge density for p-doped structures. The purpose of this paper is to point out the differences between the semi-classical and quantum mechanical charge descriptions at the insulator–semiconductor interface and the effect of the type of oxide and their position (gate oxide or buried oxide) in our structures. In particular, capacitance–voltage ($C–V$), relative position of the sub-band energies and their wavefunctions are studied to examine qualitatively and quantitatively the electron states and charging mechanisms in our devices. We find that parameters such as threshold voltage and device trans-conductance are enormously sensitive to the proper treatment of quantization effects.
• Structural and optical characteristics of silicon nanowires prepared by the Ag-assisted chemical etching method
In order to improve photovoltaic efficiency, researches have been carried out on silicon nanowires (SiNWs). In this article, we report a comparative study between silicon substrate (Si) and SiNWs developed by a metal-assisted chemical etching (Ag) method at different etching times (25, 10 and 5 min). Scanning electron microscopy (SEM), transmission electron microscopy and X-ray diffraction were used to collect the morphological and structural informationon the SiNWs. Raman spectroscopy shows that the intensity of the nanowires is 4 to 10 times higher than that of the substrate, and increases with increase in etching time. The total reflectance of SiNWs reduced to less than 5% over theentire visible range. The low reflectance and zero transmittance of SiNWs lead to higher absorbance in the visible wavelength range. The SiNW-etched nanowire structure (25 min) works best for capturing light, we believe that having longer nanowires improves the optical working of the nanostructures and may be a potential candidate for high efficiency photovoltaic solar cells and other optic devices.
• # Bulletin of Materials Science
Volume 45, 2022
All articles
Continuous Article Publishing mode
• # Dr Shanti Swarup Bhatnagar for Science and Technology
Posted on October 12, 2020
Prof. Subi Jacob George — Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bengaluru
Chemical Sciences 2020 |
# What was meant by the 'ponderomotive force' as understood by Minkowski?
Skimming through Minkowski's famous 1907 paper, he uses the term ponderomotive force.
What does he mean by this?
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He just means the Lorentz force on one in response to the field of the other. – Ron Maimon Sep 3 '11 at 2:36
@Ron: That's what I thought too, but Wikipedia has a description of ponderomotive force that's not identical to the Lorentz force. Edit: Although I now wonder if Minkowski really was referring to the Lorentz force and Ponderomotive force is not the correct translation of the German. Unfortunately I don't have Minkowski's paper. – twistor59 Sep 3 '11 at 7:18
@twistor59: I put the comment on after looking at both Wikipedia and Minkowski's paper. – Ron Maimon Sep 3 '11 at 19:07
Minkowski writes "ponderomotorische Kräfte", so there is no translation from German at all, its some kind of Latin in both cases. The gist of this new-latin "creation" is something like "force that causes movement of mass" BTW, Minknowski is a fine typo, one could call it a Freudian typo :=) – Georg Nov 2 '11 at 21:58
@Georg: The "pondermotive force" is explicitly stated to be what we call the Lorentz force today, and there are no two answers here. – Ron Maimon Nov 3 '11 at 16:50
He just means the Lorentz force. The Lorentz force is called the "pondermotive force" in his paper, for no good reason. Old papers did not have internet to standardize their terminology for them.
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Let's look at some clues as to what it probably meant at the time. The word is ponderomotive rather than pondermotive and is constructed like electromotive, magnetomotive, from ponder-o-motive. The [etymology][1] of ponder is given as
ponder early 14c., "to estimate the worth of, to appraise," from O.Fr. ponderare "to weigh, poise," from L. ponderare "to ponder, to consider," lit. "to weigh," from pondus (gen. ponderis) "weigh" (see pound (1)). Meaning "to weigh a matter mentally" is attested from late 14c.
Therefore as an initial guess, it could mean the line integral between two points of a force that acts upon substance to give it weight; perhaps the line integral of the Newtonian gravitational force?
Book Googling 'ponderomotive' turns up a quote from Energy and Empire: a biographical study of Lord Kelvin
what makes an electrified body move?
In May of 1843 Thomson published in the Cambridge Mathematical Journal a paper of a mere two pages which marks his earliest consideration of ponderomotive forces on electrified bodies. 'On the attractions of conducting and non-conducting electrified bodies' showed that, for a given distribution of electricity on the surface of a body A, the total moving force exerted on A by an arbitary electrical mass M is the same whether A be a conductor or non-conductor.
Hermann von Hermholtz and the foundations of nineteenth-centurey science by David Cahan
For he sought to orientate himself and others in the "pathless wilderness" of competing theories in electrodymanics around 1870; it was in this historical context that he promulgated his own contribution to the ongoing discussion about a fundamental potential for current elements. As already noted, those current potentials were mathematical tools used to derive further equations. Thus, the negative gradient of the potentials (the variation with repsect to changing position) furnished laws of ponderomotive forces, that is laws of mechanical forces between distant linear currents. The time derivative of the potentials furnished the electromotive force induced in systems of time-varint currents.
Page 11 of Eddington's Principle in the Philosophy of Science
In order to generate mechanical momentum, we usually need the action of a pondermotive force. Now a ponderomotive force of electromagnetic origin does act on conduction-current, but there is no conduc-tion-current in the free aether.
Page 165 of a 1922 Bulletin of the National Research Council By National Research Council (U.S.)
According to the Maxwell-Lorentz theory the fundamental equation for the calculation of all ponderomotive forces of electromagnetic origin is $f = q(E + \frac 1 c \vec v \times\vec H)$
So Minkowski meant the electromagnetic force on mass - the Lorentz force.
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Yes, this is correct, but it requires no long exegesis. – Ron Maimon May 2 '12 at 15:02
I like this answer, except for the last conclusion. In the last quotation, the Lorentz force is mentioned only as a means of explaining the ponderomotive force (which is macroscopic force on bulk) by some fundamental equation, and Minkowski says this too: – Ján Lalinský May 27 '14 at 18:23
Also the fundamental equations for electromagnetic processes in ponderable bodies are in accordance with the world-postulate throughout. I shall also show on a later occasion that even the deduction of these equations, as taught by Lorentz on the basis of the concepts of electron theory, are by no means to be given up. - en.wikisource.org/wiki/Translation:Space_and_Time#14 – Ján Lalinský May 27 '14 at 18:23
Nevertheless, however strange it seems, Minkowski did use the term "ponderomotive force" in the meaning of force acting on a charged particle, not force in bulk. Perhaps he did not see think the difference is important at the time, or who knows. – Ján Lalinský May 27 '14 at 18:27
Ron's responses are incorrect. You can see here that it was Boot and Harvie in 1957. In an inhomogenous plasma all particles regardless of charge will move toward the weaker field. This is much different from Lorentz forces, ie the motion of neutral particles do not generate a magnetic field.
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If you read the question, I do say Minkowski mentions it in 1907. – Physiks lover Mar 6 '13 at 19:00 |
### 3206
Prostate Cancer: Influence of the Diffusion Time on Diffusion Kurtosis Imaging
Tristan Anselm Kuder1, Frederik Bernd Laun2, David Bonekamp3, and Matthias Carl Röthke3,4
1Medical Physics in Radiology, German Cancer Research Center (DKFZ), Heidelberg, Germany, 2Institute of Radiology, University Hospital Erlangen, Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Erlangen, Germany, 3Radiology, German Cancer Research Center (DKFZ), Heidelberg, Germany, 4Conradia, Hamburg, Germany
### Synopsis
Diffusion MRI is routinely used in prostate cancer diagnosis. Diffusion kurtosis imaging allows measuring the kurtosis Kapp, related to deviations from free diffusion, additionally to the diffusion coefficient Dapp. Varying the diffusion time may yield additional information about the investigated tissue by probing the diffusion barriers at different length scales. Here, Dapp and Kapp were measured at three diffusion times in 27 patients with histologically confirmed prostate cancer. A reduction of Kapp was observed in tumor and normal control regions with increasing diffusion time, while a Dapp reduction was mostly seen in control regions.
### Introduction
Due to the reduction of the apparent diffusion coefficient (ADC) in tumor tissue, diffusion weighted imaging (DWI) is routinely applied for tumor detection, especially for prostate cancer [1-3]. However, tumor grading is most important for therapeutic decisions, which is not easily achievable using ADC measurements only. Therefore, it would be desirable to obtain additional parameters linked to tissue structure from diffusion measurements. Diffusion kurtosis imaging (DKI) is an approach in this regard, which measures the diffusion coefficient $D_\mathrm{app}$ and the apparent kurtosis $K_\mathrm{app}$ [4-8]. $K_\mathrm{app}$ quantifies the deviation from free Gaussian diffusion. The diffusion time $T$ is an additional experimental dimension [9]. Measuring $D_\mathrm{app}(T)$ and $K_\mathrm{app}(T)$ at varying diffusion time may yield additional information regarding the investigated tissue, since the typical length scale of the structures probed by the diffusing water molecules changes. In this work, the feasibility of measuring $D_\mathrm{app}(T)$ and $K_\mathrm{app}(T)$ for patients with histologically confirmed prostate cancer is demonstrated.
### Methods
This study was performed on a set of patients who received a diagnostic MRI which was extended by diffusion kurtosis measurements with three different diffusion times $T$ according to the institutional ethical guidelines. For the analysis, those patients were chosen, who exhibited prostate cancer confirmed by transperineal hybrid MR/ultrasound fusion image-guided biopsy in the areas previously identified as suspect on MRI by the reading radiologist. The 27 patients meeting these criteria were diagnosed with Gleason scores between 6 and 9. DKI measurements comprised three diffusion times, acquired using a spin echo (SE) EPI sequence with TE=70 ms and a stimulated echo (STEAM) EPI sequence with TE=30 ms and the mixing times TM=250 ms and TM=500 ms (3T, Siemens Magnetom Trio, body matrix coil). Additional sequence parameters: FOV 329 × 164 mm², matrix 100 × 50, slice thickness 3.3 mm, bandwidth 2632 Hz/pixel, b-values 50, 250, 500, 750, 1000, 1250, 1500, 2000 s/mm², three orthogonal diffusion directions. Additional parameters for the SE sequence: TR=2.7 s, 5 averages. For STEAM: TM=250 ms: TR=4.5 s, 4 averages; for TM=500 ms: TR=5.7 s, 4 averages. For the evaluation, regions of interest (ROIs) were placed in the histologically confirmed tumor regions as well as in normal tissue. $D_\mathrm{app}(T)$ and $K_\mathrm{app}(T)$ were calculated by fitting the equation $$S(b)=\sqrt{(S_0\exp(-bD_\mathrm{app}+b^2 D_\mathrm{app}^2 K_\mathrm{app} / 6))^2+\eta^2}$$ to the measured signal $S(b)$ with the noise level $\eta$ of the MR images [4,8].
### Results
Figure 1 shows exemplarily the influence of the different diffusion times on the fitted functions for ROI averaged signals for one patient. Differences in the slope for small b-values $(\leq500\,\mathrm{s/mm}^2)$ can be observed resulting in different $D_\mathrm{app}$ values. In the tumor region, only a different curvature is visible. Therefore, in this region, mostly variations in $K_\mathrm{app}$ are to be expected. In Fig. 2, $D_\mathrm{app}$ and $K_\mathrm{app}$ maps are depicted for one patient. A slight decrease of the measured diffusion coefficient $D_\mathrm{app}$ especially in areas with higher $D_\mathrm{app}$ values can be observed. A significant decrease of $K_\mathrm{app}$ with increasing diffusion time can be observed both in the tumor and the normal tissue. $D_\mathrm{app}$ and $K_\mathrm{app}$ values averaged over 27 patients are depicted in Fig. 3. For the control regions, a decrease of the averaged $D_\mathrm{app}$ values from 1.93 µm²/ms to 1.62 µm²/ms can be observed with increasing $T$; $K_\mathrm{app}$ decreases from 0.62 to 0.47. In tumor regions, $D_\mathrm{app}$ is mainly constant, while a decrease of the mean value of $K_\mathrm{app}$ from 1.05 to 0.77 was observed.
### Discussion
The larger packing density of diffusion restrictions in the tumor area may be a possible explanation for the lower $T$ dependence of $D_\mathrm{app}$ compared to the normal region. It may be assumed that the diffusion process is closer to the long-time limit in the tumor area resulting in smaller $T$ dependence. On the other hand, a substantial $T$ dependence of $K_\mathrm{app}$ was observed both in the normal and the tumor region, which may be interpreted as a sign of higher sensitivity of $K_\mathrm{app}$ to changing diffusion distance. Additionally, compartments with different relaxation times may contribute to the observed $T$ dependence. The general trend of decreasing diffusion coefficient was also observed using diffusion tensor imaging [9]. The larger $T$ dependence of the diffusion coefficient in tumor areas observed in [9] may be due to the use of different b-values and the fact that no kurtosis term was fitted.
### Conclusion
In this work, the possibility of measuring the diffusion time dependence of $D_\mathrm{app}(T)$ and $K_\mathrm{app}(T)$ in patients with prostate cancer was demonstrated. In future studies, a correlation with the Gleason score will be investigated with the possible aim to name an optimal $T$ for best separation. Furthermore, using similar $T$ is necessary for quantitative comparison of DKI-derived parameters from different sites or studies.
### Acknowledgements
No acknowledgement found.
### References
1. Woodfield CA, Tung GA, Grand DJ, Pezzullo JA, Machan JT, Renzulli JF, 2nd. Diffusion-weighted MRI of peripheral zone prostate cancer: comparison of tumor apparent diffusion coefficient with Gleason score and percentage of tumor on core biopsy. AJR Am J Roentgenol 2010;194(4):W316-322.
2. Kim CK, Park BK, Lee HM, Kwon GY. Value of diffusion-weighted imaging for the prediction of prostate cancer location at 3T using a phased-array coil: preliminary results. Invest Radiol 2007;42(12):842-847.
3. Gibbs P, Liney GP, Pickles MD, Zelhof B, Rodrigues G, Turnbull LW. Correlation of ADC and T2 measurements with cell density in prostate cancer at 3.0 Tesla. Invest Radiol 2009;44(9):572-576.
4. Jensen JH, Helpern JA, Ramani A, Lu H, Kaczynski K. Diffusional kurtosis imaging: the quantification of non-gaussian water diffusion by means of magnetic resonance imaging. Magn Reson Med 2005;53(6):1432-1440.
5. Rosenkrantz AB, Sigmund EE, Johnson G, Babb JS, Mussi TC, Melamed J, Taneja SS, Lee VS, Jensen JH. Prostate cancer: feasibility and preliminary experience of a diffusional kurtosis model for detection and assessment of aggressiveness of peripheral zone cancer. Radiology 2012;264(1):126-135.
6. Tamura C, Shinmoto H, Soga S, Okamura T, Sato H, Okuaki T, Pang Y, Kosuda S, Kaji T. Diffusion kurtosis imaging study of prostate cancer: preliminary findings. J Magn Reson Imaging 2014;40(3):723-729.
7. Quentin M, Pentang G, Schimmoller L, Kott O, Muller-Lutz A, Blondin D, Arsov C, Hiester A, Rabenalt R, Wittsack HJ. Feasibility of diffusional kurtosis tensor imaging in prostate MRI for the assessment of prostate cancer: preliminary results. Magn Reson Imaging 2014;32(7):880-885.
8. Roethke MC, Kuder TA, Kuru TH, Fenchel M, Hadaschik BA, Laun FB, Schlemmer HP, Stieltjes B. Evaluation of Diffusion Kurtosis Imaging Versus Standard Diffusion Imaging for Detection and Grading of Peripheral Zone Prostate Cancer. Invest Radiol 2015;50(8):483-489.
9. Lemberskiy G, Rosenkrantz AB, Veraart J, Taneja SS, Novikov DS, Fieremans E. Time-Dependent Diffusion in Prostate Cancer. Invest Radiol 2017;52(7):405-411.
### Figures
Figure 1: Exemplary signal curves in (a) a tumor region and (b) a normal control region for a patient with Gleason score 7 for the spin echo sequence (SE, TE=70ms) and the STEAM sequence for the two mixing times TM (dots: measurements, solid lines: fitted functions).
Figure 2: (a,b,c) Dapp and (d,e,f) Kapp maps for the same patient as in Fig. 1 for the three diffusion times. The decrease in Kapp with increasing T is most prominent in (d,e,f). The tumor region (arrows) exhibits low Dapp and high Kapp values.
Figure 3: Values averaged for 27 patients in tumor and control regions: Measured Dapp and Kapp values for the spin echo sequence (SE, TE=70 ms) and the STEAM sequence for the two mixing times TM. Error bars indicate the standard deviation of the 27 measurements.
Proc. Intl. Soc. Mag. Reson. Med. 26 (2018)
3206 |
# $S$-prime numbers Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource
## Suggestion
Let $S$ be the set of all positive integers that are $1$ more than a multiple of $10$, so $S = \{1, 11, 21, 31, 41, \dotsc\}$.
We say that an element $x$ of the set $S$ is $S$-prime if $x > 1$ and whenever the elements $a$ and $b$ of the set $S$ satisfy $ab = x$ we have $a = 1$ or $b = 1$.
Are there distinct $S$-prime numbers $a$, $b$, $c$ and $d$ such that $ab = cd$?
We’ve been given a new definition (of $S$-prime numbers), so it would be a good idea to try to understand that properly before trying to tackle the question. Can you come up with some examples of numbers that are $S$-prime, and also some examples of numbers that are not $S$-prime?
For example, $21$ is $S$-prime, because if $ab = 21$ and $a$ and $b$ are both $1$ more than a multiple of $10$, then $a = 1$ or $b = 1$. But $121$ is not $S$-prime, because $121 = 11 \times 11$ and $11$ is a member of $S$ but does not equal $1$. |
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# The coordinates ofi as point P are (1,2,3). Find the coordinates fothe seven pints such that the absolute vaues of their coordinates are the same as those of coordinates of P.
Updated On: 27-06-2022 |
Articles written in Pramana – Journal of Physics
• Exact travelling solutions for some nonlinear physical models by ($G'/G$)-expansion method
In this paper, we establish exact solutions for some special nonlinear partial differential equations. The ($G'/G$)-expansion method is used to construct travelling wave solutions of the twodimensional sine-Gordon equation, Dodd–Bullough–Mikhailov and Schrödinger–KdV equations, which appear in many fields such as, solid-state physics, nonlinear optics, fluid dynamics, fluid flow, quantum field theory, electromagnetic waves and so on. In this method we take the advantage of general solutions of second-order linear ordinary differential equation (LODE) to solve many nonlinear evolution equations effectively. The ($G'/G$)-expansion method is direct, concise and elementary and can be used with a wider applicability for handling many nonlinear wave equations.
• # Pramana – Journal of Physics
Volume 96, 2022
All articles
Continuous Article Publishing mode
• # Editorial Note on Continuous Article Publication
Posted on July 25, 2019 |
There is a theorem in Murphy's book on operator theory and $C^\ast$-algebras:
Let $u$ be a unitary element in a unital $C^\ast$-algebra $A$. Then if $\sigma(u) \subsetneq S^1$ then there exists a self-adjoint element $a\in A$ such that $u = e^{ia}$.
This theorem comes after a discussion of some properties of $C^\ast$-algebras and the Gelfand representaiton theorem. The theorem is followed by a proof of the existence of a functional calculus at a normal element $a$. The theorem does not appear to be used in any of the two proofs of the theorems that follow it.
I don't understand where this theorem fits into the theory: what is it used for? Why does it appear in a seemingly random place with no relation to adjacent theorems and proofs in the book?
I understand that it gives a sufficient condition for a unitary element to have a logarithm. I also understand its proof. I don't know anything about functional calculus so perhaps this is a very important theorem in functinoal calculus. But if it is this is not mentioned in the book and I'd be very grateful for context!
• It seems to be a variation on Stone's theorem on one-parameter families of unitaries, which is important in the mathematical study of quantum mechanics. (Here $u$ is the time evolution of a system after a short length of time, say, and $a$ is a multiple of the Hamiltonian of the system.) – Qiaochu Yuan Nov 1 '14 at 5:33
• Thank you for your comment. I am still hoping for a connection to functional calculus because it seems to be used in star algebra theory. – user167889 Nov 1 '14 at 7:26
As far as I can tell, the theorem is there because it allows Murphy to show an application of the Gelfand transform (2.1.10), and because it has to be somewhere in the book.
He later uses the theorem a couple times (in the proof of 7.3.2 and in the proof of 7.5.6).
• Thank you! So this theorem has nothing to do with functional calculus? I looked at the two theorems in chapter 7 and they seem to be unrelated to functional calculus. – user167889 Nov 1 '14 at 23:00
• I guess it depends on what you call "functional calculus"; the Gelfand transform is functional calculus. – Martin Argerami Nov 2 '14 at 0:44
• You seem to know the book really well. Do you think you could point me to where the theorem 2.1.13. is used later? It would help me understand what functional calculus is. – user167889 Nov 2 '14 at 4:16
• He uses it for example in 2.3.3, 2.5.6, 3.5.1, 7.2.3. Functional calculus is a way to evaluate functions (continuous, in this context; although later Murphy considers the Borel functional calculus) on a normal operator. Theorem 2.1.13 is the key because it assigns an operator in $C^*(a)$, in a natural way, to each $f\in C(\sigma(a))$. We denote this operator by $f(a)$. – Martin Argerami Nov 2 '14 at 15:17
• Of course. But $\varphi$ is indeed continuous, as any $*$-homomorphism is. – Martin Argerami Nov 3 '14 at 5:16 |
# #
Refer to the following figure.
Figure: Position vs. time graph for the following two questions.
## Part 1#
At $$t=$$ $$s$$, what is the x-component of the instantaneous velocity of the object whose position vs. time graph is shown in the figure?
For the object whose position vs time graph is shown in the figure above, the x-component of the average velocity ($$v\_{avg,x ; 0 \rightarrow 3}$$) and average speed ($$v\_{avg ; 0 \rightarrow 3}$$) over the time interval t=0s to t=3s are: |