text:
# 7.5 Electric forces in biology

 Page 1 / 5
• Describe how a water molecule is polar.
• Explain electrostatic screening by a water molecule within a living cell.

Classical electrostatics has an important role to play in modern molecular biology. Large molecules such as proteins, nucleic acids, and so on—so important to life—are usually electrically charged. DNA itself is highly charged; it is the electrostatic force that not only holds the molecule together but gives the molecule structure and strength. [link] is a schematic of the DNA double helix.

The four nucleotide bases are given the symbols A (adenine), C (cytosine), G (guanine), and T (thymine). The order of the four bases varies in each strand, but the pairing between bases is always the same. C and G are always paired and A and T are always paired, which helps to preserve the order of bases in cell division (mitosis) so as to pass on the correct genetic information. Since the Coulomb force drops with distance ( $F\propto 1/{r}^{2}$ ), the distances between the base pairs must be small enough that the electrostatic force is sufficient to hold them together.

DNA is a highly charged molecule, with about $2{q}_{\text{e}}$ (fundamental charge) per $0\text{.}3×{\text{10}}^{-9}$ m. The distance separating the two strands that make up the DNA structure is about 1 nm, while the distance separating the individual atoms within each base is about 0.3 nm.

One might wonder why electrostatic forces do not play a larger role in biology than they do if we have so many charged molecules. The reason is that the electrostatic force is “diluted” due to screening    between molecules. This is due to the presence of other charges in the cell.

## Polarity of water molecules

The best example of this charge screening is the water molecule, represented as ${\text{H}}_{2}\text{O}$ . Water is a strongly polar molecule    . Its 10 electrons (8 from the oxygen atom and 2 from the two hydrogen atoms) tend to remain closer to the oxygen nucleus than the hydrogen nuclei. This creates two centers of equal and opposite charges—what is called a dipole    , as illustrated in [link] . The magnitude of the dipole is called the dipole moment.

These two centers of charge will terminate some of the electric field lines coming from a free charge, as on a DNA molecule. This results in a reduction in the strength of the Coulomb interaction    . One might say that screening makes the Coulomb force a short range force rather than long range.

Other ions of importance in biology that can reduce or screen Coulomb interactions are ${\text{Na}}^{+}\text{,}$ and ${\text{K}}^{+}\text{,}$ and ${\text{Cl}}^{–}$ . These ions are located both inside and outside of living cells. The movement of these ions through cell membranes is crucial to the motion of nerve impulses through nerve axons.

Recent studies of electrostatics in biology seem to show that electric fields in cells can be extended over larger distances, in spite of screening, by “microtubules” within the cell. These microtubules are hollow tubes composed of proteins that guide the movement of chromosomes when cells divide, the motion of other organisms within the cell, and provide mechanisms for motion of some cells (as motors).

## Section summary

• Many molecules in living organisms, such as DNA, carry a charge.
• An uneven distribution of the positive and negative charges within a polar molecule produces a dipole.
• The effect of a Coulomb field generated by a charged object may be reduced or blocked by other nearby charged objects.
• Biological systems contain water, and because water molecules are polar, they have a strong effect on other molecules in living systems.

## Conceptual question

A cell membrane is a thin layer enveloping a cell. The thickness of the membrane is much less than the size of the cell. In a static situation the membrane has a charge distribution of $-2\text{.}5×{\text{10}}^{-6}$ C/m 2 on its inner surface and $+2\text{.}5×{\text{10}}^{-6}$ C/m 2 on its outer surface. Draw a diagram of the cell and the surrounding cell membrane. Include on this diagram the charge distribution and the corresponding electric field. Is there any electric field inside the cell? Is there any electric field outside the cell?

how can chip be made from sand
is this allso about nanoscale material
Almas
are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where is the latest information on a no technology how can I find it
William
currently
William
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
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Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
how did you get the value of 2000N.What calculations are needed to arrive at it
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# Good set of PS 2

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Senior Manager
Joined: 31 Aug 2009
Posts: 386
Location: Sydney, Australia
Re: Good set of PS 2  [#permalink]

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17 Oct 2009, 07:50
Economist wrote:
yangsta8 wrote:
Bunuel wrote:
7. If x is positive, which of the following could be the correct ordering of 1/x,2x and x^2 ?
I. x^2<2x<1/x
II. x^2<1/x<2x
III. 2x<x^2<1/x

(A) None
(B) I only
(C) III only
(D) I and II only
(E) I II and III

Two situations we need to consider:
0 < x < 1, or x > 1 (since squaring a number for the first makes the number smaller for the former and larger for the latter).

Situation 1: let X = 1/3
x^2 = 1/9
2x = 2/3
1/x = 3
Correct ordering: x^2 < 2x < 1/x
Option I is the only possibility. You could choose B based on this.

Check Situation 2: Let X = 3
x^2 = 9
2x = 6
1/x = 1/3
Correct ordering 1/x < 2x < x^2
Not in any of the possibilities.

ANS = B

I missed the word 'could' ..agree with B..

Hmmm apparrently B is still not correct, guess we can wait for OA. I couldn't find any other possibilities. I tested fractions, integers and then numbers which turn out unique properties such as 1 and 2 and still can't find one.
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Re: Good set of PS 2  [#permalink]

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17 Oct 2009, 07:52
1
2
Economist wrote:
yangsta8 wrote:
Bunuel wrote:
10. What is the lowest positive integer that is divisible by each of the integers 1 through 7, inclusive?
(A) 420
(B) 840
(C) 1260
(D) 2520
(E) 5040

Q10)
1 x 2 x 3 x 4 x 5 x 6 x 7 = 2^4 x 3^2 x 5 x 7
Option A = 420 = 2^2 x 3 x 5 x 7 (factored this out)
Option B = 2 x 420 = 2^3 x 3 x 5 x 7 (just add a 2 to A)
Option C = 3 x 420 = 2^2 x 3^2 x 5 x 7 (just add a 3 to A)
Option D = 2 x 1260 = 2^3 x 3^2 x 5 x 7 (just add a 2 to C)
Option E = 2 x 2520 = 2^4 x 3^2 x 5 x 7 (just add a 2 to D)

Actually after doing up to A you can quickly just figure out how many more 2's and 3's and determine it's E.

What does the ques ask...which of the following is divisible by EACH of the integers from 1 thru 7...so I interpret it as which is the option that is divisible by 1 and 2 and 3...It does not ask : each of the following is divisible by the product of each of the integers from 1 thru 7 ! or am I missing something badly !!

Your answer as well as Asterixmatrix's are correct according to the wording of the question.
I just assumed it would be harder, since Bunuel posts tough questions haha. Let's see what the OA says.
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Re: Good set of PS 2  [#permalink]

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17 Oct 2009, 09:42
1
1
Heres are my answers:

1. A family consisting of one mother, one father, two daughters and a son is taking a road trip in a sedan. The sedan has two front seats and three back seats. If one of the parents must drive and the two daughters refuse to sit next to each other, how many possible seating arrangements are there?
(B) 32 -
Total ways to seat them with one parent driving: 2 options for driver * 4 options for other seats = 2*4*3*2*1 = 48.
Assuming the 2 sisters do sit together, front seat we have 2 options for driver and 2 options for passenger (the other parent + son) = 4 ways.
Back seats: sisters sit together so either they sit in seats 1 and 2 or seats 2 and 3 = 2 ways. Also sisters themselves can sit in 2 ways so total = 2*2 = 4. So total ways sisters can sit together = 4*4 =16.
So ways they wont sit together = 48-16 = 32 ways.

2. What is the probability that a 3-digit positive integer picked at random will have one or more "7" in its digits?
(C) 7/25 - Total #'s between 100 and 999 inclusive = 9*10*10 = 900. # of numbers with no digit 7 = 8*9*9 = 648. So #'s with a 7 = 1 - 648/900 = 252/900 = 7/25.

3. A sphere is inscribed in a cube with an edge of 10. What is the shortest possible distance from one of the vertices of the cube to the surface of the sphere?
(D) 5( sqrt3 - 1) - Length of the diagonal of cube = ((10^2+10^2) + 10^2)^(1/2) = 10 sqrt3. Diagonal includes diameter of circle (10) + 2*length from vertex to sphere.
Length of vertex to sphere = (10*sqrt3- 10)/2 = 5( sqrt3 - 1)

4. A contractor estimated that his 10-man crew could complete the construction in 110 days if there was no rain. (Assume the crew does not work on any rainy day and rain is the only factor that can deter the crew from working). However, on the 61-st day, after 5 days of rain, he hired 6 more people and finished the project early. If the job was done in 100 days, how many days after day 60 had rain?
(C) 6 - rains for 5 days from day 56-60. So 10 guys worked for 55 days and accomplished half of the work. If 6 more guys are added to the job then the rate is 16/1100. (since one man's rate is 1/1100). Half the job left means 550/1100 is left. Therefore 550/16 = 34.375 days of more work. Since there were 40 days between day 60 and job completion, it must've rained for 40-34.375 = 5.625 or ~6 days. (I'm not sure if this is correct)

5. If s and t are positive integer such that s/t=64.12, which of the following could be the remainder when s is divided by t?
(E) 45 - 64.12 = 6412/100 or 1603/25. 1603/25 gives a remainder of 3, 3206/50 gives remainder of 6 and so on ..pattern = factors of 3. so to get remainder of 45, we multiply everything by 15: 1603*15/(25*15) = 24045/375.

6. A committee of 6 is chosen from 8 men and 5 women so as to contain at least 2 men and 3 women. How many different committees could be formed if two of the men refuse to serve together?
(E) 635 - 2 options: 2 men and 4 women -(i), or 3 men and 3 women -(ii).
Starting with women: for (i): 5C4=5 and for (ii): 5C3 = 10.
Men for (i): 2 further options if 2 men - a) neither of the problematic men chosen = 6C2 = 15 or b) one of them is chose - 2C1*6C1 = 12 so 12+15= 27 ways if (i)
(ii): same 2 options - 6C3 + 2C1*6C2 = 20 + 30 = 50.
Combining men and women - (i): 5*27 = 135 and (ii): 10*50 = 500 so total ways = 635.

7. If x is positive, which of the following could be the correct ordering of 1/x,2x and x^2 ?
I. x^2<2x<1/x
II. x^2<1/x<2x
III. 2x<x^2<1/x
(D) I and II only - if x = 1/3, then I is correct, if x = 4/5, then II is correct. I couldnt find a way to get III to work.

8. In the xy plane, Line k has a positive slope and x-intercept 4. If the area of the triangle formed by line k and the two axes is 12, What is the y-intercept of line K ?
(D) -6 - Area of bxh of triangle = 24. Given base is 4, and drawing the line, easy to see that y intercept has to be -6.

9. Of the applicants passes a certain test, 15 applied to both college X and Y. If 20 % of the applicants who applied college X and 25% of the applicants who applied college Y applied both college X and Y, how many applicants applied only college X or college Y?
(D) 105 - Let x be # who applied to only X and y who applied to only Y. Then 0.2(x+15) = 15 and 0.25(y+15) = 15. So x = 60 and y = 45 so x+y = 105.

10. What is the lowest positive integer that is divisible by each of the integers 1 through 7, inclusive?
(A) 420 - 420 is divisible by all #'s from 1 thru 7.
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Re: Good set of PS 2  [#permalink]

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17 Oct 2009, 20:02
Bunuel wrote:
ANSWERS:

1. A family consisting of one mother, one father, two daughters and a son is taking a road trip in a sedan. The sedan has two front seats and three back seats. If one of the parents must drive and the two daughters refuse to sit next to each other, how many possible seating arrangements are there?
(A) 28
(B) 32
(C) 48
(D) 60
(E) 120

As most of the combination problems this one can be solved in more than 1 way:

Sisters sit separately:
1. one of them is on the front seat (2 ways). Others (including second sister) can be arranged in: 2 (drivers seat)*3! (arrangements of three on the back seat)=12 ways. Total for this case: 2*12=24
Or
2. both by the window on the back seat (2 ways). Others can be arranged in: 2 (drivers seat)*2 (front seat)*1(one left to sit between the sisters on the back seat)=4 ways. Total for this case=8.
Total=24+8=32.

Answer: E.

In the second case, we have 1 P in the front seat, so the other can be occupied by the Son, no one else...so we have 2 ways to arrange people in front seats, and 2 ways to arrange backseaters ( with 2 daughters on window seats )...so total there are 2*2 ways...hence I think ans should be 28 and not 32.
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Re: Good set of PS 2  [#permalink]

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17 Oct 2009, 20:28
Economist wrote:
Bunuel wrote:
ANSWERS:

1. A family consisting of one mother, one father, two daughters and a son is taking a road trip in a sedan. The sedan has two front seats and three back seats. If one of the parents must drive and the two daughters refuse to sit next to each other, how many possible seating arrangements are there?
(A) 28
(B) 32
(C) 48
(D) 60
(E) 120

As most of the combination problems this one can be solved in more than 1 way:

Sisters sit separately:
1. one of them is on the front seat (2 ways). Others (including second sister) can be arranged in: 2 (drivers seat)*3! (arrangements of three on the back seat)=12 ways. Total for this case: 2*12=24
Or
2. both by the window on the back seat (2 ways). Others can be arranged in: 2 (drivers seat)*2 (front seat)*1(one left to sit between the sisters on the back seat)=4 ways. Total for this case=8.
Total=24+8=32.

Answer: E.

In the second case, we have 1 P in the front seat, so the other can be occupied by the Son, no one else...so we have 2 ways to arrange people in front seats, and 2 ways to arrange backseaters ( with 2 daughters on window seats )...so total there are 2*2 ways...hence I think ans should be 28 and not 32.

If I understood correctly: you are talking about the case when the sisters are sitting on the back seat by the window?
We have two front seats (one is driver's seat)
And we have three back seats.
Consider this:
1. 2 sisters by the windows can be arranged 2!=2 ways;
2. Drivers seat either mother or father=2 ways;
3. Second front seat either the son or the parent which is not driving=2 ways
4. Only 1 way (option) will be left between the sisters (either son, or the parent who is not driving, but only one option)=1

So, 2*2*2=8

24+8=32.

Well again if I understood your point correctly. Please let me know.
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Re: Good set of PS 2  [#permalink]

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19 Oct 2009, 09:02
3
Bunuel wrote:
ANSWERS:

4. A contractor estimated that his 10-man crew could complete the construction in 110 days if there was no rain. (Assume the crew does not work on any rainy day and rain is the only factor that can deter the crew from working). However, on the 61-st day, after 5 days of rain, he hired 6 more people and finished the project early. If the job was done in 100 days, how many days after day 60 had rain?
(A) 4
(B) 5
(C) 6
(D) 7
(E) 8

This one was solved incorrectly:
Days to finish the job for 10 people 110 days.
On the 61-st day, after 5 days of rain --> 5 days was rain, 55 days they worked, thus completed 1/2 of the job, 1/2 is left (55 days of work for 10 people).
Then 6 more people was hired --> speed of construction increased by 1.6, days needed to finish 55/1.6=34.375, BUT after they were hired job was done in 100-60=40 days --> so 5 days rained. They needed MORE than 34 days to finish the job, so if it rained for 6 days they wouldn't be able to finish the job in 100(40) days.

Answer: B.

I solved in a more easier way I think:

1) 10 man 110 days --> need for 1100 man.days
2) 55 days with 10 men --> 550 man.days
3) 40 days with 16 men --> 640 man.days

--> total man.days equals 1190 vs need for 1100 --> days of rain equals 90/16 max --> 5.625 --> rounded to 5
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Re: Good set of PS 2  [#permalink]

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19 Oct 2009, 11:23
pleonasm wrote:
1. A family consisting of one mother, one father, two daughters and a son is taking a road trip in a sedan. The sedan has two front seats and three back seats. If one of the parents must drive and the two daughters refuse to sit next to each other, how many possible seating arrangements are there?
(A) 28
(B) 32
(C) 48
(D) 60
(E) 120

Driver 1 can be taken in 2 ways ( M & F )
Front seat can be taken in 5 ways ( M, F , D1, D2 and S )
The last 3 seats can be taken in 6 ways :

D1 S D2
D2 S D1
F S D2
F S D1
M S D2
M S D1

Total = 5*2*6 = 60 ways.

Not very confident, I could be wrong.

Edit:
Front seat can be taken in 4 ways ( M or F , D1, D2 and S ) Ans is 4*2*6 = 48

I got a slightly different answer, here was my approach -
I split the scenario into two cases -
Case 1 - One of the daughters takes the front passenger seat -
Drivers seat can be occupied in 2 ways (M or F)
AND Front passenger seat can be occupied in 2 ways (D1 or D2)
AND the other three can sit in the back seat in any order ie !3 = 6
Or
Case 2 - One of the daughters doesn't take the front passenger seat
Drivers seat can be occupied in 2 ways (M or F)
AND Front passenger seat can be occupied in 2 ways (One of the parent or the Son)
AND in the back row the daughters occupy the window seats - 2 ways (the middle seat is occupied by the remaining person i.e. one of the parents or the son so only 1 way to do this).

Answer = 2*2*6+2*2*2= 32 hence B.
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Re: Good set of PS 2  [#permalink]

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19 Nov 2009, 07:08
A family consisting of one mother, one father, two daughters and a son is taking a road trip in a sedan. The sedan has two front seats and three back seats. If one of the parents must drive and the two daughters refuse to sit next to each other, how many possible seating arrangements are there?
28
32
48
60
120
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Posts: 50016
Re: Good set of PS 2  [#permalink]

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19 Nov 2009, 07:23
kairoshan wrote:
A family consisting of one mother, one father, two daughters and a son is taking a road trip in a sedan. The sedan has two front seats and three back seats. If one of the parents must drive and the two daughters refuse to sit next to each other, how many possible seating arrangements are there?
28
32
48
60
120

Refer to the question #1 in the set.

Sisters sit separately:
1. one of them is on the front seat (2 ways). Others (including second sister) can be arranged in: 2 (drivers seat)*3! (arrangements of three on the back seat)=12 ways. Total for this case: 2*12=24
Or
2. both by the window on the back seat (2 ways). Others can be arranged in: 2 (drivers seat)*2 (front seat)*1(one left to sit between the sisters on the back seat)=4 ways. Total for this case=8.
Total=24+8=32.

Another way: Total number of arrangements-arrangements with sisters sitting together=2*4*3!-2*2(sisters together)*2*2*1(arrangement of others)=48-16=32

Answer: B.
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Re: Good set of PS 2  [#permalink]

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21 Dec 2009, 07:40
whuannou wrote:
1A
Combination with one or the other of the daughters in the non-driver front seat
2 * 2 * 3! = 24
With no daughter in front seat
2 * 2 * 1(they cannot sit together so only one possibility) = 4
Total gives 28

2C
= 1 - Proba it has no digit 7 at all
Total nber of 3 digits number = 900
nbers with no 7 at all = 8*9*9= 648
yields (900-648)/900 = 252/ 900 = 7/25

3E
Draw a square with side 2 and inscribe in a circle of radius 1
Then using pythagore theorem twice and a substraction, compute the distance that goes tangently from one vertex to the circle

4E
5 I did not really understand the question wording
6E
7B
8D
9B
10A

For the rest I will type in my explanations later.
Thanks for the question sir.

The OA's are given on page 2. Some of your answers are not correct. Please refer to the solutions given along with OA's. Please ask if any question remains.

bidishabarpujari wrote:
for Q10 i am also confused! so far as i can understand, its asking for the lowest possible integer divided by each integer from 1 through 7.( i think it not asking for an integer divided by 7!). Bunuel, please clarify this.

You are right it's not asking about the integer divisible by 7!. Refer to the solution on the page 2. The answer is 420, which is LCM of the integers from 1 to 7.
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Re: Good set of PS 2  [#permalink]

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21 Dec 2009, 09:11
hi....
OA given for q7 is D..... however i feel it should be E....
III. 2x<x^2<1/x ......take x=1.1.....subst in equality 2.2<1.21<0.9... so III is also correct ....
it is just given x is +ive and not +ive int so x can be taken 1.1....
any one finds any objn ,pl let me know...
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Re: Good set of PS 2  [#permalink]

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21 Dec 2009, 09:30
chetan2u wrote:
hi....
OA given for q7 is D..... however i feel it should be E....
III. 2x<x^2<1/x ......take x=1.1.....subst in equality 2.2<1.21<0.9... so III is also correct ....
it is just given x is +ive and not +ive int so x can be taken 1.1....
any one finds any objn ,pl let me know...

How is the red part correct?

Anyway solution for this question:

7. If x is positive, which of the following could be the correct ordering of 1/x,2x and x^2 ?
I. x^2<2x<1/x
II. x^2<1/x<2x
III. 2x<x^2<1/x

(A) None
(B) I only
(C) III only
(D) I and II only
(E) I II and III

First note that we are asked "which of the following COULD be the correct ordering" not MUST be.
Basically we should determine relationship between $$x$$, $$\frac{1}{x}$$ and $$x^2$$ in three areas: $$0<1<2<$$.

$$x>2$$

$$1<x<2$$

$$0<x<1$$

When $$x>2$$ --> $$x^2$$ is the greatest and no option is offering this, so we know that x<2.
If $$1<x<2$$ --> $$2x$$ is greatest then comes $$x^2$$ and no option is offering this.

So, we are left with $$0<x<1$$:
In this case $$x^2$$ is least value, so we are left with:

I. $$x^2<2x<\frac{1}{x}$$ --> can $$2x<\frac{1}{x}$$? Can $$\frac{2x^2-1}{x}<0$$, the expression $$2x^2-1$$ can be negative or positive for $$0<x<1$$. (You can check it either algebraically or by picking numbers)

II. $$x^2<\frac{1}{x}<2x$$ --> can $$\frac{1}{x}<2x$$? The same here $$\frac{2x^2-1}{x}>0$$, the expression $$2x^2-1$$ can be negative or positive for $$0<x<1$$. (You can check it either algebraically or by picking numbers)

Answer: D.
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Re: Good set of PS 2  [#permalink]

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31 Dec 2009, 06:37
2. What is the probability that a 3-digit positive integer picked at random will have one or more "7" in its digits?
(A) 271/900
(B) 27/100
(C) 7/25
(D) 1/9
(E) 1/10

Total 3 digit numbers 900, 3 digit number with no 7 =8*9*9=648, P(at least one 7)=1-P(no 7)=1-648/900=252/900=7/25

Answer: C.

Dear Bunuel!

Could you please explain hod did you get: 3 digit number with no 7=8*9*9- where it coms from?

Thank you in advance
Math Expert
Joined: 02 Sep 2009
Posts: 50016
Re: Good set of PS 2  [#permalink]

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31 Dec 2009, 08:04
1
sher1978 wrote:
2. What is the probability that a 3-digit positive integer picked at random will have one or more "7" in its digits?
(A) 271/900
(B) 27/100
(C) 7/25
(D) 1/9
(E) 1/10

Total 3 digit numbers 900, 3 digit number with no 7 =8*9*9=648, P(at least one 7)=1-P(no 7)=1-648/900=252/900=7/25

Answer: C.

Dear Bunuel!

Could you please explain hod did you get: 3 digit number with no 7=8*9*9- where it coms from?

Thank you in advance

3 digit number with no 7, I mean without 7 = 8*9*9 = 648:

First digit can take 8 values from 1 to 9 excluding 7 (1xx, 2xx, ... 9xx, but not 7xx);
Second and third digits can take 9 values from 0 to 9 excluding 7 (eg. for second digit: x0x, x1x, ... x9x but not x7x).

Hope it's clear.
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Posts: 27
Re: Good set of PS 2  [#permalink]

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23 Feb 2010, 02:39
Bunuel wrote:
7. If x is positive, which of the following could be the correct ordering of 1/x,2x and x^2 ?
I. x^2<2x<1/x
II. x^2<1/x<2x
III. 2x<x^2<1/x

(A) None
(B) I only
(C) III only
(D) I and II only
(E) I II and III

First note that we are asked "which of the following COULD be the correct ordering" not MUST be.
Basically we should determine relationship between x, 1/x and x^2 in three areas: 0<1<2<.
x>2
1<x<2
0<x<1

When x>2 --> x^2 is the greatest and no option is offering this, so we know that x<2.
If 1<x<2 --> 2x is greatest than comes x^2 and no option is offering this.

So, we are left with 0<x<1:
In this case x^2 is least value, so we are left with:
I. x^2<2x<1/x --> can 2x<1/x? Can (2x^2-1)/x<0, the expression 2x^2-1 can be negative or positive for 0<x<1. (You can check it either algebraically or by picking numbers)
II. x^2<1/x<2x --> can 1/x<2x? The same here (2x^2-1)/x>0, the expression 2x^2-1 can be negative or positive for 0<x<1. (You can check it either algebraically or by picking numbers)

Answer: D.

@Bunuel

I tried number plugging but couldn't find values that satisfy second condition. Can you help identifying such numbers.

Also you can show how second is correct.
Math Expert
Joined: 02 Sep 2009
Posts: 50016
Re: Good set of PS 2  [#permalink]

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23 Feb 2010, 09:33
honeyrai wrote:
@Bunuel

I tried number plugging but couldn't find values that satisfy second condition. Can you help identifying such numbers.

Also you can show how second is correct.

Second condition: $$x^2<\frac{1}{x}<2x$$

Put $$0.9$$ --> $$x^2=0.81$$, $$\frac{1}{x}=1.11$$, $$2x=1.8$$ --> $$0.81<1.11<1.8$$. Hence this COULD be the correct ordering.

Basically as you can see this in the solution that second condition boils down to: can $$\frac{2x^2-1}{x}>0$$ be true for the range $$0<x<1$$? As x is positive: can $$2x^2-1>0$$ be true for the range $$0<x<1$$? For $$x>\frac{1}{\sqrt{2}}=0.71$$ it will be true and for $$0<x<\frac{1}{\sqrt{2}}$$ it won't. And vise-versa for the first condition.

Hope it's clear.
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Re: Good set of PS 2  [#permalink]

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07 May 2010, 01:49
5. If s and t are positive integer such that s/t=64.12, which of the following could be the remainder when s is divided by t?
(A) 2
(B) 4
(C) 8
(D) 20
(E) 45

I came across this problem somewhere in this club before but can't exactly remember where. ...but there was some sort of formula(trick). Does anyone have any idea????

6. A committee of 6 is chosen from 8 men and 5 women so as to contain at least 2 men and 3 women. How many different committees could be formed if two of the men refuse to serve together?
(A) 3510
(B) 2620
(C) 1404
(D) 700
(E) 635

Please explain as to why we are subtracting one and six.
Math Expert
Joined: 02 Sep 2009
Posts: 50016
Re: Good set of PS 2  [#permalink]

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07 May 2010, 03:36
bibha wrote:
5. If s and t are positive integer such that s/t=64.12, which of the following could be the remainder when s is divided by t?
(A) 2
(B) 4
(C) 8
(D) 20
(E) 45

I came across this problem somewhere in this club before but can't exactly remember where. ...but there was some sort of formula(trick). Does anyone have any idea????

6. A committee of 6 is chosen from 8 men and 5 women so as to contain at least 2 men and 3 women. How many different committees could be formed if two of the men refuse to serve together?
(A) 3510
(B) 2620
(C) 1404
(D) 700
(E) 635

Please explain as to why we are subtracting one and six.

OA's and solutions to every problem are given in my post on the second page.

5. If s and t are positive integer such that s/t=64.12, which of the following could be the remainder when s is divided by t?
(A) 2
(B) 4
(C) 8
(D) 20
(E) 45

We know that when we are told that "s divided by t gives remainder r" can be expressed by the following formula: $$s=qt+r$$, in or case $$q=64$$, --> $$s=64t+r$$, divide both parts by $$t$$ --> $$\frac{s}{t}=64.12=64+\frac{r}{t}$$ --> $$0.12=\frac{r}{t}$$--> $$\frac{3}{25}=\frac{r}{t}$$ so $$r$$ must be the multiple of 3. Only answer multiple of 3 is 45.

Or: $$\frac{s}{t}=64\frac{12}{100}=64\frac{3}{25}$$, so if the divisor=t=25 then the remainder=r=3. Basically we get that divisor is a multiple of 25 and the remainder is a multiple of 3. Only answer multiple of 3 is 45.

Answer: E.

6. A committee of 6 is chosen from 8 men and 5 women so as to contain at least 2 men and 3 women. How many different committees could be formed if two of the men refuse to serve together?
(A) 3510
(B) 2620
(C) 1404
(D) 700
(E) 635

Committee can have either: 2 men and 4 women OR 3 men and 3 women (to meet the condition of at least 2 men and 3 women).

Ways to chose 6 members committee without restriction (two men refuse to server together): $$C^2_8*C^4_5+C^3_8*C^3_5 = 700$$

Ways to chose 6 members committee with two particular men serve together: $$C^2_2*C^4_5+C2^_2*C^1_6*C^3_5=5+60=65$$

700-65 = 635

Answer: E.
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Posts: 84
Re: Good set of PS 2  [#permalink]

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10 Jun 2010, 08:12
for q 9...
20%of those who applied X = 15 (because they applied for y also)
x/5 =15
x = 75
25%of those who applied Y = 15 (because they applied for X also)
y/4 = 15
y = 60

total who applied for x and y should be x+y = 135

asterixmatrix wrote:
yangsta8 wrote:
Bunuel wrote:
9. Of the applicants passes a certain test, 15 applied to both college X and Y. If 20 % of the applicants who applied college X and 25% of the applicants who applied college Y applied both college X and Y, how many applicants applied only college X or college Y?
(A) 135
(B) 120
(C) 115
(D) 105
(E) 90

10. What is the lowest positive integer that is divisible by each of the integers 1 through 7, inclusive?
(A) 420
(B) 840
(C) 1260
(D) 2520
(E) 5040

Q9)
20% of total applied at X = 15
100% of total applied at X = 75
Only applied at X = 60

25% of total applied at Y = 15
100% of total applied at Y = 60
Only applied at Y = 45

Only applied at X + Only applied at Y = 60 + 45 = 105
ANS = D

Q10)
1 x 2 x 3 x 4 x 5 x 6 x 7 = 2^4 x 3^2 x 5 x 7
Option A = 420 = 2^2 x 3 x 5 x 7 (factored this out)
Option B = 2 x 420 = 2^3 x 3 x 5 x 7 (just add a 2 to A)
Option C = 3 x 420 = 2^2 x 3^2 x 5 x 7 (just add a 3 to A)
Option D = 2 x 1260 = 2^3 x 3^2 x 5 x 7 (just add a 2 to C)
Option E = 2 x 2520 = 2^4 x 3^2 x 5 x 7 (just add a 2 to D)

Actually after doing up to A you can quickly just figure out how many more 2's and 3's and determine it's E.

for q10 we need to find the lowest number - so should be 420
its divisible by all the integers from 1-7 inclusive
Math Expert
Joined: 02 Sep 2009
Posts: 50016
Re: Good set of PS 2  [#permalink]

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10 Jun 2010, 08:58
shalu wrote:
for q 9...
20%of those who applied X = 15 (because they applied for y also)
x/5 =15
x = 75
25%of those who applied Y = 15 (because they applied for X also)
y/4 = 15
y = 60

total who applied for x and y should be x+y = 135

Answer 135 (A) is not correct. OA's and solutions for this set of questions are given in my post on page 2.

9. Of the applicants passes a certain test, 15 applied to both college X and Y. If 20 % of the applicants who applied college X and 25% of the applicants who applied college Y applied both college X and Y, how many applicants applied only college X or college Y?
(A) 135
(B) 120
(C) 115
(D) 105
(E) 90

20%X=X&Y=15 --> X=75 --> Only X=75-15=60
25%Y=X&Y=15 --> Y=60 --> Only Y=60-15=45
Only X or Y=60+45=105

Answer: D.
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Re: Good set of PS 2 &nbs [#permalink] 10 Jun 2010, 08:58

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# Good set of PS 2

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## Wednesday, July 8, 2015

### Expected number of uniformly distributed collisions (birthday problem)

Here's an interesting mathematical problem. If you have "n" objects to be inserted into "m" available slots using a uniformly distributed random placement, how many collisions with already occupied slots should we expect to happen? This is useful for hashtables and other data structures where duplicates are not allowed.

Here is a Python 3 program that simulates inserting objects into random positions in an array and counting the average number of collisions.

def collisions(n, m):
trials = 10000
total_collisions = 0
for _ in range(trials):
slot_is_occupied = [ False for _ in range(m) ]
for _ in range(n):
slot = random.randint(0, m-1)
if slot_is_occupied[slot]:
total_collisions += 1
else:
slot_is_occupied[slot] = True

Here is a sample of the average number of collisions given by the above function for different values of "n" and "m":
n\m12345678910
10.00.00.00.00.00.00.00.00.00.0
21.00.49470.33340.25150.20540.1630.14370.12970.11180.1053
32.01.24470.88610.6850.5570.46330.40230.35370.3250.2819
43.02.12911.58121.25881.04430.90540.77540.69240.61760.5459
54.03.06172.41.9441.64571.42041.23641.11230.99840.9019
65.04.0343.2652.70782.3042.02181.76041.60041.43491.318
76.05.01684.17423.54063.04982.67162.39732.14991.94171.7744
87.06.00755.12064.4033.83633.39053.03042.73782.51512.3219
98.07.00356.07655.30524.67884.16523.7383.41683.12052.8816
109.08.00167.05266.22335.54014.96324.4934.09133.77213.5016

Basically the answer is the number of objects "n" minus the number of occupied slots. This will give us the number of objects excluding the ones which were inserted without collision, that is, in an empty slot. For example, if I insert 5 objects into an array but at the end there are only 3 occupied slots, then that must mean that 2 of those objects were inserted in the same slot as some other objects (they collided with them).

The question is how to predict the expected number of occupied slots.

Expected number of occupied slots

What is the average number of slots ending up being occupied by at least one object? This previous blog post explains that you basically just need to multiply the probability of a given slot being occupied at the end by the number of slots. So what is the probability of a slot being occupied?

Probability of a slot being occupied

What is the probability that an object is inserted into a particular slot out of "m" slots?
1/m

Therefore the probability that the slot remains empty is
1 - 1/m

What is the probability that the slot is still empty after another placement? It's the probability that the first object did not land on the slot AND that the second object did not land on the slot too. These two probabilities are independent of each other, so
(1 - 1/m)(1 - 1/m)

In general, after "n" objects have been placed, the probability that the slot is still empty is
(1 - 1/m)^n

Notice that this makes sense for n = 0 because if no objects were placed, then the probability that the slot is empty is 1.

Which means that after "n" objects have been placed, the probability that the slot is occupied is
1 - (1 - 1/m)^n

Therefore...

Therefore, the expect number of occupied slots among "m" slots after "n" objects have been inserted with uniform probability is
m(1 - (1 - 1/m)^n)

Which means that the expected number of collisions is
n - m(1 - (1 - 1/m)^n)

Here is the same table as the one at the top showing the corresponding predicted number of collisions:

n\m12345678910
10.00.00.00.00.00.00.00.0-0.00.0
21.00.50.33330.250.20.16670.14290.1250.11110.1
32.01.250.88890.68750.560.47220.40820.35940.3210.29
43.02.1251.59261.26561.0480.89350.77840.68950.61870.561
54.03.06252.39511.94921.63841.41131.23871.10330.99440.9049
65.04.03133.26342.71192.31072.00941.7761.59041.43941.3144
76.05.01564.17563.53393.04862.67452.37942.14161.94621.783
87.06.00785.11714.40053.83893.39543.03952.74892.50772.3047
98.07.00396.0785.30034.67114.16283.74813.40533.1182.8742
109.08.0027.0526.22535.53694.9694.49844.10463.77153.4868

The maximum absolute error between the two tables is 0.0179.


text:
# Making Zn in school lab

I want to make $\ce{N2O}$ in my school chem lab.

I'm thinking of making it from $\ce{Zn_{(s)}}$ in $\ce{HNO3_{(dil)}}$ :

$$\ce{ Zn_{(s)} + HNO3_{(dil)} -> N2O + Zn(NO3)2 + H2O }$$

I'm not able to get $\ce{Zn_{(s)}}$, so I would like to make it on my own in lab.

-
This might help (without zinc): chemistry.about.com/b/2013/07/27/… – Devashish Das Aug 18 '14 at 8:34
Zinc is cheap and readily available. Making it is most likely much harder than the reaction you want to use it for (even if that is the best way to achieve your goal.) Most labs and lab suppliers will have it. Unless you want to set up a metal foundry which seems overambitious to obtain something you can buy off the shelf. – matt_black Aug 19 '14 at 11:44

Ahem, forget it.

Theoretically speaking, you can get $\ce{Zn}$ from alkaline batteries (see details here), but you have no chance to get pure $\ce{N2O}$ this way and reaction may easily go out of control.

Attention! Detonation hazard !!!

The usual and easy way to produce $\ce{N2O}$ is to heat very carefully widely available $\ce{NH4NO3}$, to $180-200~^\circ\mathrm{C}$. It is an absolute imperative to keep temperature of the reaction mixture below $240~^\circ\mathrm{C}$. Decomposition of ammonia nitrate ($\ce{NH4NO3}$) is exothermic and may proceed explosively. To be on the safe side, use sand bath to heat the reaction flask and put thermometer into the sand, so its temperature never risen above safe limits. Liquid bath with appropriate liquid may be better, but I'm not sure if a school lab would have proper liquid (silicone oil with high flash point). The produced gases should be cooled in Liebig condenser to separate water, and $\ce{N2O}$ may be further dried with $\ce{H2SO4}$. Of course, all joints must be sealed using proper oil. And keep a bath of cold water to cool the reaction mixture in case if temperature raises above safe limits.

It is reported that it is possible to produce $\ce{N2O}$ by heating mixture of $5~\mathrm{mL}$ $6~\mathrm{M}~\ce{HNO3}$, $1~\mathrm{g}~\ce{NH4NO3}$ and $0.05~\mathrm{g}~\ce{NaCl}$ on water bath at $\approx 80~^\circ\mathrm{C}$. If this works, it is a safer route, as hot ammonia nitrate is diluted and autospeeding reaction cannot occur. Link


text:
# .::Armonia Chronicles::. ~The Shifting Sands~ (Profile Thread)

Discussion in 'Profile Threads' started by crazE, May 30, 2013.

1. The lush world of Armonia.
-
In the southern reaches of the gargantuan planet, a large continent shines amidst the Ocean of Glass. On this continent, the world is run by the sand.
-
Almost entirely destroyed by harsh rays of sun and atmospheric conditions, Ambutus is almost entirely inhospitable. The great Harum Desert blisters the air hanging over Ambutus, taking up all of its western and most of its eastern half. Only a few places seem decent enough to harbor anything living. However, many creatures, specifically a strong race of people, thrive here, using the world around them to their advantage.
-
The people of Ambutus are now plagued with having their continent, their only home, being entirely overrun. What little resources there are have started to deplete entirely despite the best efforts of the people and the magic they possess. The fate of Ambutus hangs in the balance of the ones it has chosen to restore its glory.
-
You have been chosen.

-
-----

With a strange and archaic magic running about, disturbing the sands and destroying the land, the natives have found panic to be a normal emotion. Hysteria plagues the medieval land as sand twisters and strange creatures of unknown origin strike through villages and desert cities. With little options, the King of Ambutus seeks to find, through the help of his sorcerers, the people he needs to conquer this dilemma. You may not know it now, but he comes for you. All you know is that the world around you calls your name. Starting in the village, you will be thrust into the brunt end of this destruction. From there, it will be your choice what the fate of everyone and everything around you becomes.

NOTE: This is a closed roleplay. An invitation has already been sent to specified members. However, if you truly wish to join, shoot a PM explaining your interest to join.

SPECIAL NOTE: Just edited at 6:45 on 5/30/2013. New game mechanics have been added that I have forgotten to add.

inner note: Re-edited at 7:35, same day.

Lifesteal and Spellvamp: Taken from a famous MMO, I bring to you Lifesteal and Spellvamp.
Lifesteal is an attribute that gives health to a player based on a percentage of the damage dealt to an opponent by basic attacks.

Spellvamp is an attribute that gives health to a player based on a percentage of the damage dealt to an opponent by skills or abilities. The normal percentage is reduced for Area of Effect abilities or basic attacks.

The mechanics between Lifesteal and Spellvamp when it comes to abilities is that they act independently from one another. If you passively have Spellvamp from an item or ability, say 10%, and use an ability that gives your next basic attack 10% Lifesteal, Spellvamp will not buff the Lifesteal bonus given from the ability. Spellvamp does not account to the damage dealt by abilities that buff basic attacks. Lifesteal, in turn, will not buff abilities that do multiple attacks and such following this. Essentially, Spellvamp handles skills, Lifesteal handles basic attacks. There are no crossroads, no grey areas.

A quick and easy note: Lifesteal and Spellvamp, due to mostly being percentage attributes, are meant to be for matches when players have higher stats. However, flat Lifesteal and Spellvamp will be allowed to even out the unfair disadvantage early-canon.

Skills with any of these two attributes will be more expensive than they would normally be. Abilities with any of these attributes will be moderated. Flat Lifesteal and Spellvamp is allowed, but must be temporal. Starting skills and items will not provide Lifesteal and Spellvamp. People who will abuse with these attributes will be sought out for questioning and horrible torture.

New change: MP to EP. EP standing for an in-game characteristic known as the "Essence of the Goddess," in which all inhabitants on Armonia do not know that Mana is known as Mana. After much whining from a peruvian, it has been changed OoC in the profile to EP. Not a big deal, it does the same thing as MP.

Code:
[b]Name:[/b]
[b]Age:[/b]
[b]Gender:[/b]
[b]Level:[/b] 1
[b]Next Level:[/b] 0/10
[b]Appearance:[/b]
[b]Profession:[/b]

[b]Weapon:[/b]
[b]Armor:[/b]
[b]Accessory:[/b]

[b]Charm-Ability:[/b] ([i]Leave blank. Charm Abilities are abilities granted by carrying a Charm on your character. Charms are items that carry certain abilities, active or passive, that improve the overall performance of a character in battle. They are dissimilar to Accessories in which that Charms work like S-abilities or R-abilities. Charms can be bought or found on maps and given or taken depending on circumstances. They can be melded with other Charms or other items and even infused with your Weapons, Armor, and Accessories to create stronger tools![/i])

[b]A-Ability:[/b]

+

+

[b]R-Ability:[/b] None.
[b]S-Ability:[/b]
[b]E-Ability:[/b] ([i]Ultimate ability in similar ratio to Shadow's X- or Masquerade's D-Ability.[/i])

[b]E-Trigger:[/b] ([i]E-triggers are a design made by Shadow, in which one can set one's E-Gauge off early in order to perform a special move. E-triggers can range from a small group heal ability to a partial resupply of mana to a brutal true damage strike. E-triggers are more powerful than normal skills considering more leeway and run on the E-gauge, but are not to be taken far. However, this should not prohibit creative thinking![/i])

[b]G-Ability:[/b] ([i]G-abilities are abilities usable by a certain party of people available on a map during a battle. G-Abilities are activated by each character providing their E-trigger or their E-gauge to deal fatal/massively helpful abilities as a group no normal E-trigger or E-ability could ever do on a singular character. G-abilities can be obtained throughout the game by completing certain tasks. G-abilities can also be secretly obtained by other certain methods, so keep an eye out![/i])

~Stats~
[b]HP:[/b]
[b]EP:[/b]
[b]SHD:[/b]
[b]ATK:[/b]
[b]DEF:[/b]
[b]INT:[/b]
[b]SPR:[/b]
[b]EVA:[/b]
[b]CRIT:[/b]
[b]MOVE:[/b]

[b]LFSTL:[/b]
[b]SPLVP:[/b]

[b]~E-Gauge~[/b] {/////}{/////}
([i]SHD is the Shield stat, a special stat in which a character gains, essentially, extra health with more durability (1 SHD per 1.5 DMG). SHD can also be given special properties in certain cases (deal damage back to attacker, regen mana, etc.), whether by natural ability or by activated ability, or even by atmospheric effects.

SHD is a liquid stat, meaning it is not a stat that can be kept (usually) and is only temporal in most cases. SHD is not usable by starting characters (unless characters are based off SHD) and is usually only obtainable by outside sources, such as abilities or atmospheres. However, it is there to provide it an option when SHD is used on characters and it is free to be used and experimented with for certain cases. All characters automatically start with a potential capacity of 10 SHD, but not have any SHD in the SHD stat. Temporal SHD is lost at the end of a chapter, while passive SHD is kept throughout the game (unless depleted in some cases))

Players:
Macintosh

#1
2. RE: .::Armonia Chronicles::. ~The Shifting Sands~

~Ally Characters~

None yet.

~Enemy Characters~

None yet.

~Neutral Characters~

None yet.

----------------------

::Map::

Coming soon!

::Inventory::

None yet.

Gold: 0

---------------------------------------
Reserved for Players:

#2
3. RE: .::Armonia Chronicles::. ~The Shifting Sands~

Name: Fataroux (fay-ter-roh)
Age: 16
Gender: Male
Appearance:
This character develops over time, so the picture will be changed every Job Promotion. You will be notified if you are using one of his appearances. Similar methods may be used by other players.
Profession: Sand Shadow- One who claims this profession names himself after the shadows of the desert, deadly creatures of the night, rightly so by professing in quick and brutal fatalities of foes. [Dual-wield] [Shadow's Curse]

*Shadow's Curse- Fataroux can move through most solid objects, save it those marked otherwise. Objects that allow Fataroux to move through require at least 2 MOVE. Fataroux cannot stop on objects he can pass through. Objects that Fataroux can pass through will be marked.

Weapon: Sand-worn Dagger- a dagger afflicted by the environment around it, it is worn but enhanced by the sanding. [+1 ATK]
Weapon: Sand-worn Dagger [+1 ATK]
Armor: "Liberated" Shoulder plates- we all know what this means. [+1 DEF]
Accessory: Sharp Stone- A small, carved, peculiar stone clipped to the side of one's waist, it provides a small boost in one's critical damage. [+5% Critical Strike Chance]

Charm-Ability: None.

+Trinity Slash- Fataroux slashes the opponent three times, each successive hit dealing more damage. 3 hits. 100% (first hit), 110% (second hit), 120% (third hit). Range: Adjacent cell. Element: Neutral. 11 EP

+Assassin's Approach- Fataroux cloaks himself for 1 turn. His next basic attack is guaranteed to deal a Critical Strike. Range: None. Element: Neutral. 8 EP

R-Ability: None.
S-Ability: Progressive Fatality- Fataroux stacks Blood Counters with every attack or ability he deals on an enemy. Each Blood Counter debuffs the enemy's DEF by 10% (rounded up). When the maximum number of Blood Counters are stacked on an enemy, the next basic attack dealt to them by Fataroux or his allies deals 10% more damage (rounded up) and the counters with the debuff are removed. Max stacks: 4.
E-Ability: Death Locust- Fataroux teleports behind an opponent, bringing his blades down and through his target mercilessly. 2 hits. 100% Piercing Attack Damage each hit. High chance of Instant Death. High Chance of Blood Loss. 25% Spellvamp from total end damage. Range: One Opponent. Element: Neutral.

E-Trigger: Sand Shield- Fataroux summons the sands around him for protection, reinforcing himself with hard, enchanted sandstone armor and enhanced offensive attacks. Gains 8 SHD and a 25% basic attack damage increase and lasts 3 turns or until SHD is depleted.

G-Ability: None.

~Stats~
HP: 20
EP: 20
SHD: 0
ATK: 4 (+1 Sand-worn Dagger)(+1 Sand-worn Dagger)
DEF: 3 (+1 "Liberated" Shoulder Plates)
INT: 1
SPR: 2
EVA: 4%
CRIT: 3% (+5% Sharp Stone)
MOVE: 2 cells (+2 Sand Shadow Job)

LFSTL: 0%
SPLVP: 0%

~E-Gauge~
{/////}{/////}

Skill Points:

#3
4. Name: Allister
Age: 16
Gender: Male
Level: 1
Next Level: 0/10
Appearance:
Profession: Meta Knight-The one who claims this profession is a loyal warrior, able to bend and manipulate metals and static electricity. [Enforcer] [Lightning Affinity I]

*Enforcer- Allister amplifies his damage based on how dense his equipment is, gaining ATK Damage equal to 10% of his DEF.

*Lightning Affinity I- Acquires Lightning Affinity. Damage dealt by Lightning Element attacks deal 10% more damage, while damage received by Lightning Element attacks is reduced by 20%.

Weapon: Rough Sword- an old sword that has been worn, but serves its job pretty well. [+1 ATK]
Armor: Thin Plates- small metal plates worn under clothing, they still provide a decent amount of protection. [+1 DEF]
Accessory: Magnetite- small metallic rocks that have magnetic properties, they always point toward the north pole. [+5 MP]

Charm-Ability: None.

A-Ability: Storm Fall

+Static Spin- Allister throws his blade directly in front of him in a saw-like motion. Using magnetic force, Allister recalls the blade back. 120% ATK Damage. Chance of immobilize. Range: 3 cells. Element: Neutral. 8 MP

+Armor bend- Morphs and manipulates the opponent's armor into a dented state. -20% DEF for three turns. Stackable up to 60%. Range: 4 cells. Element: Neutral. 7 MP

R-Ability: None.

S-Ability: Magnetic draw- Enemies begin to be pulled by a magnetic force near Allister, feeling more heavy weight to move. Enemies within a 2 cell range have their MOVE reduced to 1.

E-Ability: Rolling Thunder- Allister channels raw lightning into his blade and swings it underhand, causing a massive shockwave which blasts forward from his blade and strikes the closest enemy. Enemies nearby the first enemy are also hit. 150% piercing damage to first enemy and silence for 3 turns, the enemy is also rooted for 3 turns. Enemies within a 2 cell range of first enemy are silenced for 2 turns. Range: 4 cells Element: Lightning.

E-Trigger: Iron will- Allister reinforces armor with nearby metal for a short period of time to ignore any attack. Creates shield based on 20% maximum health. Can be cast on ally. Duration: 3 turns or until armor is broken. Range: Self or 2 cells. Health shield derives from player this E-trigger is cast upon.

G-Ability:

~Stats~
HP: 25
MP: 15 (+5 Magnetite)
SHD: 0
ATK: 4 (+1 Rough Sword)(+0 from Enforcer)
DEF: 3 (+1 Thin Plates)
INT: 1
SPR: 2
EVA: 3%
CRIT: 4%
MOVE: 2 cells (+1 Meta Knight Profession)

LFSTL: 0
SPLVP: 0

~E-Gauge~ {/////}{/////}

#4
5. Name: Eria Forcastre
Gender: Female
Age: 15
Level: 1
Progress: 0/10
Appearance:

Profession: Mage Scholar: The one who claims this Profession is dedicated in studying the branch of magiology, seeking to learn and apply theoretical and practical knowledge that can be converted into powerful magic. [Basic Attack: Magical] [Spell Counter] [Cognition I]

*Spell Counter: Eria's A-Skills, for this Profession's A-Ability, have additional SubSkills anexed to them, which function based on a Spell Counter mechanic. Parent A-Skills have an additional parameter, Spell Count, increased for each usage; Counter A-Skills cannot be used unless this value reaches Maximum Count (it is reset afterwards). A "Value" field present in Counter A-Skills determine the feasability of said A-Skill, factoring in the Parent A-Skill's EP Cost and Max Spell Count.
*Cognition I: Increases the User's Base Int by 10%. Every Level in which the Required Experience field is modified (in Progress), a free Stat Point is to be allocated specifically in MP.

Weapon: Beginner's Book: A written repository of the most basic aspects of magic followed by likewise simple spells. Its size betrays its usefulness. [+1 Int / Range: 2 Cells]
Armor: Witch's Halfrobe: Originally a full robe given to aspiring scholars as a first sign of acceptance, it seems that someone managed to rearrange it into a really lithe piece of clothing. [+1 Def]
Accessory: Chocolate Heart: Not actually made of chocolate, this heart-shaped hair accessory is actually, for some reason, neatly tied to Eria's hat. [+5 Max EP]
Charm: None.

A-Ability: Compiled Knowledge

+ Fluenta: Facilitates casting speed through coursing Essence, allowing the user to easily and rapidly perform simple attacks temporarily. Sets Basic Attack as a Free Command. 1 Stack Limit. Duration: 3 Turns. Cast Time: Instant. Accuracy: None. Variance: None. Range: Self. Element: Neutral. 6 EP.

+ Percuro: A healing incantation, archaic by modern standards, which indirectly mends the wounds of an ally. 120% Int HP Recovery. Cast Time: Instant. Accuracy: 100%. Variance: 0%. Range: 3 Cells. Element: Neutral. Spell Count: 0/2. 8 EP.

--- Afflicto: Forms orbs of energy out of resultant magic and has them repeatedly blast a target within moderate distance. 2 Hits. 110% Int Damage (Every Hit). Cast Time: Instant. Accuracy: 100%. Variance: 10%. Range: 3 Cells. Element: Neutral. Value: 10.

R-Ability: None.
S-Ability: Passive Flow I: Increases the Effectiveness of the User's Passive EP Regeneration by 1 Point. In addition, all Ally Characters (and chosen Neutral Characters) within the User's Adjacent Cells receive this effect.
E-Trigger: Counter Break: Relying on knowledge about an ancient and forgotten energy, the user allows for her own Essence to be rendered chaotic and unstable for a moment, projecting a shifting, soft-colored distortion around her. And, for this moment, the Laws of Magic begin to be overridden. Increases Magical Damage Output by 20%. Enables the usage of Count A-Skills without requirement of Spell Count from their respective Parent A-Skills. Duration: 3 Turns. After a Duration Fade, Increases the Spell Count of ALL Count A-Skills by 1. 5 E Levels.
E-Ability: Excerpt "Verum Magicae": True Magic. An excerpt from a grimoire, superimposing itself on the user's tome, which traces a complex glyph of raw SpellCode on the ground beneath her. When the user triggers it with a single word, she becomes the center of a massive pillar of Arcane Essence surging from within the glyph and shooting into the sky above, leaving behind an aftereffect of highly reactive magic in its wake. 150% Piercing Int Damage. Creates Residual Zones within Range. [Denoted { } / If Ally: Increases All Stats by 20% / If Enemy: 30% Piercing (User's) Int Damage. / Duration: 3 Turns] Cast Time: Instant. Accuracy: 100%. Variance: 0%. Range: Area of 2 Cells. Element: Arcane. 10 E Levels.

~ Statistics ~

HP: 15
EP: 25 (Chocolate Heart +5)
Shd: 0
Atk: 1
Def: 2 (Witch's Halfrobe +1)
Int: 4 (Beginner's Tome +1)(Cognition I +1)
Spr: 3
Critical: 4%
Evasion: 3%
Movement: 2 Cells (Mage Scholar Job +1)
E Gauge:
{/////}{/////}

LifeSteal: 0%
SpellVamp: 0%

Skill Points: 0


text:
# Textbook definition a c-number

1. Sep 6, 2005

### fliptomato

Hello everyone--I was hoping someone could give (or refer to) a textbook definition of a $$c$$-number, as used in the context of quantum field theory.

Does this refer to a commuting number? I've also read it referring to a classical quantity. classical quantity, from a not-very-reputable source. (Though are these the same?)

If the 'c' refers to commuting, when I read something like "...spinors are anti-commuting (c-numbers)." (Bailin and Love, SUSY book) Does this mean that spinors are anti-commuting objects composed of commuting numbers? (It certainly doesn't mean that the spinors themselves are commuting...)

Thanks,
Flip

2. Sep 6, 2005

### Norman

C-number usually refers to "complex" number. At least that is all I have ever seen it used as.

3. Sep 6, 2005

4. Sep 8, 2005

### snooper007

c-number is really defined as a classical number. it is not an operator or
spinor.!

anticommuting number is Grassmann number used in the functional integral
representation of fermionic field.

Often the Grassman number is not called c-number, maybe someone can call
Grassmann number as c-number.


text:
# Explain the concept of machine hour rate depreciation method.

Banking & FinanceFinanceFinance Management

Machine hour rate is called as service hour method. In this method, active hours of machine are taken into account for calculation of depreciation. This method is commonly used in sectors like chemicals, steel and other heavy industries.

In this method, machine hour rate is established and charged to production. Machine hour rate means the amount spent on machine to run for an hour.

## Formula

            (Depreciation rate) hourly = (Cm-SVm)/ELm
Here Cm = cost of machine, SVm = scrape value of machine and Elm = estimate life of machine
(Depreciation)Annual = WHm * (Depreciation rate) Hourly
Here WHm = hours of work of a machine in a year

Types of machine hour rate are as follows −

• Simple − Expenses that are connected directly to operation of machine.

• Composite − Expenses includes machine hour rate, standard charges, fixed charges and direct wages of machine per hour

Advantages of machine hour rate depreciation method are as follows −

• Scientific and practical method.
• Time factor is considered.
• Useful data is provided.
• Helps in decision making.
• Skilled and unskilled can be differentiated.
• Efficiency and operating cost can be compared.
• Idle time of machine can be known.
• Ideal method for machines.
• Reports made out of this are dependable.

Disadvantages of machine hour rate depreciation method are as follows −

• Additional work is carried out in calculation of working hours.
• Expenses are not considered.
• Results can be inaccurate if, both machine and manpower are required.
• Estimation of working hours is difficult.
• If manpower requirement is more than machine then, this method is not suitable.
Published on 29-Sep-2020 17:19:49


text:
Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.

I’m about to show you an R trick. Various readers may find it cool, useful and interesting, or stupid, useless and an evil deed undermining the sanctity of R’s functional programming nature (“All bow”). But I hope many of you will find the material here rather intriguing if not useful.

All this involves a trick one can employ while working in R’s top level from interactive mode, the familiar > prompt.

I’ll use as an toy example here. Let’s say I have some counting variable x that I need to increment occasionally as I work. Of course, the straightforward way to do this is

x <- x + 1

But to save typing and reduce distraction from my main thought processes, it would be nice if I were able to simply type

ix

Of course, the straightforward way to do this would be to define a function ix() (“increment x”),

ix <- function() x <<- x + 1

and then call it each time by typing

ix()

But again, I want to save typing, and don’t want to type the parentheses. How can I arrange this?

My approach here will be to exploit the fact that in R’s interactive mode, typing an expression will print the value of that expression. If I type, say

y

R will first determine the class of y, and invoke the print method for that class. If I write that method myself, I can put any R code in there that I wish, such as code to increment x above! So here goes:

> w <- list(y=3)
> class(w) <- 'ix'
> print.ix <- function(ixObject) x <<- x + 1
> x <- 88
> w
> x
[1] 89
> w
> x
[1] 90

I set up an S3 class ‘ix’, including a print method print.ix(). I created w, an instance of that class, and as you can see, each time I typed ‘w’, x did get incremented by 1.

What just happened? When I type ‘w’, the R interpreter will know that I want to print that variable. The interpreter finds that w is of class ‘ix’, so it calls the print method for that class, print.ix(), which actually doesn’t do any printing; it merely increments x, as desired.

So I don’t even need to type out even the 4 characters ‘ix()’, or even the 2 characters ‘ix’; just typing the single character ‘w’ suffices. A small thing, maybe, but very useful to me when, in the frenzy of code development and especially debugging, I am able to keep distractions from my train of thought to a minimum.

By the way, we have been doing further development on our polyreg package, with some interesting new features. More news on that soon.


text:
96 views

Five jumbled up sentences, related to a topic, are given below. Four of them can be put together to form a coherent paragraph. Identify the odd one out and key in the number of the sentence as your answer:

1. Talk was the most common way for enslaved men and women to subvert the rules of their bondage, to gain more agency than they were supposed to have.
2. Even in conditions of extreme violence and unfreedom, their words remained ubiquitous, ephemeral, irrepressible, and potentially transgressive.
3. Slaves came from societies in which oaths, orations, and invocations carried great potency, both between people and as a connection to the all-powerful spirit world.
4. Freedom of speech and the power to silence may have been preeminent markers of white liberty in Colonies, but at the same time, slavery depended on dialogue: slaves could never be completely muted.
5. Slave$-$owners obsessed over slave talk, though they could never control it, yet feared its power to bind and inspire$-$for, as everyone knew, oaths, whispers, and secret conversations bred conspiracy and revolt.


text:
Actionscript 3.0, good OOP?

I have been coding for some time (3 years I think), and have recently been wondering if my coding pattern is any good. Any kind of feedback is welcome:

EDIT: The thing that concerns me most is that I have read a lot around that getters/setters are bad for OOP. Is this true?

//Point3D.as
public class Point3D
{
private var _x:int;
private var _y:int;
private var _z:int;

public function Point3D(tx:int = 0, ty:int = 0, tz:int = 0)
{
_x = tx;
_y = ty;
_z = tz;
}

public function get x():int { return _x; }
public function set x(tv:int):void { _x = tv; }

public function get y():int { return _y; }
public function set y(tv:int):void { _y = tv; }

public function get z():int { return _z; }
public function set z(tv:int):void { _z = tv; }

}

//Geometry3D.as
public class Geometry3D
{
private var _indices:Vector.<uint>;
private var _vertices:Vector.<Number>;

public function Geometry3D(tindices:Vector.<uint>,
tvertices:Vector.<Number>)
{
_indices = tindices;
_vertices = tvertices;
}

public function get indices():Vector.<uint> { return _indices; }
public function get vertices():Vector.<Number> { return _vertices; }

}

//Primitive3D.as
public class Primitive3D
{
private var _position:Point3D;
private var _geometry:Geometry3D;

public function Primitive3D(tx:int, ty:int, tz:int, tgeometry:Geometry3D)
{
_position = new Point3D(tx, ty, tz);
_geometry = tgeometry;
}

public function get position():Point3D { return _position; }
public function get geometry():Geometry3D { return _geometry; }

public function render():void
{
}

}

//TrianglePrimitive.as
public class TrianglePrimitive extends Primitive3D
{

public function TrianglePrimitive(tx:int, ty:int, tz:int)
{
var indices:Vector.<uint> = Vector.<uint>([
0, 1, 2]);
var vertices:Vector.<Number> = Vector.<Number>([
]);

var geo:Geometry3D = new Geometry3D(indices, vertices);
super(tx, ty, tz, geo);
}

{
}

}

//Main.as
...
var tri:TrianglePrimitive = new TrianglePrimitive(0, 0, 0);
tri.position.z = -4;
tri.spin(Math.PI / 2); //90 degrees
tri.render();
...

Again, thanks a bunch for any feedback!

-
You should mention what kind of things concern you about what you wrote. –  Juan Mendes Dec 20 '12 at 0:19
I wouldn't consider using getters and setters a good practice in Actionscript, as function calls are pretty expensive there. Especially when you just create them for future changes (one of benefits of getter/setter is that you can easily replace a property with it without changing the rest of code). Also, as a side note, consider creating methods getValue() instead of getters to indicate there's some extra work and it's better to cache the value than call the getter several times in one place. –  Markus von Broady Mar 30 '13 at 16:06
• The accessor methods (get / set) in Point3D are unnecessary as they don't do any processing or restrict any access. The class would be better suited to have simple, public x, y, z properties.
• I don't see the purpose of the Geometry3D class. Maybe it's because I don't know much about your intentions or programming for 3d, but looks like things would be more straight-forward if indices and vertices properties were directly on Primitive3D. If it make sense to separate them off because they're processed as a pair by other parts of your program, you might also consider creating a IGeometry3D interface, which the Primitive3D class can implement. This would prevent there from being any ambiguity around ownership. (Not sure if I'm describing my thoughts, clearly.)


text:
# Math Help - Parametrics.....

1. ## Parametrics.....

Let x=t^3-3t
y=t^2-t

Where is the tangent parallel to the x-axis? y-axis? line y=x?

The curve intersects itself once. Find this point.

Thanks.

2. Originally Posted by zhupolongjoe
Let x=t^3-3t
y=t^2-t

Where is the tangent parallel to the x-axis? y-axis? line y=x?

The curve intersects itself once. Find this point.

Thanks.
Horizontal tangents occur when $\frac{dy}{dt}=0$

Vertical tangents occur when $\frac{dx}{dt}=0$

to find the derivative use the formula

$\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}= \frac{2t-1}{3t^2-3}$

For the last here is a hint.

For what values of t will the function have the same y-coordinate.

Find these and compare those times with the x -coordinate.


text:
# Inverse of Kleisli star, or “extension operator”

While thinking about monads in the theory of denotational semantics, I have made an observation about the Kleisli category that I would like to check

Suppose $F : \mathcal D \to \mathcal C$, $G : \mathcal C \to \mathcal D$, is an adjunction, so that $T = GF : \mathcal D \to \mathcal D$ is a monad. There is an operator $*$ on the Kleisli category that takes $f: X \to TY$ to $f^* : TX \to TY$.

I have an intuition and a sketch proof of the following, but my experience with these things is not strong, so I would like confirmation. In fact, this may just be an obvious or basic result on monads and adjunctions, but I'm not sure. The following statements I think I've proved:

$f^*$ is in fact $Gg$ for some $g : FX \to FY$ in $\mathcal C$, and moreover there is a bijection between Kleisli arrows $f : X \to TY$ and morphisms $Gg$ where $g : FX \to FY$. The inverse of the $*$ operation is given by taking $f' : TX \to TY$ to $f' \circ \eta_X$, where $\eta$ is the monad unit.

In a sense this means that the Kleisli category is "as much of $\mathcal C$ as can be represented in $\mathcal D$, given $F$. If what I say is true I suppose it must be something quite elementary, but I don't know where to look. Pointers would be appreciated.

-

Sure, you could put it that way. I think it's simpler to say that there is a natural bijection between arrows $f: x \to GFy$ in $\mathcal{D}$ and arrows $g: Fx \to Fy$ in $\mathcal{C}$, because that is obvious from the hom-formulation of adjunctions:

$$\hom_{\mathcal{C}}(Fx, Fy) \cong \hom_{\mathcal{D}}(x, GFy).$$

But it is a nice exercise to make this correspondence explicit through the use of triangular equations (see CWM). Rather than test my ability to draw commutative diagrams in MathJax, I'll just write down the equations and leave it to you to draw the diagrams. Let $\eta: 1_\mathcal{D} \to GF$ denote the unit and $\varepsilon: FG \to 1_\mathcal{C}$ the counit. Given $f: x \to GFy$, define $g = f^\wedge = (\varepsilon Fy)(Ff)$. And given $g: Fx \to Fy$, define $f = g^\vee = (Gg)(\eta x)$. Then

$$f^{\wedge \vee} = G((\varepsilon Fy)(Ff))(\eta x) = (G\varepsilon Fy)(GFf)(\eta x) = (G\varepsilon Fy)(\eta GFy)(f) = 1_{GFy} f = f$$

where the second equation uses functoriality of $G$, the third uses naturality of $\eta$, and the fourth is a triangular equation. Similarly,

$$g^{\vee \wedge} = (\varepsilon Fy)F((Gg)(\eta x)) = (\varepsilon Fy)(FGg)(F\eta x) = g(\varepsilon Fx)(F\eta x) = g 1_{Fx} = g$$

where the second equation uses functoriality of $F$, the third uses naturality of $\varepsilon$, and the fourth is the other triangular equation. All this is well-covered in CWM.

Finally, another way to think about this is that the star extension exhibits the Kleisli category as a full subcategory of the Eilenberg-Moore category of $T$-algebras: it is exactly the subcategory consisting of free algebras $Tx$.

-
That's great. Now I know I'm barking up the right tree I can start thinking harder about this without fearing I'm on the wrong track. Thanks Todd! –  Tom Ellis Oct 9 '11 at 14:06
You're welcome, Tom. By the way, if you found this answer helpful, feel free to upvote it. :-) –  Todd Trimble Oct 9 '11 at 14:15
Done. Also worth mentioning is that your second statement is stronger than my observation. I didn't claim my map was injective from $g : Fx \to Fy$ in $\mathcal C$, just from $Gg$ for such $g$, so actually what you've told me is even better for my purposes. –  Tom Ellis Oct 9 '11 at 14:50
When I wrote "second statement" I meant "statement in your second sentence". –  Tom Ellis Oct 9 '11 at 14:51


text:
## Dissolving

$\Delta G^{\circ} = -nFE_{cell}^{\circ}$

Janice Xiao 1I
Posts: 52
Joined: Thu Jul 13, 2017 3:00 am

### Dissolving

When do you know that a gold ring will dissolve based off of delta G standard?

Fatima_Iqbal_2E
Posts: 31
Joined: Fri Sep 29, 2017 7:06 am
Been upvoted: 1 time

### Re: Dissolving

When Dr. Lavelle showed us an example like this during lecture, we had not yet learned the Nernst equation, so you could use $\Delta G^{o} = -nFE^{o}$ to see if the reaction is spontaneous or not. If the $\Delta G^{o}$
is positive, the reaction would be unfavorable, and the ring would not dissolve. However, if the $\Delta G^{o}$ is negative, then we know that the reaction is favorable and spontaneous, and the ring will dissolve.

Nancy Le - 1F
Posts: 52
Joined: Fri Sep 29, 2017 7:07 am

### Re: Dissolving

Can we assume that any spontaneous reaction will dissolve? Does this apply to only this reaction?


text:
Uniformly differentiable function

I am trying to answer the following question from my book:

Propose a definition for what it should mean for $f: A \to \mathbb R$ to be uniformly differentiable on $A$.

Since the definition of uniform continuity is the following:

A function $f: A \to \mathbb R$ is uniformly continuous on $A$ if and only if for $\varepsilon > 0$ there exists $\delta$ such that $|x-y|< \delta$ implies $|f(x) -f(y)|<\varepsilon$

I was thinking of the following:

A (differentiable) function $f: A \to \mathbb R$ is uniformly differentiable on $A$ if and only if for $\varepsilon > 0$ there exists $\delta$ such that $|x-y|<\delta$ implies $|f'(x) -f'(y)|<\varepsilon$.

But this is just saying $f'$ is uniformly continuous. So the next thing I came up with is:

A (differentiable) function $f: A \to \mathbb R$ is uniformly differentiable on $A$ if and only if there exists $\varepsilon > 0$ such that for all $x\in A$ it holds that $|f'(x)|<\varepsilon$.

But this is just saying the derivative is bounded. Now I'm stuck. How do I go about finding a suitable definition?

• Perhaps, give some condition on how well the difference quotients approximate the value of the derivative. – David Mitra Feb 8 '14 at 12:16
• It is worth remembering that Cauchy's proof of the MVT (1823) contains an invalid use of this concept. – Tony Piccolo Nov 22 '15 at 16:49

The most reasonable definition:

Let $A\subset \mathbb R$ and $f: A\to \mathbb R$. We say that $f$ is uniformly differentiable on $A$, is for every $\varepsilon>0$, there exists a $\delta>0$ such that, for every $x\in A$, $$|h|<\delta\,\,\text{and}\,\,x+h\in A\quad\Longrightarrow\quad \left|\frac{f(x+h)-f(x)}{h}-f'(x)\,\right|<\varepsilon.$$

This is such a common notion in analysis that I think it is worth elaborating on, at least when I taught analysis at this level I would do so.

The first starting point is historical. Cauchy was the first to define continuity as we now know it although the word was used loosely before him. Here is his definition (that you already well know):

A function $f:[a,b]\to\mathbb{R}$ is continuous, if for each point $x$ in $[a,b]$ and for any $\epsilon>0$ there is a $\delta>0$ so that $|x-y|<\delta$ $\implies$ $|f(x)-f(y)|<\epsilon$.

Then, when he set about to use this concept he switched without noticing this to a different version:

A function $f:[a,b]\to\mathbb{R}$ is [uniformly] continuous, if for any $\epsilon>0$ there is a $\delta>0$ so that $|x-y|<\delta$ $\implies$ $|f(x)-f(y)|<\epsilon$ for each point $x$ in $[a,b]$.

All that happens, of course, is that the quantifier "for each point" moves from the beginning of the statement to the end, with dramatic consequences. The first definition we now call "pointwise continuity" and the second (much stronger) definition we now call "uniform continuity." Actually Cauchy's mistake was quite fruitful. He needed the stronger condition but ended up giving us two useful definitions (although he thought they were the same).

Here is another example which sometimes helps the intuition.

Let $A$ and $B$ be the collection of all math students at two universities. We say that University B is "smarter" than University A if for each student $s_A$ in $A$ there is a subject and a student $s_B$ in $B$ so that $s_B$ has higher grades than $s_A$ in that subject.

Let $A$ and $B$ be the collection of all math students at two universities. We say that University B is "uniformly smarter" than University A if there is a subject and a student $s_B$ in $B$ so that $s_B$ has higher grades than $s_A$ in that subject for each student $s_A$ in $A$.

Again all that is happening is that the quantifier (for each student $s_A$) has moved to the end of the statement, but the difference is again dramatic.

The exercise here (left to the reader) is to explain in words what the difference is between the uniform version and the "pointwise" version. Think, too, about what would you have to do to prove or disprove the following assertions:

Yale Math is smarter than Harvard Math.

Yale Math is uniformly smarter that Harvard Math.

When you look at the other answer to this question provided by Mr Smyrlis, make sure to see that he has reproduced the completely familiar definition of a derivative happening at each point $x$ and then moved the quantifier to the end of the definition thus obtaining "uniform differentiability." [In fact he just made sure to move it to after the choice of $\delta$.]

• If you are somewhat braver and don't mind being provocative try this version: $U$ is the set of students at a university and there is a relation $s\heartsuit t$ defined for some pairs $s$, $t$ in $U$. [Not reflexive, probably not symmetric, definitely not transitive.] We say $U$ is a happy university if for each $s\in U$ there exists a $t\in U$ so that $s\heartsuit t$. Define what would be meant by a uniformly happy university and explain in words the difference between the two concepts. – B. S. Thomson Nov 21 '15 at 18:30