text:
# Batteries in parallel vs voltage signals in parallel

From a practical point of view, when you connect two audio signals to one speaker the result is that both audio signals can be heard. That means the two voltage waves were added, correct?

However, when you connect two batteries in parallel, the resulting voltage is not the sum of both batteries, it's more like the average.

This is all from real-world observasions. I don't know if this two cases are supposed to be conceptually the same.

Why does the voltage bahave differently in this two cases?

• It's all about the frequency. – Ignacio Vazquez-Abrams Jan 25 '15 at 1:22
• @IgnacioVazquez-Abrams, can you elaborate? – GetFree Jan 25 '15 at 1:39
• They are not added, but rather interfered – Jason Pyeron Jan 25 '15 at 13:00

Output is always the weighted average (superposition) of inputs.

Consider the following circuit in which two voltage sources are connected to the same load.

simulate this circuit – Schematic created using CircuitLab

Let V1 and V2 be DC sources then by superposition theorem, $$V_O = (V_1 + V_2)/3 \tag{1}$$

Assumed that R1,R2 = RL.

If V1 and V2 be monotonic ac signal sources then we can write, $V1 = A_1 \cos (2\pi f_1 t)$ and $V2=A_2\cos (2\pi f_2 t)$. Then,

case1: $f_1 = f_2 = f$ then, $$V_O = \frac{(A_1+A_2)}{3}\cos(2 \pi f t)\tag2$$

Here both sources generates same tone when connected to a speaker (load). Then output is same tone with output amplitude = superposition (weighted average) of the two input signals.

case2: $f_1 \ne f_2$ then, $$V_O = \frac{A_1 cos(2 \pi f_1 t) +A_2 cos(2 \pi f_2 t)}{3} \tag3$$

Here both sources generates different tone when connected to a speaker (load). When both connected together, output signal will have both the tones ($f_1$ and $f_2$) as given by expression (3). Hence both the tones can be heard at the output.

If V1 and V2 are audio signals then they will have lot of frequency components in it (given by the Fourier transform). When connected to same speaker, output will also contain all these frequencies and hence one can hear both the audio signals.

• I think I get it. The voltage is always the average, but since the average is just the sum divided by a constant, the human ear doesn't notice the difference. – GetFree Jan 25 '15 at 10:32
• @GetFree Yes. the amplitude of the audio signals are also reduced in the same way. – nidhin Jan 25 '15 at 18:35

Connecting batteries in parallel is sometimes done to boost current output, but is not a good idea if you have two different voltage batteries. Unless they are exactly the same voltage--and often even those of the same kind aren't, due to discharging--one will be charging the other. Not only can this be dangerous if the batteries aren't designed to be recharged, but the voltage you measure might not be either of the two batteries' rated voltages.

If you connect two signal lines to the same speaker, you're right: you do hear both signals. However, the same principle applies; anytime one of your audio signals is at a higher voltage (i.e. is 'louder') than the other (for example, you could have vocals on one channel, and guitar+drums on the other), the louder will drive current through the other. This 'cross-talk' leads to distortion. Mixers--which are designed to add audio signals in the way you imagine, linearly--resolve this issue by using op-amps: they use a virtual ground at the point where the signals meet, so that no current wants to flow through the other signal line-- thus distortion is minimized.

• You mention the problems of doing one or the other, but my question is why voltage seems to behave differently in each case. In one case there seems to be an addition (audio signals) in the other there is not (batteries) – GetFree Jan 25 '15 at 2:35

First the bad news. There is absolutely no difference between the math performed when joining two batteries together as when mixing two audio signals. Both are a weighted average based on the impedances in the circuit.

So what about the audio signals? Well, as in all things audio, we'll start with pure sine waves.

If we take two sine waves with the same amplitude, frequency, and phase and mix them, we get an identical sine wave as the output. This is... not that exciting a revelation, nor should it be.

If we take two sine waves with the same frequency and phase but different amplitudes, the result is a weighted average of the two input waves. So we end up with a sine wave with a different amplitude (or no signal at all, if the amplitudes were the same magnitude but opposite polarities).

Now we take two sine waves with different frequencies and mix them. The result... is no longer a sine wave. But one thing remains true: the voltage of the resultant wave at any given point is the weighted average of the voltages of the input waves at the same point. This remains true if we mix three sine waves together. Ten sine waves. A hundred. A thousand. A hundred thousand.

So then the big question is why does it sound like the two original audio waves mixed together instead of one big mass of sound? That's a question for Biology.SE, but the spoiler is that the cochlea performs a Fourier transform of incoming pressure waves, and the primary auditory cortex turns the results of the transform into "hearing". TL;DR: We hear frequencies, not impulses.

• Long story short, it's just an ilusion. Two audio signals don't add to each other, they are averaged exactly like two batteries. That's it? – GetFree Jan 25 '15 at 10:20
• That is the gist of it, yes. – Ignacio Vazquez-Abrams Jan 25 '15 at 14:56

simulate this circuit – Schematic created using CircuitLab

$$V_{load} = (V_1+V_2)\frac{R_{load}}{R_{out}+2R_{load}}$$ For a simple comparison, assume Rout << Rload, as in the power supply case $$V_{load} \approx \frac{(V_1 + V_2)}{2}$$ Even though, this is the same for the audio case as the amplitudes of the individual V1 or V2 have been halved, both sources (which are AC signals) appear at the load and you can hear both of them, so the perception is that they add together.

You have to distinguish different cases when adding voltages.

• 1) The signals are DC: The sum is the algebraic sum of all voltages (Eg. U₁ = 5V, U₂ = 3V: Usum = 5V + 3V = 8V. (Eventually averaged when applicable i.e. 8/2 = 4)

• 2) The signals are pure sine waves with the exact same frequencies and no phase shift exists between them. The sum is the algebraic sum of all voltages (See 1, DC) (Eventually averaged when applicable)

• 3) The signal are sine waves with the exact same frequency, but a phase shift of φ degrees exists between them. In that case the angle-adjusted cosine rule applies to both voltages. (Eg. U₁ = 5V, u₂ = 3V, φ=30⁰: Usum = √(U1² + U2² + 2*U1*U2*cosφ) = 25 + 9 + 2*5*3.cos(30) = 7.7V (Eventually averaged when applicable)

• 4) The signals are sine waves with different frequencies or are non-sinusoidal in nature: The sum is the square root of the squared sums (RMS value): Usum = √(U1² + U2²) = √(5² + 3²) = 5.83V

Strictly speaking, it is incorrect to connect voltage sources in parallel and current sources in series since a conflict between them will appear (each of them will try to impose its "desired value" on the common output). However, these incorrect connections are widely used in electronics. For example, in a differential stage with dynamic load, both the cases can be seen - a voltage conflict and a current conflict.

In practice, voltage sources always have internal resistances, or we deliberately add external resistors in series with them. Thus we obtain the extremely useful circuit of a resistive summer with weighted inputs... that wonderfully illustrates the principle of superposition. I have created a lot of stories about this humble passive circuit and its op-amp implementations that can be useful for you:

Walking along the resistive film - a fancy interpretation of the 18th century Ohm's experiment

Parallel voltage summer - a Wikibooks story about the resistive summer

Voltage-to-voltage summer - an animated Flash story about the ubiquitous circuit

My favorite way to present this circuit is by the voltage diagram (the distribution of the local voltages along the resistors).


text:
# Are Lens Distortion Coefficients inverted for projectPoints?

Good day guys,

I am trying to simulate and image with lens barrel distortion. I create a virtual chessboard (only the corners) and then project it onto my image plane using OpenCV. The idea is to project these points with known distortion coefficients, and then attempt a lens distortion calibration (with calibrateCamera), and see if the same coefficients are obtained.

My question is about the projectPoints function which takes distCoeffs as an input. Are these coefficients the same that must be used to undistort an image (output of calibrateCamera)? This means the function will have to calculate the inverse of that operation. Or, does it use those coefficients to distort the object points directly? Meaning that the will not correlate at all at the output of e.g. calibrateCamera.

I ask, because I noticed my simulation does pincushion distortion when I expect barrel, and vica versa. Which seems that the distortion does the opposite of what I think it does.

The minimal working code that I used to simulate the image (in Python):

# The distortion matrix that I vary
distortion = np.array([0.03, 0.0001, 0.0, 0.0, 0.0001])

# Generate Grid of Object Points
grid_size, square_size = [20, 20], 0.2
object_points = np.zeros([grid_size[0] * grid_size[1], 3])
mx, my = [(grid_size[0] - 1) * square_size / 2, (grid_size[1] - 1) * square_size / 2]
for i in range(grid_size[0]):
for j in range(grid_size[1]):
object_points[i * grid_size[0] + j] = [i * square_size - mx, j * square_size - my, 0]

# Setup the camera information
f, p = [5e-3, 120e-8]
intrinsic = np.array([[f/p, 0, 0], [0, f/p, 0], [0, 0, 1]])
rvec = np.array([0.0, 0.0, 0.0])
tvec = np.array([0.0, 0.0, 3.0])

# Project the points
image_points, jacobian = cv2.projectPoints(object_points, rvec, tvec, intrinsic, distortion)

# Plot the points (using PyPlot)
plt.scatter(*zip(*image_points[:, 0, :]), marker='.')
plt.axis('equal')
plt.xlim((-4000, 4000))
plt.ylim((-4000, 4000))
plt.grid()
plt.show()

format: plain


text:
# Can an O(n) algorithm ever exceed O(n^2) in terms of computation time?

Assume I have two algorithms:

for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++) {
//do something in constant time
}
}

This is naturally O(n^2). Suppose I also have:

for (int i = 0; i < 100; i++) {
for (int j = 0; j < n; j++) {
//do something in constant time
}
}

This is O(n) + O(n) + O(n) + O(n) + ... O(n) + = O(n)

It seems that even though my second algorithm is O(n), it will take longer. Can someone expand on this? I bring it up because I often see algorithms where they will, for example, perform a sorting step first or something like that, and when determining total complexity, its just the most complex element that bounds the algorithm.

• An algorithm that belongs to O(n) can take up to c*n steps to finish, where c is some constant. So your algorithm is still O(n) for any number of constant iterations you make in the first loop. – ChronoTrigger Mar 23 '14 at 17:18
• Your understanding is correct: Big-O analysis isn't used to determine absolute resource usage, but to compare how the resource usage grows with the input size. – amon Mar 23 '14 at 17:18
• Why do you think that the second algorithm will take longer? As soon as n > 100 then the first algorithm will take longer. – aquinas Mar 23 '14 at 17:20
• See also: galactic algorithms – harold Mar 23 '14 at 17:22
• All very good answers everyone. I knew it wasn't supposed to explicitly cover runtimes, but these answers help put this dilemma in perspective. @aquinas You are right. I thought of this after posting lol. – Brian Vanover Mar 23 '14 at 17:23

Asymptotic complexity (which is what both big-O and big-Theta represent) completely ignores the constant factors involved - it's only intended to give an indication of how running time will change as the size of the input gets larger.

So it's certainly possible that an Θ(n) algorithm can take longer than an Θ(n2) one for some given n - which n this will happen for will really depend on the algorithms involved - for your specific example, this will be the case for n < 100, ignoring the possibility of optimizations differing between the two.

For any two given algorithms taking Θ(n) and Θ(n2) time respectively, what you're likely to see is that either:

• The Θ(n) algorithm is slower when n is small, then the Θ(n2) one becomes slower as n increases
(which happens if the Θ(n) one is more complex, i.e. has higher constant factors), or
• The Θ(n2) one is always slower.

Although it's certainly possible that the Θ(n) algorithm can be slower, then the Θ(n2) one, then the Θ(n) one again, and so on as n increases, until n gets very large, from which point onwards the Θ(n2) one will always be slower, although it's greatly unlikely to happen.

## In slightly more mathematical terms:

Let's say the Θ(n2) algorithm takes cn2 operations for some c.

And the Θ(n) algorithm takes dn operations for some d.

This is in line with the formal definition since we can assume this holds for n greater than 0 (i.e. for all n) and that the two functions between which the running time is lies is the same.

In line with your example, if you were to say c = 1 and d = 100, then the Θ(n) algorithm would be slower until n = 100, at which point the Θ(n2) algorithm would become slower.

(courtesy of WolframAlpha).

## Notation note:

Technically big-O is only an upper bound, meaning you can say an O(1) algorithm (or really any algorithm taking O(n2) or less time) takes O(n2) as well. Thus I instead used big-Theta (Θ) notation, which is just a tight bound. See the formal definitions for more information.

Big-O is often informally treated as or taught to be a tight bound, so you may already have been essentially using big-Theta without knowing it.

If we're talking about an upper bound only (as per the formal definition of big-O), that would rather be an "anything goes" situation - the O(n) one can be faster, the O(n2) one can be faster or they can take the same amount of time (asymptotically) - one usually can't make particularly meaningful conclusions with regard to comparing the big-O of algorithms, one can only say that, given a big-O of some algorithm, that that algorithm won't take any longer than that amount of time (asymptotically).

• As a concrete example, Insertion Sort (O(n^2)) is faster than Quicksort (O(nlogn)) for small inputs, and is often used as a "base case" for small inputs. Note also that insertion sort is an interesting example of where naive asymptotic analysis can fail. It runs in O(n^2) in the worst case, but O(n) time for "mostly" sorted lists. This is because it actually runs in O(# of inversions). – Alexis Beingessner Mar 24 '14 at 2:31
• @Raphael: In programming, O(n) often used as Θ(n) – jfs Mar 24 '14 at 13:33
• @J.F.Sebastian Then it's used in the wrong way, creating unnecessary barriers between scientists and practitioners. Just use Θ as it's usually what you (i.e. programmers) want (you should want more precise statements, but that's for another time). (The Wikipedia article suggests that this practice is wide-spread in CS. While unfortunately true, that does not make it correct: typically people who make this error are a) not aware of the facts (bad) or b) sloppy (worse).) – Raphael Mar 24 '14 at 14:00
• @Izkata The keyboard issue is a weak excuse. It's also why sites that do care about precision use MathJax. ;) (This question should probably have been migrated to Computer Science). Using a "rough guideline" is fair, but it's usually a bad idea to use a symbol that has precise meaning in a closely related discipline for basically the same thing, but construe the meaning slightly. Because then, programmers (by "socialisation") start reading CS material and things become ugly. So at least folks writing documentation or SO answers on these topics should take the time to copy-pase a Θ from somewhere. – Raphael Mar 24 '14 at 16:24
• @Raphael It's really not particularly important, and you're dealing with semantics. A programmer wants to be able to say "my algorithm is O(whatever)." To him this means, my algorithm can be no slower than this, which is what's important. Especially since requirements may often be in O format, simply because they say "We need at LEAST this, so prove that it's O(f(n)) and we're happy". Proving theta is often superfluous. – Cruncher Mar 24 '14 at 20:56

Yes, an O(n) algorithm can exceed an O(n2) algorithm in terms of running time. This happens when the constant factor (that we omit in the big O notation) is large. For example, in your code above, the O(n) algorithm will have a large constant factor. So, it will perform worse than an algorithm that runs in O(n2) for n < 10.

Here, n=100 is the cross-over point. So when a task can be performed in both O(n) and in O(n2) and the constant factor of the linear algorithm is more than that of a quadratic algorithm, then we tend to prefer the algorithm with the worse running time. For example, when sorting an array, we switch to insertion sort for smaller arrays, even when merge sort or quick sort run asymptotically faster. This is because insertion sort has a smaller constant factor than merge/quick sort and will run faster.

• Thanks Nikunj. Also a great answer. I just accepted the one that came in first. – Brian Vanover Mar 23 '14 at 17:31
• It's fine. My first answer on stackoverflow that I wrote within a few minutes of the question being posted. Guess I need to improve my typing speed. – Nikunj Banka Mar 23 '14 at 17:33
• Good answer, but shouldn't n = 100 be the crossover point in the example provided by the OP? – Derek W Mar 23 '14 at 18:03
• Big-O only gives a rough idea, and determines the asymptotic behaviour. If the difference is smaller, say O (n) vs. O (n log n), the constants are much more important - one program taking 100n nanoseconds will in practice never be faster than one taking (n ln n) nanoseconds (if you calculate the crossover point, it will be some million years in the future). – gnasher729 Mar 24 '14 at 1:15
• @SørenDebois I have not forgotten the notations. The question has been edited. Earlier the OP used 10 iterations and now it has been changed to 100. The accepted answer has also been edited accordingly as you can see just 1 hour ago. I will make the necessary changes. – Nikunj Banka Mar 24 '14 at 9:33

Big O(n) are not meant to compare relative speed of different algorithm. They are meant to measure how fast the running time increase when the size of input increase. For example,

• O(n) means that if n multiplied by 1000, then the running time is roughly multiplied by 1000.
• O(n^2) means that if n is multiplied by 1000, then the running is roughly multiplied by 1000000.

So when n is large enough, any O(n) algorithm will beat a O(n^2) algorithm. It doesn't mean that anything for a fixed n.

Long story short, yes, it can. The definition of O is base on the fact that O(f(x)) < O(g(x)) implies that g(x) will definitively take more time to run than f(x) given a big enough x.

For example, is a known fact that for small values merge sort is outperformed by insertion sort ( if I remember correctly, that should hold true for n smaller than 31)

Yes. The O() means only asymptotic complexity. The linear algorythm can be slower as the quadratic, if it has same enough large linear slowing constant (f.e. if the core of the loop is running 10-times longer, it will be slower as its quadratic version).

The O()-notation is only an extrapolation, although a quite good one.

• Something that hasn't been discussed yet is that a given algorithm might be blazingly fast once it gets going, with a small constant multiplier, but it has a huge setup time (constant, regardless of n). For very large n's, the setup time becomes insignificant, but for small n's, it could mean running more slowly than an algorithm with a "slower" O. – Phil Perry Mar 24 '14 at 18:45

The only guarantee you get is that—no matter the constant factors—for big enough n, the O(n) algorithm will spend fewer operations than the O(n^2) one.

As an example, let's count the operations in the OPs neat example. His two algoriths differ in only one line:

for (int i = 0; i < n; i++) {       (* A, the O(n*n) algorithm. *)

vs.

for (int i = 0; i < 100; i++) {     (* B, the O(n) algorithm. *)

Since the rest of his programs are the same, the difference in actual running times will be decided by these two lines.

• For n=100, both lines do 100 iterations, so A and B perform exactly the same at n=100.
• For n<100, say, n=10, A does only 10 iterations, whereas B does 100. Clearly A is faster.
• For n>100, say, n=1000. Now the loop of A does 1000 iterations, whereas, the B loop still does its fixed 100 iterations. Clearly A is slower.

Of course, how big n has to get for the O(n) algorithm to be faster depends on the constant factor. If you change the constant 100 to 1000 in B, then the cutoff also changes to 1000.

• For n<100, and n=10, how does A take 10 iterations? 10^2 is 100. Doesn't B take 100*10, so 1000? – Jose Ramirez Dec 9 '17 at 2:57


text:
# Computing Bredon Cohomology of Z/2-spheres?

Can anyone suggest me how to calculate explicitly the Bredon Cohomology of the sphere(at least for 2-dimensional case) with antipodal Z/2-action with constant coefficient system associated to integers?

EDIT: per request.

We can be pretty explicit in this case: the $n$-sphere with antipodal action has a $\Bbb Z/2$-equivariant cell structure whose $k$-skeleton is the $k$-sphere. As a result, we get a cellular chain complex of coefficient systems: $$0 \leftarrow C_0 \leftarrow C_2 \leftarrow \cdots \leftarrow C_n$$ where $C_k$ is the coefficient system associated to $(S^k,S^{k-1})$. According to Bredon's definitions, we calculate the value of the coefficient system on the trivial orbit $*$ as $H_k((S^k)^{\Bbb Z/2},(S^{k-1})^{\Bbb Z/2})$, which is trivial; we calculate the value on the orbit $\Bbb Z/2$ as $H_k(S^k,S^{k-1}) = \Bbb Z \oplus \Bbb Z$ where the generator $g$ of $\Bbb Z/2$-action swaps the two factors.

Therefore, in each degree the coefficient group is the same coefficient system $C$. More, one can calculate the cellular chain maps using the same homology calculation (one knows that it has to recover the homology of $S^n$) and find that it is a sequence of maps $$0 \leftarrow C \stackrel{1-g}{\leftarrow} C \stackrel{1+g}{\leftarrow} \cdots$$

Now we apply Hom out to any coefficient system $M$. You can show $Hom(C,M)$ is $M(\Bbb Z/2)$. The resulting chain complex is $$0 \to M(\Bbb Z/2) \stackrel{1-g}{\to} M(\Bbb Z/2) \stackrel{1+g}{\to} M(\Bbb Z/2) \stackrel{1-g}{\to} \cdots$$ Substituting $M = \Bbb Z$, we get the cochain complex computing the cohomology of $\Bbb{RP}^n$.

This is part of a more general result here when the action is free and properly discontinuous, and the group $\Bbb Z/2$ acts trivially on the value $\Bbb Z$ of the coefficient system on $\Bbb Z/2$. In these cases, the Bredon cohomology coincides with cohomology of the orbit space.

Therefore, we're getting the cohomology of $(S^n) / \Bbb Z/2 = \Bbb{RP}^n$ with coefficients in $\Bbb Z$ (hence $\Bbb Z$ in degree 0, $\Bbb Z/2$ in even positive degrees, and 0 otherwise).

• :could you please do this problem using the contractions of bredon cohomology made in J.P.May's Equivariant Homotopy and Cohomology Theory Book at pages 20-21? Jan 8, 2014 at 16:47


text:
Summer is Coming! Join the Game of Timers Competition to Win Epic Prizes. Registration is Open. Game starts Mon July 1st.

 It is currently 18 Jul 2019, 14:49

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# In ΔPQS above, if PQ =3 and PS = 4, then PR =?

 new topic post reply Question banks Downloads My Bookmarks Reviews Important topics
Author Message
TAGS:

### Hide Tags

Senior PS Moderator
Status: It always seems impossible until it's done.
Joined: 16 Sep 2016
Posts: 751
GMAT 1: 740 Q50 V40
GMAT 2: 770 Q51 V42
Re: In ΔPQS above, if PQ =3 and PS = 4, then PR =?  [#permalink]

### Show Tags

22 Dec 2018, 05:43
1
Hey there,

I am equating areas as it is the exact same triangle. For example ... if there was a triangle ABC whose area was known to be 30. Also, you knew that the area of the triangle was 1/2*AB*BC, we would equate the two as it is triangle ABC. SO above we are talking about triangle QPS only and hence can equate. The perpendicular from the right angle to the opposite side ( hypotenuse) can be a bisector only if it is an isosceles right triangle ( which is not the case here as it is a 3-4-5 triangle.

Is it clear now?

We are just equating two different ways to write the area of triangle QPS which is 1/2 * QP * PS and 1/2* PR * QS ...

Best,
dave13 wrote:

Gladiator59 thank you, so are equating areas of triangles based on the rule below ? correct ?

If the measures of the corresponding sides of two triangles are proportional then the triangles are similar. Likewise if the measures of two sides in one triangle are proportional to the corresponding sides in another triangle and the including angles are congruent then the triangles are similar.

source: https://www.mathplanet.com/education/ge ... /triangles

but how can I know that PR is not a median ?

have a look here https://www.khanacademy.org/math/geomet ... equal-area

_________________
Regards,

“Do. Or do not. There is no try.” - Yoda (The Empire Strikes Back)
VP
Joined: 09 Mar 2016
Posts: 1273
Re: In ΔPQS above, if PQ =3 and PS = 4, then PR =?  [#permalink]

### Show Tags

22 Dec 2018, 06:38
thanks Gladi, got it
Manager
Joined: 23 Apr 2018
Posts: 100
Re: In ΔPQS above, if PQ =3 and PS = 4, then PR =?  [#permalink]

### Show Tags

28 Jun 2019, 06:57
prasannar wrote:

In ΔPQS above, if PQ =3 and PS = 4, then PR =?

A. 9/4
B. 12/5
C. 16/5
D. 15/4
E. 20/3

Project PS Butler : Question #88

Subscribe to get Daily Email - Click Here | Subscribe via RSS - RSS

Attachment:
Untitled.png

the triangle can be figured out to be a 3:4:5 triangle..

when you see that there is a line from one of vertices to a side of the triangle (90deg) it cuts the triangle in half..
therefore,
triangle PSR is now 45 : 45 : and 90 (where line bisects the side QS)
and
QPR too, is a 45 : 45 : 90
as we know, the 45 : 45 sides will be of same length...
as QR is 5
we can say QR = RS = 2.5
2.5 is side RS
so PR will be 2.5

only answer closest to that is option B
feel free to correct my opinion
GMAT Tutor
Joined: 24 Jun 2008
Posts: 1728
Re: In ΔPQS above, if PQ =3 and PS = 4, then PR =?  [#permalink]

### Show Tags

03 Jul 2019, 05:49
Shrey9 wrote:
when you see that there is a line from one of vertices to a side of the triangle (90deg) it cuts the triangle in half..

You are not cutting the triangle in half here - that would only be true if the large triangle was isosceles, and it's not. That's why your answer doesn't match any of the answer choices.
_________________
GMAT Tutor in Toronto

If you are looking for online GMAT math tutoring, or if you are interested in buying my advanced Quant books and problem sets, please contact me at ianstewartgmat at gmail.com
Senior Manager
Joined: 12 Sep 2017
Posts: 295
Re: In ΔPQS above, if PQ =3 and PS = 4, then PR =?  [#permalink]

### Show Tags

07 Jul 2019, 17:02
We we cannot solve it like this?

PRS will become a 45 - 45 - 90 triangele, so $$x\sqrt{2}= 4$$ and solving it will lead us to:

PR = RS = $$2\sqrt{2}$$

What is wrong with it?

Kind regards!
GMAT Tutor
Joined: 24 Jun 2008
Posts: 1728
Re: In ΔPQS above, if PQ =3 and PS = 4, then PR =?  [#permalink]

### Show Tags

07 Jul 2019, 17:23
1
jfranciscocuencag wrote:
PRS will become a 45 - 45 - 90 triangele, so $$x\sqrt{2}= 4$$ and solving it will lead us to:

PRS is not a 45-45-90 triangle. The line PR would only cut the 90 degree angle at P perfectly in half if the big triangle was isosceles, and it is not isosceles.

You can see PRS can't be a 45-45-90 triangle by looking at the angles. In the big 3-4-5 triangle, the smallest angle is opposite the smallest side, so the smallest angle is at S. You'd need trigonometry to find that angle but it's definitely less than 45 degrees (it's actually roughly 37 degrees) if it's the smallest angle in a right triangle. But that angle at S is also one of the angles in triangle PRS, so PRS is not a 45-45-90 triangle.
_________________
GMAT Tutor in Toronto

If you are looking for online GMAT math tutoring, or if you are interested in buying my advanced Quant books and problem sets, please contact me at ianstewartgmat at gmail.com
Re: In ΔPQS above, if PQ =3 and PS = 4, then PR =?   [#permalink] 07 Jul 2019, 17:23

Go to page   Previous    1   2   [ 26 posts ]

Display posts from previous: Sort by

# In ΔPQS above, if PQ =3 and PS = 4, then PR =?

 new topic post reply Question banks Downloads My Bookmarks Reviews Important topics

 Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne


text:
## Thursday, December 30, 2004

Speak What We Feel, not what we ought to say by Frederick Buechner

The back cover blurb is actually accurate:

Reflections on literature and faith: Gerard Manley Hopkins, Mark Twain, G.K. Chesterton and William Shakespeare "wrote in his own blood about the darkness of life as he found it, and about how--for better or worse--he managed somehow to survive it, even to embrace it."

This book tries to correlate a time of darkness in the lives of these writers with the darkness expressed in one or more of their works: for Hopkins the "dark sonnets," for Twain Huckleberry Finn, for Chesterton The Man Who Was Thursday, and for Shakespeare King Lear.

If you have not yet read Huckleberry Finn or The Man Who Was Thursday, do not read Buechner's book yet. Go read Twain and Chesterton. There are some spoilers here, and though the books are great enough to stand many re-readings, I don't want you to miss the pleasant surprises of a first reading.

Hopkins became a Jesuit priest, and was assigned to teach in a Dublin school; a job for which he was temperamentally ill-equipped. Physically and mentally delicate, he burned out in the post, but not before writing a number of poems whose gramlost weldwords deutschverb sigh.*

Buechner parses Hopkins' dark sonnets to find the depths of his dark night of the soul; a dark night of self-knowledge. But it wasn't complete despair, and hints of hope remain. Buechner analyses very well.

But: I'm afraid I must disagree with Buechner and concur with Hopkins' friend Bridge's assessment of The Wreck of the Deutschland: it isn't good. Hopkins wrote much of his poetry in a language of his own, kin to English but with his own words and with grammar contorted beyond the worst of Browning. Sometimes it works in part: Spring and Fall "Margaret, are you grieving Over Goldengrove unleaving," or one of my wife's favorites Pied Beauty "Glory be to God for dappled things." I consider such private languages cheating. We have the rules of the game for communicating with the reader: you'd better have excellent reasons and consummate skill if you plan to break them and make the reader learn a new language. Some times Hopkins made it work, sometimes I find no music there.

Another illustration of cheating might be T.S. Elliott and Wasteland. It seems at first like a kitchen-sink poem with everything tossed in, but it isn't. Less-brilliant and less-workmanlike poets have used that "deeply personal" kitchen sink model disastrously since.

At the beginning of Huckleberry Finn you find this warning:

"Persons attempting to find a motive in this narrative will be prosecuted; persons attempting to find a moral in it will be banished; persons attempting to find a plot in it will be shot. By Order of the Author Per G. G., Chief Ordnance."

I'm waiting for the implications of that to sink in in English classes around the land...

Twain's life wasn't exactly a bowl of cherries. We all know the tragedies of his later life (for which he seems to have often blamed himself), but I hadn't realized how much death there was in his childhood, for which a sensitive child might well have felt responsible. Apparently he started writing Huck in a time of troubles, picked it up again in another time of trouble, and finished it during a happy era of his life. Darkness and gloom don't seem to be the keynotes of the book, but Buechner's closer inspection shows that the crisis of faith Huck endures is set in a background of lies and hypocrisy; a crisis that feels personal. Twain himself seems to have abandoned faith after his crisis—or almost abandoned it. But he preserved a humanist care for people, which Huck mirrors in the book, even for the villainous Duke and Dauphin. Or at any rate, Twain maintained that interest until the bitterness at the end of his life overwhelmed him.

Buechner's choice of Chesterton to illustrate the darkness of life seems very odd. Chesterton is far better known as one who embraced and championed the unspeakable joy of life: the mystery not of pain but of pleasure.

"You say grace before meals. All right. But I say grace before the concert and the opera, and grace before the play and pantomime, and grace before I open a book, and grace before sketching, painting, and swimming, fencing, boxing, walking, playing, dancing, and grace before I dip the pen in ink."

But Chesterton wrote The Man Who Was Thursday and subtitled it A Nightmare; and so it is. Our hero is alone in a hostile world, and the unmaskings and riddles only lead to further mysteries where everything he believed solid proves unstable. Buechner found that this reflected Chesterton's terrible years at art school, where he first became immersed in the reductionist debunking spirit of the age, finally reached a very late puberty, and first discovered the violence in his own soul—all while still an unbeliever. He made a choice for sanity and eventually became one of the champions of the faith, and later wrote Thursday. He claimed afterwards that the book did not reflect his beliefs about suffering, but then it didn't not reflect something real too. A psychiatrist once told him that he had a few patients whose lives were saved because they read The Man Who Was Thursday and understood it.

Lastly Buechner tackles Shakespeare and King Lear. Of course, Shakespeare's life is not well known in detail, but there were a few times that would have been a little tough. And we have the evidence of other plays (some far from great, like Timon of Athens) to suggest that Shakespeare spent a year or so extremely depressed about something. Buechner studies Lear in detail, and does a fine job of showing things I'd overlooked when I last read the play (I've never seen it). Every character winds up embroiled in deception and self-deception, even the loving daughter too stubborn to make the simple declarative statement that would have prevented the breach with her father. Pain rules, with the only glimmers of hope being the transient reconciliation and the pronouncement “on such sacrifices the gods themselves throw incense.” And of course, as C.S. Lewis points out, the nameless first servant who attacked Cornwall when Cornwall blinded Gloster proved that not everyone loved evil or deception. But still the play remains a vivid picture of the deceit and treachery and pain in life.

The play was given a happy ending for two centuries after Shakespeare's death: programmers thought the original too dark and painful for audiences.

Buechner's book takes its title from King Lear: “Speak what we feel, not what we ought to say.” He doesn't try to draw great conclusions, except to show people who have looked into the darkness and written honestly about both the darkness and the glimmers of light. He's a very good writer himself. Sometimes we can use reminders of the darkness when people try to pretend that all is sweetness and light. Darkness without despair here: a good book.

*

I've had complaints that my parody is unfair. Here is the entire poem Spring and Fall of which I was parodying line 8:
Margaret, are you grieving
Over Goldengrove unleaving?
Leaves, like the things of man, you
With your fresh thoughts care for, can you?
Ah! as the heart grows older
It will come to such sights colder
By and by, nor spare a sigh
Though worlds of wanwood leafmeal lie;
And yet you wíll weep and know why.
Now no matter, child, the name:
Sorrow's springs are the same.
Nor mouth had, no nor mind, expressed
What héart héard of, ghóst guéssed:
It is the blight man was born for,
It is Margaret you mourn for.

## Friday, December 24, 2004

Arabs in History by Bernard Lewis

Do I need to tell you to go read it?

As the title says, this is a history of the Arabs, not of Islam as such. I learned a lot about things I'd never heard of before, such as the great revolts of negro slaves around Basra.

This was the 1966 edition of a book first published in 1950, and I suspect Bernard would have had his hands on new material by now. It might be, for example, that the three century long devastation in agriculture in Tunisia in the wake of the Hilal and Sulaim tribes invasions owed something to climate change.

I'd be quite interested in knowing if he would revise passages like this:

The word "atomistic" is often used to describe a habit of mind and outlook, recognizable in many aspects of the civilization of the Arab and dominant in the later stages of his history. By this is meant the tendency to view life and the universe as a series of static, concrete and disjunct entities, loosely linked in a sort of mechanical or even casual association by circumstances or the mind of an individual, but having no organic interrelation of their own. Though by no means universal, this tendency affects the life of the Arab in many different ways. He conceives his society not as an organic whole, compounded of interrelated and interacting parts, but as an association of separate groups--religions, nations, classes--held together only by the ground beneath and the government above. His town is an agglomeration of quarters, guilds, clans, houses, only rarely with any corporate civic identity of its own. In contrast to the scientists and philosophers on the one hand and the mystics on the other, the ordinary orthodox theologian, scholar or litterateur shows the same quality in his attitude to knowledge. The various disciplines are not different ways of reaching out towards the same heart, pooling their findings in an integrated whole, but separate and self-contained compartments, each holding a finite number of pieces of knowledge, the progressive accumulation of which constitutes learning. Arabic literature, devoid of epic or drama, achieves its effects by a series of separate observations or characterizations, minute and vivid, but fragmentary, linked by the subjective associations of author and reader, rarely by an overriding plan. The Arabic poem is a set of separate and detachable lines, strung pearls that are perfect in themselves, usually interchangeable. Arabic music is modal and rhythmic, developed by fantasy and variation, never by harmony. Arabic art--mainly applied and decorative--is distinguished by its minuteness and perfection of detail rather than by composition or perspective. The historians and biographers, like the fiction writers, present their narrative as a series of loosely connected incidents. Even the individual is drawn as a sum of attributes, often listed, as a recent writer remarks, like the description on a passport.

Is that oversimplified? Or perhaps an accurate portrayal of the better documented years of decay?

## Thursday, December 23, 2004

The Killing of History by Keith Windschuttle

Perhaps I can best summarize the book by quoting from the preface:

The structure of the book is designed to examine how both the general and the specific versions of these theories have been applied to the writing of history. The principal targets of the investigation and the places where they are discussed are:
• Cultural relativism: Chapters Two and Nine
• Semiotics: Chapter Two
• Structuralist theory: Chapters Two, Three and Nine
• Poststructuralist theory: Chapters Four and Five
• Anti-humanism, genealogy and discourse theory: Chapter Five
• Hegelian and Marxist philosophy of history: Chapter Six
• Postmodernist philosophy of history: Chapter Six
• Radical skepticism and scientific relativism: Chapter Seven
• Hermeneutics: Chapter Seven
• Historical fiction and theory of peotics: Chapter Eight

Executive summary: Any time you find a person using the currently fashionable forms of the above theories to model the world, you are looking at a fool.

In each case above, Keith explains the theory, doing his best to make it as clear and plausible as possible, then shows it in action in some historical analysis, and then shows the author's self-contradictions and the failure of the theory.

Perhaps you have heard of the widely used fake "Chinese taxonomy" (actually invented by poet Jorge Luis Borges) which Foucault quotes as dividing the world into things "(a) belonging to the Emperor, (b) embalmed, (c) tame, (d) sucking pigs ... (n) that from a long way off look like flies." If this was a real taxonomy then some of the more radical propositions about human consciousness would have a leg to stand on, but on close inspection all such evidence for these solipsistic philosophies dissolves into fog.

Keith's background doesn't include enough physics to allow him to hammer Kuhn's The Structure of Scientific Revolutions as hard as it deserves (a physicist like Kuhn should have known better than to write such muddy nonsense). Those chapters (on non-cumulative meaning) I read were, in Pauli's famous phrase "not even wrong." But Keith has a grand old time taking Popper and Hume to pieces in defense of induction.

Keith is a reasonably good writer, but there is only so much nonsense I can take at one time, and it got to be a struggle to get through the last chapters. I applaud his work to refute nonsense and lies, and I recommend the book.

I have to take some detailed notes of the who goes with what theory, and I'll post these later. A painful duty...

## Friday, December 17, 2004

The Peace of Dives
Rudyard Kipling 1903
THE WORD came down to Dives in Torment where he lay:
“Our World is full of wickedness, My Children maim and slay,
“And the Saint and Seer and Prophet
“Can make no better of it
“Than to sanctify and prophesy and pray.

“Rise up, rise up, thou Dives, and take again thy gold,
“And thy women and thy housen as they were to thee of old.
“It may be grace hath found thee
“In the furnace where We bound thee,
“And that thou shalt bring the peace My Son foretold.”

Then merrily rose Dives and leaped from out his fire,
And walked abroad with diligence to do the Lord’s desire;
And anon the battles ceased,
And the captives were released,

........

Then Satan said to Dives:—“Declare thou by The Name,
“The secret of thy subtlety that turneth mine to shame.
“It is known through all the Hells
“How my peoples mocked my spells,
“And my faithless Kings denied me ere I came.”

........

“Their nearest foes may purchase, or their furthest friends may lease,
“One by one from Ancient Accad to the Islands of the Seas.
“And their covenants they make
“For the naked iron’s sake,
“But I—I trap them armoured into peace.

“The flocks that Egypt pledged me to Assyria I drave,
“And Pharaoh hath the increase of the herds that Sargon gave.
“Not for Ashdod overthrown
“Will the Kings destroy their own,
“Or their peoples wake the strife they feign to brave.

“Is not Carchernish like Calno? For the steeds of their desire
“They have sold me seven harvests that I sell to Crowning Tyre;
“And the Tyrian sweeps the plains
“With a thousand hired wains,
“And the Cities keep the peace and—share the hire.

...........

“So I make a jest of Wonder, and a mock of Time and Space.
“The roofless Seas an hostel, and the Earth a market-place,
“Each is surety for his foe,
“And none may thrive without his fellows’ grace.

“Now this is all my subtlety and this is all my wit,
“God give thee good enlightenment, My Master in the Pit.
“But behold all Earth is laid
“In the Peace which I have made,
“And behold I wait on thee to trouble it!”

Kipling hoped he foresaw a world where the nations, bound by trade, would study war no more. He dreamed that self-interest would keep nations at peace. Note the date: 1903. Kipling lost his son in World War I.

I hear the same sort of refrain, based on as as much hope and as little history as Kipling's. No democratic nations have ever gone to war against each other.

Aside from the fact that you can't demonstrate this without shading the meanings of "democratic" and "nations," it is wishful thinking to project this very far. Anatole France was similarly skeptical in Penguin Island: he feared that economic interests would be freer to start wars in a democracy! What in the nature of a democracy (I assume that it values freedom--this excludes most alleged democracies in the world) intrinsically forbids aggressive war?

I said you have to fudge the meaning of democratic and nation to make the slogan work. Think of the American Civil war: a democracy fighting itself! And democracy is not an eternal possession: think of Hitler's rise to power; a popular rise that lost the Germans their democracy in only a few years. Or think of Algeria and the threatened Islamist "One man, one vote, one time." And, if it isn't too extreme to consider a ship as a microcosm of a state, recall that historians now say that many of the Caribbean pirate ships were run on democratic lines.

I don't see any magic potion to solve all the world's problems. Some things help: systems in which people participate in their own government I judge will almost always work better than those that don't. But I think it requires a certain blindness to imagine that democracy is a cure-all to eliminate wars, or that free trade will make men act like saints.

This isn't Kipling's best work by a long shot. But I thought the bitter irony of it made it memorable.

## Monday, December 13, 2004

Bumper Sticker

"Spiritual People Inspire Me, Religious People Scare Me"

The difference between "spiritual" and "religious" is who is in charge. The spiritual person wants to enjoy, the religious person to obey the spiritual power.

Recall Ezekiel 33:30-32

"As for you, son of man, your countrymen are talking together about you by the walls and at the doors of the houses, saying to each other 'Come and hear the message that has come from the Lord.' My people come to you, as they usually do, and sit before you to listen to your words, but they do not put them into practice. With their mouths they express devotion, but their hearts are greedy for unjust gain. Indeed, to them you are nothing more than one who sings love songs with a beautiful voice and plays an instrument well, for they hear your words but do not put them into practice."

His listeners were spiritual. They enjoyed the call of God, but never let it bother them.

## Saturday, December 11, 2004

Serenata

Some pieces of musics touch springs in us that no words reach.

One passage of Tchaikovsky sends a shiver up my spine--it connects me to myself at 5, listening to the radio with my father in California, when the world was younger and less full.

And as long as I can remember, Serenata by Leroy Anderson has been a sad piece. It is a merry work--a gallop and a sparkling dance--so why should I find it sad?

It sings of a path irrevocably passed by; a part of life many share, but not I. I took the Apollonian road rather than the Dionysian, and things like dance are utterly opaque to me. Pep rallies leave me cold. (And how can one socialize in a crowd with the band so loud you can't converse? Some can, and love it. Not I. Something's missing.)

I've always felt this way about Serenata, so this isn't the nostalgic regret of a midlife crisis. Unless, of course, my sister is right. She says I was born old and have been growing younger.

The hood of our big white dinosaur Aerostar has a few duct tape flaps hanging off it. #1 Daughter wrinkled the hood and radiator a month ago. The hood latches work, but I don't want to push my luck.

We've had an assortment of heaps, fossils, and one pretty new car that lasted long beyond its expected lifetime.

Somebody honored our wedding day by writing our initials in shaving cream on the hood of our little blue Comet. Our initials stood bleached into the hood for the rest of the car's life. We were still picking rice out of it the day we traded it in 7 years later.

We bought an extended warranty on the only new car we ever had, a dainty light blue hatchback. #1 Son, age 5 at the time, heard the name "Plymouth Reliant" and called the car "Timothy Lion." Four years into the car's life, a fifty cent part broke deep in the engine. The dealer needed two days to take the engine apart, replace the fifty cent part, and reassemble the engine. We suspect that the reboring the dealer had to give the engine while making the repair extended the life of "Timothy Lion" by forty thousand miles.

My mother had a big, heavy,1976 Chevy Nova that needed repairs. The mechanic in Chicago said $600 for the repairs and another$300 if she wanted the air conditioning to work again. So she sold it to James for $1 and he took it to Randy, the grumpy Madison Mechanic, who couldn't say three words without cussing. I dropped the car off at Randy's shop and told him what the Chicago mechanic had said. When I came back, the air was blue with Randy's opinions about Chicago mechanics. Randy had fixed the Nova for$104, air conditioning included.

The Nova had a sound engine and ran well. The bottom of the cab had rusted out, however, so the only thing between our feet and the roadway was the carpet. Without the carpet, it would have been a Flintstone Mobile. Actually, I painted some slabs of plywood and laid them under the carpet so our feet wouldn't go through the holes in the floor. Most of the time you wouldn't notice a thing, but if you drove through a deep puddle you got quite a surprise!

The Nova saved our lives and our friend's lives. One morning in 1986, when US 51 just south of 12-18 was still 2 lanes, some southbound bonehead tried to pass 10 cars in a line, at 100 mph and counting. As I came northbound over a bridge he was at 100 yards and closing rapidly. I steered for the shoulder. The Nova fishtailed for about 400 feet and came to rest in the only place where the ditch was shallow. Because I was wearing my seatbelt, I was able to keep my hands on the wheel. Because our kids were in tubular steel reinforced seats, the only thing thrown around the car was the diaper bag. Because I was driving the Nova, I didn't flip.

A few years later, when we lived on a bus line, we didn't need the second car. We sold it to our neighbor Joe, an excellent backyard mechanic. Joe and his family had to take evasive action on Christmas Eve, and the car veered into the brush. The sticks tore the bottom out of the car, but the car stayed upright. Joe's brother took the engine out of the Nova and put it into another car. For all I know, that sturdy V8 is still running.

We had just finished paying off "Timothy Lion" when we learned we were expecting one more kid than the car had seats for. James found a massive 1986 Olds, originally owned by a carpet installer who had Arrived in this world, and wanted something big enough to carry his tools and carpet rolls, but with Class. We had it from 87,000 miles until Reverse quit on it, many miles later. It rode like an ocean liner; smooth and unhurried. I once drove it in a 60 mph crosswind and it didn't so much as twitch. We made the mistake of trading it instead of fixing the transmission. Whoever said, "Never put $1000 into a$300 car" probably owned stock in an auto loan concern. I'm sure I saw the Olds cruising majestically down East Wash last summer.

Mrs James (and Mr)

## Monday, December 06, 2004

Christmas Spectacular

My youngest son and I went to the Madison Symphony's Christmas Spectacular, with the Madison Youth Choir, Symphony Choir, and the Handbell Choir as well. Overture Hall is very pretty, and the concert was great.

We were in the nosebleed seats (row M, third from the top), and it was rather curious to watch the tympanist hit his drum and hear the boom half a beat later (and hear it coming from center stage!). I could hear everything quite clearly, though.

They bathed the orchestra with blue light. The effect reminded me a bit of the ghosts in Lord of the Rings: probably not quite the associations they had in mind. My youngest likes organs, and loved the organ and handbell version of "Oh Tanenbaum." The tenor did a great job with "Oh Holy Night," one of my favorites. I wish I had the range to sing it myself.

A wonderful feature of the hall: there is a separate exit for the high altitude seats! Down 7 stories into the basement and out the back, without standing in line forever as the hundred rows in front of you slowly drain out.

Movie Theaters in Monrovia

I've forgotten much of Monrovia. I remember bits and peices--the beggars in front of the old post office, for instance.

The Rivoli had street parking, boys with candy and cigarettes for sale in trays outside, a concession stand inside (I never bought there), dark seats, the President's box, and movies we'd heard about six months before.

There were sometimes older boys outside who would offer to watch your car for a fee. . .

The President's box was supposed to be reserved for President Tubman and his guests, though of course we never saw him here. The seats were bigger and better cushioned in that central box, walled in with a front ledge.. I went in several times, but didn't stick around--I saw few enough movies to risk getting thrown out. Some bolder souls did try to watch from the sacred premises, without ill effect. Apparently nobody was deeply offended . . . I didn't learn until later that a lot of the "We love President Tubman" chorus was insincere. (I didn't talk politics with people.)

Sometimes someone would buy a tube of M&M's from the concession stand, or (ever so rarely) a Toblerone bar. I coveted those (still do), but not enough to borrow money to buy one. One night I sweltered while waiting for a friend hesitating in front of a persistent candy boy. I kept saying "They're cheaper inside," goading the candy boy into shouting "They are not!" But perhaps I wasn't quite unjustified in my estimate: unaffordable inside equals unaffordable outside. 75 cents was a big deal for me then.

I don't recall ever going to the Roxy--perhaps once. I'm not sure why--perhaps my parents saw things there that a child might not notice. Maybe they didn't like the selection (though I didn't see any difference in kind from the Rivoli). In later years I hear it deteriorated badly, both in the physical plant and in the selection offered. The Relda drew the better customers away.

When the Relda was built it actually had a parking lot, and a mottled green statue of a naked woman outside. The selection was like the Rivoli's: popular US movies about 6 months after their US release. For some reason we didn't go to the movies with the most interesting previews, but I can't complain that the ones we saw were dull.

One such was a biopic of some old English king who apparently wanted to be a saint, but wound up in a lot of battles anyway. I remember talking with missionary who drove us, wondering why on earth the king didn't want to have sex with his wife (or anybody else--he seemed to want to be a married monk). It didn't reassure me of the basic sanity of the world to be told that the king's attitude wasn't all that uncommon.

I still remember those interminable Benson and Hedges commercials. They seemed to be 10 minutes long, but were probably only 2. I remember looking around the theater, and seeing almost all black faces; and looking back at the ads with all rich white faces and wondering "Why?" (The ads had no effect--I don't smoke.)

I'm told that these days one-room "movie theaters" are popular, with a TV and VCR or DVD player as the centerpeice; playing martial arts movies or less noble fare.

Correction: "The Roxy . . . specialized in karate and Indian movies."

## Wednesday, December 01, 2004

Green slogans

In Madison the Republicans didn't even bother to field candidates for many offices, so the contest was between Democrats and Greens.

On two triangular pillars of the Humanities Building, shielded from the rain, you may see chalked

VOTE
LOCAL
GREE
NS

Both messages are in the same handwriting, and both show the craftsmanship, attention to detail, and ability to learn from experience that we have come to expect from the Green Party.


text:
# Existence of set $\sigma$ with $|\sigma| > 0$ such that $\forall\,x\in\sigma\,\forall\varepsilon > 0$ we have $|B_\varepsilon(x)\setminus\sigma| > 0$

Is there a Borel set $\sigma\subset [0,1]$ of positive Lebesgue measure such that for all $x\in\sigma$ and all $\varepsilon > 0$ we have that $|B_\varepsilon(x)\setminus\sigma| > 0$?

Here, $|\cdot|$ denotes the Lebesgue measure and $B_r(x)$ is the interval with center $x$ and length $r > 0$.

It seems to me that the Smith-Volterra-Cantor set might be a candidate for this, but I cannot prove it.

• What about $\sigma$ equals the irrational numbers in $(0,1)$? – Gregory Grant Jul 15 '17 at 3:16
• Have you tried using a Cantor set of positive measure? – Mustafa Said Jul 15 '17 at 3:16
• @Gregory Grant: That doesn't work because the rational numbers have measure zero. – carmichael561 Jul 15 '17 at 3:17
• @carmichael561 Ok I see – Gregory Grant Jul 15 '17 at 3:18
• Also called a "fat Cantor set", a certain kind of closed nowhere dense subset of [0,1] that has positive measure. – DanielWainfleet Jul 15 '17 at 6:18


text:
# Recall that the volume of a sphere of radius r is V(r) = \frac{4\pi r^{3}}{3}. Find L, the...

## Question:

Recall that the volume of a sphere of radius r is {eq}V(r) = \frac{4\pi r^{3}}{3} {/eq}.

Find L, the linearisation of V(r) at r = 30.

L(r) = A

sphere of radius 30 centimeters is covered with a layer of paint of thickness 0.45 millimeters. Use the linearisation of V at r = 30 to estimate the volume of paint that is used. cubic centimeters of paint, help (numbers)

## Linearization of a function

One of the applications of differentiation is Linearization.

Any function can be approximated as a linear function using derivative. Usually linearization is calculated at a given value of x.

If {eq}f(x) {/eq} is the given function, then it's linearization at {eq}x = a {/eq} is given by,

{eq}L(x) = f(a) + f'(a)(x-a) {/eq}

Become a Study.com member to unlock this answer! Create your account

Volume of a sphere with radius r is given by,

{eq}\displaystyle V(r) = \frac{4 \pi r^3}{3} \\ \therefore V'(r) = 4 \pi r^2 {/eq}

Linearization of...

Linearization of Functions

from

Chapter 10 / Lesson 1
3.5K

Over the river and through the woods to Grandmother's house we go ... Are we there yet? In this lesson, apply linearization to estimate when we will finally get to Grandma's house!


text:
Does a messier room actually have higher entropy?

When my chemistry teacher was explaining entropy, she used the analogy of a clean vs. messy room. What I'm wondering is whether a messy room really does have higher entropy. Does this still work on a larger scale? If not, can I trust the analogy?

• Rooms don't have entropy. Other than that, the analogy is good. – Ivan Neretin Aug 25 '16 at 20:16
• Calculating the entropy of a small number of objects is not very meaningful, neither is calculating the entropy of an assortment of macroscopic objects.You cannot predict anything sensible from the result. – Karl Aug 25 '16 at 20:38
• Fun fact: Every letter in a common sentence contains roundabout one bit of information. Is the rest noise (=entropy), or is it structure, predetermined by the information? – Karl Aug 25 '16 at 20:50

A room can be a complicated system, (at least mine can be) so I shall assume a simplified version: an enclosed space bound by 4 walls, a ceiling and a floor and the only piece of furniture inside the room is a desk, and two books on it -- book 1 and book 2.

In the this scenario, I would define a clean room as one with both the books neatly stacked on the desk, and in a messy room you can find one of the books on the floor.

Now, if I were to try enumerate the number of ways of organising the books in a clean and a messy room, I would find that there is exactly one way in which I can achieve a "clean room configuration", and that is by having both books on the desk. On the other hand if you wish to achieve a "messy room configuration", you can do so in two ways-- either have book 1 on the floor and book 2 on the desk, or have book 2 on the floor and book 1 on the desk.

The no. of ways in which you can achieve a configuration is called its weight and is defined as:

$$W = \frac{N!}{N_0!N_1!N_2!....}$$

In the first case, $W=1$ (clean) and in the second $W=2$ (messy)

The boltzmann formula for entropy is: $S = k_b\ln W$

Plugging in $W$, we get $$S_\text{clean} = 0$$ and $$S_\text{messy} = k_b\ln2$$

Thus, the messier room has a higher entropy. This somewhat "silly" argument can be reproduced for a room with more furniture, and more items in it. Feel free to toss in some clothes, papers, chairs, book cases, toys, shoes, pets...go nuts.

So, I believe the analogy holds good. But again, what I did here is a silly exercise to illustrate a point, and it isn't meaningful as such.

If you wish to learn more, watch this lecture by Leonard Susskind

This is a tricky question. The answer is no.

From Wikipedia

In thermodynamics, entropy (usual symbol S) is a measure of the number of microscopic configurations Ω that correspond to a thermodynamic system in a state specified by certain macroscopic variables.

It can not be defined the messy and clean states thermodynamically. (Which would be the reasonable thermodynamic variables?) If it would be possible, after further assumptions, the problem is reduced to the @getafix answer. Well, it is also needed that all the book remains stacked in the desk (all of them with the same propertied but distinguishable).

It is more important to note that the non technical concept of messy is not related to entropy.

EDIT:

I have decided to make it explicit (although at first I though it was) as the content of the answer was qualified as arbitrary and it was down-voted.

The OP said:

What I'm wondering is whether a messy room really does have higher entropy

The other questions turns to be meaning less if we can not assert that the messy room has higher entropy.

1. It is clear from the context ("When my chemistry teacher was explaining entropy...") that we are talking about entropy in thermodynamics.

2. It is also clear from the context that we are talking at an introductory level.

3. In order to compare the entropy of two systems, it is required that the states of both systems can be defined in terms of state variables.

4. The system state variables can not be established for messy nor clean rooms.

5. As the system states can not be specified, then it is not true that a messy room has higher entropy.

The usage of analogies is not just matter of taste. At first, there is a goal when an analogy is used while teaching a concept $X$, clearly it should be to help the students to learn $X$. In order to describe $X$, a well known $Y$ is used. $X$ and $Y$ must share a common set of properties. Aided by $Y$, some features of $X$ are inferred by inductive reasoning.

It is not evident which are the common features that links entropy with the level of cleanliness of a room, so a valid analogy can not be built. Of course there could be established some analogy when a lot of considerations are taking into account. But, if the student have reached a conceptual level to understand "why" they are made, then, the analogy will be of little use. If they did not, take for sure that they won't understand the concept (because they won't know when they can trust the analogy).

Does this still work on a larger scale? If not, can I trust the analogy?

I think I was no clear when I said "no". I really should have said "sometimes, entropy has nothing to do with cleanliness".

• It's your wilful decision to drive the analogy to the point where it breaks. I don't like it very much either. But saying it is just wrong is, well, arbitrary. – Karl Sep 4 '16 at 19:10
• @Karl , I expanded my answer in order to address the points of our comments. Thanks for sharing your thoughts. It help to improve the contents and clarity of the answers. – user1420303 Sep 4 '16 at 22:31

"whether a messy room really does have higher entropy".

1. One messy room left untouched does not have a higher entropy than a clean one. Both represent one way of arranging the things in the room (a microstate), so the entropy is zero for both.

2. If you have a messy person living in the room then, yes, over time the messy room will have a high entropy. Things will move around but never be in the same place (many microstates), so you will have many different arrangements of things that always will be characterised as "messy" (macrostate).

3. A collection of several messy rooms, e.g. identically furnished but unmade rooms in a hotel, has a higher entropy than the same collection of rooms after house keeping has been to them all and arranged the things in exactly the same way.

• +1 You touched an interesting point where an analogy can be drawn, if the internal structure of macroscopic objects is not considered. I would like to read "ergodic" scattered somewhere in point 3, to turn the statement a little more accurate. – user1420303 Sep 19 '16 at 15:20


text:
# Generating equation for Diophantine equation

Given a triangular tower of bricks where the bottom row has $N$ bricks, the next row has $N-1$ bricks, etc, we can ask the question: For which $N$ does there exist a smaller tower with $M \lt N$ bricks (somewhere above the bottom row) such that the number of bricks in the $M$ tower is half the bricks in the $N$ tower?

The Diophantine equation to solve here is:$$N(N+1)=2M(M+1)$$

and the first few solutions are $(N,M)$:

$(3,2)$

$(20,14)$

$(119,84)$

$(696,492)$

Searching OEIS for the $M$ solutions we find it here. In the Formula section of that link it gives the generating function of this sequence as: $$M(k)=\frac{1}{8}\left(-4+(2+\sqrt{2})(3+2\sqrt{2})^k +(2-\sqrt{2})(3-2\sqrt{2})^k\right)$$ My question is: How in the world was this generating function arrived at?

It seems incredible to me that a closed formula exists for this Diophantine equation. Is this always the case?

EDIT

As @Sil pointed out in a comment, the formula at the OEIS link is not a generating function, but a closed formula. My question though, is basically the same. How was this closed formula arrived at?

• @Sil My mistake. I'll edit the question to reflect this. But my basic question is the same: How is the closed form arrived at? – Jens Jul 12 '18 at 21:36

By viewing the links, formulas, and other information of sequences A001652, A001653, and A053141 what will be found is that, in terms of Pell, $P_{n}$, and Pell-Lucas, $Q_{n}$, numbers is that $$N_{k} = \frac{Q_{2k+1} - 2}{4} \hspace{10mm} M_{k} = \frac{P_{2k+1} - 1}{2}.$$ When these values are placed into the equation as a check it will be found that the resulting identity is a known value for these number sets.
Since the first few values of $$\binom{n+1}{2} = 2 \, \binom{m+1}{2}$$ in terms of $(n,m)$ are $(n,m) \in \{ (3,2), (20,14), (119,84), (696,492), \cdots \}$. For the "$m$" values $m \in \{2, 14, 84, 492, \cdots \}$ it can be determined that $m_{k} = 6 \, m_{k-1} - m_{k-2} + 2$. The solution of this difference equation is $2 \, m_{k} = P_{2k+1} -1$, where $P_{n}$ are the Pell numbers. A similar process can be taken for the "$n$" values and yields the formula as stated in terms of the Pell-Lucas numbers.
• @Jens Since now all you need is closed form for Pell's numbers, and those satisfy $P_0=0, P_1=1, P_n=2P_{n-1}+P_{n-2}$... Can you find a closed form from linear recurrence relation? – Sil Jul 13 '18 at 3:26
Multiply the equation by $4$ and complete the square to put it into Pell's equation \begin{eqnarray*} 4N^2+4N+1=2(4M^2+4M+1)-1 \\ (2N+1)^2-2(2M+1)^2=-1 \\ n^2-2m^2=-1. \end{eqnarray*}


text:
# Work and Kinetic Energy of a sled

140
First, let us consider a sled of mass m being pulled by a constant, horizontal force of magnitude F along a rough, horizontal surface. The sled is speeding up.

how many major forces are acting on the sled? would it be 4 forces? gravity, normal, tension and kinetic?

2. ### dextercioby

12,314
Yes,there are 4 forces acting on the sled.That 'kinetic' hopefully stands for 'kinetic friction force'...What tension...?There's the force that pulls the sled,F,there's no tension.

Daniel.

3. ### Jameson

787
Gravity - yes. Kinetic (friction force) - yes. No tension.. well none that's really on the sled. Normal force - yes. And finally, the horizontal force. You get a nice diagram of 4 forces all on every side of the sled.

Last edited: Mar 2, 2005


text:
# American Institute of Mathematical Sciences

May  2014, 34(5): 1747-1774. doi: 10.3934/dcds.2014.34.1747

## Elliptic problems with nonlinear terms depending on the gradient and singular on the boundary: Interaction with a Hardy-Leray potential

 1 Laboratoire D'Analyse Nonlinéaire et Mathématiques Appliquées, Université Aboubekr Belkaïd, Tlemcen, Tlemcen 13000, Algeria 2 Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Università di Roma Sapienza, via Scarpa 16, 00161 Roma, Italy 3 Departamento de Matemáticas, U. Autónoma de Madrid, 28049 Madrid 4 Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain

Received  July 2013 Revised  August 2013 Published  October 2013

In this article we consider the following family of nonlinear elliptic problems,
$-\Delta (u^m) - \lambda \frac{u^m}{|x|^2} = |Du|^q + c f(x).$
We will analyze the interaction between the Hardy-Leray potential and the gradient term getting existence and nonexistence results in bounded domains $\Omega\subset\mathbb{R}^N$, $N\ge 3$, containing the pole of the potential.
Recall that $Λ_N = (\frac{N-2}{2})^2$ is the optimal constant in the Hardy-Leray inequality.
1.For $0 < m \le 2$ we prove the existence of a critical exponent $q_+ \le 2$ such that for $q > q_+$, the above equation has no positive distributional solution. If $q < q_+$ we find solutions by using different alternative arguments.
Moreover if $q = q_+ > 1$ we get the following alternative results.
(a) If $m < 2$ and $q=q_+$ there is no solution.
(b) If $m = 2$, then $q_+=2$ for all $\lambda$. We prove that there exists solution if and only if $2\lambda\leq\Lambda_N$ and, moreover, we find infinitely many positive solutions.
2. If $m > 2$ we obtain some partial results on existence and nonexistence.
We emphasize that if $q(\frac{1}{m}-1)<-1$ and $1 < q \le 2$, there exists positive solutions for any $f \in L^1(Ω)$.
Citation: Boumediene Abdellaoui, Daniela Giachetti, Ireneo Peral, Magdalena Walias. Elliptic problems with nonlinear terms depending on the gradient and singular on the boundary: Interaction with a Hardy-Leray potential. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 1747-1774. doi: 10.3934/dcds.2014.34.1747
##### References:
 [1] B. Abdellaoui, A. Dall'Aglio and I. Peral, Some Remarks on Elliptic Problems with Critical Growth in the Gradient, J. Diff. Eq., 222 (2006), 21-62. doi: 10.1016/j.jde.2005.02.009.  Google Scholar [2] B. Abdellaoui, D. Giachetti, I. Peral and M. Walias, Elliptic problems with nonlinear terms depending on the gradient and singular on the boundary, Nonlinear Analysis, 74 (2011), 1355-1371. doi: 10.1016/j.na.2010.10.008.  Google Scholar [3] B. Abdellaoui and I. Peral, Nonexistence results for quasilinear elliptic equations related to Caffarelli-Kohn-Nirenberg inequalities, Communications in Pure and Applied Analysis, 2 (2003), 539-566. doi: 10.3934/cpaa.2003.2.539.  Google Scholar [4] B. Abdellaoui and I. Peral, The Equation $-\Delta u - \lambda \frac u{|x|^2} = |D u|^p + c f(x)$, the optimal power, Ann. Scuola Norm. Sup. Pisa, VI, (2007), 159-183.  Google Scholar [5] B. Abdellaoui and I. Peral, Existence and nonexistence results for quasilinear elliptic equations involving the p-Laplacian with a critical potential, Annali di Matematica Pura e Applicata, 182 (2003), 247-270. doi: 10.1007/s10231-002-0064-y.  Google Scholar [6] B. Abdellaoui and I. Peral, A note on a critical problem with natural growth in the gradient, J. Eur. Math. Soc., 8 (2006), 157-170. doi: 10.4171/JEMS/43.  Google Scholar [7] N. E. Alaa and M. Pierre, Weak solutions of some quasilinear elliptic equations with data measures, SIAM J. Math. Anal., 24 (1993), 23-35. doi: 10.1137/0524002.  Google Scholar [8] D. Arcoya, L. Boccardo, T. Leonori and A. Porretta, Some elliptic problems with singular natural growth lower order terms, J.diff. equation, 249 (2010), 2771-2795. doi: 10.1016/j.jde.2010.05.009.  Google Scholar [9] P. Baras and M. Pierre, Singularités éliminables pour des équations semi-linéaires, Ann. Inst. Fourier, 34 (1984), 185-206. doi: 10.5802/aif.956.  Google Scholar [10] L. Boccardo, Dirichlet problems with singular and gradient quadratic lower order terms, ESAIM - Control, Optimisation and Calculus of Variations, 14 (2008), 411-426. doi: 10.1051/cocv:2008031.  Google Scholar [11] L.Boccardo and F. Murat, Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlinear Anal., 19 (1992), 581-597. doi: 10.1016/0362-546X(92)90023-8.  Google Scholar [12] L. Boccardo, L. Orsina and I. Peral, A remark on existence and optimal summability of solutions of elliptic problems involving Hardy potential, Discrete Contin. Dyn. Syst., 16 (2006), 513-523. doi: 10.3934/dcds.2006.16.513.  Google Scholar [13] H. Brezis and X. Cabré, Some simple nonlinear PDE's without solution, Boll. Unione. Mat. Ital. Sez. B, 8 (1998), 223-262.  Google Scholar [14] H. Brezis and S. Kamin, Sublinear elliptic equations in $\mathbbR^N$, Manuscripta Math., 74 (1992), 87-106. doi: 10.1007/BF02567660.  Google Scholar [15] H. Brezis and A. Ponce, Kato's inequality when $\Delta u$ is a measure, C.R. Math. Acad. Sci. Paris, 338 (2004), 599-604. doi: 10.1016/j.crma.2003.12.032.  Google Scholar [16] L. Caffarelli, R. Kohn and L. Nirenberg, First Order Interpolation Inequality with Weights, Compositio Math., 53 (1984), 259-275.  Google Scholar [17] G. Dal Maso, F. Murat, L. Orsina, Luigi and A. Prignet, Renormalized solutions of elliptic equations with general measure data, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 28 (1999), 741-808.  Google Scholar [18] D. Giachetti and F. Murat, An elliptic problem with a lower order term having singular behavior, Boll. Unione Mat. Ital., 2 (2009), 349-370.  Google Scholar [19] T. Kato, Schrödinger operators with singular potentials, Israel J. Math., 13 (1972), 135-148. doi: 10.1007/BF02760233.  Google Scholar [20] F. Murat, L'injection du cone positif de $H^{-1}$ dans $W^{-1,q}$ est compacte pour tout $q<2$, J. Math. Pures Appl., 60 (1981), 309-322.  Google Scholar

show all references

##### References:
 [1] B. Abdellaoui, A. Dall'Aglio and I. Peral, Some Remarks on Elliptic Problems with Critical Growth in the Gradient, J. Diff. Eq., 222 (2006), 21-62. doi: 10.1016/j.jde.2005.02.009.  Google Scholar [2] B. Abdellaoui, D. Giachetti, I. Peral and M. Walias, Elliptic problems with nonlinear terms depending on the gradient and singular on the boundary, Nonlinear Analysis, 74 (2011), 1355-1371. doi: 10.1016/j.na.2010.10.008.  Google Scholar [3] B. Abdellaoui and I. Peral, Nonexistence results for quasilinear elliptic equations related to Caffarelli-Kohn-Nirenberg inequalities, Communications in Pure and Applied Analysis, 2 (2003), 539-566. doi: 10.3934/cpaa.2003.2.539.  Google Scholar [4] B. Abdellaoui and I. Peral, The Equation $-\Delta u - \lambda \frac u{|x|^2} = |D u|^p + c f(x)$, the optimal power, Ann. Scuola Norm. Sup. Pisa, VI, (2007), 159-183.  Google Scholar [5] B. Abdellaoui and I. Peral, Existence and nonexistence results for quasilinear elliptic equations involving the p-Laplacian with a critical potential, Annali di Matematica Pura e Applicata, 182 (2003), 247-270. doi: 10.1007/s10231-002-0064-y.  Google Scholar [6] B. Abdellaoui and I. Peral, A note on a critical problem with natural growth in the gradient, J. Eur. Math. Soc., 8 (2006), 157-170. doi: 10.4171/JEMS/43.  Google Scholar [7] N. E. Alaa and M. Pierre, Weak solutions of some quasilinear elliptic equations with data measures, SIAM J. Math. Anal., 24 (1993), 23-35. doi: 10.1137/0524002.  Google Scholar [8] D. Arcoya, L. Boccardo, T. Leonori and A. Porretta, Some elliptic problems with singular natural growth lower order terms, J.diff. equation, 249 (2010), 2771-2795. doi: 10.1016/j.jde.2010.05.009.  Google Scholar [9] P. Baras and M. Pierre, Singularités éliminables pour des équations semi-linéaires, Ann. Inst. Fourier, 34 (1984), 185-206. doi: 10.5802/aif.956.  Google Scholar [10] L. Boccardo, Dirichlet problems with singular and gradient quadratic lower order terms, ESAIM - Control, Optimisation and Calculus of Variations, 14 (2008), 411-426. doi: 10.1051/cocv:2008031.  Google Scholar [11] L.Boccardo and F. Murat, Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlinear Anal., 19 (1992), 581-597. doi: 10.1016/0362-546X(92)90023-8.  Google Scholar [12] L. Boccardo, L. Orsina and I. Peral, A remark on existence and optimal summability of solutions of elliptic problems involving Hardy potential, Discrete Contin. Dyn. Syst., 16 (2006), 513-523. doi: 10.3934/dcds.2006.16.513.  Google Scholar [13] H. Brezis and X. Cabré, Some simple nonlinear PDE's without solution, Boll. Unione. Mat. Ital. Sez. B, 8 (1998), 223-262.  Google Scholar [14] H. Brezis and S. Kamin, Sublinear elliptic equations in $\mathbbR^N$, Manuscripta Math., 74 (1992), 87-106. doi: 10.1007/BF02567660.  Google Scholar [15] H. Brezis and A. Ponce, Kato's inequality when $\Delta u$ is a measure, C.R. Math. Acad. Sci. Paris, 338 (2004), 599-604. doi: 10.1016/j.crma.2003.12.032.  Google Scholar [16] L. Caffarelli, R. Kohn and L. Nirenberg, First Order Interpolation Inequality with Weights, Compositio Math., 53 (1984), 259-275.  Google Scholar [17] G. Dal Maso, F. Murat, L. Orsina, Luigi and A. Prignet, Renormalized solutions of elliptic equations with general measure data, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 28 (1999), 741-808.  Google Scholar [18] D. Giachetti and F. Murat, An elliptic problem with a lower order term having singular behavior, Boll. Unione Mat. Ital., 2 (2009), 349-370.  Google Scholar [19] T. Kato, Schrödinger operators with singular potentials, Israel J. Math., 13 (1972), 135-148. doi: 10.1007/BF02760233.  Google Scholar [20] F. Murat, L'injection du cone positif de $H^{-1}$ dans $W^{-1,q}$ est compacte pour tout $q<2$, J. Math. Pures Appl., 60 (1981), 309-322.  Google Scholar
 [1] Xiaomei Sun, Wenyi Chen. Positive solutions for singular elliptic equations with critical Hardy-Sobolev exponent. Communications on Pure & Applied Analysis, 2011, 10 (2) : 527-540. doi: 10.3934/cpaa.2011.10.527 [2] Xiaomei Sun, Yimin Zhang. Elliptic equations with cylindrical potential and multiple critical exponents. Communications on Pure & Applied Analysis, 2013, 12 (5) : 1943-1957. doi: 10.3934/cpaa.2013.12.1943 [3] Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037 [4] Yinbin Deng, Qi Gao, Dandan Zhang. Nodal solutions for Laplace equations with critical Sobolev and Hardy exponents on $R^N$. Discrete & Continuous Dynamical Systems, 2007, 19 (1) : 211-233. doi: 10.3934/dcds.2007.19.211 [5] Massimiliano Berti, M. Matzeu, Enrico Valdinoci. On periodic elliptic equations with gradient dependence. Communications on Pure & Applied Analysis, 2008, 7 (3) : 601-615. doi: 10.3934/cpaa.2008.7.601 [6] Dumitru Motreanu, Viorica V. Motreanu, Abdelkrim Moussaoui. Location of Nodal solutions for quasilinear elliptic equations with gradient dependence. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 293-307. doi: 10.3934/dcdss.2018016 [7] Boumediene Abdellaoui, Ahmed Attar. Quasilinear elliptic problem with Hardy potential and singular term. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1363-1380. doi: 10.3934/cpaa.2013.12.1363 [8] Yanfang Peng, Jing Yang. Sign-changing solutions to elliptic problems with two critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2015, 14 (2) : 439-455. doi: 10.3934/cpaa.2015.14.439 [9] Fengshuang Gao, Yuxia Guo. Multiple solutions for a critical quasilinear equation with Hardy potential. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 1977-2003. doi: 10.3934/dcdss.2019128 [10] Jing Zhang, Shiwang Ma. Positive solutions of perturbed elliptic problems involving Hardy potential and critical Sobolev exponent. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1999-2009. doi: 10.3934/dcdsb.2016033 [11] Naoki Hamamoto, Futoshi Takahashi. Sharp Hardy-Leray inequality for three-dimensional solenoidal fields with axisymmetric swirl. Communications on Pure & Applied Analysis, 2020, 19 (6) : 3209-3222. doi: 10.3934/cpaa.2020139 [12] Fengshuang Gao, Yuxia Guo. Infinitely many solutions for quasilinear equations with critical exponent and Hardy potential in $\mathbb{R}^N$. Discrete & Continuous Dynamical Systems, 2020, 40 (9) : 5591-5616. doi: 10.3934/dcds.2020239 [13] Xiaorong Luo, Anmin Mao, Yanbin Sang. Nonlinear Choquard equations with Hardy-Littlewood-Sobolev critical exponents. Communications on Pure & Applied Analysis, 2021, 20 (4) : 1319-1345. doi: 10.3934/cpaa.2021022 [14] Mousomi Bhakta, Debangana Mukherjee. Semilinear nonlocal elliptic equations with critical and supercritical exponents. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1741-1766. doi: 10.3934/cpaa.2017085 [15] Lucio Boccardo, Luigi Orsina, Ireneo Peral. A remark on existence and optimal summability of solutions of elliptic problems involving Hardy potential. Discrete & Continuous Dynamical Systems, 2006, 16 (3) : 513-523. doi: 10.3934/dcds.2006.16.513 [16] Emmanuel Hebey, Jérôme Vétois. Multiple solutions for critical elliptic systems in potential form. Communications on Pure & Applied Analysis, 2008, 7 (3) : 715-741. doi: 10.3934/cpaa.2008.7.715 [17] Jinhui Chen, Haitao Yang. A result on Hardy-Sobolev critical elliptic equations with boundary singularities. Communications on Pure & Applied Analysis, 2007, 6 (1) : 191-201. doi: 10.3934/cpaa.2007.6.191 [18] Yinbin Deng, Shuangjie Peng, Li Wang. Infinitely many radial solutions to elliptic systems involving critical exponents. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 461-475. doi: 10.3934/dcds.2014.34.461 [19] Yu Zheng, Carlos A. Santos, Zifei Shen, Minbo Yang. Least energy solutions for coupled hartree system with hardy-littlewood-sobolev critical exponents. Communications on Pure & Applied Analysis, 2020, 19 (1) : 329-369. doi: 10.3934/cpaa.2020018 [20] Minbo Yang, Fukun Zhao, Shunneng Zhao. Classification of solutions to a nonlocal equation with doubly Hardy-Littlewood-Sobolev critical exponents. Discrete & Continuous Dynamical Systems, 2021, 41 (11) : 5209-5241. doi: 10.3934/dcds.2021074

2020 Impact Factor: 1.392


text:
$$\require{cancel}$$

# Voltage

[ "article:topic" ]

### The Volt

Electrical circuits can be used for sending signals, storing information, or doing calculations, but their most common purpose by far is to manipulate energy, as in the battery-and-bulb example of the previous section. We know that lightbulbs are rated in units of watts, i.e., how many joules per second of energy they can convert into heat and light, but how would this relate to the flow of charge as measured in amperes? By way of analogy, suppose your friend, who didn't take physics, can't find any job better than pitching bales of hay. The number of calories he burns per hour will certainly depend on how many bales he pitches per minute, but it will also be proportional to how much mechanical work he has to do on each bale. If his job is to toss them up into a hayloft, he will got tired a lot more quickly than someone who merely tips bales off a loading dock into trucks. In metric units,

equation

Similarly, the rate of energy transformation by a battery will not just depend on how many coulombs per second it pushes through a circuit but also on how much mechanical work it has to do on each coulomb of charge:

equation

or

equation

Units of joules per coulomb are abbreviated as volts, 1 V=1 J/C, named after the Italian physicist Alessandro Volta. Everyone knows that batteries are rated in units of volts, but the voltage concept is more general than that; it turns out that voltage is a property of every point in space.

To gain more insight, let's think again about the analogy with the haybales. It took a certain number of joules of gravitational energy to lift a haybale from one level to another. Since we're talking about gravitational energy, it really makes more sense to talk about units of mass, rather than using the haybale as our measure of the quantity of matter. The gravitational version of voltage would then be joules per kilogram. Gravitational energy equals mgh, but if we calculate how much of that we have per kilogram, we're canceling out the m, giving simply gh. For any point in the Earth's gravitational field, we can assign a number, gh, which tells us how hard it is to get a given amount of mass to that point. For instance, the top of Mount Everest would have a big value of gh, because of the big height. That tells us that it's expensive in terms of energy to lift a given amount of mass from some reference level (sea level, say) to the top of Mount Everest.

h / Alessandro Volta (1745-1827).

Voltage does the same thing, but using electrical energy. We can visualize an electrical circuit as being like a roller-coaster. The battery is like the part of the roller-coaster where they lift you up to the top. The height of this initial hill is analogous to the voltage of the battery. When you roll downhill later, that's like a lightbulb. In the roller-coaster, the initial gravitational energy is turned into heat and sound as the cars go down the hill. In our circuit, the initial electrical energy is turned into heat by the lightbulb (and the hot filament of the lightbulb then glows, turning the heat into light).

##### Example 3: Energy stored in a battery

i / Example 3.

• The 1.2 V rechargeable battery in figure i is labeled 1800 milliamp-hours. What is the maximum amount of energy the battery can store?
• An ampere-hour is a unit of current multiplied by a unit of time. Current is charge per unit time, so an ampere-hour is in fact a funny unit of charge:

equation

1800 milliamp-hours is therefore 1800×10-3× 3600 C=6.5×103 C. That's a huge number of charged particles, but the total loss of electrical energy will just be their total charge multiplied by the voltage difference across which they move:

equation

Using the definition of voltage, V, we can rewrite the equation <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow><mtext>power</mtext><mo lspace="0.278em" rspace="0.278em">=</mo><mtext>current</mtext><mo lspace="0.222em" rspace="0.222em">×</mo><mtext>work per unit charge</mtext></mrow> [/itex] more concisely as P=IV.

##### Example 4: Units of volt-amps

◊ Doorbells are often rated in volt-amps. What does this combination of units mean?

◊ Current times voltage gives units of power, P= I V, so volt-amps are really just a nonstandard way of writing watts. They are telling you how much power the doorbell requires.

##### Example 5: Power dissipated by a battery and bulb
• If a 9.0-volt battery causes 1.0 A to flow through a lightbulb, how much power is dissipated?
• The voltage rating of a battery tells us what voltage difference Δ V it is designed to maintain between its terminals.

equation

The only nontrivial thing in this problem was dealing with the units. One quickly gets used to translating common combinations like A⋅V into simpler terms.

##### Discussion Questions
• In the roller-coaster metaphor, what would a high-voltage roller coaster be like? What would a high-current roller coaster be like?
• Criticize the following statements:
• “He touched the wire, and 10000 volts went through him.”
• “That battery has a charge of 9 volts.”
• “You used up the charge of the battery.”
• When you touch a 9-volt battery to your tongue, both positive and negative ions move through your saliva. Which ions go which way?
• I once touched a piece of physics apparatus that had been wired incorrectly, and got a several-thousand-volt voltage difference across my hand. I was not injured. For what possible reason would the shock have had insufficient power to hurt me?


text:
Bayesian linear regression - Defining hyperparameters

I am currently working on a linear regression task from a textbook and I have some questions. In the following you will find the task as well as some thoughts of mine to possible results.

Short description: A quadratic regression analysis shall be made for temperature data from France. The explained variable is given by: $temperature = \beta_0 + \beta_1 * time + \beta_2 * time^2 + \epsilon, \epsilon \sim N(0,\sigma^2)$

The conjugate prior is:

$\\ p(\beta, \sigma^2) = N-Inv-\chi^2(\beta_0, \sigma^2 \Omega^{-1}; v_0, s_0^2) \\ p(\beta | \sigma^2) = N(\beta_0, \sigma^2 \Omega^{-1}) \\ p(\sigma^2) = Inv-\chi^2(v_0, s_0^2)$

a) In the first task I shall determine the prior hyperparameters to suitable values:

$\mu_0 = Prior \ mean \\ \Omega_0 = Prior \ precision \\ v_0 = Degrees \ of \ freedom \\ \sigma^2 = Prior \ variance$

I can think about the prior mean as a guess of the average temperature in France (lets assume 20°C). But I can not make a sense out of the other parameters. How do I define the prior precision as well as the degrees of freedom and what are they telling me? I know that $\sigma^2$ can be derived from the Inv-$\chi^2$ distribution but how do I estimate $s_0^2$?


text:
# Suppose the independent number of a graph is bounded. How small the clique number can be?

Suppose the independent number of a graph is bounded. How small the clique number can be? linear? It seems to be a natural problem to ask. but I could not find any reference. Thanks.

-

This is basically a question in Ramsey theory. The Ramsey number $R(s, t)$ is the minimum integer $n$ for which every red-blue coloring of the edges of a complete $n$-vertex graph induces either a red complete graph of order $s$ or a blue complete graph of order $t$. So for example Kim (Random Structures and Algorithms 7 (1995), 173-207) showed that $R(3,t)\asymp t^2/\log t$. Roughly speaking this means that a graph with $t^2/\log t$ vertices with independence number at most 2 must have a clique of size $t$. So in this case the clique number grows like the square root of the number of vertices of the graph. I'm not sure what the best known results are in general but this should point you to the right literature.


text:
# What is 1/3 of 24 1/4 of 24 1/6 of 24

What is
1/3 of 24
1/4 of 24
1/6 of 24

### What does Diondra want to avoid revealing to others?

What does Diondra want to avoid revealing to others?...

### Two technicians regularly make repairs when breakdowns occur on an automated production line. the first technican, who services 40% of the breakdowns, has 5% chance of making incomplete repair. the second technican, who services 60% of the breakdowns, has 3% chance of making an incomplete repair. given that there is a problem with the production line due to an incomplete repair, what is the probability that thids intial repair was made by the first technican

two technicians regularly make repairs when breakdowns occur on an automated production line. the first technican, who services 40% of the breakdowns, has 5% chance of making incomplete repair. the second technican, who services 60% of the breakdowns, has 3% chance of making an incomplete repair. gi...

### A triangle has side lengths of (2t + 2) centimeters, (6t – 7) centimeters, and (u – 2) centimeters. Which expression represents the perimeter, in centimeters, of the triangle?

A triangle has side lengths of (2t + 2) centimeters, (6t – 7) centimeters, and (u – 2) centimeters. Which expression represents the perimeter, in centimeters, of the triangle?...

### What is an equation of the line that passes through 7,3 and a slope of 4/7

What is an equation of the line that passes through 7,3 and a slope of 4/7...

### Calculate the slope of the line on the graph using the formula: 72-V1 m X2 - X The slope of the line is _____m/s.

Calculate the slope of the line on the graph using the formula: 72-V1 m X2 - X The slope of the line is _____m/s....

### Compare and contrast. (Easy)

Compare and contrast. (Easy)...

### . In China, the Great Leap Foward was an attempt to ⦁ promote democratic reform ⦁ end the private ownership of land ⦁ strengthen economic ties with Europe ⦁ increase agricultural and industrial production

. In China, the Great Leap Foward was an attempt to ⦁ promote democratic reform ⦁ end the private ownership of land ⦁ strengthen economic ties with Europe ⦁ increase agricultural and industrial production...

### SCIENCE IS SO MEEN RN HALP

SCIENCE IS SO MEEN RN HALP...

### More points hope they help c:

More points hope they help c:...

### How does a reader determine the credibility of a narrator? 1)by evaluating whether or not the narrator is actually the speaker 2)by determining whether or not the narrator is the same as the author 3)by examining the balance and consistency of information provided by the narrator 4)by judging if the narrator is realistic enough to trust

How does a reader determine the credibility of a narrator? 1)by evaluating whether or not the narrator is actually the speaker 2)by determining whether or not the narrator is the same as the author 3)by examining the balance and consistency of information provided by the narrator 4)by judging if the...

### On December 1, Year 7, Gelt Corporation declared a dividend and distributed to its sole shareholder a parcel of land that was not an inventory asset. On the date of the distribution, the following data were available: Adjusted basis of land $6,500 Fair market value of land 14,000 Mortgage on land 5,000 For the year ended December 31, Year 7, Gelt had earnings and profits of$30,000 without regard to the dividend distribution. If the mortgage on the land was assumed by the sole shareholder, by h

On December 1, Year 7, Gelt Corporation declared a dividend and distributed to its sole shareholder a parcel of land that was not an inventory asset. On the date of the distribution, the following data were available: Adjusted basis of land \$ 6,500 Fair market value of land 14,000 Mortgage on land 5...

### Why was the PLO originally created? O to serve as the government of the Gaza Strip to promote the "right to return" of Israelis O to represent the interests of Palestinians O to advance resettlement of the West Bank

Why was the PLO originally created? O to serve as the government of the Gaza Strip to promote the "right to return" of Israelis O to represent the interests of Palestinians O to advance resettlement of the West Bank...

### Explain the process for finding the product of two integers.

Explain the process for finding the product of two integers....

### PLEASE HELP I ONLY HAVE LIMITED TIME!!! Choose all the phenotypes you see a bb b pink c short d Cc e wW

PLEASE HELP I ONLY HAVE LIMITED TIME!!! Choose all the phenotypes you see a bb b pink c short d Cc e wW...

### A triangular prims cover with a fold label. the label dose not cover the 2 ends. how much foil is necessary to cover the prism.ples help I need to pass

a triangular prims cover with a fold label. the label dose not cover the 2 ends. how much foil is necessary to cover the prism.ples help I need to pass ...

### How many valence electrons does each atom of arsenic have?

how many valence electrons does each atom of arsenic have?...

### Help me pleaseee and thank youWhich pronoun signals a first-person narration? ○ his ○ you ○Our ○they

Help me pleaseee and thank youWhich pronoun signals a first-person narration? ○ his ○ you ○Our ○they...


text:
# 유유아(乳幼兒) 및 성장기 아동을 위한 영양식품 개발에 관한 연구 -흰쥐 성장에 미치는 영향-

• Ho, Jin-Hee (Dept. of Food and Nutrition, College of Home Economics, Ewha Woman's University) ;
• Kim, Sook-He (Dept. of Food and Nutrition, College of Home Economics, Ewha Woman's University)
• 호진희 (이화여자대학교 가정대학 식품영양학과) ;
• 김숙희 (이화여자대학교 가정대학 식품영양학과)
• Published : 1970.06.30

#### Abstract

Korea is one of the typical rice eating countries where availability of animal protein is limited even for the growing generation. Nutritive food products for infants and children are not available commercially in large scale at the present time, although a limited amount of expensive milk products are produced. The present study deals with a pioneering attempt on a new food mixture to meet this demand. Several food mixtures, possibly produced in the country, consisting of rice, soybean, FPC, vitamin, minerals and other food additives are developed in this work Sixty female and male rats aged $30{\sim}40$ days were divided into five groups, twelve rats each. The rats had been maintained with the six different diets, as follow. Formulation of proposed infant food mixturesComposition F-R-1 F-S-2 F-F-3 F-P-4 S Rice 100% 40% 37% 46% 70% Sugar - 12 13 10 70 Casein - - - - 20 Bean - 40 37 24 - Yeast - 3 2 3 - Mineral Vitamain - 2 4 2 4.3 F.P.C - 3 4 7 - Fat - - 4 8 4 Cod liver Oil - - - - 3 The findings of this study presented in FER, PER, final organ weights, body weights growth, percentage of nitrogen retention in the body. F-P-4 group tended to remain the heaviest and F-R-1 the lightest in the body weight among three groups throughout the experimental period. In terms of FER (Feed Efficiency Ratio) and PER (Prortein E. R.) value, F-P-4 group kept the highest record throughout. F-P-4 group showed the highest value of nitrogen retention in the body. In comparision between F-P-4 groups and casein 20% group (standard) in all respects of this experiment, F-P-4 group which is the most superior among experimental group, revealed statisfically no significant inferior than that of standard group. This fact could be interpreted that most limiting a. a., methionine in the soy protein produced little inferiority of the experimental groups in the study.