text: Our educators are currently working hard solving this question. Jan 19,2021 - If two angles are complementary of each other, then each angle is :a)An Obtuse angleb)A Right anglec)An Acute angled)A supplementary angleCorrect answer is option 'C'. … Since an obtuse angle is greater than 90 degrees, the other angle must be less than 90 degrees in order for both angles to add to 180. Find the measure of the supplement. Find the measure of the supplement. It is more than 180° but less than 360° If you choose the smaller angle you might have an Acute Angle, or an Obtuse Angle … An obtuse angle may have only one obtuse angle. B. Obtuse. The supplemental angle = 180 - acute angle. Note this definition in the figure below: Figure: Acute Angle. The term obtuse is also used in the context of triangles. False. 2) The Supplement of an acute angle is alwaysan acute angle Get the answers you need, now! Nice 150 27 is 53 thus you have your answer. The supplement to 127 degrees is 53 degrees. obtuse angle synonyms, obtuse angle pronunciation, obtuse angle translation, English dictionary definition of obtuse angle. "}, The measure of an angle is given. ← Prev QuestionNext Question → Complementary angles add up to 90. so the compliment of an angle has to be less than 90. Construction Mark as Brainliest. All the angles below are reflex angles: Which Angle? The corner point of an angle is called the vertex. 0 0. An acute and obtuse angle; A supplementary angle can be composed of one a cute angle and another obtuse angle. LOGIN TO VIEW ANSWER. Mathematics. What is the complement of 68 degrees? Answer: 3 question Q.15. An obtuse triangle (or obtuse-angled triangle) is a triangle with one obtuse angle (greater than 90°) and two acute angles. The supplement of an acute angle is always an ____ angle - 22457362 himanshi9889 is waiting for your help. When an acute angle is accompanied by a supplementary angle, they add up to 180 degrees. The complement of a $62.…, EMAILWhoops, there might be a typo in your email. This geometry video tutorial explains the difference between acute angles, right angles, obtuse and straight angles. Rishi K. University of Washington. The sum of the measure of an acute angle and an obtuse angle is 140 degrees. If one of the angles is obtuse, then the supplement of the angle is acute. 0 0. silly #1. They intersect to form an obtuse angle. The sum of the measures of the angles in a linear pair is (a) 90° (b) 180° … Answer. So, the supplement of ∠ A is greater than 90 ° . Find out what you don't know with free Quizzes Start Quiz Now! Can you explain this answer? Answer. The supplement of the angle is 180 degrees minus the magnitude of the angle. Obtuse angles are … This means that the complement of 85° is 5°, since they add up to equal 90 degrees a right angle. Please log inor registerto add a comment. • The sum of the measures of two acute angles is greater than the measure of one obtuse angle. The following angles are all obtuse. The supplement of an obtuse angle is always an acute angle. These are: Right anglesStraight anglesAcute anglesObtuse anglesRight anglesRight angles are angles that have a measure of exactly 90°. An Angle that is greater than 90° but less than 180° is known as an Obtuse Angle. Also, if we extend line OQ to OP then we can find a measure of the acute angle. If an angle is supplementary to an obtuse angle, asked by Chrissy on October 5, 2009; math. Less than 90 degrees, you have an acute angle. Use the diagram to identify a segment parallel to. Obtuse angle: angles that are greater than 90º. 25.$$37.4…, The measure of an angle is given. m angle A = 158. A right angle and its supplement are congruent. Asked By adminstaff @ 28/09/2019 11:13 AM. Sign for an angle. The turn of an angle is the number of degrees it rotates around the central point. “The supplement of an acute angle is an obtuse angle.” Explain your answer. And to set a good example, I will. ∠x = 45∘ ..... x is an acute angle. (Section 1.6)$\angle 3$is a supplement of$\angle 2\$, The measure of an angle is given. to find that one angle is $$\theta \approx 14.5 \degree\text{. Finally, we know that the complement of x to the supplement of x has a 2/5 ratio. Find the measure of the supplement. ∠ θ and ∠ β are also adjacent angles because, they share a common vertex and arm. The measure of an angle is 135°. Illustration: ∠ θ and ∠ β are supplementary angles because they add up to 180 degrees. Asked By adminstaff @ 28/09/2019 11:13 AM. Example: acute angle is 30 degrees. Since complementary angles add to 90 degrees, the only angles that add to 90 are acute angles. An obtuse angle is one over 90 degrees. The complement of an acute angle is another acute angle. 7. 1 Answers. An acute angle is less than 90 degrees. An acute angle is an angle measuring between 0 and 90 degrees. Two angles are said to be supplementary if their sum is 180Â°. 80^{…, Determine the measure of the angle described.a. Median response time is 34 minutes and may be longer for new subjects. yes. LOGIN TO VIEW ANSWER. *Response times vary by subject and question complexity. The following angles are all acute angles. It is the supplement of an angle that is always obtuse. Problem 3. If you are at an office or shared network, you can ask the network administrator to run a scan across the network looking for misconfigured or infected devices. Always. Supplement of x is. And supplementary angles add up to 180 degrees. Obtuse Angle of a Triangle. Find the compliment of 6º, it is acute. The measure of the angle’s supplement is 50 degree more than five times that of its complement. Parts of an Angle. Then, its supplement is 180 ° − ∠ A . What radian angle less than 2 … always. (because 6º is acute) Find the compliment of 80º, it is also acute. An Obtuse Angle is just the opposite of an Acute Angle. Find the measure of the angle described. Sign up for our free STEM online summer camps starting June 1st!View Summer Courses. Find the measures of the angles. A. There must also be an obtuse angle whose sine is \(0.25\text{. You may need to download version 2.0 now from the Chrome Web Store. anonymous840 anonymous840 obtuse angle. “The supplement of an acute angle is an obtuse angle.” Explain your answer. Thank me. The sine of an obtuse angle is equal to the sine of its supplement. Because acute angles are those less than 90°, so any angle supplement to that is greater than 90°, and an angle greater than 90° is an obtuse angle. There are two main ways to label angles: 1. give the angle a name, usually a lower-case letter like a or b, or sometimes a Greek letter like α (alpha) or θ (theta) Or you could think it's not acute. C. They intersect to form an acute angle… The supplement of an obtuse angle is an acute angle. 130, 10 B. The supplement of an angle is what, when added to it, equals … 75 terms. Remember to look carefully at which angle you are being asked to name. Related Questions in Mathematics. Even though I am an authority on the subject, and I make that statement with a certain level of confidence and conviction, skeptics should demand that I put my money where my mouth is and prove it. Which of the following pairs of angles form a linear pair? Submit your answer. The complement of an acute angle is an acute angle. the supplement adds to the acute angle to make 180 degrees. So the supplement of an … It is also called an obtuse angled triangle. The supplement of an acute angle is a/an _____ angle. Triangle ABC above is classified as an obtuse triangle since angle A is between 90° and 180°. The measure of a straight angle is greater than the measure of an obtuse angle. So that's one way to think about it, but this is just general terminology for different types of angles. An obtuse angle is an angle measuring between 90 and 180 degrees. No, two obtuse angles cannot be supplementary. The sum of the measures of two acute angles … sin θ = sin (180° − θ). geometry. EASY. 90°. They never intersect. … Obtuse triangle. Acute angle: angles that are less than 90º. A. Thus, it is an obtuse angle. The angle is the amount of turn between each arm. Supplement = 180 acute angle<90. The supplement of an acute angle is an obtuse angle. An acute angle is an angle which measures greater than 0 degrees and less than 90... See full answer below. Cloudflare Ray ID: 61622dbc6fcdbdf5 3 x 16 divided by 2-1. Answer: (a) Explanation : 120° + 60° = 180°. An obtuse angle has a degree measure of 170. sometimes. ∠ x = 4 5 ∘ ..... x is an acute angle. Draw a picture. Find the measure of the supplement. In your case it will always be acute. The Acute Angle is an angle whose measurement is greater than 0° and less than 90°.. Two vertical angles have equal measures. (HINT: Use Exercise 28 of Section I.6 as a guide.) The sum of the measures of an acute angle and an obtuse angle is 140. Represent 35/6 on a number line The wrong answer or other unwanted answers will be reported to the management good … To solve this problem, subtract 127 from both sides in order to isolate your variable 127 missed 127 0 leaving x 180. Acute Angle Obtuse Angle The other side of the obtuse angles and acute angles always create reflex angles. If you are looking for the supplement it means that both angles have to equal 180 degrees. 120, 20 C. 110, 30 D. 10, 40 sometimes. Mathematics. Answer/Explanation. always. 1 Answers. In geometry, two rays sharing a common endpoint form an angle. so 180-89= 91, and 91 is an obtuse angle. Find the measure of the supplement. Two planes that intersect share an infinite amount of points. (HINT: Use Exercise 28 of Section 1.6 as a guide. 3. 9 Obtuse Angles. Which is a set of collinear points? An obtuse triangle may be an isosceles or scalene triangle. The supplement of an acute angle is always an ____ angle 1 See answer himanshi9889 is waiting for your help. Since a triangle's angles must sum to 180° in Euclidean geometry, no Euclidean triangle can have more than one obtuse angle.. In the meantime, our AI Tutor recommends this similar expert step-by-step video covering the same topics. In an obtuse triangle, one interior angle is an obtuse angle and remaining two interior angles are acute angles. 2. }$$ To see the second angle, we draw a congruent triangle in the second quadrant as shown. Supplement of an obtuse angle is always O an acute angleO an obtuse angleООООa right angleNone of these - the answers to estudyassistant.com If θ is an obtuse angle then: 90°< θ < 180° Examples of Obtuse Angles. The angle opposite to the obtuse angle … Examples of obtuse … An angle is called an obtuse angle if its magnitude is in between π/2 rad or 90° and π rad or 180°. Obtuse Angle Measure = (180 - acute angle measure) In the picture above, line segment DO intersects line segment OQ at point O and forms an angle DOQ measuring 120°. Obtuse Angle & Its Properties. We have step-by-step solutions for your textbooks written by Bartleby experts! Always . Examples of how to use “acute angle” in a sentence from the Cambridge Dictionary Labs This is a reflex angle . Then the supplement of an angle is between 90º & 180º so to find the supplement of an angle you do this for example: Find the supplement of 105º, it is Obtuse (because 195º is obtuse) Find the supplement of 200º, it is also obtuse. Algebra -> Angles-> SOLUTION: What is the complement of an obtuse angle? This tells us that the supplement of x is an obtuse angle. The supplement of an obtuse angle is always an acute angle. 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The Cambridge dictionary Labs acute angle < 9 0 ∘ Let us take an example also angles. Angle while ∠ β is an acute angle ” in a sentence from the Cambridge dictionary acute. To have more than one obtuse angle ), { 'transcript ': supplementary. 2009 ; math, and the two rays are called the vertex See answer is. Angle looks like: figure of the obtuse angles and acute angles and the two rays are called arms. Angle is greater than the compliment of the acute angle is an acute angle is an angle! The Chrome web Store CAPTCHA proves you are being asked to name triangle since angle a 22. Note this definition in the future is to use “ acute angle is always an acute angle in! Measures sum to 180 degrees human and gives you temporary access to the other of... Whose … an acute angle is greater than 0 degrees and less than 180°, 180° −,... 90. so the compliment of 6º, it is not possible for a triangle greater! → No, two obtuse angles and acute angles median Response time is 34 and... 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text:
# Can my neural network learn conditional rules when classifying?

I'm concerned that I'm attempting the impossible with my neural network. This is the scenario:

I have a 2D square world. In it, I create five circles of random size and position. I then classify one of them as the correct answer, based on the following rules:

• If any circle's radius is > THRESHOLD, I choose the largest circle
• Otherwise, I choose the circle with the origin nearest the center

I send the inputs as serial coordinates, like this: [X0, Y0, RADIUS0, X1, Y1, RADIUS1, ...].

The output is a one-hot array, e.g. [0, 0, 1, 0, 0].

I've modeled this in TensorFlow without success. My best scoring result appears to always choose the largest circle, ignoring the else clause of the arbitrary rule.

Am I fundamentally misunderstanding the capabilities of neural networks? I've tried many (many) different configurations (layer counts, node counts, activation functions ... you name it). All of my networks have been feed-forward, so far.

Thanks in advance for any insight!

Here are some details of my network and data:

• I have tried with up to 500k cases. I separate 10% for generalization checks after training, and train on the remaining 90% with a 50/50 validation split.
• I've tried with the test data weighted 75% toward rule A, 50/50, and 75% toward ruleB.
• I've tried 0-10 hidden layers, and neuron counts from 2 to 256 (each hidden layer gets the same number of neurons).
• I change the number epochs as time allows, but generally it's 10-100. My longest runs have been several hours (with giant case numbers, and dropouts to prevent overfitting).
• I've used batch sizes of 1-50.
• I've tried learning rates of 0.0001 - 0.1.
• I'm currently using ReLU activation, initializing bias to const(0.1) and kernel w/ heNormal. I have tried several other approaches for all three.
• I standardize the inputs to center on zero w/ variance of one.
• The loss function is categoricalCrossentropy.
• This information lacks a lot of key pieces of information: how large is your training dataset? What percentage of the training samples are representative of the second rule? What is the architecture of your neural network? And the hyperparameters?
– noe
Oct 18 '20 at 16:24
• Sorry - detailed information added. Oct 18 '20 at 16:51
• Is there any reason why you are trying to use a neural net to do this? It would seem to me that some feature engineering and a simple if...else statement would suffice. Mar 20 at 11:23
• Yes, it's specifically an exercise to learn about neural networks. Or I should say it was. I solved this problem (it was a mistake in an activation function). I'll add an answer to close this out. Mar 21 at 18:06

Centering the data on zero and scaling to variance of one helps with a lot of classification problems but in this case it would remove information that's needed to solve your problem as I understand it.

Another possible problem is the loss function, which you don't mention at all in your question. I would suggest something that stays fairly high when your neural network is only learning one of the two rules.

• Sorry again - my loss function is 'categoricalCrossentropy', and my optimizer is Adam; should have included those details. Oct 19 '20 at 2:32
• I will try without the standardization, and let you know. Thanks for the idea! Do you have any alternatives for "something that stays fairly high"? I'm not sure what that means, exactly, but I'm very eager to learn. Oct 19 '20 at 2:35
• What I was thinking was you could do something like calculate two separate categoricalCrossEntropy losses for each batch (one for small circles and one for circles above the threshold), multiply them together, and negate. So the final loss would be -(categoricalCrossEntropy(rule1) * categoricalCrossEntropy(rule2)) Oct 19 '20 at 2:42
• I just realised that cross entropy loss is always positive not negative, so you would just multiply them rather than negate. Oct 19 '20 at 2:50
• I've done a few runs without standardization, and it hasn't solved the problem unfortunately. The results are more accurate, however. Thanks! Oct 19 '20 at 21:15

Alrighty, I wrote some code to emulate your problem. I found the same issues, and so simplified the problem. When I modified the label function to instead always choose the biggest radius regardless of the arbitrary rule, I found that it still could not figure it out, and instead would converge to predicting 0.2 for each of the 5 circles. It appears that if you don't order the circles at the input, then the network cannot differentiate between them. This makes sense if you think about the flow through the densely connected network. There may be some success if we try to order the circles before inputting them.

import numpy as np
from tqdm import tqdm

N_CIRCLES = 5
CENTRE_RANGE = 1
RMIN, RMAX = 0.1, 0.5
THRESHOLD = 0.45

def label(x):
# If above threshold, then choose largest circle
if np.any(x[:5] > THRESHOLD):
return np.argmax(x[:5])

# Else, choose the circle nearest to (0, 0)
return np.argmax([np.linalg.norm(x[i:i+2]) for i in range(N_CIRCLES, 3*N_CIRCLES, 2)])

def generate_sample():
# {r0, r1, r2, r3, r4, x0, y0, x1, y1, x2, y2, x3, y3, x4, y4}
x = np.concatenate((np.random.uniform(RMIN, RMAX, N_CIRCLES),
np.random.uniform(-CENTRE_RANGE, CENTRE_RANGE, 2*N_CIRCLES)))

return x, label(x)

def generate_samples(n):
x = np.zeros((n, N_CIRCLES*3))
y = np.zeros(n)

for i in range(n):
x[i], y[i] = generate_sample()

return x, y

import torch
import torch.nn as nn
import torch.nn.functional as F

class Net(nn.Module):
def __init__(self):
super().__init__()
# Kernel size 5
self.fc1 = nn.Linear(3*N_CIRCLES, 32)
self.fc2 = nn.Linear(32, 64)
self.fc3 = nn.Linear(64, N_CIRCLES)

def forward(self, x):
x = F.relu(self.fc1(x))
x = F.relu(self.fc2(x))
x = F.relu(self.fc3(x))
return F.softmax(x, dim=1)

net = Net()

import torch.optim as optim

loss_function = nn.MSELoss()

BATCH_SIZE = 100
EPOCHS = 1_000

losses = []
for epoch in tqdm(range(EPOCHS)):
X, y = generate_samples(BATCH_SIZE)
y = np.array(y, dtype=int)

ohe = np.zeros((y.size, y.max()+1))
ohe[np.arange(y.size), y] = 1

X = torch.Tensor(X).view(-1, 3*N_CIRCLES)
y = torch.Tensor(ohe)

yhat = net(X)
loss = loss_function(yhat, y)
loss.backward()
optimizer.step()

losses.append(float(loss.detach().numpy()))

import matplotlib.pyplot as plt
%matplotlib inline
import seaborn as sns

fig, ax = plt.subplots(figsize=(20, 10))
ax.plot(losses)
plt.show()
$$$$
`
• Wow, thanks for the code sample! Regarding ordering - how do you determine order? Do you mean to sort them by size or distance-from-center? That also feels like a form of 'cheating' to me. The network wouldn't really learn the rule, but rather learn to always choose the first (or last) sorted entry. Oct 19 '20 at 21:29
• A bit of background - on my first attempts at this I actually did sort the circles so that the correct answer was always first (as a side effect of my answer-choosing preprocessing). And the network indeed learned to always choose the first entry, almost immediately. Oct 19 '20 at 21:31

Yes, they absolutely can 'learn' conditional rules.

It turns out I had a bug in an activation function, and I (eventually) got this network to learn this problem. It learned it incredibly well, and very quickly. Amazing things, these NNs :) Big thanks to those that answered and commented!

Yes, the Universal Approximation Theorem states that a neural network can learn any function in $$R^n$$ with one hidden layer and a finite number of neurons with a non-linear activation function. There are many things that can go wrong with training a network, for one, have you tried graphing its performance over time and seeing if it is converging?

• This is wrong. Please have a look at the correct theorem. Oct 18 '20 at 12:12
• Yes, I watch the loss and accuracy values very closely. They attenuate fairly quickly (depending on the config of the run). Oct 18 '20 at 16:36
• Sure there are some nuances, but I don't think what I've written is wrong. Check out en.wikipedia.org/wiki/Universal_approximation_theorem Oct 18 '20 at 23:13
• As I understand it, that theorem pertains to continuous functions. I wonder if I am not using a continuous function? To be honest, I don't know how to visualize a function with 15 inputs and three outputs. I have no idea whether it's continuous, or whether it's solvable via NN (thus the question). Oct 19 '20 at 2:18
• You're right that it's for continuous functions, and although your problem is not continuous, the approximations of the NN should certainly be good enough for this problem. Oct 19 '20 at 2:36

format: plain


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### Finally!

05:51 pm

It took them quite a bit longer than it should have, but my doctoral dissertation is finally live on the internet. Be sure to grab your limited-time only special edition before the downloads expire! (there is no special edition, I'm just trying to gin up downloads to make whatever internal stats they have look impressive to someone :p)

08:34 pm

### Success!

12:16 pm
I may feel like I'm the walking dead, but I've survived my defense. Still got a lot of work left to go, but with that hurdle out of the way I feel that nothing can stop me now.

### Doomsday

11:00 pm
Going to bed early. Got my graphs done today, but I haven't really had a chance to do a trial performance of my dissertation. I've done it so many times in my head, though, that I don't think I'll actually need a trial-run of the whole thing.

I wanted to be done with my slides around, say, 4 or 5 PM, but I didn't finish them until a few minutes before the timestamp on this post. Part of this is due to needing to go up to school to make a graph (and confirm some results) which took far longer (a couple hours) than it should have.

### Slides, slides everywhere, and not a single graph to drink

08:10 pm
Said impending feeling of doom has carried over to today. Especially since I was rather paralyzed (and frustrated) last night and managed to create only a couple of the 30 or so slides I'll need. Fortunately today has been somewhat rather more productive and I'm doing pretty good. I'll probably have to finish up tomorrow but I don't think it'll take all day.

### Preparing a presentation

04:06 pm
Got some nice graphs made today so I can put them into my powerpoint presentation for the defense on Monday. I have this overwhelming feeling of doom hanging over me: it's rather crippling.

09:25 pm
science... q.q

whimper

### woo crunch time is here again

11:18 pm
Time to put my nose to the grind for a week and crank out this thesis.

### It's been a good day at work...

03:58 pm
And not just because I've got something to work on that isn't making graphs (it's editing code to make more slightly different graphs!):

<elizibar_atwork> Behold, nature's most perfect snac--- Eldritch Horror. http://img412.imageshack.us/i/1297886901203.jpg/
<coywintau> SHE SAID ALL EYES ON HER.
<phimini> FFFF
<coywintau> AND SO I LOOKED AT HER.
<phimini> DAMMIT COYWIN
<cougardraven> lulz
<cougardraven> Also, eli, what the fuck.
<sickslickman> ...wha
<phimini> YOU DAMN EYE UNDRESSER
<phimini> ... WHAT IN THE NAME OF GOD IS THAT THING
<elizibar_atwork> It's jus' a l'il fella
<coywintau> Links to 4chan don't work for me.
<sickslickman> Me neither
<coywintau> I'm actually confused about that.
<elizibar_atwork> Aww you guys are missing awesome
<elizibar_atwork> Just look at CD and Phimini's responses
<phimini> If I find that in my bed tonight...
<sickslickman> Alright, class time
<sickslickman> later folkas
<sickslickman> *folks
<cougardraven> If I find that in my apartment at all, I'm burning the entire city to the ground to kepp that from spreading.
<coywintau> Wow, I had to drag it to a new tab.
<phimini> Bye Slick!
<sickslickman> peace
<phimini> Have a good class!
<coywintau> But it wouldn't open as a new tab.
<sickslickman> I shall. It's fuckin' Casablanca :D
<coywintau> SEE YA, SLICK
* SickSlickMan has quit (Quit: http://www.mibbit.com ajax IRC Client)
<coywintau> WHY
<coywintau> WHY DID I FIND A WAY TO OPEN THAT.
<coywintau> I"M FUCKING TERRIFIED OF SPIDERS.
<phimini> If that's in my bed tonight I will scream like a little girl.
<phimini> And I don't scream.
<coywintau> I would do many things, screaming being minor.
<cougardraven> Okay, almost caught up on my MH archive.
<cougardraven> Now to continue EMH.
<phimini> I already have waking dreams about spiders in my bed. I don't need a real one D:
* IRuleHyRule118 (Mibbit@mib-F9F5B72C.sbr9.nsw.optusnet.com.au) has joined #habitsangora
<lethargic> irulehyrule118!
<coywintau> Hey Hyrule!
<cougardraven> Sup Hyrule.
<elizibar_atwork> To misquote the classic song "Asbestos on my balls": Spiders in my bed, Spiders in my bed, Why the fuck Why the fuck, Spiders in my bed"
<irulehyrule118> Hello
<coywintau> Hey, Elizibar, I mean this in the nicest way, but fuck you.
<cougardraven> I lol'd.
<phimini> XDDD
<irulehyrule118> So how we all been?
<cougardraven> Depends. You watch Entry #35 yet?
<elizibar_atwork> Apparently people don't like me q.q
<irulehyrule118> Yep
* CoywinTau has arachnophobia he read, "Spiders on my balls" and jumped and cringed.
<coywintau> I like you enough Elizibar, just not when you talk of spiders.
<elizibar_atwork> I'm pretty much done now anyway, I need to get back to editing this java program.

I threw the image on imageshack instead of leaving the link to /tg/ on 4chan, since that'll expire.

### It should go without saying that you deserve this

11:16 pm
*murders Gullwhacker*
*murders the crap out of him, a lot*

### The Book of E, chapter 25

04:31 pm
1 From the tops of the hills a voice rang out, calling forth triumphantly, "It is completed,
2 lo' though the path was long and the results not what 3 we expected or hoped.
3 Yea the great labor is finished and pain has left the earth."
4 It was the afternoon then, upon the time of the late meal.
5 "Truly, the forgotten taste of victory is more savory than I can recall.
6 It is like Chick-fil-A nuggets, fries cut and shaped as if waffles, and the fresh green dew of the mountain, sweet and invigorating."
7 The tools of Fortran and Java proved their worth, having struck down the followers of Jacobian,
8 that false idol of computations who rules over the Implicit and the Semi-implicit domains.
9 Freedom reigned for a day as the darkness was driven forth.

### *slowly reforms*

02:42 am
*crows scream in terror at the amazing reforming astrophysicist*

11:41 pm
*self destructs*

### Bleach

11:39 pm
And not the anime.

Apparently they're like ... bleaching the entire physics building or something. I don't know man, but the chlorine fumes kind of make it hard for me to be up there working in the evening.

Edit: and I totally forgot to hit post earlier lolme

### *Sciences at everyone*

11:53 pm
Fear the guy that's still in his office at midnight! Rawr!

### Two! Minutes! To midniiiiight!

11:58 pm
The hands that threaten doom!

Yeah I altered the timestamp on this post, because I've been at my office way too late tonight and am getting silly.

### \section{Post}

11:40 pm
type type type type words words refer to graph type type number number $T=3\times 10^{9}$ words words type type explanation in Ref. \cite{citation}

### Progress

07:08 pm
I had a calculation going today that I was just about ready to give up on (my estimate was 225 more days to run to completion). ...then it went woosh and finished in about 5 minutes.

I have another calculation going that I estimate will take more than 300 years to finish. I don't hold out hope for a miracle on that one. :p

### Whew

11:40 pm
Long day at the office doing that science thing.

### SCIENCE!

11:03 pm
Went out to the lab today. It was pretty great for once \m/

demonicgerbil


text:
# Directory `macros/latex/contrib/xoptarg`

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optional arguments cannot be expandable. However, it is possible to
make them expandable if there is at least one mandatory argument (see
`Limitations').

This material is subject to the LaTeX Project Public License.


text:
# How to spot right $a$ value satisfying equation in $\mathbb Z/ \mathbb Z_p$ when prime $p$ is a large number

I need to factorize $f=8x^3+x^2+2x+6 \in \mathbb Z_{13}[x]$. I figure I can divide $f$ by $(x-a)$ and then determine for which $a$ value the reminder is equal to $0$.

So the resulting quotient is $8x^2+(8a+1)x+(3+8a)$ and the reminder is $6+a(3+8a)$. As previously said I have to find a suitable $a$ value solution to \begin{align} 6+a(3+8a)\equiv_{13}0 \Leftrightarrow a(3+8a)\equiv_{13}7\end{align}

at this point I have to manually substitute the $a$ to $0,1,2,\,...$ and verify for every value which among them satisfies the equation. My question is for a small number like $13$ it is acceptable to run tests and see, but what if instead of $13$ there was $31$, is there any way that lets me spot that value without embarking in a pretty time consuming quest?

Edit: I'm not in the math field, but I have to take the exam, maybe this question is going beyond the contents of my class, so I was wondering if there is any resonably simple answer to that.

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You want a general method for all polynomials ? or just this polynomial in different fields –  Amr Nov 25 '12 at 12:18
Preferably a general one or using this particular example a way to generalize to other polynomials in different fields, if possible. –  haunted85 Nov 25 '12 at 12:21
I would be a little careful with writing $\mathbb Z_p$. Some take it to mean the ring of $p$-adic integers. Alternative notations for integers mod $p$ are $\mathbb Z/p\mathbb Z = \mathbb Z/(p)$ (quotienting out by the ideal $p\mathbb Z=(p)$). –  kahen Nov 25 '12 at 12:44
Thank you @kahen I wasn't aware of that. My textbook uses freely this notation and so did I. –  haunted85 Nov 25 '12 at 12:48

Instead of dividing $f(x)$ with $x-a$ and then trying to find $a$ s.t. the remainder is zero sometimes is easier to first find $a$ s.t. $f(a)\equiv0$ and then divide with $x-a$.
Of course this method to factorize $f(x)$ is sufficient only if $\deg f\leq3.$

For this particular example:
write $f(x)\equiv_{13}-5x^3+x^2+2x+6$. Then calculate $f(0),f(\pm1),f(\pm2),\ldots,f(\pm6)$ until you find a zero. Here $f(2)\equiv_{13}0$.
Next we divide $f(x)$ with $x-2$ in $\mathbb Z_{13}[x]$. We get $f(x)\equiv_{13}(x-2)(-5x^2+4x-3)$.
Then we repeat for $g(x)=-5x^2+4x-3$. Since $g(a)\not \equiv_{13} 0, \ \forall a \in \mathbb Z_{13} \Rightarrow g(x)$ is irreducible in $\mathbb Z_{13}[x]$.

Also check this, this and this.

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You want to find a solution to $a x^2 + bx + c =_p 0$ for a prime $p$ and integers $a, b,$ and $c$, or show no solution exists. My first inclination is to reduce this to quadratic residues.

If $a =_p 0$, this is just linear, so assume $a \ne_p 0$. Multiplying by $1/a$, this becomes $x^2+Bx+C =_p 0$. Completing the square, this becomes $(x+B/2)^2 =_p -C-(B/2)^2$. If $-C-(B/2)^2$ is a quadratic residue (this can be done in time $O(\ln p)$), then you can find $x$. If not, there is no solution.

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You mean something like Berlekamp's algorithm?

Also, Chapter 4 of the Lidl - Niederreiter book (Introduction to finite fields and their applications) is dedicated on this subject -- if you can access it through a library or so.


text: . Papers on the Intrinsic Flat Distance between Riemannian Manifolds ordered according to availability as a preprint with abstracts and comments Anyone interested in working on SWIF convergence should email Sormani to ensure no one else is working on their planned project. "The Intrinsic Flat Distance between Riemannian Manifolds and Integral Current Spaces" by Christina Sormani and Stefan Wenger Journal of Differential Geometry, Vol 87, (2011) (arxiv preprint) (early preprint distributed in Fall 2008) (reprint) Abstract: Inspired by the Gromov-Hausdorff distance, we define the intrinsic flat distance between oriented m dimensional Riemannian manifolds with boundary as:                                   dF(M,N) = inf dZF ( [f(M)] , [h(N)] ) where f,h are isometric embeddings from M and N respectively into a complete metric space Z and where [f(M)] and [h(N)] denote their images viewed as integral currents in Z in the sense of Ambrosio-Kirchheim and the infimum is taken over all $f$, $h$ and $Z$. We show the intrinsic flat distance between oriented Riemannian manifolds is zero iff they have an orientation preserving isometry between them. Using the theory of Ambrosio-Kirchheim, we study converging sequences of manifolds and their limits, which are in a class of metric spaces that we call integral current spaces. We describe the properties of such spaces including the fact that they are countably Hm rectifiable spaces and give numerous examples. Comments on prior work leading to this paper: Whitney first introduced the flat and sharp norms for submanifolds of Euclidean space viewed as chains in his 1957 book Geometric Integration Theory.    In Federer-Flemming’s “Normal and Integral Currents” Annals of Math 2 72 (1960) 458-520 the notion of a submanifold in Euclidean space as a current acting on differential forms was introduced.  They proved  flat and weak compactness theorems and slicing theorems.    In Ambrosio-Kirchheim’s “Currents in Metric Spaces” Acta Math Vol 185 2000 no 1, 1--80, integral currents on a complete metric space were introduced using DeGiorgi tuples. They proved compactness theorems for sequences of currents with respect to weak convergence and much more.  In  “Flat Convergence for Integral Currents” Wenger studied flat convergence of Ambrosio-Kirchheim’s integral currents in a fixed metric space and proved when flat and weak convergence agree. The Intrinsic Flat distance is a distance between pairs of distinct Riemannian manifolds and, more generally, integral current spaces.  It is defined by imitating Gromov’s definition of his “Intrinsic Hausdorff distance” (now called the Gromov-Hausdorff distance) using Gromov’s idea of taking an infimum over all isometric embeddings into all possible common metric spaces and then taking the Hausdorff distance between the images.  See Gromov’s 1981 book Structures Metriques pour les Varietes Riemannians.   Instead of taking the Hausdorff distance, Sormani and Wenger take the flat distance between the images. "Compactness for manifolds and integral currents with bounded diameter and volume" by Stefan Wenger Calculus of Variations and Partial Differential Equations, 40 (2011). (arxiv preprint) Abstract: By Gromov's compactness theorem for metric spaces, every uniformly compact sequence of metric spaces admits an isometric embedding into a common compact metric space in which a subsequence converges with respect to the Hausdorff distance. Working in the class or oriented $k$-dimensional Riemannian manifolds and, more generally, integral currents in metric spaces (in the sense of Ambrosio-Kirchheim) and replacing the Hausdorff distance with the filling volume or flat distance, we prove an analogous compactness theorem in which we replace equicompactness with uniform upper bounds on volume and diameter for the sequence.  Comments: In particular, Wenger has shown that any sequence of oriented $k$ dimensional Riemannian manifolds with a uniform upper bound on volume and on diameter has a subsequence which converges in the intrinsic flat sense to an integral current space. If the manifolds have boundary, he only needs to add an assumption that the boundaries have a uniform upper bound on volume. "Cancellation under weak convergence" (prior name: "Cancellation under flat convergence") by Christina Sormani and Stefan Wenger Appendix by Raanan Schul and Stefan Wenger Calculus of Variations and Partial Differential Equations, Vol 38, No. 1-2, May 2010. (arxiv preprint) (reprint) Abstract: This paper concerns cancellation and collapse when a sequence of manifolds or integral currents is converging in the flat norm. Applying Gromov's filling paper and imitating a theorem of Greene-Petersen, we show that the flat limits and Gromov-Hausdorff limits of linearly locally contractible manifolds agree. As a consequence the limits of these spaces are countably Hm rectifiable spaces. Applying Cheeger-Colding and Perelman, we show that the flat limits and Gromov-Hausdorff limits of noncollapsing sequences of manifolds with nonnegative Ricci curvature agree. Cheeger-Colding had already shown that limits of such sequences are as rectifiable as current spaces. We give examples of sequences with positive scalar curvature where they do not agree. These examples have lots of local topology. Within our proofs we also describe the sets of limits of flat converging sequences of integral currents using the theory of Ambrosio-Kirchheim.  Typo: In statement of Theorem 4.1 it must explicitly require spt of the boundary to avoid the ball. This is the case in all our applications within the paper.    "The pointed flat compactness theorem for locally integral currents" by Urs Lang and Stefan Wenger  Communications in Analysis and Geometry. 19 (2011), no. 1, 159-189. (arxiv preprint) Abstract: Recently, an embedding/compactness theorem for integral currents in a sequence of metric spaces has been established by the second author. We present a version of this result for locally integral currents in a sequence of pointed metric spaces. To this end we introduce another variant of the Ambrosioâ€“Kirchheim theory of currents in metric spaces, including currents with ï¬nite mass in bounded sets.  Comments: this is not an intrinsic paper, as the balls employed to define the pointed convergence sit in the extrinsic space $Z$ into which the spaces are isometrically embedded. It is built on Lang's local theory of currents (rather than Ambrosio - Kirchheim's theory) to handle currents of infinite mass.   An intrinsic way of defining pointed intrinsic flat convergence of Riemannian manifolds imitating Gromov's pointed convergence, is to restrict to balls in the manifolds and study their intrinsic flat limits as balls in the limit space.  "Stability of the Positive Mass Theorem for Rotationally Symmetric Riemannian Manifolds" by Dan Lee and Christina Sormani Journal fur die Riene und Angewandte Mathematik (Crelle's Journal), Vol 686 (January 2014)  (arxiv preprint) Abstract: We study the stability of the Positive Mass Theorem using the Intrinsic Flat Distance. In particular we consider the class of complete asymptotically flat rotationally symmetric Riemannian manifolds with nonnegative scalar curvature and no interior closed minimal surfaces whose boundaries are either outermost minimal hypersurfaces or are empty. We prove that a sequence of these manifolds whose ADM masses converge to zero must converge to Euclidean space in the pointed Intrinsic Flat sense. In fact we provide explicit bounds on the Intrinsic Flat Distance between annular regions in the manifold and annular regions in Euclidean space by constructing an explicit filling manifold and estimating its volume. In addition, we include a variety of propositions that can be used to estimate the Intrinsic Flat distance between Riemannian manifolds without rotationally symmetry. Conjectures regarding the Intrinsic Flat stability of the Positive Mass Theorem in the general case are proposed in the final section.     "Almost Equality in the Penrose Equality for Rotationally Symmetric Riemannian Manifolds" by Dan Lee and Christina Sormani Annales Henri Poincare November 2012, Volume 13, Issue 7, pp 1537-1556. (arxiv preprint) Abstract: This article is the sequel to our previous paper [LS] dealing with the near-equality case of the Positive Mass Theorem. We study the near-equality case of the Penrose Inequality for the class of complete asymptotically flat rotationally symmetric Riemannian manifolds with nonnegative scalar curvature whose boundaries are outermost minimal hypersurfaces. Specifically, we prove that if the Penrose Inequality is sufficiently close to being an equality on one of these manifolds, then it must be close to a Schwarzschild space with an appended cylinder, in the sense of Lipschitz Distance. Since the Lipschitz Distance bounds the Intrinsic Flat Distance on compact sets, we also obtain a result for Intrinsic Flat Distance, which is a more appropriate distance for more general near-equality results.   "Smooth Convergence Away from Singular Sets" by Sajjad Lakzian and Christina Sormani Communications in Analysis and Geometry, Volume 21 (2013) No 1, 39-104 (arxiv preprint) Abstract: This paper applies intrinsic flat convergence to understand smooth convergence away from singular sets of codimension two and to determine when the metric completion of a smooth limit agrees with the Gromov-Hausdorff or Intrinsic Flat limit of the manifolds. Many examples are given. "Diameter Controls and Smooth Convergence away from Singular Sets" by Sajjad Lakzian (arxiv preprint) Journal of Differential Geometry and Applications, to appear Abstract: This paper builds upon and extends prior results of the author and C. Sormani considering both the Intrinsic Flat and Gromov-Hausdorff limits of sequences of manifolds with metrics that converge smoothly away from singular sets $S\subset M^m$ of Hausdorff measure $H^{m-1}(S)=0$. Continuity of Ricci flow through Neck Pinch Singularities" by Sajjad Lakzian   Geometriae Dedicata, December 2015, Volume 179, Issue 1, pp 69–89 (arxiv preprint) Abstract: In this article, we consider the Angenent-Caputo-Knopf's Ricci Flow through neckpinch singularities. We will explain how one can see the A-C-K's Ricci flow through a neckpinch singularity as a flow of integral current spaces. We then prove the continuity of this weak flow with respect to the Sormani-Wenger Intrinsic Flat (SWIF) distance. Dirac and Plateau Billiards in Domains with Corners" by Misha Gromov (posted preprint, (Sept 30, 2013))   Central European Journal of Mathematics August 2014, Volume 12, Issue 8, pp 1109-1156 Abstract: Groping our way toward a theory of singular spaces with positive scalar curvatures we look at the Dirac operator and a generalized Plateau problem in Riemannian manifolds with corners. Using these, we prove that the set of C^2 -smooth Riemannian metrics g on a smooth manifold X, with scalar curvature bounded from below by a continuous function is closed under C^0 limits of Riemannian metrics for all continuous functions on X. Apart from that our progress is limited but we formulate many conjectures. All along we emphasize geometry, rather the the topology of the manifolds with their scalar curvature bounded from below.  Comments: Within this paper, Gromov suggests that a sequence of tori with Riemannian metrics g_k whose scalar curvature is bounded below by -1/k might be shown to converge in the intrinsic flat sense to a flat torus as k diverges to infinity. The geometric properties of scalar curvature that Gromov studies in this paper may be useful towards proving such a theorem. "Semicontinuity of eigenvalues under intrinsic flat convergence" by Jacobus W. Portegies (arxiv preprint) Calculus of Variations and Partial Differential Equations October 2015, Volume 54, Issue 2, pp 1725-1766 Abstract: We use the theory of rectifiable metric spaces to define a normalized Dirichlet energy of Lipschitz functions defined on the support of integral currents. This energy is obtained by integration of the square of the norm of the tangential derivative, or equivalently of the approximate local dilatation, of the Lipschitz functions. We define min-max values based on the normalized energy and show that when integral current spaces converge in the intrinsic flat sense without loss of volume, the min-max values of the limit space are larger than or equal to the upper limit of the min-max values of the currents in the sequence. In particular, the infimum of the normalized energy is semicontinuous. On spaces that are infinitesimally Hilbertian, we can define a linear Laplace operator. We can show that semicontinuity under intrinsic flat convergence holds for eigenvalues below the essential spectrum, if the total volume of the spaces converges as well.    The appendix contains a decomposition theorem for one dimensional integral currents and a Poincare-like inequality. "Plateau Stein Manifolds" by Misha Gromov, ihes preprint  Central European Journal of Mathematics July 2014, Volume 12, Issue 7, pp 923-951 Abstract: We study/construct (proper and non-proper) Morse functions f on complete Riemannian manifolds $X$ such that the hypersurfaces f(x) = t for all âˆ’âˆž < t < +âˆž have positive mean curvatures at all non-critical points x âˆˆ X of f. We show, for instance, that if an X admits no such (not necessarily proper) function, then it contains a (possibly, singular) complete (possibly, compact) minimal hypersurface of finite volume.  Comments: In this paper Gromov suggests the intrinsic flat distance might be useful to study sequences of manifolds with nonnegative scalar curvature. "Intrinsic Flat Arzela-Ascoli Theorems" by Christina Sormani, arxiv preprint, Communications in Analysis and Geometry Vol. 27, No 1, 2019 Abstract: In this paper two Arzela-Ascoli Theorems are proven: one for uniformly Lipschitz functions whose domains are converging in the intrinsic flat sense, and one for sequences of uniformly local isometries between spaces which are converging in the intrinsic flat sense. A basic Bolzano-Weierstrass Theorem is proven for sequences of points in such sequences of spaces. In addition it is proven that when a sequence of manifolds has a precompact intrinsic flat limit then the metric completion of the limit is the Gromov-Hausdorff limit of regions within those manifolds. Open problems with suggested applications are provided throughout the paper. "Nonlinear stability of rotationally symmetric spaces with low regularity" by Philippe LeFloch and Christina Sormani,  Journal of Functional Analysis  Vol. 268 (2015), no. 7, 2005–2065.  arxiv preprint  Abstract: Our main theorem states that if one has a sequence of rotationally symmetric regions of nonnegative scalar curvature with no closed interior minimal surfaces that have spherical boundaries of fixed area with uniform upper bounds on: the depth of the region and the Hawking mass of the boundary, then a subsequence converges in the intrinsic flat sense to a region whose metric h\ as $H^1$ regularity with nonnegative scalar curvature (in the generalized sense). \ In addition the area of the boundary is continuous, the volume of the region is continuous and the Hawking mass of the boundary is continuous. In order to obtain the continuity of these values we show the metric tensors converge in the Sobolev sense. In addition to this main theorem, we provide a thorough analysis of these limit spaces, proving the Hawking mass is monotone and providing precise estimates on the intrinsic flat distance, the Sobolev distance, and a new notion, the D flat distance, between a region of small Hawking mass and a disk in Euclidean space extending results of Sormani with Dan Lee. It should be noted that the ideas involving the generalized notions of curvature on a manifold with $H^1$ metric tensors has been explored by LeFloch and collaborators Mardare, Stewart and Rendall. Ordinarily intrinsic flat limit spaces are only countably H^m rectifiable.    "On Lower Semicontinuity of the ADM mass" by Jeffrey L. Jauregui,   Communications in Analysis and Geometry Vol 26 issue 1 (2018) 18 pages arxiv preprint  Abstract: The ADM mass, viewed as a functional on the space of asymptotically flat Riemannian metrics of nonnegative scalar curvature, fails to be continuous for many natural topologies. In this paper we prove that lower semicontinuity holds in natural settings: first, for pointed Cheeger--Gromov convergence (without any symmetry assumptions) for n=3 and , second, assuming rotational symmetry, for weak convergence of the associated canonical embeddings into Euclidean space, and, finally, also assuming rotational symmetry, for pointed intrinsic flat convergence (applying work of LeFloch-Sormani).  We provide several examples, one of which demonstrates that the positive mass theorem is implied by a statement of the lower semicontinuity of the ADM mass.     "Volumes and Limits of Manifolds with Ricci Curvature and Mean Curvature Bounds" by Raquel Perales, arxiv preprint Journal of Differential Geometry and Applications  Vol 48, (2016) pp 23-37 (reprint) Abstract: This paper proves a number of theorems including the following intrinsic flat compactness theorems building upon Wenger's Compactness Theorem: A sequence of n dimensional oriented manifolds with boundary that have nonnegative Ricci curvature and an upper bound on mean curvature and area of the boundary and an upper bound on diameter, have a subsequence which converges in the intrinsic flat sense. The diameter bound on the manifold may be replaced by a diameter bound on the boundary if the mean curvature is uniformly strictly negative in this theorem. The necessity of various conditions have been presented as examples in this paper as well. It is unknown if such a sequence has a subsequence converging in the GH sense or whether GH and intrinsic flat limits agree in this setting. "Intrinsic Flat Convergence with bounded Ricci curvature" by Michael Munn, arxiv preprint  Abstract: In this paper we address the relationship between Gromov-Hausdorff limits and intrinsic flat limits of complete Riemannian manifolds. In prior work, Sormani-Wenger show that for a sequence of Riemannian manifolds with nonnegative Ricci curvature, a uniform upper bound on diameter, and non-collapsed volume, the intrinsic flat limit exists and agrees with the Gromov-Hausdorff limit. This can be viewed as a non-cancellation theorem showing that for such sequences, points don't cancel each other out in the limit. Here we prove a similar no-cancellation theorem, replacing the assumption of nonnegative Ricci curvature with a two-sided bound on Ricci curvature.  "Intrinsic flat stability of the positive mass theorem for graphical hypersurfaces of Euclidean space" by Lan-Hsuan Huang, Dan Lee, Christina Sormani, arxiv preprint  to appear in Crelle (arxiv preprint)  (reprint) Abstract: The rigidity of the Positive Mass Theorem states that the only complete asymptotically flat manifold of nonnegative scalar curvature and zero mass is Euclidean space. We study the stability of this statement for spaces that can be realized as graphical hypersurfaces in Euclidean space. We prove (under certain technical hypotheses) that if a sequence of complete asymptotically flat graphs of nonnegative scalar curvature has mass approaching zero, then the sequence must converge to Euclidean space in the pointed intrinsic flat sense. The appendix includes a new Gromov-Hausdorff and intrinsic flat compactness theorem for sequences of metric spaces with uniform Lipschitz bounds on their metrics.   "Intrinsic Flat Convergence of Covering Spaces" by Zahra Sinaei and Christina Sormani,  Geometriae Dedicata, Vol 184 (2016) pages 83–114  (arxiv preprint) (reprint) DOI 10.1007/s10711-016-0158-0 Abstract: We examine the limits of covering spaces and the covering spectra of oriented Riemannian manifolds, Mj, which converge to a nonzero integral current space, M∞, in the intrinsic flat sense. We provide examples demonstrating that the covering spaces and covering spectra need not converge in this setting. In fact we provide a sequence of simply connected Mj diffeomorphic to S4 that converge in the intrinsic flat sense to a torus S1×S3. Nevertheless, we prove that if the delta covers have finite order N, then a subsequence of these delta covers converge in the intrinsic flat sense to a metric space which is the disjoint union of covering spaces of M∞.   This is related to prior work of the the second author with Guofang Wei. "On the Sormani-Wenger Intrinsic Flat Convergence of Alexandrov Spaces" by Nan Li and Raquel Perales arxiv preprint Abstract: We study noncollapsing sequences of integral current spaces with no boundary that have lower Alexandrov curvature bounds and weight 1.   We prove that for such sequences Sormani-Wenger Intrinsic Flat convergence and Gromov-Hausdorff convergence agree. "Properties of the Intrinsic Flat Distance" by Jacobus Portegies and Christina Sormani, arxiv preprint, (completed in April 2015) 56 pages  to appear in a special volume in honor of Y Burago edited by D Burago and S Buyalo Abstract: Here we explore a variety of properties of intrinsic flat convergence. We introduce the sliced filling volume and interval sliced filling volume and explore the relationship between these notions, the tetrahedral property and the disappearance of points under intrinsic flat convergence. We prove two new Gromov-Hausdorff and intrinsic flat compactness theorems including the Tetrahedral Compactness Theorem. Much of the work in this paper builds upon Ambrosio-Kirchheim's Slicing Theorem combined with an adapted version Gromov's Filling Volume. "Convergence of Manifolds and Metric Spaces with Boundary" by Raquel Perales,  arxiv preprint, (completed in May 2015)  Abstract: We study sequences of oriented Riemannian manifolds with boundary and, more generally, integral current spaces and metric spaces with boundary. We prove theorems demonstrating when the Gromov-Hausdorff [GH] and Sormani-Wenger Intrinsic Flat [SWIF] limits of sequences of such metric spaces agree. Thus in particular the limit spaces are countably H^n rectifiable spaces. From these theorems we derive compactness theorems for sequences of Riemannian manifolds with boundary where both the GH and SWIF limits agree. For sequences of Riemannian manifolds with boundary we only require nonnegative Ricci curvature, upper bounds on volume, noncollapsing conditions on the interior of the manifold and diameter controls on the level sets near the boundary "Intrinsic Flat and Gromov-Hausdorff Convergence for Manifolds with Ricci curvature bounded below" by Rostitslav Matveev and Jacobus Portegies, arxiv preprint, (completed in Oct 2015) (reprint) The Journal of Geometric Analysis Vol 27 issue 3 (July 2017) 1855-1873. Abstract: We show that for a noncollapsing sequence of closed, connected, oriented Riemannian manifolds with Ricci curvature uniformly bounded from below and diameter uniformly bounded above, Gromov-Hausdorff convergence essentially agrees with intrinsic flat convergence.   Comments: This paper provides a new proof for the Ricci nonnegative case as well. "Scalar Curvature and Intrinsic Flat Convergence" by Christina Sormani, (preprint) These are the notes for the Como School. (reprint) published as a chapter in Measure Theory in Non-Smooth Spaces edited by Nicola Gigli de Gruyter Press. Abstract:  Herein we present open problems and survey examples and theorems concerning sequences of Riemannian manifolds with uniform lower bounds on scalar curvature and their limit spaces. Examples of Gromov and of Ilmanen which naturally ought to have certain limit spaces do not converge with respect to smooth or Gromov-Hausdorff convergence. Thus we focus here on the notion of Intrinsic Flat convergence, developed jointly with Wenger. This notion has been applied successfully to study sequences that arise in General Relativity. Gromov has suggested it should be applied in other settings as well. We first review intrinsic flat convergence, its properties, and its compactness theorems, before presenting the applications and the open problems.  Comments: In this paper the term "almost rigidity" is used in a more general way than it was used in the original 1996 Cheeger-Colding paper coining the term.  In April 2018, Cheeger stated that he was not happy with this and  requested that the term not be used in this way in future work. "Ricci Curvature and Orientability" by Shouhei Honda, arxiv preprint, to appear in CVPDE Abstract:  In this paper we define an orientation of a measured Gromov-Hausdorff limit space of Riemannian manifolds with uniform Ricci bounds from below. This is the first observation of orientability for metric measure spaces. Our orientability has two fundamental properties. One of them is the stability with respect to noncollapsed sequences. As a corollary we see that if the cross section of a tangent cone of a noncollapsed limit space of orientable Riemannian manifolds is smooth, then it is also orientable in the ordinary sense, which can be regarded as a new obstruction for a given manifold to be the cross section of a tangent cone. The other one is that there are only two choices for orientations on a limit space. We also discuss relationships between L2-convergence of orientations and convergence of currents in metric spaces. In particular for a noncollapsed sequence, we prove a compatibility between the intrinsic flat convergence by Sormani-Wenger, the pointed flat convergence by Lang-Wenger, and the Gromov-Hausdorff convergence, which is a generalization of a recent work by Matveev-Portegies to the noncompact case. Moreover combining this compatibility with the second property of our orientation gives an explicit formula for the limit integral current by using an orientation on a limit space. Finally dualities between de Rham cohomologies on an oriented limit space are proven.   "Sewing Riemannian Manifolds with Positive Scalar Curvature" by Jorge Basilio, Jozef Dodziuk and Christina Sormani, arxiv preprint Abstract: We explore to what extent one may hope to preserve geometric properties of three dimensional manifolds with lower scalar curvature bounds under Gromov-Hausdorff and Intrinsic Flat limits. We introduce a new construction, called sewing, of three dimensional manifolds that preserves positive scalar curvature. We then use sewing to produce sequences of such manifolds which converge to spaces that fail to have nonnegative scalar curvature in a standard generalized sense. Since the notion of nonnegative scalar curvature is not strong enough to persist alone, we propose that one pair a lower scalar curvature bound with a lower bound on the area of a closed minimal surface when taking sequences as this will exclude the possibility of sewing of manifolds.    "Alexandrov Spaces with Integral Current Structure" by  Abstract: We endow each closed, orientable Alexandrov space (X,d) with an integral current T of weight equal to 1, $\partial T = 0 and \set(T) = X$, in other words, we prove that (X,d,T) is an integral current space with no boundary. Combining this result with a result of Li and Perales (arxiv preprint), we show that non-collapsing sequences of these spaces with uniform lower curvature and diameter bounds admit subsequences whose Gromov-Hausdorff and intrinsic flat limits agree.   This builds upon a paper of Mitsuishi (arxiv preprint) which does not involve integral current spaces but does apply Ambrosio-Kirchheim theory to Alexandrov spaces. (NSF DMS 1309360) (team website) "Almost Rigidity of the Positive Mass Theorem for Asymptotically Hyperbolic Manifolds with Spherical Symmetry" by Anna Sakovich and Christina Sormani, (arxiv preprint) 23 pages,  General Relativity and Gravitation, selected to be an Editor's Choice, September 2017 49:125. (reprint) We use the notion of intrinsic flat distance to address the almost rigidity of the positive mass theorem for asymptotically hyperbolic manifolds. In particular, we prove that a sequence of spherically symmetric asymptotically hyperbolic manifolds satisfying the conditions of the positive mass theorem converges to hyperbolic space in the intrinsic flat sense, if the limit of the mass along the sequence is zero.  (NSF DMS 1612409)  Comments: In this paper the term "almost rigidity" is used in a more general way than it was used in the original 1996 Cheeger-Colding paper coining the term.  In April 2018, Jeff Cheeger stated he was not happy with this and requested that the term not be used in this way in future work. "Geometrostatic manifolds of small ADM Mass" by C Sormani and I Stavrov, 42 pages, (arxiv preprint)  We bound the locations of minimal surfaces in Geometrostatic manifolds of small ADM mass and then prove the Intrinsic Flat Stability of the Positive Mass Theorem in the Geometrostatic setting. (NSF DMS 1309360)   "A generalized tetrahedral property for spaces with conical singularities" by P Nunez-Zimbron and R Perales (arxiv preprint) We extend the definition of Sormani's Tetrahedral Property so that conical metric spaces satisfy our new definition. We prove that our generalized definition retains all the properties of the original tetrahedral property proven by Portegies-Sormani: it provides a lower bound on the sliced filling volume and Gromov filling volume of spheres, and a lower bound on the volumes of balls. Thus, sequences with uniform bounds on our generalized tetrahedral property also have subsequences which converge in both the Gromov-Hausdorff and Sormani-Wenger Intrinsic Flat sense to the same limit space. (NSF DMS 1309360) "Contrasting Various Notions of Convergence in Geometric Analysis" by Brian Allen and Christina Sormani,   (figures by Pen Chang) 43 pages, (arxiv preprint) We explore the distinctions between $L^p$ convergence of metric tensors on a fixed Riemannian manifold versus Gromov-Hausdorff, uniform, and intrinsic flat convergence of the corresponding sequence of metric spaces.   We provide a number of examples which demonstrate these notions of convergence do not agree even for two dimensional warped product manifolds with warping functions converging in the $L^p$ sense.  We then prove a theorem which requires $L^p$ bounds from above and $C^0$ bounds from below on the warping functions to obtain enough control for all these limits to agree. (NSF DMS 1612049) “Warped Tori of Almost Nonnegative Scalar Curvature" originally named "Almost Rigidity of Warped Tori" by Brian Allen, Lisandra Hernandez-Vazquez, Davide Parise, Alec Payne, Shengwen Wang; 21 pages, (arxiv preprint)to appear in Geometriae Dedicata We show that for warped products on a 3-torus, there is almost rigidity of the Scalar Torus Rigidity Theorem: for sequences of warped product metrics on a 3-torus satisfying the scalar curvature bound Rj≥−1j, uniform upper volume and diameter bounds, and a uniform lower area bound on the smallest minimal surface, we find a subsequence which converges in both the Gromov-Hausdorff and the Sormani-Wenger Intrinsic Flat sense to a flat 3-torus.  (NSF DMS 1612049) (Team website).  Comments: Cheeger was not happy with the use of the term “almost rigidity” both in title and within the paper and requested the change. The term was first used in his 1996 paper with Colding and had a more restricted meaning there.  Although the team offered to clarify the distinction between “Cheeger Colding Almost Rigidity” and the more general modern meaning, he did not agree to this compromise.  He requested that everyone stop using “Almost Rigidity” in any setting other than his original meaning.  In deference to his seniority we should stop.               “Oberwolfach Report: 2012 Spacetime Intrinsic Flat Convergence”                                 by C Sormani (arxiv) This page is maintained by Christina Sormani who has National Science Foundation funding to conduct research on the Intrinsic Flat Distance: DMS 1006059 and DMS 1309360 and DMS 1612409..   The comments are by Christina Sormani.


text:
Android : The Flexible Operating System
/ August 4, 2018

Android is leading the smartphone market as a champion with the leadership rate of 84 %, over 3.3 billion smartphones are running on the Android operating system because of it’s efficiency and performance.   Android : It is an open source operating system which is made on Linux-kernel. Today Android is featured in almost 70% of smart electronics globally but the major sector that has hooked with this operating system is the smartphone market. On average, almost every fourth of the smartphone is running on the Android operating system. And due to the fact that it is open source, it has a huge community of developers across the globe.   The First Boot: The journey of this revolutionary operating system has begun in October 2003 [1] by Andy Rubin and his three teammates. Rubin described the this project as “tremendous potential in developing smarter mobile devices that are more aware of its owner’s location and preferences” [1]. Later on, one of the members of Google estimated the potential of the futuristic operating system because it has the efficiency to become a revolutionary OS and they desired to buy it but the first meeting didn’t work well, but later on July 2005 Google…

Insert math as
$${}$$


text:
# Contour plot for a likelihood function

For the likelihood associated with the mixture model

1/4*Normal(mu1,1)+3/4*Normal(mu2,1)

I would like to have a figure as like as following one for a simulated sample of 500 observation from mu1=0 and mu2=2.5. My R code is:

da=rbind(rnorm(10^2),2.5+rnorm(4*10^2))
like=function(mu1,mu2){
sum(log((.25*dnorm(da-mu1)+.75*dnorm(da-mu2))))}
mmu1=seq(-2,5,length.out=length(da))
mmu2=seq(-2,5,length.out=length(da))
post=outer(mmu1,mmu2,like)
image(mmu1,mmu2,post,xlab=expression(mu1),ylab=expression(mu2))

which contains the error:

"Error in dnorm(da-mu1):
dims [product 800] do not match the length of object"

I've changed the length of mmu1 & mmu2, but nothing seemed to be happening. Could someone help me?

The function outer does not work the way you want. When you call outer it passes to the function like the vectors and expects the vector as a result. In your case your function expects the scalars and returns the scalar. Try changing it to something similar to this:

like <- function(mu1,mu2) {
apply(cbind(mu1,mu2),1,function(x)sum(log((.25*dnorm(da-x[1])+.75*dnorm(da-x[2])))))
}

As outer() expects that the function you pass in is vectorized, change your line to

post=outer(mmu1,mmu2,Vectorize(like))

to get


text:
The spreadsheet in GeoGebra has most of the regular spreadsheet-features. When it comes to just doing numerical calculations, regular spreadsheet software is more advanced than the GeoGebra spreadsheet. The object-oriented way of doing things in GeoGebra however, makes the spreadsheet a very useful tool. Apart from manipulating numbers and formulas, you can also manipulate all GeoGebra-objects in the spreadsheet view.

Whenever you need many objects that follow some regular pattern, you can use the spreadsheet.

## Relative copies

A spreadsheet is a table of so-called cells. Each cell is denoted by a letter (the column) and a number (the row).

The basic feature of any spreadsheet is that you can make relative copies by dragging a cell, or by dragging many cells. After selecing one or many cells, a filled-in blue rectangle is shown in the lower right corner of the selection. By dragging the filled in rectangle, you make so called relative copies.

If cell A1 contains the value 5 and A1 + 1 is written in cell A2, then the value of A1 is picked from the relative position, i.e. from the cell above. When A2 is dragged downwards to make relative copies, each cell vill get the value from the cell above and add one to this value.

### Absolute references

You can use the dollar-sign to make copies using an absolute reference.

By writing $A1, you make an absolute reference to A1 when dragging along a row. By writing A$1, you make an absolute reference to A1 when dragging along a column.

By writing $A$1, you make an absolute reference to A1 when dragging both along a column and along a row.

Absolute references can be used for producing a multiplication table.

## Creating objects using the spreadsheet

You can create objects by writing commands in a cell in the spreadsheet. If you need help writing a command, write it in the input bar of the spreadsheet to get help with code completion.

### Dynamic colors

If you right-click on an object to see the properties-window, there is a heading Dynamic Colors under the tab Advanced. Here you can specify a color by giving the red, green, and blue value. Each value must be between 0 and 1. You can write logical expressions for what colors to use, or let the color depend on some variable, hence the name dynamic colors. In the spreadsheet you can let the color of an object depend on the value of some cell. When making relative copies, the values of the colors will also be relatively copied.

## Creating functions using the spreadsheet

If you write an expression using the variable $$x$$, GeoGebra will create a function of $$x$$. By writing such expressions in cells in the spreadsheet, you can create functions.

## Exercise

#### Exercise 1

Linear and exponential pay raise

• Make a slider $$a$$ of integer values between 0 and 10,000 representing Anna's annual raise of her monthly salary in Swedish kronor.

• Make a slider $$b$$ of integer values between 0 and 100 representing Bertil's annual percentage raise of his monthly salary.

• Enter year and monthly salaries for Anna and Bertil in the spreadsheet and make relative copies showing their salaries over a number of years. Let them start with the same salary year 0.

• Create a list of points for Anna and one for Bertil. Select cells from two columns, then choose Create - > List of points.

Drag the $$y$$-axis until you see the points in the graphics view.

• Drag the sliders to study the salaries.

#### Exercise 2

Parabola

The ancient Greeks didn't have the Cartesian coordinate system or functions. For them a parabola was a geometric object that could be defined like this:

Given a line (called the directrix) and a point (called focus) a parabola is the set of points having the same distance to the point and to the line.

A point $$P$$ having the same distance to the point and to the line must lie on the perpendicular bisector of the point and some point on the line. It can be shown that this perpendicular bisector is a tangent to the parabola through $$P$$. Many perpendicular bisectors can be used to show the shape of the parabola.

Part 1

Make a point $$A$$ and a line through two points $$B$$ and $$C$$. Place a fourth point $$D$$ on the line and create the perpendicular bisector between $$A$$ and $$D$$.

Put a trace on the perpendicular bisector and drag $$D$$.

Part 2

Instead of the using trace you will now create a number of points on the line and a number of perpendicular bisectors.

• Remove the trace from the perpendicular bisector.
• Place a point $$E$$ on the line and make a vector $$\vec{u}$$ from $$D$$ to $$E$$.
• Rename $$D$$ to $$A1$$ and $$E$$ to $$A2$$. In that way they will be shown in the spreadsheet.
• Write A2 + u in cell A3 and make relative copies along column A.
• Write PerpendicularBisector(A1, A) in cell B1 and make relative copies along column B.
• Change the focus and the directrix, all perpendicular bisectors should also change dynamically.

Comment: In a similar way you can make perpendicular bisectors between a point and points on a circle. Such an example is shown at the top of this page.

#### Exercise 3

Discrete model of dog chasing rabbit

A rabbit is running along a river. In the model the rabbit starts at the origin and runs upwards along the $$y$$-axis.

A dog sees the rabbit and starts chasing it. In the model the dog starts somewhere on the positive $$x$$-axis.

The dog runs a short distance along a straight line. Then it looks up, sees the rabbit and changes its direction. Then it runs another short distance along a straight line. And so on.

Make four sliders.

1. Two sliders $$u$$ and $$v$$ representing the speeds of the rabbit and the dog.
2. A slider $$d$$ representing the initial distance between the dog and the rabbit.
3. A slider $$t$$ representing the time step during which the dog runs along a straight line.

Create all other objects needed for the model.

You may need the command UnitVector( <Vector> ) and the command Vector( <Point>, <Point> ).

If your model assumes that the dog never runs into the river, i.e. never crosses the $$y$$-axis, you can make an if-statement to achieve this. You make an if-statement using the command If( <Condition>, <Then>, <Else> ). It is up to you whether you let the dog run into the river or not.

When you are done you should be able to drag the sliders to see different trajectories.

# further info:

Fourier Series - Square Wave from Wolfram MathWorld

RGB colour model


text:
Local limit approximations for Markov population processes

Barbour, A D; Socoll, S (2009). Local limit approximations for Markov population processes. Journal of Applied Probability, 46(3):690-708.

Abstract

We are concerned with the equilibrium distribution $\prod _n$ of the $n$th element in a sequence of continuous-time density-dependent Markov processes on the integers. Under a $(2+\alpha )$th moment condition on the jump distributions, we establish a bound of order $O(n^{-(\alpha +1)/2}\sqrt{ \log n})$ on the difference between the point probabilities of $\prod n$ and those of a translated Poisson distribution with the same variance. Except for the factor $\sqrt{ \log n}$, the result is as good as could be obtained in the simpler setting of sums of independent, integer-valued random variables. Our arguments are based on the Stein-Chen method and coupling.

Abstract

We are concerned with the equilibrium distribution $\prod _n$ of the $n$th element in a sequence of continuous-time density-dependent Markov processes on the integers. Under a $(2+\alpha )$th moment condition on the jump distributions, we establish a bound of order $O(n^{-(\alpha +1)/2}\sqrt{ \log n})$ on the difference between the point probabilities of $\prod n$ and those of a translated Poisson distribution with the same variance. Except for the factor $\sqrt{ \log n}$, the result is as good as could be obtained in the simpler setting of sums of independent, integer-valued random variables. Our arguments are based on the Stein-Chen method and coupling.

Statistics

Citations

Dimensions.ai Metrics


text:
University of Groningenfounded in 1614  -  top 100 university

# Colloquium Mathematics - Dr. Max Lein

 When: Th 13-09-2018 16:00 - 17:00 Where: 5171.0151 (Zernike, Linnaeusborg)

Title:

Maxwell-type Operators — A New Class of Operators Describing Classical Waves

Abstract:

Over the last few decades lot research has been dedicated to the study of Schrödinger operators such as $H = -\Delta + V$, and researchers nowadays have access to a rich toolbox of mathematical techniques. Recently, physicists have managed to realize analogs of quantum phenomena with classical waves such as topological insulators. The mathematical basis for these \emph{quantum-wave analogies} is that many classical wave equations can be recast in the form of a Schrödinger equation. Here,\emph{Maxwell-type operators} $M = W \, D$ take the place of a Schrödinger operator $H$. However, since these operators are the \emph{product} of weights $W$ and a free operator $D$, techniques initially developed for Schrödinger operators do not extend without significant work. In this talk I will give an overview over some recent rigorous results and detail some of the complications that arise when adapting e. g. pseudodifferential and semiclassical techniques.

Colloquium coordinators are Prof.dr. A.J. van der Schaft (a.j.van.der.schaft rug.nl),

Dr. A.V. Kiselev (e-mail: a.v.kiselev rug.nl)


text:
# How Gravitational Potential due to the spherical shell of radius R varies with the distance r from its center (For r > R)

then For Outside the surface , I.e For    $r>R$
$V=-\frac{GM}{r}$


text:
# Q.3 A coin is flipped to decide which team starts the game. What is the probability that your team will start?

Therefore Required probability = $$\frac{n(E)}{n(S)}=\frac{1}{2}$$


text:
# Classical mechanics, initial conditions question

1. Mar 11, 2010

### fluidistic

I just started CM (I had 2 classes until now) and the professor said that if you know the position and velocity of say all the particles, then you know how the system will evolve.
This, I already read and knew. I've probably a common question so feel free to redirect me to any similar thread. Here it comes:
Suppose you're given the position and velocity of 2 particles. They are making contact and their velocity is 0 m/s. How do you know if a moment later they will start to move away from each other because they had a collision or if they will stay motionless, i.e. will remain stuck together? (Suppose the universe is constituted by these only 2 particles)

If I knew v(t) it would be a different story, but only knowing v at a certain moment makes me wonder.

Maybe it's like having a second order differential equation (representing the motion of each particles) that is totally unknown, but you have y'(0) and y(0). But I'm not sure at all on this.

Could you answer my first question in words? Or maybe mathematically? Thanks for any help.

2. Mar 11, 2010

### tiny-tim

Hi fluidistic!
Two particles can't make contact … particles don't work like that.

Two electrons, say, could get very close before they started to repel each other. But they could only be close because there had been a "collision", ie one had been heading for the other.

When they repel each other, the repulsion will be predictable from their positions when stationary.

3. Mar 11, 2010

### fluidistic

Ok I see. But I thought CM was a theory that is a "simplified" way of describing the universe (for example, it does not consider electric and magnetic forces, as far as I know). I thought it would work pretty well assuming particles collide (even though as you said they don't make contact).

4. Mar 11, 2010

### tiny-tim

But if they're "simplified" bodies, stationary and in contact, they'd have no reason to separate, would they?

To put it another way, if there's no electric etc forces, how did they get to be stationary, if there was a collision?

5. Mar 11, 2010

### elect_eng

I think the point is that you will know v(t) if you have the dynamical equations for the system. The dynamical equations plus the initial conditions tell you the complete future history. This is what makes the system deterministic. This is true even in chaotic systems, although calculations become problematic due to sensitivity to intial conditions, in that case.

Anyway if there is a colision occuring at the point of your initial conditions, then your dynamic equations will (or, at least should) include the forces in action (compression, electrostatic, electromagnetic, gravitational, etc.).

6. Mar 11, 2010

### fluidistic

Hmm I think I was trying to say something else, my English is maybe too limited, hahaha!
Yes, if the 2 particles or "perfect" balls are in contact with a 0 velocity, there is no reason they would separate. However if you pitch one ball into another one and take a photo at the moment of the collision (in CM I guess this moment is not even a differential amount of time, it's only 1 moment; it's instantaneous. But I could be wrong in this assumption. I think it would hold if we assume no deformation, hence the word "perfect"), you'd see both balls making contact with each other and you'd see they both have a 0 velocity. Thus you are in the same case as when the balls are initially quiet and making contact. The difference is that in the first case there will be or there is a transfer of linear momentum and if the ball you throw was less heavy than the quiet one, the ball would bounce back.
My question is, how do you determine whether you are in the first case or in the second case, with only 1 photo taken? Namely, the photo showing the 2 balls making contact and with velocity =0.
An equivalent problem would be to let fall a ball on the floor and take a photo at the moment it touches the ground. The second case would to put the ball on the ground and take a photo. Both photos would show exactly the same thing to my eyes: a ball touching the ground and with velocity worth 0 m/s. How do you distinguish whether the ball will bounce off or not? I repeat, I've been told that knowing only the position and the velocity is enough to determine how the system will evolve.

The problem is that at first I only know $$\vec r(0)$$ and $$\vec v (0)$$. I've been told it's enough to determine the equation of motion of the whole system if I know this information for each particle. So I can't start with the dynamical equations of the system, I do not have them.

7. Mar 11, 2010

### elect_eng

You must have misunderstood, or you must be dealing with a special case. A special case would be particles that have no forces on them. Otherwise, if you don't know the forces, how can you know what the particles will do after their initial condition?

Once you know the system constraints and forces, you can use physics (Newtonian, special relativity, etc as the case may be) to develop dynamical equations. These equations are often put in terms of state space equations which would be first order differential equations for time rate of change of the position and velocity vectors, which are the state variables. This is why initial conditions are needed for position and velocity, because you need to use them as the starting values of the system state variables. The rate equations then tell you how those state variables change in time.

8. Mar 12, 2010

### fluidistic

Ok thanks a lot, it makes sense.


text:
Next: Total Internal Reflection Up: Multi-Dimensional Waves Previous: Laws of Geometric Optics

# Fresnel Relations

The theory described in the previous section is sufficient to determine the directions of the reflected and refracted waves, when a light wave is obliquely incident on a plane interface between two dielectric media. However, it cannot determine the fractions of the incident energy that are reflected and refracted, respectively. In order to calculate the coefficients of reflection and transmission, it is necessary to take into account both the electric and the magnetic components of the various different waves. It turns out that there are two independent wave polarizations that behave slightly differently. The first of these is such that the magnetic components of the incident, reflected, and refracted waves are all parallel to the interface. The second is such that the electric components of these waves are all parallel to the interface.

Consider the first polarization. Let the interface correspond to the plane , let the region be occupied by material of refractive index , and let the region be occupied by material of refractive index . Suppose that the incident, reflected, and refracted waves are plane waves, of angular frequency , whose wavevectors lie in the - plane. (See Figure 44.) The equations governing oblique electromagnetic wave propagation through a uniform dielectric medium are a generalization of Equations (494)-(495), and take the form (see Appendix C)

 (578) (579) (580)

where

 (581)

is the electric displacement, the characteristic wave speed, and the refractive index. Suppose that, as described in the previous section,

 (582)

in the region , and

 (583)

in the region . Here, is the vacuum wavenumber, the angle of incidence, and the angle of refraction. (See Figure 44.) In writing the above expressions, we have made use of the law of reflection (i.e., ), as well as the law of refraction (i.e., ). The two terms on the right-hand side of Equation (582) correspond to the incident and reflected waves, respectively. The term on the right-hand side of Equation (583) corresponds to the refracted wave. Substitution of Equations (582)-(583) into the governing differential equations, (578)-(580), yields

 (584) (585)

in the region , and

 (586) (587)

in the region .

According to standard electromagnetic theory (see Appendix C), both the normal and the tangential components of the magnetic intensity must be continuous at the interface. This implies that

 (588)

which yields

 (589)

Standard electromagnetic theory (ibid.) also implies that the normal component of the electric displacement, as well as the tangential component of the electric field, are continuous at the interface. In other words,

 (590)

and

 (591)

The former of these conditions again gives Equation (589), whereas the latter yields

 (592)

Here,

 (593) (594)

It follows that

 (595) (596)

The electromagnetic energy flux in the -direction (i.e., normal to the interface) is (see Appendix C),

 (597)

[cf. Equation (486)]. Thus, the mean energy fluxes associated with the incident, reflected, and refracted waves are

 (598) (599) (600)

respectively. The coefficients of reflection and transmission are defined

 (601) (602)

respectively. Hence, it follows that

 (603) (604)

These expressions are known as Fresnel relations, and are the generalizations of expressions (510) and (511) for the case of oblique incidence (with the polarization in which the magnetic field is parallel to the interface).

Let us now consider the second polarization, in which the electric components of the incident, reflected, and refracted waves are all parallel to the interface. In this case, the governing equations are (see Appendix C)

 (605) (606) (607)

If we make the transformations , , , then we can reuse the solutions that we derived for the other polarization. We find that

 (608) (609) (610)

in the region , and

 (611) (612) (613)

in the region . The first two matching conditions at the interface are that the normal and tangential components of the magnetic intensity are continuous. (See Appendix C.) In other words,

 (614) (615)

The first of these conditions yields

 (616)

whereas the second gives

 (617)

The final matching condition at the interface is that the tangential component of the electric field is continuous. (See Appendix C.) In other words,

 (618)

which again yields Equation (616). It follows that

 (619) (620)

The electromagnetic energy flux in the -direction is (see Appendix C),

 (621)

Thus, the mean energy fluxes associated with the incident, reflected, and refracted waves are

 (622) (623) (624)

respectively. Hence, the coefficients of reflection and transmission are

 (625) (626)

respectively. These expressions are the Fresnel relations for the polarization in which the electric field is parallel to the interface.

It can be seen that, at oblique incidence, the Fresnel relations (603) and (604) for the polarization in which the magnetic field is parallel to the interface are different to the corresponding relations (625) and (626) for the polarization in which the electric field is parallel to the interface. This implies that the coefficients of reflection and transmission for these two polarizations are, in general, different.

Figure 45 shows the coefficients of reflection and transmission for oblique incidence from air ( ) to glass ( ). In general, it can be seen that the coefficient of reflection rises, and the coefficient of transmission falls, as the angle of incidence increases. However, for the polarization in which the magnetic field is parallel to the interface, there is a particular angle of incidence, know as the Brewster angle, at which the reflected intensity is zero. There is no similar behavior for the polarization in which the electric field is parallel to the interface.

It follows from Equation (603) that the Brewster angle corresponds to the condition

 (627)

or

 (628)

where use has been made of Snell's law. The previous expression reduces to

 (629)

or . Hence, the Brewster angle corresponds to , where

 (630)

If unpolarized light is incident on an air/glass (say) interface at the Brewster angle then the reflected light is linearly polarized. (See Section 8.6.)

The fact that the coefficient of reflection for the polarization in which the electric field is parallel to the interface is generally greater that that for the other polarization (see Figure 45) implies that sunlight reflected from a horizontal water or snow surface is partially linearly polarized, with the horizontal polarization predominating over the vertical one. Such reflected light may be so intense as to cause glare. Polaroid sunglasses help reduce this glare by blocking horizontally polarized light.

Next: Total Internal Reflection Up: Multi-Dimensional Waves Previous: Laws of Geometric Optics
Richard Fitzpatrick 2013-04-08


text:
Fundamental theorem of arithmetic applications

The Fundamental Theorem of Arithmetic

Fundamental Theorem of Calculus application Stack. The Fundamental Theorem of Arithmetic states that every natural number greater than 1 can be factored into prime numbers in exactly one way (the order of the factors, Class-10 Andhra Pradesh Board - Fundamental Theorem of Arithmetic - LearnNext offers animated video lessons with neatly explained ….

The fundamental theorem of arithmetic Khan

The Fundamental Theorem of Arithmetic Divisibility. Fundamental Theorem of Arithmetic. Let us begin by noticing that, in a certain sense, there are two kinds of natural number: composite numbers and prime numbers. Composite numbers we get by multiplying together other numbers. For example, $$6=2\times 3$$. We say that 6 factors as 2 times 3, and that 2 and 3 are divisors of 6., I just learned about the fundamental theorem of calculus and I am doing some applications of the theorem. I have been given this problem: If ….

I just learned about the fundamental theorem of calculus and I am doing some applications of the theorem. I have been given this problem: If … Fundamental Theorem of Arithmetic. The assertion that prime factorizations are unique. That is, if you have found a prime factorization for a

Fundamental Theorem of Arithmetic states that every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart 2016-09-22 · The fundamental theorem of arithmetic states that every positive integer (except the number 1) can be represented in exactly one way apart from

Fundamental Theorem of Arithmetic. The assertion that prime factorizations are unique. That is, if you have found a prime factorization for a 6-14-2008 T h e F u n d a m en ta l T h eore m o f A rith m etic ¥ T h e F u n d a m e n ta l T h e o re m o f A rith m e tic say s th at every integer greater th an

2016-09-22 · The fundamental theorem of arithmetic states that every positive integer (except the number 1) can be represented in exactly one way apart from The primes and the fundamental theorem of arithmetic Modular arithmetic and applications Polynomial arithmetic and factorization Rings, Domains, and Fields

Mathematics Semester 1 (AUG) Fundamental theorem of Arithmetic, Congruences, Fermat's Seifert-van Kampen theorem, applications. This section contains lecture video excerpts, lecture notes, a problem solving video, and a worked example on the fundamental theorem of calculus.

The primes and the fundamental theorem of arithmetic Modular arithmetic and applications Polynomial arithmetic and factorization Rings, Domains, and Fields “In mathematics, and in particular number theory, the fundamental theorem of arithmetic is the statement that every positive integer can be written as a product of prime numbers in a unique way. For instance, we can write

3.5 The Division Theorem. 3.6 Applications of gcd and lcm. 4.The Euclidean Algorithm. The Fundamental Theorem of Arithmetic. (Historical figure: Mersenne). The fundamental theorem of arithmetic states that every positive integer (except the number 1) can be represented in exactly one way apart from rearrangement as a product of one or more primes (Hardy and Wright 1979, pp. 2-3).

The 'fundamental theorem of arithmetic' states "every natural number greater than 1 can be written as a unique product of prime numbers". If 'product' implies multiplication, what are the factors of a prime number? The fundamental theorem of calculus is a theorem that links the concept of differentiating a function This part of the theorem has key practical applications,

9 The Fundamental Theorem of Arithmetic

Syllabus Math 417 Mathematics at Illinois. Theorem. Fundamental Theorem of Arithmetic Every integer n > 1 can be represented as a product of prime factors in only one way, apart from the order of the factors. proof. The proof is by induction on n: The theorem is true for n = …, The primes and the fundamental theorem of arithmetic Modular arithmetic and applications Polynomial arithmetic and factorization Rings, Domains, and Fields.

The Fundamental Theorem of Arithmetic NCERT MATH. 2012-01-04 · application of fundamental theorem of maths Application Of The Fundamental Theorem Of By applying the fundamental theorem of arithmetic to, $\require{AMSsymbols}$Before diving into the Fundamental Theorem of Arithmetic, we need to first dispatch with an old claim. Namely, that if $p$ is prime and $p \mid. Fundamental Theorem of Arithmetic The fundamental theorem of arithmetic Khan. We are ready to prove the Fundamental Theorem of Arithmetic. Recall that this is an ancient theorem—it appeared over 2000 years ago in Euclid's Elements. https://en.wikipedia.org/wiki/Euclid%27s_lemma Class-10 CBSE Board - Fundamental Theorem of Arithmetic - LearnNext offers animated video lessons with neatly explained examples, Study …. Fundamental theorem of arithmetic. Proof:- P(n): n is a prime or can written uniquely as a product of primes. P(2) is true, and assume true for n=3,4,...,k. Now if k+1 is not a prime it can be written as k+1=r×s . Now r The Fundamental Theorem of Arithmetic 1. Applications of The Fundamental Theorem The Fundamental Theorem can also be … The fundamental theorem of arithmetic states that every positive integer (except the number 1) can be represented in exactly one way apart from rearrangement as a product of one or more primes (Hardy and Wright 1979, pp. 2-3). Chapter 1 The Fundamental Theorem of Arithmetic 1.1 Prime numbers If a;b2Zwe say that adivides b(or is a divisor of b) and we write ajb, if b= ac for some c2Z. Fundamental Theorem of Arithmetic. The assertion that prime factorizations are unique. That is, if you have found a prime factorization for a ... the Fundamental Theorem of Calculus. Weekly Prime numbers and Fundamental Theorem of arithmetic; of Fermat's little theorem; Applications: Class-10 CBSE Board - Fundamental Theorem of Arithmetic - LearnNext offers animated video lessons with neatly explained examples, Study … The 'fundamental theorem of arithmetic' states "every natural number greater than 1 can be written as a unique product of prime numbers". If 'product' implies multiplication, what are the factors of a prime number? of the fundamental theorem of arithmetic in sorne'irnportant domains theory having a very large number of applications in arithmetic as well as modern algebra. Theorem of Theaetetus: Square root of 2 is$ The fact which is known as The Fundamental Theorem of Arithmetic. So that the repeated applications … ... the Fundamental Theorem of Calculus. Weekly Prime numbers and Fundamental Theorem of arithmetic; of Fermat's little theorem; Applications:

2016-09-22 · The fundamental theorem of arithmetic states that every positive integer (except the number 1) can be represented in exactly one way apart from Syllabus Math 417 Math 417. Intro to Fundamental theorem of arithmetic. Applications of group actions, eg. to Sylow's theorem or Polya counting.

2012-01-04 · application of fundamental theorem of maths Application Of The Fundamental Theorem Of By applying the fundamental theorem of arithmetic to Fundamental Theorem of Arithmetic: Statement: Every composite number can be decomposed as a product prime numbers in a unique way, except for the order in which the prime numbers occur. For example: (i) 30 = 2 × 3 × 5, 30 = 3 × 2 × 5, 30 = 2 × 5 × 3 and so on. (ii) 432 = 2 × 2 × 2 × 2 × 3 × 3 × 3 = 2 4 × 3 3 or 432 = 3 3 × 2 4.

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