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# How many different prime numbers are factors of the positive

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How many different prime numbers are factors of the positive [#permalink]  29 Jan 2012, 17:31
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How many different prime numbers are factors of the positive integer n?

(1) Four different prime numbers are factors of 2n.
(2) Four different prime numbers are factors of n^2.
[Reveal] Spoiler: OA

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Re: Factors of positive integer n [#permalink]  29 Jan 2012, 17:42
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How many different prime numbers are factors of the positive integer n?

(1) 4 different prime numbers are factors of 2n --> if $$n$$ itself has 2 as a factor (eg $$n=2*3*5*7$$) than its total # of primes is 4 but if $$n$$ doesn't have 2 as a factor (eg $$n=3*5*7$$) than its total # of primes is 3. Not sufficient.

(2) 4 different prime numbers are factors of n^2 --> $$n^x$$ (where $$x$$ is an integer $$\geq{1}$$) will have as many different prime factors as integer $$n$$, exponentiation doesn't "produce" primes. So, 4 different prime numbers are factors of $$n$$. Sufficient.

Similar question: different-prime-factors-ds-95585.html

Hope it helps.
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Re: Factors of positive integer n [#permalink]  29 Jan 2012, 19:57
Bunuel wrote:
How many different prime numbers are factors of the positive integer n?

(1) 4 different prime numbers are factors of 2n --> if $$n$$ itself has 2 as a factor (eg $$n=2*3*5*7$$) than its total # of primes is 4 but if $$n$$ doesn't have 2 as a factor (eg $$n=3*5*7$$) than its total # of primes is 3. Not sufficient.

(2) 4 different prime numbers are factors of n^2 --> $$n^x$$ (where $$x$$ is an integer $$\geq{1}$$) will have as many different prime factors as integer $$n$$, exponentiation doesn't "produce" primes. So, 4 different prime numbers are factors of $$n$$. Sufficient.

Similar question: different-prime-factors-ds-95585.html

Hope it helps.

Hi Bunnel,
for statement 1: if n has 2 as a factor(e.g.: 2*3*4*5) then 2n becomes 2*2*3*4*5 then n has 4 factors, but if n dosn't has 2 as a factor (e.g.: 3*4*5) then 2 n becomes 2*3*4*5 then also n has 4 factors.

so both statement alone are sufficient.
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Re: Factors of positive integer n [#permalink]  29 Jan 2012, 20:16
subhajeet wrote:
Bunuel wrote:
How many different prime numbers are factors of the positive integer n?

(1) 4 different prime numbers are factors of 2n --> if $$n$$ itself has 2 as a factor (eg $$n=2*3*5*7$$) than its total # of primes is 4 but if $$n$$ doesn't have 2 as a factor (eg $$n=3*5*7$$) than its total # of primes is 3. Not sufficient.

(2) 4 different prime numbers are factors of n^2 --> $$n^x$$ (where $$x$$ is an integer $$\geq{1}$$) will have as many different prime factors as integer $$n$$, exponentiation doesn't "produce" primes. So, 4 different prime numbers are factors of $$n$$. Sufficient.

Similar question: different-prime-factors-ds-95585.html

Hope it helps.

Hi Bunnel,
for statement 1: if n has 2 as a factor(e.g.: 2*3*4*5) then 2n becomes 2*2*3*4*5 then n has 4 factors, but if n dosn't has 2 as a factor (e.g.: 3*4*5) then 2 n becomes 2*3*4*5 then also n has 4 factors.

so both statement alone are sufficient.

1) 4 different numbers are prime factors of 2n.
2n could equal 2*3*5*7 in which case n would equal 3*5*7. So in this case n would have only 3 prime factors.
But it is also possible that 2n = 2*2*3*5*7. In this case n=2*3*5*7 and so n has 4 prime factors.
So Bunuel is right. Statement 1 is not sufficient
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Re: Factors of positive integer n [#permalink]  30 Jan 2012, 01:50
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subhajeet wrote:
Bunuel wrote:
How many different prime numbers are factors of the positive integer n?

(1) 4 different prime numbers are factors of 2n --> if $$n$$ itself has 2 as a factor (eg $$n=2*3*5*7$$) than its total # of primes is 4 but if $$n$$ doesn't have 2 as a factor (eg $$n=3*5*7$$) than its total # of primes is 3. Not sufficient.

(2) 4 different prime numbers are factors of n^2 --> $$n^x$$ (where $$x$$ is an integer $$\geq{1}$$) will have as many different prime factors as integer $$n$$, exponentiation doesn't "produce" primes. So, 4 different prime numbers are factors of $$n$$. Sufficient.

Similar question: different-prime-factors-ds-95585.html

Hope it helps.

Hi Bunnel,
for statement 1: if n has 2 as a factor(e.g.: 2*3*4*5) then 2n becomes 2*2*3*4*5 then n has 4 factors, but if n dosn't has 2 as a factor (e.g.: 3*4*5) then 2n becomes 2*3*4*5 then also n has 4 factors.

so both statement alone are sufficient.

In your own example "if n dosn't has 2 as a factor (e.g.: 3*4*5)" n has only 3 factors not 4.

(1) says: 4 different prime numbers are factors of 2n. Now, if 2 is not a prime factor of n then 2n would have one more prime than n (this same exact 2), thus n has 3 prime factors. But if 2 is already a prime factor of n then 2n has the same number of prime factors as n.

Hope it's clear.
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Re: Number Properties [#permalink]  24 Apr 2013, 22:27
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mannava189 wrote:
How many different prime numbers are factors of the positive integer n ?
(1) Four different prime numbers are factors of 2n.
(2) Four different prime numbers are factors of n^2.

From F.S 1, for n = 3*5*7, we have 3 different prime factors for n. However, for n = 2*3*5*7, we would have 4 different prime factors for n. Insufficient.

From F.S 2, we know that $$n^2 = a*b*c*d$$,where a,b,c,d are primes.Thus, as n is also a positive integer, n = $$\sqrt{a*b*c*d}$$ = integer. As square root of any prime number is not an integer, thus, for n to be an integer, all the primes have to be in some even power form.

Thus, the number of prime factors for n = 4.Sufficient.

B.
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Re: Number Properties [#permalink]  25 Apr 2013, 00:52
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mannava189 wrote:
How many different prime numbers are factors of the positive integer n ?
(1) Four different prime numbers are factors of 2n.
(2) Four different prime numbers are factors of n2.

The question will be very straight forward if you just understand the prime factorization of a number.

When we say that n has 4 distinct prime factors, it means that

$$n = 2^a * 3^b * 5^c * 7^d$$ or $$n = 2^a * 5^b * 11^c * 17^d$$ or $$n = 23^a * 31^b * 59^c * 17^d$$ etc

What if we square n?
We get $$n^2 = 2^{2a} * 3^{2b} * 5^{2c} * 7^{2d}$$ etc
Notice that the number of prime factors will not change.
So if we know that n^2 has 4 prime factors, we can say that n MUST have 4 prime factors too. So statement 2 alone is sufficient.

(2) Four different prime numbers are factors of 2n.

There is a complication here. 2n introduces a new prime number 2. We don't know whether n had 2 before or not.

2n has 4 different prime factors implies n can have either 3 different prime factors or 4 different prime factors. Hence this statement alone is not sufficient. Let's look at examples:

Case 1:
n has 3 different prime factors.
Say, $$n = 3*5*7$$ (3 prime factors)
$$2n = 2*3*5*7$$ (4 prime factors)

Case 2:
n has 4 different prime factors
Say,$$n = 2*3*5*7$$
$$2n = 2^2*3*5*7$$ (still 4 prime factors)
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Re: Number Properties [#permalink]  13 May 2013, 09:02
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How many different prime numbers are factors of the positive integer n?
(1) four different prime numbers are factors of 2n
(2) four different prime numbers are factors of n^2.

I feel the answer should be [B]. Please find below my explanation:

Statement 1. Since its mentioned, there are 4 different prime factors under 2n, 2 must be one of them, so the question arises can be simply subtract 1 from 4 and claim that n has 3 different prime factors?! But in case n already has 2 as a factor, simply subtracting would not give you the exact count. For example is 2*n = 2*2*3, the total different prime factors of 2n would be 2 but when it comes to n, the value is still 2!

Hence, since we are not sure if n is even or not, we cannot say for sure that n has 3 prime factors or not. Hence, Insufficient.

Statement 2. Since its n^2, all the prime factors under n must be doubled. No exponent will be 1. Hence if the term n^2 has 4 different prime factors, it can be assumed that n too has 4 different prime factors. Hence, sufficient!

Therefore the answer should be [B]. Hope I am correct!

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Re: Number Properties [#permalink]  13 May 2013, 09:46
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How many different prime numbers are factors of the positive integer n?
(1) four different prime numbers are factors of 2n
(2) four different prime numbers are factors of n^2

St1: If 2 is one of the primes, then factors of 2N = {2,3,5,7} or 2N = 210. N then becomes 105 and N has 3 distinct prime factors {3,5,7}
But if 2 is not one of the primes, and if {3,5,7,11} are 4 distinct prime factors of 2N, then 2N = 1155. N becomes 577.5 and N has distinct 4 prime factors {3,5,7,11}.
Hence INSUFFICIENT

TAkeAwAY: If 2 is one of the prime factors of 2N, then N has x-1 primes (excluding 2) but if 2 is not one of the prime factors of 2N, then N has as many distinct primes factors as 2N itself.

ST2: 4 different prime numbers are factors of n^2
N^2 = (2*3*5*7)^2 then n is a multiple of (2*3*5*7)
thus # of primes in N will be the same as # of primes in n^2
Another example if n^2 = (3*5*13*19)^2, then n must be a multiple of (3*5*13*19)
Hence sufficient. There will be 4 different prime factors of N.

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Re: Number Properties [#permalink]  13 May 2013, 09:50
My answer is B.

from Stmt 1.
four different prime numbers are factors of 2n.
Let 2n = 2*3*5*7. in this case n= 3*5*7..meaning n has three prime factors.
let 2n = 2^3 *3*5*7. in this case n= 2^2 *3*5*7..meaning n has 4 prime factors.

hence its insufficient.

from stmt 2:
four different prime numbers are factors of n^2

let n^2 = a^2p * b^2q*c^2r*d^2s..where a,b,c,d are prime numbers and p,q,r,s are integers(as it is given that n is an integer).. taking square root on both sides
n = a^p * b^q*c^r*d^s...meaning n will have 4 prime factors.

sufficient.
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Re: How many different prime numbers are factors of the positive [#permalink]  03 Aug 2014, 00:13
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Re: How many different prime numbers are factors of the positive [#permalink]  24 Feb 2015, 17:18
I do not understand your explanation for statement one. Also 1 is a prime number too therefore 2n can have the following prime factors 1, 3, 5, 7. Please correct me if I am wrong. I just don't understand what makes you assumed that 2n can have less than four prime factors when the question states that 2n has four prime factors.
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Re: How many different prime numbers are factors of the positive [#permalink]  24 Feb 2015, 18:06
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mawus wrote:
I do not understand your explanation for statement one. Also 1 is a prime number too therefore 2n can have the following prime factors 1, 3, 5, 7. Please correct me if I am wrong. I just don't understand what makes you assumed that 2n can have less than four prime factors when the question states that 2n has four prime factors.

1 is most definitely NOT a prime number. As for your other question please refer to the example in my solution above.
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Re: How many different prime numbers are factors of the positive [#permalink]  02 Mar 2015, 08:12
mawus wrote:
I do not understand your explanation for statement one. Also 1 is a prime number too therefore 2n can have the following prime factors 1, 3, 5, 7. Please correct me if I am wrong. I just don't understand what makes you assumed that 2n can have less than four prime factors when the question states that 2n has four prime factors.

If it is difficult for you to understand or remember why 1 is not a prime number, remember that 1 can be deleted only to one number. However, prime numbers must be able to be deleted only by two numbers.
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Re: How many different prime numbers are factors of the positive   [#permalink] 02 Mar 2015, 08:12
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# How many different prime numbers are factors of the positive

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# Antifragile Random Walks

A million timesteps with a Pareto distribution with $latex \alpha =1$ and mode shifted down to $latex -11$ from $latex 1$. Notice how for most time steps, the walk moves downward. However, the rarer upticks are large, orders of magnitude larger than downward movements.

In [1]:
T = 1000000
X = np.zeros(T)

for i in range(5):
for t in range(T-1):
X[t+1] = X[t] + np.random.pareto(1) - 12

plt.plot(X)
plt.show()

# Cauchy Random Walks, 2D and 3D

1 million steps, with step size determined by a Cauchy distribution, and angle(s) by a flat distribution.

In [1]:
T = 1000000
X = np.zeros((T,2))

figsize(10,10)

for t in range(T-1):
stepSize = np.random.standard_cauchy()
direction = np.random.rand()*2*math.pi
xStep, yStep = cos(direction)*stepSize, sin(direction)*stepSize
X[t+1,0] = X[t,0] + xStep
X[t+1,1] = X[t,1] + yStep

plt.plot(X[:,0],X[:,1],lw=2)
plt.show()

In [2]:
X = np.zeros((T,3))

for t in range(T-1):
stepSize = np.random.standard_cauchy()
direction1 = np.random.rand()*2*math.pi
direction2 = np.random.rand()*2*math.pi

xStep = cos(direction2)*cos(direction1)*stepSize
yStep = sin(direction1)*stepSize
zStep = sin(direction2)*cos(direction1)*stepSize

X[t+1,0] = X[t,0] + xStep
X[t+1,1] = X[t,1] + yStep
X[t+1,2] = X[t,2] + zStep

from mpl_toolkits.mplot3d import Axes3D
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.plot(X[:,0],X[:,1],X[:,2])
frame1 = plt.gca()
frame1.axes.set_xticklabels([])
frame1.axes.set_yticklabels([])
frame1.axes.set_zticklabels([])

plt.show()

format: plain


text:
### Constraint Streams get some more love

We have recently merged a lot of improvements to OptaPlanner‘s Constraint Streams API
in order to make it faster and even easier to use.
Let’s take a closer look at some of them.

## Constraint Streams by default

I find it hard to believe that it’s been over two years already since we’ve introduced Constraint Streams into OptaPlanner; how time flies!

To get you started, OptaPlanner has always brought a wide selection of examples and more recently quickstarts.
While quickstarts have always used Constraint Streams and Constraint Streams only, the examples being older in age, they typically bring more scoring implementations, such as DRL or incremental Java.
To clearly state that Constraint Streams are now the scoring method of choice, we have recently converted all OptaPlanner examples to use Constraint Streams by default.
That said, we do not intend to deprecate any of the other scoring methods any time soon.

## New constraint collectors

Seeing increased adoption of Constraint Streams, we have been expanding the selection of constraint collectors available out of the box.
Recently, we have added the following new constraint collectors:

• average() constraint collector allows you to calculate an average of a group of items.
• compose() collector allows you to merge results of several constraint collectors. (For example, the average() collector is a composite of count() and sum().)
• conditionally() constraint collectors allows you to only delegate to another collector if a given condition is met first.

Especially with the latter two collectors, the expressive power of Constraint Streams has grown significantly.

## Faster constraint collectors

Until recently, constraint collectors toList(), toSet(), toSortedSet(), toMap() and toSortedMap() have been comparatively slow.
We have now changed the underlying implementation to be much more friendly to incremental calculation, and the end result is a performance improvement on the order of magnitudes on large enough data sets.

## Experimental constraint collectors

As we were implementing constraint providers for all the various examples, we noticed some constraints (which were already hard to implement in DRL) were impossible with Constraint Streams as it stands.
Consider the following Nurse Rostering DRL-based constraint to penalize too many consecutive shifts:

    rule "insertEmployeeConsecutiveAssignmentStart" when
... // Omitted for brevity.
then
insertLogical(new EmployeeConsecutiveAssignmentStart($employee,$shiftDate));
end

rule "insertEmployeeConsecutiveAssignmentEnd" when
... // Omitted for brevity.
then
insertLogical(new EmployeeConsecutiveAssignmentEnd($employee,$shiftDate));
end

rule "insertEmployeeWorkSequence" when
EmployeeConsecutiveAssignmentStart($employee : employee,$firstDayIndex : shiftDateDayIndex)
EmployeeConsecutiveAssignmentEnd(employee == $employee, shiftDateDayIndex >=$firstDayIndex, $lastDayIndex : shiftDateDayIndex ) not EmployeeConsecutiveAssignmentEnd(employee ==$employee, shiftDateDayIndex >= $firstDayIndex && <$lastDayIndex)
then
insertLogical(new EmployeeWorkSequence($employee,$firstDayIndex, $lastDayIndex)); end rule "minimumConsecutiveWorkingDays" when$contractLine : MinMaxContractLine(
contractLineType == ContractLineType.CONSECUTIVE_WORKING_DAYS, minimumEnabled == true,
$contract : contract,$minimumValue : minimumValue
)
EmployeeWorkSequence(getEmployee().getContract() == $contract, dayLength <$minimumValue, $dayLength : dayLength) then scoreHolder.addSoftConstraintMatch(kcontext, ($dayLength - $minimumValue) *$contractLine.getMinimumWeight());
end

As you can see, this is a lot of DRL which fundamentally does this:

1. Infer the first shift in a sequence of consecutive shifts.
2. Infer the last shift in a sequence of consecutive shifts.
3. Infer all the non-overlapping sequences.
4. Penalize sequences longer than what is prescribed by the contract.

Now consider how the same constraint is accomplished with Constraint Streams, using the experimental consecutive constraint collector:

    Constraint consecutiveWorkingDays(ConstraintFactory constraintFactory) {
return constraintFactory.from(MinMaxContractLine.class)
.filter(minMaxContractLine -> minMaxContractLine
.getContractLineType() == ContractLineType.CONSECUTIVE_WORKING_DAYS &&
minMaxContractLine.isEnabled())
.join(ShiftAssignment.class,
Joiners.equal(ContractLine::getContract, ShiftAssignment::getContract))
.groupBy((contract, shift) -> shift.getEmployee(),
(contract, shift) -> contract,
ExperimentalConstraintCollectors.consecutive((contract, shift) -> shift.getShiftDate(),
ShiftDate::getDayIndex))
.flattenLast(ConsecutiveInfo::getConsecutiveSequences)
.filter((employee, contract, shiftList) -> contract.isViolated(shiftList.getLength()))
.penalize("consecutiveWorkingDays", HardSoftScore.ONE_SOFT,
(employee, contract, shiftList) -> contract.getViolationAmount(shiftList.getLength()));
}

This constraint uses the experimental consecutive constraint collector to get us a list of all sequences of consecutive shifts.
This list is then flattened, giving us each sequence individually.
As you can see, this is a much more concise implementation of the same behavior, with the brunt of the logic hidden inside the constraint collector itself.

This constraint collector is not part of our public API and we consider it experimental.
Before we make it part of the public API, we are looking for your feedback to make sure it fits the various use cases that are out there.
If you have any questions or see issues in applying this pattern to your own constraints, do not hesitate to reach out to us.

## Conclusion

All of these improvements are available as of OptaPlanner 8.11.0.Final, coming soon to a mirror near you.

We will continue improving Constraint Streams as we find more and more problems to solve.
If you want to make sure we can solve your problems too, share them with us.

")

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# Related to channel power gain

I am having a doubt that why channel power gain is denoted by $$|h|^2$$. For example, if the signal received is $$y(n) = hx(n)+w(n)$$

where $$x(n)$$ is transmitted signal with power $$P$$ , $$w(n)$$ is AWGN with power $$N$$ and $$h$$ rayleigh fading channel.

The SNR from above equation is given as

$$\mathrm{SNR} = \frac{|h|^2P}{N}\tag{1}$$

My query is why $$|h|^2$$ is written in equation $$(1)$$?

Any help in this regard is highly appreciated.

• SNR stands for signal to noise ratio, i.e., it is the ratio of the power of the input signal to the power of the noise. In this case, the signal power is given as $h \times x[n] \times x^*[n] \times h^*$, which is equal to $|h|^2 P$. The noise power is given as $N$. Take the ratio, and you get the SNR which includes the absolute value of $h$. Sep 2, 2021 at 6:52
• Ok... Thank You so much...
– paru
Sep 2, 2021 at 8:35

In general, the channel coefficient $$h$$ is a complex number so it has magnitude and phase. Physically, $$|h|^2$$, is the attenuation/power-loss due to the channel and that is why you include it in the signal power calculation. The signal power is:
$$\mathbb{E}\big[hx[n](hx[n])^* \big]=|h|^2\mathbb{E}\big[|x[n]|^2\big]=|h|^2P$$
• The $|h|^2$ is taken out of Expectation because it is not depending on $n$. Is my understanding correct?


text:
# Thread: Avg. Value of a function: Word Problem

1. ## Avg. Value of a function: Word Problem

1. The problem statement, all variables and given/known data

The velocity v of blood that flows in a blood vessel with radius R and length l at a distance r from the central axis is v(r), where P is the pressure difference between the ends of the vessel and η is the viscosity of the blood.

(a)Find the average velocity v_ave (with respect to r) over the interval 0 ≤ r ≤ R.

(b)Compare the average velocity v_ave with the maximum velocity v_max.(v_ave/v_max)

So I need to set the problem up, into an integral form, to find the avg velocity for part A. The integration limits for the integral should be from zero to Big R, multiplied by (1 over big R) to get our average value. Now the problem is, every thing in the equation except the (1/4) is a variable. How would I even begin to integrate this?

Thanks.

2. Originally Posted by RedBarchetta
1. The problem statement, all variables and given/known data

The velocity v of blood that flows in a blood vessel with radius R and length l at a distance r from the central axis is v(r), where P is the pressure difference between the ends of the vessel and η is the viscosity of the blood.

(a)Find the average velocity v_ave (with respect to r) over the interval 0 ≤ r ≤ R.

(b)Compare the average velocity v_ave with the maximum velocity v_max.(v_ave/v_max)

So I need to set the problem up, into an integral form, to find the avg velocity for part A. The integration limits for the integral should be from zero to Big R, multiplied by (1 over big R) to get our average value. Now the problem is, every thing in the equation except the (1/4) is a variable. How would I even begin to integrate this?

Thanks.
$\int_{0}^{R}\frac{P}{4\eta l}(R^2-r^2)dr$

The only variable in the integrand is little r the rest are parameters(measured constants) so we are integrating with repsect to little r.

$\frac{P}{4\eta l} \int_{0}^{R} (R^2-r^2)dr=\frac{P}{4\eta l}\left( rR^2-\frac{r^3}{3}\right)_{0}^{R}=\frac{P}{4\eta l}\left( R^3-\frac{R^3}{3}\right)=\frac{2R^3P}{12\eta l}=\frac{R^3P}{6\eta l}$

3. Ahhh, so the other letters are just constants. Well that certainly makes the intergrating _a lot_ easier.

For part B, where do you get the v_max from?

4. Originally Posted by RedBarchetta
Ahhh, so the other letters are just constants. Well that certainly makes the intergrating _a lot_ easier.

For part B, where do you get the v_max from?
You need to maximize your velocity function

$v(r)=\frac{P}{4\eta l}(R^2-r^2)$

so taking the derivative we get

$\frac{dv}{dr}=\frac{P}{4\eta l}(-2r)$

setting equal to zero and solving for r, we get r=0. The maximum velocity occurs when r=0.

$v_{max}=v(0)=\frac{P}{4\eta l}(R^2-0^2)=\frac{PR^2}{4\eta l}$


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Revision history [back]

We write \begin{align} x=75\;000+X\ ,\\ y=200\;000+Y\ ,\\ \end{align} and the new function to maximize on $X,Y\ge 0$ is the following function $Q(X,Y)$:

sage: var( 'x,y,X,Y' );
sage: P(x,y) = 5/100*x*(1-x/150000-8/10^7*x*y) + 8/100*y*(1-y/200000-8/10^7*x*y)
sage: Q(X,Y) = P(X+75000, Y+200000).expand()
sage: Q(X,Y)
-1/25000000*X^2*Y - 1/15625000*X*Y^2 - 24001/3000000*X^2 - 79/2500*X*Y - 12001/2500000*Y^2 - 3760*X - 53627/25*Y - 236998125

The maximal value for Q is taken at the "corner" $(X,Y)=(0,0)$, because all monomials appearing in total degree $\ge 1$ have negative coefficients, so the maximal value for both P and Q is - 236998125, taken at "the one or the other corner".

Note: The theory tells us to search for the absolute extremal values

• in the inner of the closed definition domain, by searching all local extremal values, obtained as a solution of $P'_x=P'_y=0$, and

• at the (finite and/or infinite) boundary of it, by setting $x=75\;000$, $x=\infty$, $y=200\;000$, $y=\infty$, and computing here, in each of the four cases the supremum.

In our case there are no inner extremal points:

sage: for sol in solve( [diff(P,x) == 0, diff(P,y) == 0], [x,y] ):
....:     print sol
....:
[x == 816.8214285714286, y == 506.0324074074074]
[x == -816.2645502645503, y == -514.5402298850574]
[x == (13.33265402759081 + 1414.015105182754*I), y == (3.906674551362281 - 883.7900965450355*I)]
[x == (13.33265402759081 - 1414.015105182754*I), y == (3.906674551362281 + 883.7900965450355*I)]

and no solution is in the inner domain.

Also there are no maximal values at the infinite boundaries, since $P\to -\infty$ for either $x$ or $y$ going to $+\infty$.

For the remained boundaries, we can set $x=75\;000$, respectivel $y=200\;000$, getting two new maximization problems in the simpler lower dimension...


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# Family (math)

In mathematics, the concept of family is derived directly from the basic concept of function . The two terms agree in many ways. The difference between the two lies on the one hand in the form, i.e. in the way of writing and speaking, and on the other hand in the use and the meaning suggested by it. The representation of the family as a set of value pairs is particularly common, with the independent variable (s) being noted as the index (indices) of the dependent variable. If the function represented in this way is not injective , the set representation contains elements that only differ in pairs by the index.

Deviating from this, a “family of sets” or “set family” is sometimes understood to mean a set of sets (a so-called set system ), or a set family .

## properties

The notation consists of

• an indexed element symbol in round brackets,
• the specification of the domain of the index in the subscript (i.e. bottom right) of this expression in brackets and
• the indication of the source set of the elements of the family (informal in context or formal).

Example: or equivalent to for all . It corresponds to the function . ${\ displaystyle (a_ {i}) _ {i \ in I}}$${\ displaystyle (a_ {i} \ mid i \ in I)}$${\ displaystyle a_ {i} \ in A}$${\ displaystyle i \ in I}$${\ displaystyle f \ colon \, I \ to A, \, i \ mapsto a_ {i}}$

The called the members or the terms of the family and they are elements from the source volume or the indexed amount , ie index and the index set or the index range . A speech for this example would be: "A family of elements from with index from the index set ." The specification of the domain of the index if it does not matter or is apparent from the context, sometimes omitted: ${\ displaystyle a_ {i}}$ ${\ displaystyle A}$${\ displaystyle i}$${\ displaystyle I}$${\ displaystyle a_ {i}}$${\ displaystyle A}$${\ displaystyle i}$${\ displaystyle I}$

Example: . ${\ displaystyle (a_ {i}) _ {\}}$

To be distinguished from this (which is not always done) is the set of all members of the family, which is a subset of the source set and corresponds to the image set : ${\ displaystyle f (I)}$

Example: . ${\ displaystyle \ {a_ {i} \ mid i \ in I \} \ subseteq A}$

Some authors write families in this form , but there is a risk that the reader might mistake this for the crowd . ${\ displaystyle \ {a_ {i} \} _ {i \ in I}}$${\ displaystyle \ {a_ {i} \ mid i \ in I \}}$

The characteristics of families are as follows:

Two families and are exactly the same if and for each applies. ${\ displaystyle (a_ {i}) _ {i \ in I}}$${\ displaystyle (b_ {j}) _ {j \ in J}}$${\ displaystyle I = J}$${\ displaystyle a_ {i} = b_ {i}}$${\ displaystyle i \ in I}$

Schematically, the spellings for functions and families can be compared as follows:

function family
${\ displaystyle f \ colon \, I \ to A, \, i \ mapsto a_ {i}}$  ${\ displaystyle (a_ {i}) _ {i \ in I}}$with for everyone${\ displaystyle a_ {i} \ in A}$${\ displaystyle i \ in I}$
Picture or value ${\ displaystyle f (i) = a_ {i}}$  Term or member with index${\ displaystyle a_ {i}}$${\ displaystyle i}$
Domain of definition ${\ displaystyle I}$  Index amount ${\ displaystyle I}$
Image or value range ${\ displaystyle A}$  Source quantity or indexed quantity ${\ displaystyle A}$
Restriction with${\ displaystyle f | _ {J}}$${\ displaystyle J \ subseteq I}$  Partial family with${\ displaystyle (a_ {j}) _ {j \ in J}}$${\ displaystyle J \ subseteq I}$

More generally speaking, there are three interpretations of left total and right unambiguous relations, namely as:

• Function (mapping from I to A ),
• Occupancy (from I through A ),
• Indexing ( A indexed by I ).

A family is the indexing interpretation of a function with a special notation that does not use a special function symbol like the mapping notation.

The emphasis here is on interpretation . No new mathematical terms are introduced here, only alternative points of view of the same formal situation are given. The purpose of these alternative perspectives is to make them easier to use in special application situations, especially in calculating arithmetic.

For the set of families indexed with the index set I, whose members are all in A, one writes . If A and I are finite sets, then the following applies to their cardinality : ${\ displaystyle A ^ {I}}$

${\ displaystyle | A ^ {I} | = | A | ^ {| I |}}$.

## Examples of families and application situations

### Examples and cases

• Families with finite index sets, usually or , are called lists and the empty family is called empty list , but the indexed source set can be any. A list is also called a finite sequence and for is written in the same way , or the tuple , for the empty sequence . A list can also be understood as a word above the respective indexed source set.${\ displaystyle \ {1,2,3, \ ldots, n \}}$${\ displaystyle \ {0,1,2, \ ldots, n-1 \}}$${\ displaystyle (a_ {i}) _ {i \ in \ {\, \}} = \ {\, \}}$ ${\ displaystyle (a_ {i}) _ {i \ in \ {1, \ ldots, n \}}}$${\ displaystyle (a_ {i}) _ {i = 1, \ ldots, n}}$${\ displaystyle (a_ {i}) _ {i = 1} ^ {n}}$ ${\ displaystyle (a_ {1}, \ ldots, a_ {n})}$ ${\ displaystyle (\,)}$
• An infinite sequence , often simply called a sequence , is a family whose index set is countably infinite , usually the set of natural numbers or . Analogous to lists, infinite sequences can also be written in the form or like endless tuples . The members of infinite and finite sequences are called members .${\ displaystyle \ mathbb {N} = \ {1,2,3, \ ldots \}}$${\ displaystyle \ mathbb {N} _ {0} = \ {0,1,2, \ ldots \}}$${\ displaystyle (a_ {i}) _ {i \ in \ mathbb {N}}}$${\ displaystyle (a_ {i}) _ {i = 1} ^ {\ infty}}$${\ displaystyle (a_ {1}, a_ {2}, a_ {3}, \ ldots)}$
• In topology , a network is a generalization of a sequence; the family notation is also used here.
• Matrices are lists with index sets that are the Cartesian product of two finite sets. Has z. For example, if a list is the index set, it is called amatrix and it has a representation, the sublistsare then called rows and the sublists are called columns of the matrix.${\ displaystyle \ {(1,1), \ ldots, (1, n), \ ldots, (m, 1), \ ldots, (m, n) \}}$${\ displaystyle m \ times n}$${\ displaystyle {\ bigl (} a _ {(1,1)}, \ ldots, a _ {(1, n)}, \ ldots, a _ {(m, 1)}, \ ldots, a _ {(m, n )} {\ bigr)}}$${\ displaystyle {\ bigl (} a _ {(j, 1)}, \ ldots, a _ {(j, n)} {\ bigr)}}$${\ displaystyle {\ bigl (} a _ {(1, k)}, \ ldots, a _ {(m, k)} {\ bigr)}}$

The family notation is typically used for:

A lot is often mistakenly spoken of when a family is meant and required. For example , if one were to define the term linear independence for sets instead of families of vectors in the theory of vector spaces , one could not even formulate that two vectors different from the zero vector, etc. a. then are linearly dependent if they are equal. In that case they would together form only a one-element set, which is then linearly independent. Conversely, if necessary, a set can be understood as a family at any time by indexing it by yourself using the identical mapping to : ${\ displaystyle A}$${\ displaystyle \ operatorname {id} _ {A}}$${\ displaystyle A}$

${\ displaystyle (a) _ {a \ in A}}$.

### Families of pairwise disjoint subsets

If a family of sets with the properties and is to be, then according to the representation presented in this article it is a function with very special properties, and . An alternative representation in this case is a function . With this, and the pairwise disjointness results automatically. ${\ displaystyle (A_ {i}) _ {i \ in I}}$${\ displaystyle \ forall i \ in I \ colon \; A_ {i} \ subseteq A}$${\ displaystyle \ forall i, j \ in I \ colon \; i \ neq j \ implies A_ {i} \ cap A_ {j} = \ emptyset}$${\ displaystyle f \ colon \; I \ to {\ mathcal {P}} A}$${\ displaystyle A_ {i} = f (i)}$${\ displaystyle (A_ {i}) _ {i \ in I}}$${\ displaystyle g \ colon \; A \ to I}$${\ displaystyle A_ {i} = g ^ {- 1} (i)}$

## Individual evidence

1. for example Serge Lang : Algebra. Addison-Wesley, 1965.


text:
# Can I track grants against contributions?

Is there a way with CiviGrant to match money received as Contributions with money disbursed?

E.g. if I receive a donation for $100K for a scholarship fund, and I give out 3 scholarships of$20K each, is there a report that can show me that I have \$40K left in the fund?


text:
## nanoVNA – evaluation of a voltage balun – W2AU 1:1

In this article, I will outline an evaluation of a ‘classic’ voltage balun, the 1:1 W2AU voltage balun, specified for 1.8-30MHz.

These were very popular at one time, but good voltage baluns achieve good current balance ONLY on very symmetric loads and so are not well suited to most wire antennas.

Above is W2AU’s illustration of the internals.

Mine barely saw service before it became obvious that it had an intermittent connection to the inner pin of the coax connector. That turned out to be a poor soldered joint, a problem that is apparently quite common and perhaps the result of not properly removing the wire enamel before soldering.

Having cut the enclosure to get at the innards and fix it (they were not intended to be repaired), I rebuilt it in a similar enclosure made from plumbing PVC pipe and caps, and took the opportunity to fit some different output terminals and an N type coax connector.

Above is the rebuilt balun which since that day has been reserved for test kit for evaluating the performance of a voltage balun in some scenario or another. Continue reading nanoVNA – evaluation of a voltage balun – W2AU 1:1

## nanoVNA – RG6/U with CCS centre conductor MLL measurement

In my recent article RG6/U with CCS centre conductor – shielded twin study I made the point that it is naive to rely upon most line loss calculators for estimating the loss of this cable type partly because of their inability to model the loss at low HF and partly because of the confidence one might have in commonly available product. In that article I relied upon measured data for a test line section.

I have been asked if the nanoVNA could be bought to bear on the problem of measuring actual matched line loss (MLL). This article describes one method.

The nanoVNA has been OSL calibrated from 1-299MHz, and a 35m section of good RG6 quad shield CCS cable connected to Port 1 (Ch0 in nanoVNA speak).

A sweep was made from 1-30MHz with the far end open and shorted and the sweeps saved as .s1p files.

Above is a screenshot of one of the sweeps. Continue reading nanoVNA – RG6/U with CCS centre conductor MLL measurement

## RG6/U with CCS centre conductor – shielded twin study

Some online experts advise the use of synthesised shielded twin instead of ordinary two wire line for HF antennas claiming it is vastly superior.

Now it could be vastly superior for several reasons in all, but let’s focus on just one important parameter, loss under mismatch conditions.

The scenario then is the very popular 132′ multi band dipole:

• the famous 40m (132′) centre fed dipole;
• 20m of feed line being parallel RG6/U CCS quad shield with shields bonded at both ends;
• 7MHz where we will assume dipole feed point impedance is ~4000+j0Ω.

We will consider the system balanced and only deal with differential currents.

Now rather than depend on loss calculators, most of which don’t reconcile with measurement of CCS RG6/U, I will used measured loss. RG6/U with CCS centre conductor at HF gives a chart of measured loss of a sample of commercial grade CCS quad shield coax.

Above is a comparison of matched line loss (MLL) based on measurement of a length of RG6/U Quad Shield CCS cable and prediction from Simsmith of Belden 8215 (also CCS). The ripple is due to measurement system error, measurements were made quite some years ago with a AIMuhf. Continue reading RG6/U with CCS centre conductor – shielded twin study

## A dipole centre insulator from HDPE cutting board

I made a small dipole centre insulator from 10mm HDPE cutting board on the CNC router. HDPE is moderately UV resistant so should survive for some years outdoors.

The insulator is 100mm across its widest points. No provision is made for supporting coax, it is for use with home made open wire line which will fall from the dipole leg ends. If you want to use it with coax or ribbon feedline, then incorporate a tab to secure those lines.

## A model of current distribution in copper clad steel conductors at RF – capturing conductor curvature

A model of current distribution in copper clad steel conductors at RF laid out a model for current distribution, though ignoring curvature of the conductor in calculating current density vs depth.

A model for current distribution in a conductor is that for a homogenous conducting half space with surface current parallel to the interface. Current density at depth d is given by the expression $$J_r=J_R\frac{J_0(kr)}{J_0(kR)}$$ where δ is the skin depth $$δ=(ω \cdot µ \cdot σ)^{0.5}$$ and $$k=\frac{1-\jmath}{\delta}$$, σ is the conductivity). This takes into account curvature of the conductor surface, albeit with slower compute time.

Let’s compare the two algorithms on a test case at 1.8MHz being copper cladding of 67µm copper over a steel core for an overall diameter of 1.024mm (#18).

Above is a stacked image, the simpler algorithm is the feint plot.

There is a quite small difference in this case. When the expected loss of 400Ω line using the conductor is calculated, the result with the simpler algorithm is 1.3% less than the later one using the Bessel distribution.

## A Smith chart view of EFHW transformer compensation

I have written several articles on design of high ratio ferrite cored transformers for EFHW antennas.

Having selected a candidate core, the main questions need to be answered:

• how many turns are sufficient for acceptable InsertionVSWR at low frequencies and core loss; and
• what value of shunt capacitance best compensates the effect of leakage inductance at high frequencies?

Lets look at a simplified equivalent circuit of such a transformer, and all components are referred to the 50Ω input side of the transformer.

Above is a simplified model that will illustrate the issues. For simplicity, the model is somewhat idealised in that the components are lossless. Continue reading A Smith chart view of EFHW transformer compensation

## Stacking of Yagis and antenna effective aperture

The matter of stacking Yagis for improved gain is it seems a bit of a black art (and it should not be).

A common piece of advice is to visualise the capture area of the individual Yagi, and to stack them so that their capture areas just touch… with the intimation that if they overlap, then significant gain is lost.

Above is a diagram from F4AZF illustrating the concept. Similar diagrams exist on plenty of web sites, so it may not be original to F4AZF. Continue reading Stacking of Yagis and antenna effective aperture

## Desk review of the AAA-1C as an active dipole antenna

The AAA-1C is an amplifier for small receiving antennas by LZ1AQ. The amplifier is designed for use with one or two small loops or a short dipole (possibly comprising two small loops).

The datasheet contains some specifications that should allow calculation of S/N degradation (SND) in a given ambient noise context (such as ITU-R P.372). Of particular interest to me is the frequency range 2-30MHz, but mainly 2-15MHz.

The specifications would appear to be based on models of the active antenna in free space, or measurements of the device using a dummy antenna. So, the challenge is to derive some equivalent noise estimates that can be compared to P.372 ambient noise, and with adjustment for the likely effects of real ground.

Key specifications:

• plot of measured output noise of the amplifier, and receiver noise in 1kHz ENB;
• Antenna Factor (AF) from a simulation.

Above is the published noise measurements at the receiver input terminals. The graph was digitised and then a cubic spline interpolation used to populate a table. Continue reading Desk review of the AAA-1C as an active dipole antenna

## Small untuned loop for receiving – a design walk through #4

Small untuned loop for receiving – a design walk through #1 arrived at a design concept comprising an untuned small loop loaded with a broadband amp with input Z being a constant resistive value and with frequency independent gain and noise figure.

In that instance, the design approach was to find a loop geometry that when combined with a practical amplifier of given (frequency independent) NoiseFigure (NF), would achieve a given worst case S/N degradation (SND). Whilst several options for amplifier Rin were considered in the simple analytical model, the NEC mode of the antenna in presence of real ground steered the design to Rin=100Ω.

A question that commonly arises is that of Rin, there being two predominant schools of thought:

• Rin should be very low, of the order of 2Ω; and
• Rin should be the ‘standard’ 50Ω.

Each is limiting… often the case of simplistic Rules of Thumb (RoT).

Let’s plot loop gain and antenna factor for two scenarios, Rin=2Ω and Rin=100Ω (as used in the final design) from the simple model of the loop used at Small untuned loop for receiving – a design walk through #2.

Above, loop gain is dominated by the impedance mismatch between the source with Zs=Rr+Xl and the load being Rin. We can see that the case of Rin=100Ω achieves higher gain at the higher frequencies by way of less mismatch loss than the Rin=2Ω case. Continue reading Small untuned loop for receiving – a design walk through #4

## Small untuned loop for receiving – a design walk through #3

Small untuned loop for receiving – a design walk through #1 arrived at a design concept comprising an untuned small loop loaded with a broadband amp with input Z being a constant resistive value and with frequency independent gain and noise figure.

Small untuned loop for receiving – a design walk through #2 developed a simple spreadsheet model of the loop in free space loaded by the amplifier andperformed some basic SND calculations arriving at a good candidate to take to the next stage, NEC modelling.

The simple models previously used relied upon a simple formula for predicting radiation resistance Rr in free space, and did not capture the effects of proximity of real ground. The NEC model will not be subject to those limitations, and so the model can be run from 0.5-30MHz.

The chosen geometry was:

• loop perimeter: 3.3m;
• conductor diameter: 20mm;
• transformer ratio to 50Ω amplifier: 0.7; and
• height of the loop centre: 2m;
• ground: average (σ=0.005 εr=13).

## NEC-5.0 model results

The effect of interaction with nearby real ground is to modify the free space radiation pattern. The pattern at low frequencies has maximum gain at the zenith, and above about 15MHz the pattern spreads and maximum gain is at progressively lower elevation. For the purposes of a simple comparison, the AntennaFactor was calculated for external plan wave excitation at 45° elevation in the plane of the loop.

Above is a plot of loop Gain and AntennaFactor at 45° elevation along the loop axis. The frequency range is 0.5-30MHz as the NEC model is not limited by the simple Rr formula. Additionally there is some ‘ground gain’ of around 5dB due to lossy reflection of waves from the ground interface. Continue reading Small untuned loop for receiving – a design walk through #3


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# R or Cpp for some finance work involved complex numbers?

I need to implement some pricing functions which involve complex numbers. The equations involve various expressions such as $$Re$$ and $$Img$$ (i.e the real and imaginary part of the complex number), and I need to do some Fast Fourier Transforms and other things. Obviously the end result is always real, but the intermediate calculations are complex.

I have never used complex numbers before when I have programmed, so is there something particular I should be aware of? I want to use either R or C++: which would be more suitable for handling complex numbers? Are there any computational difficulties when dealing with complex numbers?

• A very warm welcome to Quant.SE! Thank you for question, please see my answer below. – vonjd Oct 5 '19 at 14:51
• ...for "your" question, of course ;-) – vonjd Oct 5 '19 at 16:24
• If my answer helped you it is good practice here to upvote and accept it - Thank you – vonjd Oct 7 '19 at 11:46

Why choose? C++ is perfectly integrated into R via the excellent Rcpp package (on CRAN). And you can use complex numbers there too:

library(Rcpp)
cppFunction("ComplexVector doubleMe(ComplexVector x) { return x+x; }")
doubleMe(1+1i)
## [1] 2+2i

doubleMe(c(1+1i, 2+2i))
## [1] 2+2i 4+4i

I would suggest starting out with R and if you run into performance problems, use C++ via Rcpp.

Just for reference, the same as above in Base R:

doubleMeR <- function(x) x+x
doubleMeR(1+1i)
## [1] 2+2i

doubleMeR(c(1+1i, 2+2i))
## [1] 2+2i 4+4i


text:
Today I've been playing with the induced subgraphs of the Clebsch graph. Several other interesting and well-known graphs can be obtained from it by deleting a small number of vertices and forming the induced subgraph of the remaining vertices.

To begin with, one simple construction of the Clebsch graph is to take all length-four binary strings as vertices, and to make two strings neighbors when they differ either by a single bit or by all four bits. So it has sixteen vertices and 40 edges, and can be visualized as a four-dimensional hypercube with the long diagonals added.

It can be split into two three-dimensional cubes, by partitioning the binary strings according to the value of their first bit. Since the cube is bipartite, we can color the whole Clebsch graph with four colors, by using two colors within each cube. This is optimal, because the largest independent sets in the Clebsch graph (the sets of neighbors of a single vertex) have five vertices, not enough for three of them to cover the whole graph.

The Clebsch graph can also be split in a different way into two copies of the Wagner graph:

Removing a vertex and its five neighbors from the Clebsch graph (a $$K_{1,5}$$ subgraph) leaves a copy of the Petersen graph as the remaining graph. Similarly removing a maximum independent set leaves a Petersen graph together with a single isolated vertex.

Removing a five-vertex cycle (such as the central cycle of the Petersen graph above) leaves a copy of the Grötzsch graph:

Removing a four-vertex maximal independent set (four independent vertices that do not have a single common neighbor) leaves a subdivision of the complete bipartite graph $$K_{4,4}$$, in which four edges forming a matching have been subdivided:

Removing a four-cycle leaves a twelve-vertex torus graph with 24-way symmetry:

Here are a couple of less-symmetric large planar induced subgraphs.

The second of these planar graphs can be formed by removing all binary strings with equal numbers of zeros and nonzeros, a six-vertex subset that induces a three-edge matching. If we instead partition the vertices into strings with even and odd parity, we get a partition of the Clebsch graph into two eight-vertex induced matchings.

The Schläfli graph (or its complement) has many induced Clebsch graphs inside it, so presumably it also has an interesting collection of symmetric induced subgraphs. But that's getting a bit too large for the sort of by-hand analysis I did to get the list here; it calls for automation instead.


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Acta Metall Sin  2018, Vol. 54 Issue (5): 727-741    DOI: 10.11900/0412.1961.2018.00027
 Special Issue for the Solidification of Metallic Materials Current Issue | Archive | Adv Search |
Directionally Solidified Porous Metals: A Review
Yanxiang LI1,2(), Xiaobang LIU1
1 School of Materials Science and Engineering, Tsinghua University, Beijing 100084, China
2 Key Laboratory for Advanced Materials Processing Technology, Ministry of Education, Tsinghua University, Beijing 100084, China
Abstract

This paper reviews the recent development of porous metals with directional pores, from the aspects of the solidification principle, fabrication method, properties and applications. This kind of porous metals is fabricated by a directional solidification process in pressurized gas atmosphere, utilizing a metal/gas eutectic reaction (Gasar). By controlling solidification direction, not only lotus-type porous structure but also radial-type porous structure can be produced. The coupled growth of solid/gas phases is discussed by applying a solution procedure similar to that in the classical Jackson-Hunt eutectic growth model. The working window considering hydrogen escape and the formation of directional solidification porous structure has been given. Three fabrication techniques including mold casting, continuous casting techniques and Bridgman-type directional solidification method are introduced. Two new progresses about the fabrication of directionally solidified porous structure are described in details: porous alloy with uniform directional pores and high-porosity directionally solidified porous aluminum. Since directionally solidified porous metals exhibit peculiar physical and mechanical properties such as light-weight, air and water permeability, and anisotropy of thermal and mechanical properties, they are suitable for applications in heat sinks, filters, biomaterials and so on.

 ZTFLH: TG249
Fund: Supported by National Natural Science Foundation of China (No.51371104)
 Fig.1  Cu-rich portion of Cu-H phase diagram under a hydrogen pressure of 0.1 MPa (T—temperature; XH—composition of hydrogen; Tm—melting temperature; Pext—external pressure; TE—eutectic temperature; XH,E—eutectic composition of hydrogen; XH,S—composition of hydrogen in solid phase)[18] Fig.2  Temperature dependence of hydrogen solubility in solid and liquid metals at 0.1 MPa hydrogen pressure[45] Fig.3  A schematic diagram for metal-gas eutectic unidirectional solidification and the corresponding coordinate system for solving the solute field (L—liquid phase; G—gas phase; S—solid phase; v—solidification velocity; rG—radius of the gas pore; rS—one half of the interpore spacing)[17] Fig.4  Comparison between predicted porosities and experimental values on Mg-H system at different hydrogen (a) and argon (b) gas pressures (Ptotal—total gas pressure;$PH2$—hydrogen pressure; PAr—argon pressure)[34] Fig.5  A comparison between experimental results and theoretical values of pore diameter and interpore spacing at different total gas pressures (v=0.4 mm/s, T=1023 K)[17] Fig.6  The working window of the partial pressure ratio and superheat (ΔT′) for the formation of lotus-type porous magnesium (L1—critical line for the formation of lotus-type porous magnesium; L2—critical line for the hydrogen escaping; L3—optimized parameter)[50] Fig.7  A schematic diagram for the effect of argon partial pressure on metal-hydrogen phase diagram[50] Fig.8  Typical structures of radial porous Mg/Cu[51](a) three-dimensional structure of radial po-rous Mg(b) cross section of radial porous Mg(c) cross section of radial porous Cu Fig.9  Three apparatus schematics for fabrication of directionally solidified porous metals by mold casting method(a) common device (1—graphite stopper; 2—graphite crucible; 3—heating coil; 4—high pressure chamber; 5—molten metal; 6—mold; 7—copper chiller) (b) 90° rotation device (c) 180° rotation device Fig.10  A schematic drawing of Bridgman directional solidification apparatus[37] (a) melting and temperature holding (b) unidirectional solidification Fig.11  Cross sections parallel (upper row) and perpendicular (lower row) to solidification direction of directionally solidified porous Cu-Mn alloy[61] Fig.12  Relation between porosity and solidification velocity in directionally solidified porous metals Fig.13  Experimental results of heat transfer coefficient of directionally solidified porous copper heat sink under different heat power densities (ΔP—pressure drop; U—flow rate)[97]


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# [plt-scheme] Using LaTeX in Scribble

 From: Matthew Flatt (mflatt at cs.utah.edu) Date: Mon Jun 22 07:34:20 EDT 2009 Previous message: [plt-scheme] Using LaTeX in Scribble Next message: [plt-scheme] Using LaTeX in Scribble Messages sorted by: [date] [thread] [subject] [author]

At Sun, 21 Jun 2009 13:55:14 +0100, "Paulo J. Matos" wrote:
> I have started to write the docs for my model checker in LaTeX,
> however, after looking at Scribble I might give it a try.
> The issue is that some parts of the docs contain examples of the
> formal language I use and to display it in LaTeX I have a special mode
> for the listings package. Is it possible for me to use this listings
> package inside scribble or to \include a latex document in the middle
> of a scribble document?

Yes.

Overall, support for non-documentation uses of Scribble is still
evolving, but it already works well enough for many purposes. I've used
Scribble to write a grant proposal and a few papers. Naturally, we used
Scribble to write the Scribble ICFP'09 paper:

http://www.cs.utah.edu/plt/publications/icfp09-fbf.pdf

The bridge between Scribble and Latex or HTML is documented in Chapter
18 of the Scribble manual:

http://docs.plt-scheme.org/scribble/config.html

The documentation includes a small example. For a big example, see the
source of the Scribble paper, which is available as the
`mflatt/scribble' Planet package.

 Posted on the users mailing list. Previous message: [plt-scheme] Using LaTeX in Scribble Next message: [plt-scheme] Using LaTeX in Scribble Messages sorted by: [date] [thread] [subject] [author]


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# Feedback on new package for creating on-line quizzes from latex

WebQuiz has now been released as a package on ctan. It will be included in TeXLive 2019. Accordingly, the links below have been updated to point to ctan, or disabled.

If you want to install WebQuiz from the ctan zipfile then run:

webquiz --tex-install

from an administrator account, or use

 sudo webquiz --tex-install

on unix-like systems (including macosx). This will install the package into the "standard" locations in the TeXLive tree. (The webquiz executable is in the webquiz/scripts directory of the zipfile.) I am not sure what this will do on a windows PC.

Arguably this question is either too broad or outside of the scope of TeX.SX, so I accept that it may be closed. As I only want feedback I'd also be happy if this was made a community wiki.

I have a close-to-finished latex package, WebQuiz , documentation, that can be used to write on-line quizzes starting from "normal" latex files. The package is basically ready for uploading to ctan but before I do this I would be really grateful for feedback from TeX.SX, both from TeX experts and from people who might use such a package. My hope is to fix any existing problems before the package appears on ctan rather than having to put out a new release soon after the code first appears.

Using an example from a "real" quiz, the idea of the package is that with a latex file like this:

\documentclass{webquiz}
\title{Quiz 1: Numbers and sets}
\begin{document}

\begin{question}
Which of the following are correct ways of writing the set
$A=\{x \in \mathbb{R} \mid -3 < x \leq -1 \text{ or } x\geq 0\} ?$
\begin{choice}
\incorrect $(-3,\infty)$
The interval \response $(-3,\infty)$ includes the real numbers between $-1$ and $0$,
which do not belong to $A$.

\incorrect $[-3,\infty)$
\response The interval $[-3,\infty)$ includes $-3$, as well as
the real numbers between $-1$ and $0$, none of which belong to $A$.

\incorrect $[-3,-1]\cap[0,\infty)$
\response The interval $[-3,-1]\cap[0,\infty)$ is the empty set $\emptyset$. As $A$ is not empty (for example, $A$ includes $-1$), this option cannot be correct.

\incorrect $(-3,-1)\cup[0,\infty)$
\response The interval $-1$ is not in $(-3,-1)\cup[0,\infty)$, but $-1$ is in $A$.

\correct $(-3,-1]\cup[0,\infty)$
\end{choice}
\end{question}

...more questions...
\end{document}

you can produce an on-line quiz that looks something like this (after some has unwisely answered option b):

WebQuiz displays one question at a time. It is a formative assessment tool. It does not collect assessment data or worry about any of the associated "security issues" this would entail. The quizzes made with the package are ideal for use as a revision resource. One key feature is that the quiz writer can give feedback to the students based on each possible answer that they can give to questions.

To write quizzes using WebQuiz you only need working knowledge of LaTeX. Nothing more. Behind the scenes the conversion of the quiz from LaTeX into HTML is done using using tex4ht (and make4ht) and some python code. In more detail, tex4ht, make4ht, xml, python, HTML, mathjax, css and javascript all play a role but the LaTeX user only needs to know LaTeX.

As tex4ht is the underlying engine there certainly exists LaTeX code that WebQuiz cannot cope with. This said, the system has already been used to write literally thousands of quiz questions without any problems. In particular, it is possible to use tikz and pstricks in WebQuiz quizzes, although (very?) complicated drawings with these packages may fail.

I am interested in any feedback people are willing to give me. I am particularly interested in knowing the following:

• Is the zip file packaged correctly for ctan?
• Is the installation process clear and does it work? (The installation process works for me but it may fail for others. Once the package is on ctan this will be easier)
• Are there any bugs or obvious improvements?
• Is there anything missing from, or unclear in, the documentation?
• Are there any font encoding issues? Perhaps unwisely everything is processed as UTF-8
• In order to keep settings when the tex system is updated the package saves a system configuration file in \$TEXMFLOCAL/scripts/webquiz/webquizrc. Is this the appropriate place and will this work under windows and MikTeX?
• Are there any problems under windows? Preferably with a fix because I do not have access to a windows computer so I am unable to test such issues.

The first complete version of WebQuiz was written around 2004. This year I have almost completely rewritten it whilst making it compatible with HRML5 and, more significantly, changing it from in-house code to a more fully featured package. The bulk of the code has been used extensively, so it is well tested. Some of the new fewer features, such as the "programmable" \BreadCrumbs command and the installation procedure, are relatively untested. I would not be too surprised if I have broken something with the many recent changes that I have made whilst packaging it. This is why I'd like some feedback before putting the package on ctan.

WebQuiz requires an update-to-date version of LaTeX, including TeX4ht and make4ht, a recent version of python 3 and, of course, a web server. Page 5 of the manual, which is the file webquiz.pdf in the doc directory of the zip file, describes the steps necessary to get the code working. The short version is:

• the files in the latex directory need to be put in the LaTeX search path (after which you have to run texhash or equivalent)
• the MathQuiz program is in the scripts directory. On unix like systems the best option is to make a link like ls -s webquiz.py /usr/local/bin/webquiz
• ideally the files in the scripts/www directory need to be put somewhere on your web server, which can be done using the command webquiz --initialise. In fact, WebQuiz will still work even if you don't do this but you will get (harmless) warning messages about initialising whenever you run the program and similar warning messages on every web page you create
• documentation can be found in the doc directory and example code from the manual in the scripts/www/examples directory.

Once WebQuiz is installed, you can test your quiz file quiz.tex first using (pdf)LaTeX, to create a DVI or PDf file, and then use

webquiz quiz.tex

from the command-line to produce the file quiz.html, which you can view using your web browser.

I have used WebQuiz on linux and mac operating systems. Several groups have used it under windows, but I don't have access to a windows machine so I have not done this myself.

Finally, I apologise for the length of this post, any bugs in the package and for the many typos here and in the documentation that I am sure are there...

• This looks like a very useful package to me! I haven't tried installing yet, but I will. Meanwhile, is there anything in the package that restricts its application to math questions? Could it not also be used for multiple choice quizes that contain mostly text? If so, I would consider changing the name to something that suggests a broader scope. Can graphics be included via \includegraphics? Do chemical formula packages like mhchem or chemformula work? Sep 26, 2017 at 13:20
• Do you remember which line was the permissions error on? You can see the python traceback, and a lot of other info that you probably don't want to see, if you use mathquiz --debugging ...
– user30471
Sep 26, 2017 at 22:52
• The permission error was my own - I had created some directory as root and then trying to put files in it as myself. However, my point is that this catch-all-and-tell-nothing clause gets in the way. Trying to run mathquiz on pst2pdf.tex, I get LaTeX Warning: File pst2pdf/pst2pdf-fig-1.svg' not found on input line 11. I see neither pstoedit nor dvi2svgm mentioned in the console output. I do have ghostscript installed and use it all the time. Sep 26, 2017 at 23:00
• quizmaker would be better than mathquiz, but exercisemaker or something would be better, I think. Even if you include assessment capabilities later, the package still won't be limited to this kind of use. Not very inspirational, but it is quite descriptive. What about exercisewizard or exwizz or ...? I don't know. I'm hopeless at this. Why don't you ask in chat?
– cfr
Sep 28, 2017 at 0:41
• Oh no, "possibly without ducks"! I wanted to suggest quizquaz! Sep 29, 2017 at 1:50

## 1 Answer

It has been quite some time since I posted this question so I am happy to say that the package is finally available on ctan. As suggested in the comments above, the name of the package has been changed to WebQuiz. The package has expanded a little since I posted the question. In particular, it now includes support for 11 languages (in principle, support for other languages is easy to add) and different colour themes (some of which are, arguably, horrible -- sorry!).

Apart from asking about the functionality of the package, the main question that I asked in the post was about how best to package the code for ctan (and TeXLive). I was told by the ctan and TeXLive maintainers that they strongly preferred a zipfile (and not a TDS file) with:

• a README file in the top directory
• separate directories for the latex, doc and script components of the package
• TeXLive asked for a UNIX man page, since the package contains a standalone executable
• I initially had all all of the web files, which need to be installed when the program is initialised, in the scripts directory. TeXLive asked for these files to be moved to the doc directory as files in the scripts` are in the runtime search tree
• if a file needs to be included twice in different directories then the second copy should be a symbolic link


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# get 2nd and 3rd control points of a Cubic Bezier Curve fit in a rectangle

I would like to fit a Cubic Bezier Curve in a rectangle, and wondering how to get the 2nd and 3rd control points's Y value, illustrated below:

(Sorry the rectangle is bit distorted.) Basically, given the position of first point (x1, y1), fourth point (x4, y4), and the height of middle point h, how can I get the y2 and y3 from this curve?

curve generated using: http://demofox.org/bezcubic.html

$$\vec{B}(1/2)=\Sigma_{i=0}^3 \phi_{3,i}\vec{b_i}$$ $$B_y(t=1/2)=\phi_{3,0}*b_{0,y}+\phi_{3,1}*b_{1,y}+\phi_{3,2}*b_{2,y}+\phi_{3,3}*b_{3,y}\\ =\phi_{3,1}*b_{1,y}+\phi_{3,2}*b_{2,y}\\ =\binom{3}{1}t^1(1-t)^{3-1}*b_{1,y}+\binom{3}{2}t^2(1-t)^{3-2}*b_{2,y} \\ =3*(1/2)^3*b_{1,y}+3*(1/2)^3*b_{2,y}=3*(1/2)^2*b_{1,y}=50$$ $$b_{1,y}=b_{2,y}=200/3$$


text:
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After we have explained what Boolean functions and what truth tables are, it is time to use these concepts to introduce important examples of connectives.

## Definition: Negation

Let $$L$$ be a set of propositions with the interpretation $I$ and the corresponding valuation function $[[]]_I$. A negation “$$\neg$$” is a Boolean function

$\neg :=\begin{cases}L & \mapsto L \\ x &\mapsto \neg x \\ \end{cases}$

defined by the following truth table:

Truth Table of the Negation
$[[x]]_I$ $[[\neg x]]_I$
$$1$$ $$0$$
$$0$$ $$1$$

### Notes

• The negation of any proposition changes from one truth value to the other.
• By the axiom of bivalence of truth, the negation of any negated proposition $\neg x$ must be the proposition itself: $\neg(\neg x)=x.$
• Later in the text, it will turn out that negation is one of the most important connectives: We can construct successfully logical systems even if we dispense some connectives, but we cannot do without the negation.

| | | | | created: 2014-06-22 20:51:07 | modified: 2018-05-04 23:32:35 | by: bookofproofs | references: [7838]


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# How do I define the domain of a function?

I have a function of two variables, f[x_, y_], and I would like to restrict the domain to values of x and y greater than zero. How do I do this?

I also want to plot the function for restricted values. I have tried to use RegionFunction -> Function[{x, y}, ...], but this doesn't work.

-
f[x_?Positive, y_?Positive] := ... –  rm -rf Apr 21 '13 at 21:30
3D plots, by the basic syntax of Plot3D, are restricted to a rectangular domain x ∈ {x_min, x_max} and y ∈ {y_min, y_max}. If doesn't work for you, please tell us what is the shape of the domain you have in mind. –  m_goldberg Apr 21 '13 at 23:09

## 1 Answer

You can also use Boole ... provided you are happy for your func to return 0 when the domain conditions are not met. For example:

f[x_, y_] := Boole[x > 0 && y > 0]  Cos[x] Sin[y]

Plot3D[f[x, y], {x, -1, 2}, {y, -1, 2}]

Alternatively, you can set up a Piecewise function with explicit settings for what is to be returned (e.g. Indeterminate) when your conditions are not satisfied.

-


text: Lemma. There must be some pattern that can be described in a certain way. Sequential Convergence. It is fundamental but important tools in analysis. Examples It is used in the analysis process, and it always concerns about the behaviour of the function at a particular point. A sequence in Xis a function from N to X. Should it be necessary that sequence values are never equal to its limit? Example 267 Let S= [a;b] and x2[a;b]. Section 3 Sequences and Limits Deﬁnition A sequence of real numbers is an inﬁnite ordered list a 1,a 2,a 3, a 4,... where, for each n ∈ N, a n is a real number. Examples are always helpful to understand the key points about a new concept. Example of a Loop Sequence. As another example, consider the sequence. Definition. Example The sequence 1 n ∈N is convergent with limit 0. Thus, the sequence is … They also crop up frequently in real analysis. Example 1 1 n n 1 Deﬁnition. The starting value of the sequence is given. It could be that x2Aor that x=2A. Because to be a limit point you have to have another element in the set in the neighborhood. If we look at the subsequence of odd terms we have that its limit is -1… In words, a sequence is a function that takes an input from N and produces an output in X. Our limit in this case, or our claim of a limit, is 0. We say a sequence is divergent if it does not converge to a real limit. Let {f n} be the sequence of functions on R deﬁned by f … Show Step-by-step Solutions. We note that our definition of the limit of a sequence is very similar to the limit of a function, in fact, we can think of a sequence as a function whose domain is the set of natural numbers $\mathbb{N}$ . Scroll down the page for more examples and solutions. Examples. 2. for every x in D, is called the pointwise limit of the sequence {f n}. Limit sup and limit inf. Deﬁnition 2. The two notations for the limit of a sequence are: lim n→∞ {an} = L ; an → L as n → ∞ . Deﬁne the sign function sgn : R → R by sgnx = 1 if x > 0, 0 if x = 0, −1 if x < 0, Then the limit lim x!0 sgnx doesn’t exist. Example 1.6. So this would be zero minus epsilon. It could be that x2Aor that x=2A. 3.3. A number $$l$$ is said to be a limit point of a sequence $$u$$ if every neighborhood $${N_l}$$ of $$l$$ is such that $${u_n} \in {N_l}$$, for infinitely many values of $$n \in \mathbb{N}$$, i.e. Count limit of sequence $$\lim_{n \to \infty} \frac{1}{n}+3$$. The set Z R has no limit points. any convergent sequence of points in L has limit in L. A sequence is bounded above if and only if supL < ∞. For example, if … for any ε > 0, u n ∈ ( l – ε, l + ε), for finitely many values of n ∈ N. Evidently, if l = u n for infinitely many values of … Example 2.4Prove that the sequence, s n= n+ 1 n+ 2 does not converge to 0: Proof. Now, this is saying for any epsilon, we need to find an M such that if n is greater than M, the distance between our sequence and our limit is going to be less than epsilon. Then, a formula to generate the nth term from one or more previous terms. In a topological space $${\displaystyle X}$$, a point $${\displaystyle x\in X}$$ is said to be a cluster point (or accumulation point) of a sequence $${\displaystyle (x_{n})_{n\in \mathbb {N} }}$$ if, for every neighbourhood $${\displaystyle V}$$ of $${\displaystyle x}$$, there are infinitely many $${\displaystyle n\in \mathbb {N} }$$ such that $${\displaystyle x_{n}\in V}$$. Approaching Infinity. We then say that zero is the limit (or sometimes the limiting value) of the sequence and write, lim n → ∞an = lim n → ∞n + 1 n2 = 0. lim n → ∞ a n = lim n → ∞ n + 1 n … sequence. f(x) = x 2 as x → 3 from below. Sometimes when calculating a limit, the answer varies depending on the path taken toward $$(a,b)$$. ); but its range, $$\{0,1\}$$, has no cluster points (being finite). For anyϵ >0, there are at most … The sequence f1;1;1;:::ghas 1 as the limit, but it has no accumulation point. This sequence does not converge, however, if we look at the subsequence of even terms we have that it's limit is 1, and so $1$ is an accumulation point of the sequence $((-1)^n)$. Example 3 Three examples: 1. More Example Problems for Limit Point of a Sequence Example problem 2: Find out the 5th term of a geometric sequence if a1 = 70 and the common ratio (C.R) r = 2 Solution: Use the formula a_n = a_1 * r^(n-1) that gives the nth term to find a_5 as follows a_5 = a_1 * r^(5-1) = 70 * (2)4 = 70 * 16 After simplify this, we get = 1120. R with the usual metric Sets sometimes contain their limit points and sometimes do not. If such an L exists, we say {an} converges, or is convergent; if not, {an} diverges, or is divergent. Here are some more sequence diagram templates and examples that you can edit right away.. Sequence Diagram Tutorial – SlideShare Presentation If $$\varepsilon > 0$$, $${u_n} = 1 \in \left( {1 – \varepsilon ,1 + \varepsilon } \right)$$ for only a finite number of values of $$n$$ then $$l$$ is not a limit point of $$u$$. A point x2R is a limit point of Aif every -neighborhood V (x) of xintersects A at some point other than x, i.e. Try the free Mathway calculator and problem solver below to practice various math topics. Example 1: If a sequence $$u$$ is defined by $${n_n} = 1$$, then $$1$$ is the only limit point of limit point in . For example, any sequence in Z converging to 0 is eventually constant. And it is written in symbols as: limx→1 x 2 −1x−1 = 2. A sequencehas limitif infL= supL. is an example of a -space: every finite set is closed in (and finite sets are the only closed sets except for the space itself), but every sequence having infinite number of different points converges to every point. Then for all $$n$$, $$\left| {{u_n} – \alpha } \right| = \left| {1 – \alpha } \right|\not < \varepsilon$$. So this is negative epsilon, 0 minus epsilon, 0 plus epsilon. R with the usual metric Sets sometimes contain their limit points and sometimes do not. If for every $$\varepsilon > 0{\text{ }}\exists m \in \mathbb{N}$$ such that $${u_n} \in \left( {l – \varepsilon ,l + \varepsilon } \right)$$, $$\forall n \geqslant m$$ or equivalently $$\left| {{u_n} – l} \right| < \varepsilon$$ $$\forall n \geqslant m$$, then $$l$$ is a limit point of the sequence $$u$$ .In such a case it can be easily seen that $$l$$ is the only limit point of the sequence. If $${\displaystyle X}$$ is a metric space or a first-countable space (or, more generally, a Fréchet–Urysohn space), then $${\displaystyle x}$$ is cluster point of $${\displaystyle (x_{n})_{n\in \mathbb {N} }}$$ if and only if $${\displaystyle x}$$ is a limit of some subsequence of $${\displaystyle (x_{n})_{n\in \mathbb {N} }}$$. Example: Find the limit of the sequence {n-n 3. Step 2: Enter your x-values into the given function. There is absolutely no reason to believe that a sequence will start at $$n = 1$$. Note 1: Limit Points Notice that the de nition of a limit point xof Adoes not say anything about whether or not x2A. }{(n+1)!\cdot (n+2)-(n+1)! Count limit of sequence $$\lim_{n \to \infty} \frac{1}{n}-\sqrt{2}$$. If $$\eta$$ is an arbitrary small positive number and given any $$k > 0$$ then $$k\eta$$ is also an arbitrary small positive number, this follows immediately if we take $$0 < \eta < \varepsilon /2k$$ for an $$\varepsilon > 0$$. The same is true for b. The limit of (x 2 −1) (x−1) as x approaches 1 is 2. Limit Points of a Sequence. Given >0, the interval (a ;a+ ) contains in–nitely many points of (a;b) nfagthus showing ais a limit point of (a;b). The Limit Inferior and Limit Superior of a Sequence Proof The superscript 1 plays the role of adding another level of subscripting which is pretty ugly! Find a pointwise convergent sequence of differentiable functions such that the limit function is continuous but not differentiable. We call a n the n-th term of the sequence. Non-example: If a subset of a metric space is not closed, this subset can not be sequentially compact: just consider a sequence converging to a point outside of the subset! For this example, I’ll use the sequence values for x from above: 2.9, 2.99, 2.999, 2.9999, 2.99999. 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If it is bounded and a limit pint of the sequence 1 n ∈N is convergent with limit.... You have to have another element in the neighborhood } \ ) the starting point is the. Aand bare limit points and sometimes do not, b ) \ ), has no cluster points the. Certain way the infimum interval ( 0 ; 1 ; 1 ;:. Is continuous but not differentiable closer to zero on limit points two integers count limit of sequence \ 0,1,0,1. Set in the sequence { n-n 3 examples are always helpful to understand the key points about a concept! Number in the analysis process, and so on denoted by lim sup or s n= n+ 1 n+ does... The next two examples of a limit point is called the limit of the sequence (! The steps, then we will draw a general conclusion the infimum Let  other $... Taken toward \ ( \lim_ { n } +3\ ) real numbers, the sequence approaches as general. The numerator and denominator n ( x ) = x 0 definitions, thier names upper! 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