text:
# 12.5 Sound interference and resonance: standing waves in air columns  (Page 5/11)

 Page 5 / 11

We have emphasized sound applications in our discussions of resonance and standing waves, but these ideas apply to any system that has wave characteristics. Vibrating strings, for example, are actually resonating and have fundamentals and overtones similar to those for air columns. More subtle are the resonances in atoms due to the wave character of their electrons. Their orbitals can be viewed as standing waves, which have a fundamental (ground state) and overtones (excited states). It is fascinating that wave characteristics apply to such a wide range of physical systems.

Describe how noise-canceling headphones differ from standard headphones used to block outside sounds.

Regular headphones only block sound waves with a physical barrier. Noise-canceling headphones use destructive interference to reduce the loudness of outside sounds.

How is it possible to use a standing wave's node and antinode to determine the length of a closed-end tube?

When the tube resonates at its natural frequency, the wave's node is located at the closed end of the tube, and the antinode is located at the open end. The length of the tube is equal to one-fourth of the wavelength of this wave. Thus, if we know the wavelength of the wave, we can determine the length of the tube.

## Phet explorations: sound

This simulation lets you see sound waves. Adjust the frequency or volume and you can see and hear how the wave changes. Move the listener around and hear what she hears.

## Section summary

• Sound interference and resonance have the same properties as defined for all waves.
• In air columns, the lowest-frequency resonance is called the fundamental, whereas all higher resonant frequencies are called overtones. Collectively, they are called harmonics.
• The resonant frequencies of a tube closed at one end are:
${f}_{1}$ is the fundamental and $L$ is the length of the tube.
• The resonant frequencies of a tube open at both ends are:

## Conceptual questions

How does an unamplified guitar produce sounds so much more intense than those of a plucked string held taut by a simple stick?

You are given two wind instruments of identical length. One is open at both ends, whereas the other is closed at one end. Which is able to produce the lowest frequency?

What is the difference between an overtone and a harmonic? Are all harmonics overtones? Are all overtones harmonics?

## Problems&Exercises

A “showy” custom-built car has two brass horns that are supposed to produce the same frequency but actually emit 263.8 and 264.5 Hz. What beat frequency is produced?

0.7 Hz

What beat frequencies will be present: (a) If the musical notes A and C are played together (frequencies of 220 and 264 Hz)? (b) If D and F are played together (frequencies of 297 and 352 Hz)? (c) If all four are played together?

What beat frequencies result if a piano hammer hits three strings that emit frequencies of 127.8, 128.1, and 128.3 Hz?

0.3 Hz, 0.2 Hz, 0.5 Hz

Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
what is the Synthesis, properties,and applications of carbon nano chemistry
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
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Berger describes sociologists as concerned with
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text:
# Discrete topology, product topology

1. Dec 2, 2008

### mathsss2

For each $$n \in \omega$$, let $$X_n$$ be the set $$\{0, 1\}$$, and let $$\tau_n$$ be the discrete topology on $$X_n$$. For each of the following subsets of $$\prod_{n \in \omega} X_n$$, say whether it is open or closed (or neither or both) in the product topology.

(a) $$\{f \in \prod_{n \in \omega} X_n | f(10) = 0 \}$$
(b) $$\{f \in \prod_{n \in \omega} X_n | \text{ }\exists n \in \omega \text{ }f(n) = 0 \}$$
(c) $$\{f \in \prod_{n \in \omega} X_n | \text{ }\forall n \in \omega \text{ }f(n) = 0 \Rightarrow f(n + 1) = 1 \}$$
(d) $$\{f \in \prod_{n \in \omega} X_n | \text{ }|\{ n \in \omega | f(n) = 0 \}| = 5 \}$$
(e)$$\{f \in \prod_{n \in \omega} X_n | \text{ }|\{ n \in \omega | f(n) = 0 \}|\leq5 \}$$

2. Dec 4, 2008

### mathsss2

Recall that $$\omega = \mathbb{N} \cup \{0\}$$

3. Dec 5, 2008

### morphism

And what are your thoughts on the problem?

4. Dec 5, 2008

### mathsss2

Here is what I know:

So remember that open sets in the infinite product topology is really just having all but finitely many the whole space and the rest are open. Since the individual factors are discrete, you only need to check that all but finitely many are the whole space.

e.g. in (a) the 10th coordinate has a specific value, but all other coordinates can be whatever, so this is certain open.

$$f$$ is a function. Here $$\omega = \mathbb{N} \cup \{ 0 \}$$ (the reason for using $$\omega$$ is because he is using it to refer to the natural as an ordinal, but whatever that is not important). If it helps you can think of $$\prod_{n\in \omega} X_n$$ as $$\prod_{n=0}^{\infty} X_n$$. We define $$\prod_{n=0}^{\infty}X_n$$ to be the set of all functions $$f: \mathbb{N} \to \{ 0 , 1\}$$ that satisfies $$f(n) \in \{ 0 , 1\}.$$

This is as far as I've gotten.

5. Dec 5, 2008

### morphism

There's a nice graphical representation of the product topology on Y^X (i.e. the product of the space Y |X| times). Namely, if we draw X as an "x-axis" and Y as a "y-axis", then elements in X^Y are "graphs of functions" in the X-Y "plane". An open nbhd of an element f is the set of all functions g whose graphs are close to the graph of f at finitely points. We get different nbhds by varying the closeness to f and/or the set of finite points.

In our case the product space is 2^w=2^N, whose "plane" looks like two copies of the naturals N. In other words, if you were to imagine this as a 'subset' of R^2, it's just the set $\{(n,i) \colon n \in \bN, i \in \{0,1\}\}$.

6. Dec 7, 2008

### mathsss2

Progress:

Take set (b). Let $$B = \{f \in \prod_{n \in \omega} X_n | \;\exists n \in \omega \; f(n) = 0 \}$$. If $$f\in B$$ then there exists m such that f(m)=0. Then the set $$\{g \in \prod_{n \in \omega} X_n |\; g(m) = 0\}$$ is an open neighbourhood of f contained in B. Therefore B is open.

It's usually more difficult to check when a set is closed. You have to look at its complement and decide whether that is open. Sometimes this is straightforward. For example, the complement of set (a) is the set of all f such that f(10)=1. That is open, so set (a) is closed as well as open.

For a slightly less easy example, look at set (c). Let $$C = \{f \in \prod_{n \in \omega} X_n | \text{ }\forall n \in \omega \text{ }f(n) = 0 \Rightarrow f(n + 1) = 1 \}$$. If $$f\notin C$$ then there exists m such that f(m)=f(m+1)=0. Then $$\{g \in \prod_{n \in \omega} X_n |\; g(m) = g(m + 1) = 0\}$$ is an open neighbourhood of f containing no points of C. Therefore the complement of C is open and so C is closed.

I still do not know how to do parts (d.) and (e.)

format: plain


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# Complex Numbers

## Complex Numbers

Complex Number: If a, b are two real numbers, then a number of the form a + ib is called a complex number.

Examples:

• 7 + 3i
• -1 + i
• 3 + 2i
• 0 + 3i

etc.  are complex numbers.

Real and Imaginary Parts of a Complex Number: If z = a + ib is a complex number, then ‘a’ is called the real part of z and b is known as the imaginary part of z. The real part of z is denoted by Re(z) and the imaginary part by Im(z).

Example:

If z = 6 + 6i then Re(z) = 6 and Im (z) = 6.

Purely Real and Purely Imaginary Complex Numbers: A complex number z is purely real if it its imaginary pat is zero i.e., Im (z) = 0 and purely imaginary if its real part is zero i.e., Re(z) = 0

Set of Complex Numbers: The set of all complex numbers is denoted by C.

i.e., C = [ a+ i b | a, b ϵ R]

Since a real number ‘a’ can be written as a+i0 therefore every real number is a complex number. Hence,  where R is the set of all real numbers.

Equality of Complex Numbers: Two complex numbers z₁ = a₁ + ib₁ and z₂ = a₂ + ib₂ are equal if a₁ = a₂ and b₁ = b₂ i.e., Re(z₁) = Re(z₂) and Im(z₁) = Im(z₂).

Thus,  and Im(z₁) = Im(z₂).


text:
On the argument functions of simple closed curves and simple arcs
Compositio Mathematica, Tome 4 (1937) , pp. 271-275.
@article{CM_1937__4__271_0,
author = {Van Kampen, E. R.},
title = {On the argument functions of simple closed curves and simple arcs},
journal = {Compositio Mathematica},
pages = {271--275},
publisher = {Johnson Reprint Corporation},
volume = {4},
year = {1937},
zbl = {0016.13803},
language = {en},
url = {http://www.numdam.org/item/CM_1937__4__271_0/}
}
Van Kampen, E. R. On the argument functions of simple closed curves and simple arcs. Compositio Mathematica, Tome 4 (1937) , pp. 271-275. http://www.numdam.org/item/CM_1937__4__271_0/


text:
# Are these two questions really duplicates?

First, I'd like to appologize if this similar questions were already made. I've tried to look for something similar and the nearest I found was this. I've even looked the two examples given in the mentioned question but didn't find them too similar to my own: In my case, I guess the question is different due to the intentions of both askers.

Recently, I've made this question which was voted to close as a duplicate of this other question. In my question, I gave every information in my trial to answer and it was quite correct, I missed only the condition $x>0$ which was pointed by some users. In the second question, the user had a very faint approach to solving it. I knew beforehand that I could use the search (this must be a classic problem in the beginning of the study of inequalities), but I was more interested to show my work and receive criticism for it than to find an answer to it.

I feel like my question is like "is my answer correct?" with a subtle sense of correctness (I've read in the earlier pages of Courant's book that $x^2 +1\geq 2x$ and this was due to a property of the quadratic function so I knew it depended on it) and his question is "how can I do it?" with no clue on how to do it.

So are they really duplicates? This is important to me because I do a lot of this kind of questions in which I'm interested in showing what I thought - which - at least for me, seems a totally different kind of intention and hence, of question.

• Are you asking whether solution verification questions should be closed as duplicates? This was discussed here and in several other threads. The consensus seems to be that they should be closed as duplicates after the OP received sufficient feedback on their solution, see this answer. – Martin Sleziak Aug 6 '16 at 4:19
• It is worth mentioning that a possibility of disallowing solution verification questions completely was discussed a few times, I give some links in my answer here, In case of your question, you ask about the inequality $x+\frac1x\ge2$, which is a very frequent question. This might have been a factor, too. – Martin Sleziak Aug 6 '16 at 4:25
• You should probably clarify in your post whether the linked questions are only an illustration of a more general issue you wish to discuss or whether you are interested in this particular closure. If it is the latter, you should add (specific-question) tag, see the tag info. – Martin Sleziak Aug 6 '16 at 4:34
• Don't you think studying the solutions on the existing questions would have allowed you to decide the correctness of your approach? – quid Aug 6 '16 at 9:41
• Per Martin Sleziak's first Comment, the key here (if "proof verification" was your goal) is that you received feedback about the omission of requirement $x\gt 0$, needed in the first step of your proof. A posted Answer points out how far one can go without this assumption, essentially deferring its use until the end when a comparison of signs for numerator and denominator has to be made. In short I think you got the desired feedback, and you should not look upon closing as a duplicate as a bad thing since it did not deprive you of learning opportunity or reputation. – hardmath Aug 6 '16 at 17:49


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## Made a single hit cook torrence raytracer

A picture is worth a thousand words.
rouncer
Posts: 18
Joined: Wed Jun 19, 2013 11:04 am

### Made a single hit cook torrence raytracer

Getting organic surfaces with a cook torrence bounce... its on the books to add multibounce by recursing the procedure, but itll be alot slower.
Interesting things that make my framework different->

a) I render to a binary dithering map, and then i reduce it to the screens resolution.
b) I only use a low colour and a high colour, then the midrange develops as the dots join.
c) i then use add and multiply and clamp, segregated channels, to get my finished 'illumination'

my ior is 1.3 the consistency of cotton wool.

this lecture spills the beans on how to get these 'charactered' surfaces.

note i dont understand the lecture that well, i just plug and played the code, its only 5 lines long blew my mind.

and the demo->

my current weapon is a gtx980. 64x32 stream processors.

rouncer
Posts: 18
Joined: Wed Jun 19, 2013 11:04 am

### Re: Made a single hit cook torrence raytracer

Ill give myself a deadline for a miniature finished product on thursday, 11th feb, and ill give an executable, and the source for people to play with,
its very small, its one file only 711 lines so far, including all the comments and spaces and my newbie indents.

it has a fixed function material section that fits on the screen, its very simple.

The environment making process, is meshless volumetric functions, of course, i use a binary honing technique plus a feedback system to keep the occlusions when it finds them.
But it needs alot of improvement if i want to see volume functions in general. (which mathematicians would love. especially when they are all furry lookin hehe)

The scripter system will use fxc.exe as a commandline system call. from the program, which it communicates to, to update the scene. which actually has to recompile the entire
kernel to fit the new level in, because its a function call from the binary honing kernel, and it does take about 5 seconds to compile on my computer, because direct compute is a slow girly compiler, but its suited me, I cant be bothered learning cuda optimizations, im not so lowlevel with conventional hardware. (but i have crazy hardware ideas of my own regularly as well.)

The crazy stuff you see in the demo, are all little stretch variations of a common theme, some glitched bilinear filtering (which must be coded in yourself! or you can just borrow mine.)
and waves of harmonics on top of this.

//geometry function->

Code: Select all

//this has the bilinear filter in it.
int compute_altitude( int walkpos_x, int walkpos_z, int rpy)
{
int alt=-rpy-22100;
int k;
int idx,idy;
int w0,w1;

int h=96;

for(k=0;k<h;k+=16)
{

int nhh[4];
int nhv[4];

int k2=k;
if(k2>3) k2=0;

#define HG >>8)*(k+1))
#define NM  3

{
{
nhh[2]=abs(((Bufferui4[(0)+((0)<<NM)].amp)HG;
nhh[3]=abs(((Bufferui4[(0)+((0)<<NM)].amp)HG;
nhv[3]=abs(((Bufferui4[(0)+((0)<<NM)].amp)HG;
}
else
{
}
}
else
{
{
}
else
{
}
}

w0=idx;
w1=idy;

if(w0>1<<8 || w0<0
|| w1>1<<8 || w1<0)
{
int bp=1;
}

//bilinear average->
+((((nhh[2]*(255-w0))+(nhh[3]*(w0)))/255*(w1))))/255;
+((((nhv[2]*(255-w0))+(nhv[3]*(w0)))/255*(w0))))/255;

}

return alt;
}

material bar done. i just have a few things to fix then the demo will go here.
I worked out how to get the powderyish surface, and its roughness as close as possible to 0, and gauss as high as you can get it, and you get the fat material.

heres a little microworld thing i did with it, with silver snail trails ->


text:
# How to use the rotation matrix calculated by Matlab in Blender?

Good morning,

For a project, I created in MATLAB a scene composed of an object and twelve cameras looking at it. With the tool box of MATALB, I got the rotation matrix and the translation vector for each camera.

Now I would like to use these information (rotation matrix and translation vector) to create the same scene in Blender.

So I want to use the translation vector to position correctly my cameras in the space, so I will modify the location parameters in Blender with it. I want to use the rotation matrix to get the correct rotation of my cameras in the space. So I will modify the rotation parameters with it. The panel I am using is this one:

For the rotation, I know in Blender, I can use the quaternion, the euler rotation angles (XYZ, YXZ, ZYX) or the axis vector. I calculated all of them with MATLAB but unfortunately they don't give the correct result. The cameras don't look in the correct direction as shown on the following pictures:

Camera's view:

What I would like is the camera looking at the object like on the following picture but with the parameters calculated by MATLAB. In this example, I use the object constraint properties:

So, in the view of the camera, we can see the object:

The rotation matrix of the first camera is:

Its translation is:

The XYZ angle calculated with this matrix by MATLAB is:

I read that MATLAB and Blender don't have the same coordinate system. How can I do that?

• We are going to need a lot more info in order to address your issue. Could you please add some screenshots of your setup and an example of what is different between the two? – PGmath Mar 10 at 13:13
• Hi, I would like to use the parameters calculated by MATALB to define my model to built it with Blender. Sorry, I will ask a stupid question but how I can share here the screenshots from my computer? – Noella Mar 11 at 14:59
• See this meta post. Use the add image button when editing your post. – PGmath Mar 11 at 15:08
• Thanks for your help. Let me know if it is still unclear. – Noella Mar 13 at 10:15
• See blender.stackexchange.com/a/176762/15543 the camera in blender looks down its -Z axis and pitches on Y. If you know the MATLAB camera FORWARD and UP can use blender.stackexchange.com/questions/102293/… – batFINGER Mar 13 at 11:39

Some rattling on.

Not sure one arbitrary rotation matrix and one location vector with very little more info is enough. Instead of a very long commment, here is some ways to try and work this out.

The matrices given.

>>> print(R)
<Matrix 3x3 (-0.4877,  0.8728,  0.0190)
( 0.4404,  0.2648, -0.8578)
(-0.7537, -0.4100, -0.5136)>

T = Matrix.Translation((1.5581, 1.2190, 1.4871))

In blender (row order matrix) the columns of a 3x3 rotation matrix are the vectors of local axes. If the camera was placed such that it focused directly on (0, 0, 0) (In blender a camera looks down its $$-Z$$ axis)

For example in default blend, with a track to constraint on camera to look at cube at (0, 0, 0)

>>> C.object
bpy.data.objects['Camera']

>>> print(C.object.matrix_world.to_3x3())
<Matrix 3x3 (0.6563, -0.3579,  0.6642)
(0.7545,  0.3114, -0.5778)
(0.0000,  0.8803,  0.4744)>

>>> print(C.object.matrix_world.translation.normalized())
<Vector (0.6642, -0.5778, 0.4744)>

ie its local $$Z$$ axis matches its normalized location

if it was same for given data would expect

>>> T.translation.normalized()[:]
(0.6295621991157532, 0.4925462305545807, 0.6008740663528442)

to match a column (or row, not sure of order) of rotation matrix, unfortunately it does not,.

Using angles given to get this matrix by setting order to ZYX and transposing

>>> print(Euler((map(radians, (141.4, -48.9, -137.9))), 'ZYX').to_matrix().transposed())
<Matrix 3x3 (-0.4878,  0.8728,  0.0187)
( 0.4407,  0.2647, -0.8577)
(-0.7536, -0.4101, -0.5138)>

Reversed order suggests pitch, roll, yaw type rotation.

Transposing suggests either column order of matrices, it is also a cheap way of inverting a rotation matrix.

>>> print(Euler((map(radians, (141.4, -48.9, -137.9))), 'ZYX').to_matrix().inverted())
<Matrix 3x3 (-0.4878,  0.8728,  0.0187)
( 0.4407,  0.2647, -0.8577)
(-0.7536, -0.4101, -0.5138)>

Which also suggests there could be a difference in +/- for CW / CCW rotation.

>>> print(Euler((map(radians, (-141.4, 48.9, 137.9))), 'XYZ').to_matrix())
<Matrix 3x3 (-0.4878,  0.8728,  0.0187)
( 0.4407,  0.2647, -0.8577)
(-0.7536, -0.4101, -0.5138)>

Seeing what happens, set the matrix world to

C.object.matrix_world = T @ R.to_4x4()

for the hell of it in this one have used

C.object.matrix_world = T @ R.transposed().to_4x4()

since it is easy to see in image that (if the object of interest to focus is around (0, 0, 0) it's looking the wrong way and is upside down.

Can make a conversion matrix via

>>> bpy_extras.io_utils.axis_conversion(
...         from_forward='Z',
...         from_up='-Y',
...         to_forward='-Z',
...         to_up='Y')

Matrix(((1.0, 0.0, 0.0),
(0.0, -1.0, 0.0),
(0.0, 0.0, -1.0)))

Or can see that flipping 180 degrees around $$X$$ axis would give same result.

>>> Matrix.Rotation(pi, 3, 'X')
Matrix(((1.0, 0.0, 0.0),
(0.0, -1.0, -8.742277657347586e-08),
(0.0, 8.742277657347586e-08, -1.0)))

So with one of the conversion matrices above as S

>>> S = Matrix.Rotation(pi, 4, 'X')
>>> C.object.matrix_world @= S

In as much as it doesn't exactly answer your question, I hope it gives enough info that if you have two locations and a rotation from each that points directly at a known point (eg the origin) then it would be a relatively easy task to work out the space.

• Hi, Thanks for your answer. I tried to write the script step by step, according to your answer. I wrote this script (I put it in my next comment) but I have an error when I try to calculate Camera.matrix_world = T @ R.transposed().to_4x4() I searched on the internet, but I found nothing. Why I have the following message: expected a sequence of float, not float. Otherwise, I will try this script when the problem on this line will be sorted out, and see if it is the solution of my problem or the start of a possible solution. Thanks a lot for your help. – Noella Mar 17 at 11:06
• Sounds like you have missed some brackets.a = ((1, 0), (0, 1)) as opposed to b = (1, 0, 0, 1) first element of a is a sequence of floats, first element of b is a float. – batFINGER Mar 17 at 11:11
• Thanks, it was that. I tried what you told me. The position of my camera does not match with your screenshots. Is it possible by comment to share what I have on my screen and the code? – Noella Mar 17 at 16:00


text:
"Since his

It looks like I've found part of a puzzle? I wonder where the rest of it is! It should be around here somewhere...

Once you do find it, you might find this transcription of some of the text in the puzzle helpful:

"Since his

now that it has been TWEAKED,
when spoken aloud.

Numbers 1:3
One who might work after the

Numbers 13:
In a delirio

Genesis 40:
Dutch
and semi-

Numbers 7:7
In order to finish putting on this garment, you have to

Joshua 19:1
It could be the source of a lot of

Numbers 1:1
Place where a salvager might

Genesis 46:2
Minty offering that might cause a thirst to be

Genesis 49:19
Increase

Luke 2:25
Divide

1 Chronicles 12:26
Squid's cousin

Genesis 35:18
Original

Joshua 15:20
Render harmless, as a cat

Exodus 1:1-5
(3 5 4) → "(3 6)"

• Oh, this is beautiful! What a clever collaboration - well done both of you :) It's midnight here so I'll have to leave this to the 'night crew' but I shall look forward to having a go myself when I wake up. Lovely job!
– Stiv
May 5 at 23:06
• This is, in my opinion, the most brilliant puzzle ever crafted on this site. My hat's off to you, @noneuclideanisms. May 6 at 0:38
• @Stiv collaboration, or two accounts from the same person? (Which is allowed, with some limitations.) Note that the usernames are anagrams of each other. May 6 at 14:40
• @Glorfindel That is an excellent spot - not something I'd ever noticed before! Either way, wonderfully coordinated and so satisfying to experience :)
– Stiv
May 6 at 14:47

Now that this puzzle has been tweaked,

IT'S HOLIER

First you have to recognize that

the blank area in this puzzle's image is this other puzzle Then you combine them to make this image:

The clues are solved like so

Notice that the yellow cells now spell the months of the year.

Each clue has a Bible verse.

Each verse talks about one of the sons of Israel that went to Egypt with Jacob. These are shown in the image above.

Finally, Exodus 1:1-5

lists them in order. Taking the circled letters in the order indicated by the names yields the string ITSWHOLEYEAR.

This makes sense because we have all the months of the year.

(3 5 4) → "(3 6)" shows the enumeration for the literal solution and its homophone when spoken aloud.

When spoken aloud

It's holier!


text:
Is the function f(x)=1/(x^3+1) even, odd or neither?

May 7, 2018

It is neither .

Explanation:

A function f(x) is even if f(-x)=f(x) and odd if f(-x)=-f(x)
Putting x=-x we get
f(x)=$\frac{1}{- {x}^{3} + 1}$
which is not equal to either f(x) or f(-x).
So its neither of the two.

Hope it helps!!


text:
# How do you solve the system of equations and enter the solution as an ordered pair. 4x-7y=-23 & 8x+5y=49?

Nov 8, 2017

$y = 0.16$

$x = 6.06$

#### Explanation:

$4 x - 7 y = 23$ Eqn 1
$8 x + 5 y = 49$ Eqn 2
Multiply Eqn 1 by 2

$8 x - 14 y = 46$
Subtract Eqn 2,

$- 19 y = - 3 , y = \frac{3}{19} = 0.16$

Substituting value of y in Eqn 1
$4 x - \left(\frac{21}{19}\right) = 23$
$x = \frac{24 \left(\frac{2}{19}\right)}{4} = 6.06$


text:
Volume 17, issue 4 (2017)

 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Ethics Statement Author Index To Appear ISSN (electronic): 1472-2739 ISSN (print): 1472-2747 Other MSP Journals
The $C_2$–spectrum $\mathrm{Tmf}_1(3)$ and its invertible modules

Michael A Hill and Lennart Meier

Algebraic & Geometric Topology 17 (2017) 1953–2011
Abstract

We explore the ${C}_{2}$–equivariant spectra ${Tmf}_{1}\left(3\right)$ and ${TMF}_{1}\left(3\right)$. In particular, we compute their ${C}_{2}$–equivariant Picard groups and the ${C}_{2}$–equivariant Anderson dual of ${Tmf}_{1}\left(3\right)$. This implies corresponding results for the fixed-point spectra ${TMF}_{0}\left(3\right)$ and ${Tmf}_{0}\left(3\right)$. Furthermore, we prove a real Landweber exact functor theorem.

Keywords
topological modular forms, real homotopy theory, Picard group, Anderson duality
Mathematical Subject Classification 2010
Primary: 55N34, 55P42


text:
# DIMACS Workshop on Distributed Optimization, Information Processing, and Learning

My colleague Waheed Bajwa, Alejandro Ribeiro, and Alekh Agarwal are organizing a Workshop on Distributed Optimization, Information Processing, and Learning from August 21 to August 23, 2017 at Rutgers DIMACS. The purpose of this workshop is to bring together researchers from the fields of machine learning, signal processing, and optimization for cross-pollination of ideas related to the problems of distributed optimization, information processing, and learning. All in all, we are expecting to have 20 to 26 invited talks from leading researchers working in these areas as well as around 20 contributed posters in the workshop.

Registration is open from now until August 14 — hope to see some of you there!

# Signal boost: DPCOMP.ORG is live

I got the following email from Gerome Miklau:

Dear colleagues:

We are writing to inform you of the launch of DPCOMP.ORG.

DPCOMP.ORG is a public website designed with the following goals in mind: (1) to increase the visibility and transparency of state-of-the-art differentially private algorithms and (2) to present a principled and comprehensive empirical evaluation of these algorithms. The intended audience is both researchers who study privacy algorithms and practitioners who might deploy these algorithms.

Currently DPComp includes algorithms for answering 1- and 2-dimensional range queries. We thoroughly study algorithm accuracy and the factors that influence it and present our findings using interactive visualizations. We follow the evaluation methodology from the paper “Principled Evaluation of Differentially Private Algorithms using DPBench”. In the future we plan to extend it to cover other analysis tasks (e.g., higher dimensional data, private regression).

Our hope is that the research community will contribute to improving DPCOMP.ORG so that practitioners are exposed to emerging research developments. For example: if you have datasets which you believe would distinguish the performance of tested algorithms, new algorithms that could be included, alternative workloads, or even a new error metric, please let us know — we would like to include them.

Please share this email with interested colleagues and students. And we welcome any feedback on the website or findings.

Sincerely,

Michael Hay (Colgate University)
Ashwin Machanavajjhala (Duke University)
Gerome Miklau (UMass Amherst)

# IHP “Nexus” Workshop on Privacy and Security: Day 3

I’m doggedly completing these notes because in a fit of ambition I actually started posts for each of the workshop days and now I feel like I need to finish it up. Day 3 was a day of differential privacy: Adam Smith, Cynthia Dwork, and Kamalika Chaudhuri.

Adam gave a tutorial on differential privacy that had a bit of a different flavor from tutorials I have seen before (and given). He started out by highlighting a taxonomy of potential attacks on released data to make a distinction between re-identification, reconstruction, membership, and correlation inferences before going into the definitions, composition theory, Bayesian interpretation, and so on. With the attacks, he focused a bit more on the reconstruction story. The algorithms view of things (as I get it) is to think of, say, an LP relaxation of a combinatorial problem: you solve the LP and round the solution to integers and prove that it’s either correct or close to correct. This has more connections to things we think about in information theory (e.g. compressed sensing) but the way of stating the problem was a bit different. He also described the Homer et al. attack on GWAS. The last part of his talk was on multiplicative weights and algorithms for learning distributions over the data domain, which I think got a bit hairy for the IT folks who hadn’t seen MW before. This made me wonder if these connections between mirror descent on the simplex, information projections, and other topics can be taught in a “first principles” way that doesn’t require you to have a lot of familiarity with one interpretation of the method before bridging to another.

Cynthia gave a talk on false discovery control and how to use differential privacy ideas in a version of the Benjamini-Hochberg BHq procedure for controlling the false discovery rate. A key primitive is the the report noisy argmax procedure, which gives the index of the argmax but not its value (which would entail a further privacy loss). Since most people are not familiar with FDR control, she spent a lot of her talk on that and so the full details of the private version were deferred to the paper. I covered FDR in my detection and estimation class partly from some of the extra attention it has received in the privacy workshops over the last few years.

Kamalika’s talk was on a model for privacy when data may be correlated between individuals. This involves using the Pufferfish model for privacy in which there is an explicit class of probability distribution on parameters and a set of explicit secrets which the algorithm wants to obfuscate: the differential privacy guarantee should hold for the output distribution of the mechanism conditioned on any valid data distribution and any pair of secrets. Since the class of data distributions is arbitrary, we can also consider joint distributions on individuals’ data — if the distribution class has some structure, then there might be a hope to efficiently produce an output of a function. She talked about using the $\ell_{\infty}$ Wasserstein distance to measure the sensitivity of a function, and that adding noise that scales with this sensitivity would guarantee privacy in the Pufferfish model. She then gave an example for Bayesian networks and Markov chains. As we discussed, it seems like for each dependence structure you need to come up with a sort of covering of the dependencies to add noise appropriately. This seems pretty challenging in general now, but maybe after a bit more work there will be a clearer “general” strategy to handle dependence along these lines.

# Signal boost: IBM Social Good Fellowship for data science

This announcement came via Kush Varshney. IBM is launching a new fellowship program. This came out of his work on DataKind and Saška Mojsilović’s work on Ebola. It’s open to students and postdocs!

I am pleased to let you know that Saška Mojsilović and I are launching a new fellowship program at IBM Research related to data science for social good. We are offering both 3-month summer fellowships for PhD students and full-year fellowships for postdocs. The fellowship webpage and link to apply may be found here.

Fellows will come work with research staff members at our Yorktown Heights laboratory to complete projects in partnership with NGOs, social enterprises, government agencies, or other mission-driven organizations that have large social impact. We are currently in the process of scoping projects across various areas, such as health, sustainability, poverty, hunger, equality, and disaster management. The program is intended to allow students to develop their technical skills and produce publishable work while making a positive impact on the world.

I request that you spread the word to students in your respective departments and the broader community.

# Postdoctoral opportunity at NYU CS/Global Health

Post-doctoral opportunity in developing novel computational approaches for disease surveillance. The laboratory of Dr. Rumi Chunara in Computer Science & Engineering, and the College of Global of Public Health at New York University is seeking highly motivated researchers to develop and study crowdsourced and point-of-care data for understanding infectious and chronic disease in populations worldwide.

Ideal postdoctoral candidates will have a Ph.D. with a strong background in bioinformatics, biostatistics, computer science or related field. Expertise in statistical machine learning and/or data mining are required. Preferred requirements for this position include experience designing software applications and/or storing, retrieving, and analyze large datasets. Experience with R, Python, SQL, JavaScript is preferred. Experience in hacking with cloud technologies (e.g., AWS, Hadoop) is a big plus. You must demonstrate an interest or experience in working with biological data such as genomic sequence, syndromic surveillance or physiological data.

This is an exciting research area and New York City provides great opportunities for networking and support of innovative work. Our group is engaged in many high-profile studies in collaboration with startups and other groups. The selected post-doc will be supported and encouraged to generate high impact publications, gain experience in supervising students and in grant writing if interested. All applicants should send an updated CV to Rumi Chunara (rumi.chunara@nyu.edu).

# ISIT 2015 : statistics and learning

The advantage of flying to Hong Kong from the US is that the jet lag was such that I was actually more or less awake in the mornings. I didn’t take such great notes during the plenaries, but they were rather enjoyable, and I hope that the video will be uploaded to the ITSOC website soon.

There were several talks on entropy estimation in various settings that I did not take great notes on, to wit:

• OPTIMAL ENTROPY ESTIMATION ON LARGE ALPHABETS VIA BEST POLYNOMIAL APPROXIMATION (Yihong Wu, Pengkun Yang, University Of Illinois, United States)
• DOES DIRICHLET PRIOR SMOOTHING SOLVE THE SHANNON ENTROPY ESTIMATION PROBLEM? (Yanjun Han, Tsinghua University, China; Jiantao Jiao, Tsachy Weissman, Stanford University, United States)
• ADAPTIVE ESTIMATION OF SHANNON ENTROPY (Yanjun Han, Tsinghua University, China; Jiantao Jiao, Tsachy Weissman, Stanford University, United States)

I would highly recommend taking a look for those who are interested in this problem. In particular, it looks like we’re getting towards more efficient entropy estimators in difficult settings (online, large alphabet), which is pretty exciting.

QUICKEST LINEAR SEARCH OVER CORRELATED SEQUENCES
Javad Heydari, Ali Tajer, Rensselaer Polytechnic Institute, United States
This talk was about hypothesis testing where the observer can control the samples being taken by traversing a graph. We have an $n$-node graph (c.f. a graphical model) representing the joint distribution on $n$ variables. The data generated is i.i.d. across time according to either $F_0$ or $F_1$. At each time you get to observe the data from only one node of the graph. You can either observe the same node as before, explore by observing a different node, or make a decision about whether the data from from $F_0$ or $F_1$. By adopting some costs for different actions you can form a dynamic programming solution for the search strategy but it’s pretty heavy computationally. It turns out the optimal rule for switching has a two-threshold structure and can be quite a bit different than independent observations when the correlations are structured appropriately.

MISMATCHED ESTIMATION IN LARGE LINEAR SYSTEMS
Yanting Ma, Dror Baron, North Carolina State University, United States; Ahmad Beirami, Duke University, United States
The mismatch studied in this paper is a mismatch in the prior distribution for a sparse observation problem $y = Ax + \sigma_z z$, where $x \sim P$ (say a Bernoulli-Gaussian prior). The question is what happens when we do estimation assuming a different prior $Q$. The main result of the paper is an analysis of the excess MSE using a decoupling principle. Since I don’t really know anything about the replica method (except the name “replica method”), I had a little bit of a hard time following the talk as a non-expert, but thankfully there were a number of pictures and examples to help me follow along.

SEARCHING FOR MULTIPLE TARGETS WITH MEASUREMENT DEPENDENT NOISE
Yonatan Kaspi, University of California, San Diego, United States; Ofer Shayevitz, Tel-Aviv University, Israel; Tara Javidi, University of California, San Diego, United States
This was another search paper, but this time we have, say, $K$ targets $W_1, W_2, \ldots, W_K$ uniformly distributed in the unit interval, and what we can do is query at each time $n$ a set $S_n \subseteq [0,1]$ and get a response $Y_n = X_n \oplus Z_n$ where $X_n = \mathbf{1}( \exists W_k \in S_n )$ and $Z_n \sim \mathrm{Bern}( \mu(S_n) + b )$ where $\mu$ is the Lebesgue measure. So basically you can query a set and you get a noisy indicator of whether you hit any targets, where the noise depends on the size of the set you query. At some point $\tau$ you stop and guess the target locations. You are $(\epsilon,\delta)$ successful if the probability that you are within $\delta$ of each target is less than $\epsilon$. The targeting rate is the limit of $\log(1/\delta) / \mathbb{E}[\tau]$ as $\epsilon,\delta \to 0$ (I’m being fast and loose here). Clearly there are some connections to group testing and communication with feedback, etc. They show there is a significant gap between the adaptive and nonadaptive rate here, so you can find more targets if you can adapt your queries on the fly. However, since rate is defined for a fixed number of targets, we could ask how the gap varies with $K$. They show it shrinks.

ON MODEL MISSPECIFICATION AND KL SEPARATION FOR GAUSSIAN GRAPHICAL MODELS
Varun Jog, University of California, Berkeley, United States; Po-Ling Loh, University of Pennsylvania, United States
The graphical model for jointly Gaussian variables has no edge between nodes $i$ and $j$ if the corresponding entry $(\Sigma^{-1})_{ij} = 0$ in the inverse covariance matrix. They show a relationship between the KL divergence of two distributions and their corresponding graphs. The divergence is lower bounded by a constant if they differ in a single edge — this indicates that estimating the edge structure is important when estimating the distribution.

CONVERSES FOR DISTRIBUTED ESTIMATION VIA STRONG DATA PROCESSING INEQUALITIES
Aolin Xu, Maxim Raginsky, University of Illinois at Urbana–Champaign, United States
Max gave a nice talk on the problem of minimizing an expected loss $\mathbb{E}[ \ell(W, \hat{W}) ]$ of a $d$-dimensional parameter $W$ which is observed noisily by separate encoders. Think of a CEO-style problem where there is a conditional distribution $P_{X|W}$ such that the observation at each node is a $d \times n$ matrix whose columns are i.i.d. and where the $j$-th row is i.i.d. according to $P_{X|W_j}$. Each sensor gets independent observations from the same model and can compress its observations to $b$ bits and sends it over independent channels to an estimator (so no MAC here). The main result is a lower bound on the expected loss as s function of the number of bits latex $b$, the mutual information between $W$ and the final estimate $\hat{W}$. The key is to use the strong data processing inequality to handle the mutual information — the constants that make up the ratio between the mutual informations is important. I’m sure Max will blog more about the result so I’ll leave a full explanation to him (see what I did there?)

More on Shannon theory etc. later!

# AISTATS 2015: a few talks from one day

I attended AISTATS for about a day and change this year — unfortunately due to teaching I missed the poster I had there but Shuang Song presented a work on learning from data sources of different quality, which her work with Kamalika Chaudhuri and myself. This was my first AISTATS. It had single track of oral presentations and then poster sessions for the remaining papers. The difficulty with a single track for me is that my interest in the topics is relatively focused, and the format of a general audience with a specialist subject matter meant that I couldn’t get as much out of the talks as I would have wanted. Regardless, I did get exposed to a number of new problems. Maybe the ideas can percolate for a while and inform something in the future.

Computational Complexity of Linear Large Margin Classification With Ramp Loss
Søren Frejstrup Maibing, Christian Igel
The main result of this paper (I think) is that ERM under ramp loss is NP-hard. They gave the details of the reduction but since I’m not a complexity theorist I got a bit lost in the weeds here.

A la Carte — Learning Fast Kernels
Zichao Yang, Andrew Wilson, Alex Smola, Le Song
Ideas like “random kitchen sinks” and other kernel approximation methods require you to have a kernel you want to approximate, but in many problems you in fact need to learn the kernel from the data. If I give you a kernel function $k(x,x') = k( |x - x'| )$, then you can take the Fourier transform $K(\omega)$ of $k$. This turns out to be a probability distribution, so you can sample random $\{\omega_i\}$ i.i.d. and build a randomized Fourier approximation of $k$. If you don’t know the kernel function, or you have to learn it, then you could instead try to learn/estimate the transform directly. This paper was about trying to do that in a reasonably efficient way.

Learning Where to Sample in Structured Prediction
Tianlin Shi, Jacob Steinhardt, Percy Liang
This was about doing Gibbs sampling, not for MCMC sampling from the stationary distribution, but for “stochastic search” or optimization problems. The intuition was that some coordinates are “easier” than others, so we might want to focus resampling on the harder coordinates. But this might lead to inaccurate sampling. The aim here twas to build a heterogenous sampler that is cheap to compute and still does the right thing.

Tradeoffs for Space, Time, Data and Risk in Unsupervised Learning
Mario Lucic, Mesrob Ohannessian, Amin Karbasi, Andreas Krause
This paper won the best student paper award. They looked at a k-means problem where they do “data summarization” to make the problem a bit more efficient — that is, by learning over an approximation/summary of the features, they can find different tradeoffs between the running time, risk, and sample size for learning problems. The idea is to use coresets — I’d recommend reading the paper to get a better sense of what is going on. It’s on my summer reading list.

Averaged Least-Mean-Squares: Bias-Variance Trade-offs and Optimal Sampling Distributions
Alexandre Defossez, Francis Bach
What if you want to do SGD but you don’t want to sample the points uniformly? You’ll get a bias-variance tradeoff. This is another one of those “you have to read the paper” presentations. A nice result if you know the background literature, but if you are not a stochastic gradient aficionado, you might be totally lost.

Sparsistency of $\ell_1$-Regularized M-Estimators
Yen-Huan Li, Jonathan Scarlett, Pradeep Ravikumar, Volkan Cevher
In this paper they find a new condition, which they call local structured smoothness, which is sufficient for certain M-estimators to be “sparsistent” — that is, they recover the support pattern of a sparse parameter asymptotically as the number of data points goes to infinity. Examples include the LASSO, regression in general linear models, and graphical model selection.

Some of the other talks which were interesting but for which my notes were insufficient:

• Two-stage sampled learning theory on distributions (Zoltan Szabo, Arthur Gretton, Barnabas Poczos, Bharath Sriperumbudur)
• Generalized Linear Models for Aggregated Data (Avradeep Bhowmik, Joydeep Ghosh, Oluwasanmi Koyejo)
• Efficient Estimation of Mutual Information for Strongly Dependent Variables (Shuyang Gao, Greg Ver Steeg, Aram Galstyan)
• Sparse Submodular Probabilistic PCA (Rajiv Khanna, Joydeep Ghosh, Russell Poldrack, Oluwasanmi Koyejo)


text:
## Differential and Integral Equations

### Fractional heat equations on a segment

Elena I. Kaikina

#### Abstract

We study the initial-boundary-value problem for the fractional heat equation on a segment $( 0,a )$ $$\left \{ \begin{array}{c} u_{t}+\lambda \vert u \vert ^{\rho }\text{ }u+C_{\alpha }\partial _{x}^{\alpha }u=0,\text{ }t>0, \\ u(x,0)=u_{0}(x), \\ u(a,t)=h_{1}(t),u_{x}(0,t)=h_{2}(t),t>0, \end{array} \right . \tag*{(0.1)}$$ where $\lambda \in R$ , $\rho \geq 0$ , $\alpha \in ( 1,\frac{3}{2} ]$ $,$ the constant $C_{\alpha }$ is chosen by a dissipative condition, such that $\ \text{Re}C_{\alpha }p^{ [ \alpha ] +1- \{ \alpha \} }>0$ for $\text{Re}p=0$ and \begin{equation*} \partial _{x}^{\alpha }u=\int_{0}^{x}\frac{\partial _{s}^{ [ \alpha ] +1}u(s,t)}{ ( x-s ) ^{ \{ \alpha \} }}ds. \end{equation*} Here $[ \alpha ]$ and $\{ \alpha \}$ are integer and fractional parts of $\alpha .$ The aim of this paper is to prove the global existence of solutions to the initial-boundary-value problem (0.1) and to find the main term of the asymptotic representation of solutions.

#### Article information

Source
Differential Integral Equations, Volume 19, Number 8 (2006), 891-918.

Dates
First available in Project Euclid: 21 December 2012


text:
scroll identifier for mobile
main-content

## Über dieses Buch

This book is based on the 18 tutorials presented during the 25th workshop on Advances in Analog Circuit Design. Expert designers present readers with information about a variety of topics at the frontier of analog circuit design, including low-power and energy-efficient analog electronics, with specific contributions focusing on the design of continuous-time sigma-delta modulators, automotive electronics, and power management. This book serves as a valuable reference to the state-of-the-art, for anyone involved in analog circuit research and development.

## Inhaltsverzeichnis

### Chapter 1. WiFi Receiver Evolution in a Dense Blocker Environment

This paper presents three different WiFi receiver chain lineups using a high dynamic range continuous time Sigma Delta A/D converter (SD-ADC). The chains tradeoffs are analyzed from the perspective of the ADC design. The key performance indexes are performance, blocker rejection, area and power consumption. Two lineups implement an active filter stage between the mixer and the ADC: one uses a trans-impedance amplifier and the other one a Gm stage. A third lineup avoids the active filtering stage and replaces it with a passive pole. The measurements show that the three lineups achieve the same chain performance and can therefore be directly and fairly compared in the same environment.

Patrick Torta, Antonio Di Giandomenico, Lukas Dörrer, Jose Luis Ceballos

### Chapter 2. High-Resolution Wideband Continuous-Time ΣΔModulators

The signal bandwidth of delta-sigma analog-to-digital converters has been greatly extended during the last few decades from a few tens of kHz bandwidth for audio to several hundred MHz to date for wireless applications. To enable a wideband and filterless RF radio front-end, high dynamic range and very high linearity of the wideband delta-sigma modulator is required as well. In this chapter the design aspects of high-resolution and wideband continuous-time delta-sigma modulators are presented, from architectural choices to implementation and circuit design examples.

Lucien Breems, Muhammed Bolatkale

### Chapter 3. Sigma-Delta ADCs with Improved Interferer Robustness

This article reviews the application and the requirements of Sigma-Delta ADCs with improved interferer robustness. While most power efficient state-of-the-art architectures for Sigma-Delta ADCs are based on forward compensated loop filters, these architectures are known to feature gain in their signal transfer function at out-of-band frequencies. In applications, where interferer and blocker signals are present outside the band of interest, gain at these frequencies consequently increases the required dynamic range or the necessary pre-processing complexity, e.g. within the preceding filter stages. More recently, increasing attention was dedicated to improved signal transfer functions of Sigma-Delta ADC’s, such that—by intrinsically attenuating the interferers within the Sigma-Delta loop filter—pre-processing stages can be relaxed or even omitted, and the dynamic range can be reduced. In this contribution, the state-of-the-art of Sigma-Delta ADCs is reviewed with respect to interferer robustness from system level on.

Rudolf Ritter, Jiazuo Chi, Maurits Ortmanns

### Chapter 4. Design Considerations for Filtering Delta Sigma Converters

Many signal chains need to digitize signals in the presence of interferers. The usual way of accomplishing this is to use an analog anti-alias filter followed by a Nyquist ADC. Continuous-time ΔΣ $$\varDelta \varSigma$$ modulators (CTΔΣ $$\varDelta \varSigma$$ M) are compelling alternatives to Nyquist rate ADCs. While oversampling relaxes the requirements of the anti-alias filter, it is natural to wonder if the built-in filtering of a CTΔΣ $$\varDelta \varSigma$$ M, characterized by its Signal Transfer Function (STF), can be used to attenuate interferers so that an explicit filter up front can be dispensed with altogether. It turns out that the Signal Transfer Function (STF) of a conventional CTΔ ΣM is a by-product of NTF synthesis, with the designer having little control over it. The STF can be independently controlled by using a filter up front—however, this increases power dissipation and degrades linearity of the signal chain. Embedding the filter in the modulator achieves the same objective in a power efficient manner, while improving out-of-band linearity and reducing active area. This work reviews filtering ΔΣ $$\varDelta \varSigma$$ conversion and circuit techniques for such converters.

Shanthi Pavan, Radha Rajan

### Chapter 5. Blocker and Clock-Jitter Performance in CT ΣΔ ADCs for Consumer Radio Receivers

The smallest, most efficient CT ΣΔ ADCs have typically been feed‐forward (FF) compensated single-bit implementations. However, this has not translated in consumer radio receivers because of poor performance in the presence of out‐of‐band near-blockers and clock jitter. As a result SAR ADCs, filtering, and multi-bit CT ΣΔ ADCs have become increasingly prevalent at the cost of receiver power and area. This paper presents the challenges and design techniques used to restore single-bit FF CT ΣΔ ADC performance and efficiency in the context of a modern consumer radio receiver.

Sebastián Loeda

### Chapter 6. Continuous-Time MASH Architectures forWideband DSMs

BW = fs/(2 OSR). As this equation indicates, wideband ΣΔ ADCs having bandwidths in the hundreds of MHz require clock frequencies in the GHz range even to obtain a relatively low OSR of ten. In such low-OSR systems, MASH architectures achieve better power efficiency than traditional single-loop ΣΔ ADCs. Nanometer CMOS process technologies enable continuous-time ΣΔ ADCs operating at GHz clock frequencies. However, the combination of continuous-time and low-OSR at a GHz clock frequency presents new challenges. In this paper, ΣΔ ADCs including the traditional single-loop and MASH, are reviewed in the context of wideband wireless applications with out-of-band blockers. A unique circuit block in continuous-time MASH, a continuous-time residue generation circuit, is discussed in detail. Two wideband MASH implementations in a 28 nm CMOS process are compared and their properties and performances are discussed based on the architectural differences.

Hajime Shibata, Yunzhi Dong, Wenhua Yang, Richard Schreier

### Chapter 7. Trends and Characteristics of Automotive Electronics

Electronics are rapidly becoming the major driver in new car developments. Since their humble beginnings as entertainment and comfort features for the car driver, they have gradually been used to optimize the car behavior and are now more and more taking over the operation of the human driver itself. This is a disruptive evolution since the fully autonomous car will ultimately function as a personal public transport with high comfort and flexibility. Many technical, safety and legal issues have to be solved but this evolution can also have a large impact on traffic and city organization. Due to the large impact of mobility on the human society, it is clear that the future automotive electronics in the autonomous car will have to be very predictable and accurate and must remain extremely reliable and safe under the harsh automotive environment conditions and disturbances.

Herman Casier

### Chapter 8. Next Generation of Semiconductors for Advanced Power Distribution in Automotive Applications

Power distribution is the overall term for all items bringing electrical power from source to loads. For carmakers power distribution is “the” hot topic in power network (power net) architectures, this is due to the fact that complexity is increasing from year to year driven by new functionality, safety and legal requirements.This leads to many new high power loads which must be served by the power net. To overcome the current power net limitations a change from relays and fuse to semiconductor is required. The target of new power net architectures is a reduction of cost, power losses and weight of the vehicle.Semiconductors instead of relays and fuses are “the enabler” for these architecture changes. Optimization on semiconductor or wiring harness level only is not sufficient anymore. To gain more system benefit wiring harness, switches and loads have to be optimized together. Main challenges for semiconductors are increased power capability, precise diagnosis, high reliability and low idle/quiescent currents.

Andreas Kucher, Alfons Graf

### Chapter 9. High-Voltage Fast-Switching Gate Drivers

This article covers the design of highly integrated gate drivers and level shifters for high-speed, high power efficiency and dv/dt robustness with focus on automotive applications. With the introduction of the 48 V board net in addition to the conventional 12 V battery, there is an increasing need for fast switching integrated gate drivers in the voltage range of 50 V and above. State-of-the-art drivers are able to switch 50 V in less than 5 ns. The high-voltage electrical drive train demands for galvanic isolated and highly integrated gate drivers. A gate driver with bidirectional signal transmission with a 1 MBit/s amplitude modulation, 10/20 MHz frequency modulation and power transfer over one single transformer will be discussed. The concept of high-voltage charge storing enables an area-efficient fully integrated bootstrapping supply with 70 % less area consumption. EMC is a major concern in automotive. Gate drivers with slope control optimize EMC while maintaining good switching efficiency. A current mode gate driver, which can change its drive current within 10 ns, results in 20 dBuV lower emissions between 7 and 60 MHz and 52 % lower switching loss compared to a conventional constant current gate driver.

Bernhard Wicht, Jürgen Wittmann, Achim Seidel, Alexis Schindler

### Chapter 10. A Self-Calibrating SAR ADC for Automotive Microcontrollers

This paper presents a low complexity digital self-calibration technique for a successive-approximation-register analog-to-digital converter (SAR ADC) for automotive applications. The ADC includes a digital-to-analog converter (DAC) having a binary, not calibrated, lsb part and a calibrated thermometric msb part. Very few extra elements are required in the analog part due to the self-calibration capability and only a limited amount of logic is added to the digital section.The calibration sequence is executed once, during the testing process, and the results are stored on the device itself. The calibration process does not need any special precise external equipment.The converter, realized in a 40 nm CMOS technology, achieves an INL of less than 0.25 lsb @ 12 bits thanks to the calibration procedure that brings a factor reduction of eight of the INL.

Carmelo Burgio, Mauro Giacomini, Enzo Michele Donze, Domenico Fabio Restivo

### Chapter 11. Advanced Sensor Solutions for Automotive Applications

Sensors are widely used in automotive applications and they are among the most used electronic components in a car. Within the advanced sensing solutions for automotive, the capacitive sensors play a significant role because they can indirectly monitor many different physical quantities with the advantage of a contactless implementation. The design challenge is to implement high performance capacitive sensing solutions, with relatively easy and cheap construction, which can tolerate the harsh automotive electrical and environmental conditions. After a general introduction about sensors in automotive, this work presents several architectures for capacitive sensor interfaces and especially the design aspects to be considered for achieving excellent electromagnetic immunity and emission performance.

Paolo D’Abramo, Alberto Maccioni, Giuseppe Pasetti, Francesco Tinfena

### Chapter 12. A Low-Power Continuous-Time Accelerometer Front-End

This paper presents a low power analog front-end circuit for a 3-axis MEMS capacitive accelerometer. The circuit includes an analog preamplifier to sense the signal coming from the sensor and a successive-approximation A/D converter. Power minimization is achieved by using a continuous-time preamplifier for realizing constant-charge capacitance-to-voltage conversion and a SAR A/D converter with split capacitive array. To cope with potentially large sensor leakage current we used a specific sensor biasing technique, that guarantees ac coupling above 0.1 Hz. The complete analog front-end circuit consumes 90 μW from a single 1.2-V power supply voltage, achieving 65.1 dB of SNR over a 4-kHz bandwidth and − 43. 4 dB of THD.

Piero Malcovati, Marcello De Matteis, Alessandro Pezzotta, Marco Grassi, Marco Croce, Marco Sabatini, Andrea Baschirotto

### Chapter 13. Switched-Capacitor Power-Converter Topology Overview and Performance Comparison

Switched-capacitor power converters are interesting candidates to realize integrated power converters with acceptable power efficiencies. Depending on the input and output voltage ranges to be accommodated at a desired efficiency, certain voltage conversion ratio(s) need(s) to be implemented. Though the theoretical minimum number of floating capacitors to realize a desired voltage conversion ratio is known, how to actually synthesize the corresponding topologies and what impact these topologies have on circuit performance is less trivial. Besides two-clock-phase topologies, multiple-clock-phase topologies have recently been introduced. This paper gives an overview of various methods to implement desired voltage conversion ratios with two or multiple clock phases and compares their performance under given boundary conditions.

Ravi Karadi, Gerard Villar Piqué, Henk Jan Bergveld

### Chapter 14. Resonant and Multimode Switched Capacitor Converters for High-Density Power Delivery

A variety of modern applications spanning portable computing, communications, and even renewable energy demand low-cost, small form-factor power delivery solutions that remain efficient and reliable. This paper provides an overview of resonant and multimode switched capacitor DC-DC converters and discusses their prospects for meeting these challenges. An overview of the constraints that limit performance scaling of conventional magnetic and switched-capacitor converters is provided. A comparison and analytical framework for the resonant multimode approach is given that helps identify where the approach is most practical and effective. Implementation and performance details of recently published work is highlighted, both at the chip and board level.

Jason T. Stauth, Christopher Schaef, Kapil Kesarwani

### Chapter 15. Heterogeneous Integration of High-Switching Frequency Inductive DC/DC Converters

Ultimate integration of a power converter either means embedding all components in silicon or stacking them closely in a 3D arrangement. High switching frequency is a straightforward approach for reducing the values of passive devices. Silicon integration is the natural approach for switched-capacitor converters but still fails for inductive converters because of the magnetic devices, especially when a magnetic material is involved, what is the case for high power density. These latter components may be fabricated in an integrated approach but the fabrication process is not compatible in essence and cost with the silicon process. Heterogeneous integration is then a cost-effective solution. The article details an example of such an heterogeneous fabrication of a step-down converter. A suitable architecture is first selected for high switching frequency and high efficiency operation. Using an advanced silicon node complicates the architecture if the input voltage is much higher than the technology nominal voltage. Experimental measurements are presented that corroborate the simulation. 91 % peak efficiency is demonstrated for a 100 MHz switching frequency buck converter at 3.3 V input voltage, 1.2 V to 2.4 V output voltage and 0–300 mA load current.

Bruno Allard, Florian Neveu, Christian Martin

### Chapter 16. Electrical Compensation of Mechanical Stress Drift in Precision Analog Circuits

Mechanical stress has a notable effect on the parameters of most micro-electronic devices. This leads to inaccuracies and drifts, which are relevant for precision analog circuits. In contrast to built-in stress during wafer manufacturing processes, mechanical stress caused by packaging process shows a marked short and long term drift mainly due to moisture absorption. For economic plastic encapsulated packages with high reliability for automotive applications it seems impossible to avoid the drift of mechanical stress. A cost efficient way to tackle this problem is to measure the mechanical stress on-chip and to compensate for its effects by way of an electronic circuit. The paper explains the main concepts and challenges of these mechanical stress compensation circuits. It shows which classes of circuits are suited for this technique, which are the costs and benefits.

Mario Motz, Udo Ausserlechner

### Chapter 17. Power Electronics for LED Based General Illumination

This work presents a 7.5-W output power 96 % efficiency capacitor-free single-inductor 4-channel all-digital integrated DC-DC LED driver in a 0.18 µm technology, with up to 24 V input voltage and 1-A LED current for general lighting. It is based on the use of laterally double-diffused DMOS transistors, which offer LED string isolation without any other external component compared to LDMOS. The single-inductor approach is the smallest possible form factor and lowest BOM for multi-channel LED drivers. More than 90 % efficiency in a wide range of 20 mA in DCM to 500 mA in CCM is presented.

Stefan Dietrich, Stefan Heinen

### Chapter 18. An Ultra-Low-Power Electrostatic Energy Harvester Interface

This paper presents an ultra-low-power (1 μW-to-1 mW) and high voltage (5 V-to-60 V input voltage) inductive DC-DC converter to efficiently interface electrostatic energy harvesters. The chip is fully autonomous and includes start-up, shunt regulation, and maximum power point tracking. All these functions are realized with a static consumption of less than 500 nW, while being able to efficiently interface harvesters with an equivalent source resistance between 0.5 MΩ and 25 MΩ.

Stefano Stanzione, Chris van Liempd, Chris van Hoof
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text:
# Talk:Graphical projection

## Oblique Projection

Note that a page needs creating on Oblique Projection and added as a link after "oblique Projection" on this page. Pat 23:17, 20 Feb 2004 (UTC)

Isn't Axonometric projection a sub-category under Orthographic projection? If so, it should be properly nested.67.170.168.214 06:43, 17 October 2006 (UTC)

## Planometric Projection

${\displaystyle {\begin{bmatrix}x'\\y'\\z'\\1\end{bmatrix}}={\begin{bmatrix}\cos {\frac {\pi }{4}}&-\cos {\frac {\pi }{4}}&0&0\\\sin {\frac {\pi }{4}}&-\sin {\frac {\pi }{4}}&1&0\\0&0&0&0\\0&0&0&1\end{bmatrix}}{\begin{bmatrix}x\\y\\z\\1\end{bmatrix}}}$

Me121121 04:35, 4 September 2007 (UTC)

Got a reference for that? It looks like a kind of Oblique projection to me, in fact it's this one I think: --Allefant 09:19, 4 September 2007 (UTC)
Yes. That's it. But it's not the same as oblique. But I don't have any references, and I don't know how widely used it is. Yet I still learn it at school so I thought it should be included. Can't you use wherever you got that image as a reference? Me121121 09:42, 6 September 2007 (UTC)
So the fact that each 3d dimention is distorted equally (45 degree) means that accurate scale drawings can be made, because a 3d line of length x will always be projected as a 2d line of length cx, irrespective of the line's orientation. This feature makes Planometric images look ugly as sin, but also makes the projection notable. —Preceding unsigned comment added by Stestagg (talkcontribs) 14:26, 14 May 2008 (UTC)

The very first sentence of this article links to protocol. That is (of course!) a disambiguation page. I'd disambiguate if I were sure of which article to link to. Michael Hardy (talk) 20:16, 20 November 2008 (UTC)

## "General" Projection?

I don't see any reference to "general" projection. All I mean by the term is the act of projecting a point in n dimensions onto a space of m dimensions, where m < n. Standard 3D graphics pipelines (as described in this article, also in games, ray-tracing, etc) are the case where n = 3 and m = 2. Most likely, I'm not finding it because I just don't know the proper term to search for. -LesPaul75talk 00:33, 16 May 2009 (UTC)

Found it: Projection (linear algebra). Is it worth mentioning in the introduction (to "graphical projection") that that graphical projection is subset of this more general field? -LesPaul75talk 00:45, 16 May 2009 (UTC)
This seems like a personal point of view, that graphical projection is subset of this more general field? As far as I know graphical projection is a collection of methods, which principles have been developed in the projective geometry. -- Marcel Douwe Dekker (talk) 13:34, 16 May 2009 (UTC)
I don't think it's a matter of opinion, but I could be wrong. This page talks about different ways to take a 3-dimensional object and "project" it to form a 2-dimensional image. Projection (linear algebra) is about how to do exactly the same thing, but in N dimensions. Quote from Projection (linear algebra): "For example, the function which maps the point (x, y, z) in three-dimensional space R3 to the point (x, y, 0) is a projection onto the x-y plane." That's the mathematical description of all the projections discussed here. -LesPaul75talk 18:52, 18 May 2009 (UTC)

I just undid this edit which removed most of tyhe content of this article. I think this article shouldn't be reduced but expended and explain all items allready mentioned in the template views. -- Marcel Douwe Dekker (talk) 21:39, 16 August 2009 (UTC)

Do you mean 'expanded' instead of 'expended'? If you meant expanded, I agree. However, expanding the article doesn't mean that it shouldn't be split. The template provides the necessary guide to the material. It is confusing to the reader to include material that is stated elsewhere. The text was essentially moved, not deleted.
Trihoiseachaithne (talk) 02:01, 17 August 2009 (UTC)
Yes, I mean expanded. Sorry. There are a lot of separate articles here, and I don't think dividing the current text in even more articles helps here. What is needed is one overview article, which explains all different aspects. I have written one for the Dutch Wikipedia, which expressed this ideal. And I rearranged the whole category structure in Wikicommons making the Graphical projection category the top category in the field. I didn't get to it to rearrange this article yet, but I would like it to become an overvied article. -- Marcel Douwe Dekker (talk) 02:24, 17 August 2009 (UTC)

## Stereographic projection?

There is a discussion at Talk:Stereographic projection about whether stereographic projection is truly a "graphical projection", according to the definition in this article. Your advice would be appreciated. Mgnbar (talk) 11:25, 11 September 2015 (UTC)

## Merger

I suggest merging 3D projection to here, as they are basically the same topic. The only difference is that this article is for techniques "without the aid of numerical calculation". Which I don't think is an important criteria, considering most of the small summaries and articles this page links to ignore it. Thoughts? 03:05, 13 May 2017 (UTC)


text:
×

# Reaction between Sulphur and Oxygen

I had been asked by one of my chemistry teachers to find out about an equation. The equation is:

$$3S + 4O_{2} \to SO_{2} + 2SO_{3}$$

$$4S + 5O_{2} \to 2SO_{2} + 2SO_{3}$$

How is it possible than an equation can be balanced in two ways even when it is in the simplest form ?

Also this equation cannot be balanced by P.O.A.C. method.

Please anyone help me out and enlighten me on this topic.

Note by Rishabh Tripathi
1 year, 11 months ago

Sort by:

Sulphur is going into different oxidation states in sulphur dioxide and sulphur trioxide. So in the first equation one of the sulphur atoms is getting oxidised into sulphur dioxide and the other two into sulphur trioxide. In the second one two sulphur atoms each are going into sulphur dioxide and troxide. Any such reaction could be made with some $$x$$ sulphur atoms getting oxidised into dioxide and $$y$$ into trioxide. Now the question is how is this possible that different number of sulphur atoms are getting oxidised in the reactions. The answer is simple. It is because the above reaction is actually a combination of two reactions - one being $$S + O_2 \rightarrow SO_2$$ and the other being $$2 S + 3 O_2 \rightarrow 2 SO_3$$. So if the first reaction is added two the second one you would obtain the first reaction given by your teacher. Figure out what happens when you add $$2$$ times the first reaction to the second reaction.

Realize the idea?

Hope this helps. · 1 year, 11 months ago

yeah.. after doing it gives another balanced equation that too in simplest form. So, this reaction as a whole cannot be possible without breaking it into two different reactions ? · 1 year, 11 months ago

The reaction is actually a linear combination of two reactions. The question isn't whether the reaction is possible by breaking it into two different reactions. · 1 year, 11 months ago

I see. Well Thank you.. :) · 1 year, 11 months ago

Nice explanation. · 1 year, 11 months ago

Thanks. · 1 year, 11 months ago

Mathematically, this can be explained as follows. Suppose that you can balance the equation as follows.

$$aS+bO_2\rightarrow cSO_2+dSO_3$$

Now, comparing the number of sulfur and oxygen atoms on both sides,

$$a=c+d$$ and $$2b=2c+3d$$

Now suppose you want $$a,b,c,d$$ to be integers. It is clear from the second equation that $$d$$ must be even. Hence $$d=2D$$. So $$(a,b,c,d)=(c+2D,c+3D,c,2D)$$ for any pair of positive integers $$c$$ and $$D$$. In your first reaction you have $$(c,D)=(1,1)$$ and in the second reaction, you have $$(c,D)=(2,1)$$. You can find other solutions for other values of $$c$$ and $$d$$.

I don't know if there is a chemical reason behind this. · 1 year, 11 months ago

Your explanation doesn't make it clear why this actually happens. · 1 year, 11 months ago

Well, I find Sudeep Salgia's answer to be more insightful, especially because he divides the equation into two separate equations. My answer basically shows you how to convert this into a number theoretic problem. · 1 year, 11 months ago

IMO, You explained it better. · 1 year, 11 months ago

IMO ?? What it means here ? · 1 year, 11 months ago


text:
# Should we [join] this tag with the other burninated ones?

I've just again realized that we have a tag, which corresponds to SQL. It doesn't appear that the questions have anything else in common, either. Do we still need it?

No, your statement that is related to is a faulty assumption.

There are also questions about performing joins with , , , or just joining data the hard way in regular programming languages.

While it is true that most interesting queries involve some kind of JOIN, that is not universally true. (Many INSERT and UPDATE queries do not involve JOIN, for example.) It is debatable whether those uses of the tag are worth eliminating. I'd say that they are harmless to leave there.

# Burn it.

Pretty much all SQL queries use joins in one form or another. I think that having is on its own sufficient and it should be assumed that any SQL question could have joins, making the use of together redundant.


text:
mersenneforum.org Has anyone investigated this before??
 Register FAQ Search Today's Posts Mark Forums Read

 2012-10-14, 17:09 #133 mart_r     Dec 2008 you know...around... 23×3×52 Posts I was running out of ideas as of late, but here's a small suggestion on how to treat the uniqueness problem, i.e. to prove that, if such a constant exists, then there are infinitely many of them. Essentially, the number of "immortal" primes after each stage with the corresponding prime p and the current number of primes n should be asymptotic to $\frac{2n}{\sqrt p}$ as described in my document on page 5. Under the assumption that, for any $\eps>0$, $\lim_{i\to\infty}\frac{\sum_{x=2}^i x^{\frac 1 2 -\eps}}{\sum_{x=2}^i x-\theta(x)}<1$and $\lim_{i\to\infty}\frac{\sum_{x=2}^i x^{\frac 1 2 +\eps}}{\sum_{x=2}^i x-\theta(x)}>1$ (*), if $\lim_{p\to\infty}\frac{2n}{\sqrt p}=\infty$, then y is not unique. (*) One of my pet ideas, sorry for bringing it up all the time despite it may still be wrongly phrased. Under average conditions (nextprime(p+1)=p+log(p) and such), the absolute growth of the numerator of the last mentioned fraction after each step is $\frac{2n}{\sqrt p}$, and for the denominator this is $\frac{\frac 1 2 \log p}{\sqrt p}$. So even if the number of primes n drops below $\frac 1 2 \sqrt p$ at several points (for n=$\frac 1 2 \sqrt p$ the sequence is in limbo, so to speak), in the long run $\frac{2n}{\sqrt p}$ will by far outnumber $\frac{\frac 1 2 \log p}{\sqrt p}$, and n
2014-08-31, 17:28   #134
mart_r

Dec 2008
you know...around...

23·3·52 Posts
Gravedigging

Some consolidation work and major improvements on minor details.

This version should be closer to something publicly acceptable.

I guess I should ask about the most important flaws that this "paper" indubitably still contains, and maybe some hints how to try to fix them.

Attached Files
 Project p#Y.zip (244.7 KB, 54 views)

2014-08-31, 19:26   #135
NBtarheel_33

"Nathan"
Jul 2008
Maryland, USA

3·7·53 Posts

Quote:
 For y >= 1.5, floor (2#*y) must be an odd prime 2m-1...
Not necessarily! What if $y = 2$, or indeed what if $y$ is any other integer greater than 2?

2014-08-31, 20:21   #136
mart_r

Dec 2008
you know...around...

23×3×52 Posts

Quote:
 Originally Posted by NBtarheel_33 Not necessarily! What if $y = 2$, or indeed what if $y$ is any other integer greater than 2?
Floor(p#*y) must be prime by definition. And apart from 2, all other primes are odd. If y>=2 then 2#*y is already >= 4, thus an odd prime.

2014-08-31, 22:12   #137
NBtarheel_33

"Nathan"
Jul 2008
Maryland, USA

3×7×53 Posts

Quote:
 Originally Posted by mart_r Floor(p#*y) must be prime by definition. And apart from 2, all other primes are odd. If y>=2 then 2#*y is already >= 4, thus an odd prime.
Floor(p#*y) is the largest integer less than or equal to the product of p# and y, correct? Or is the * not multiplication?

2014-09-02, 15:33   #138
mart_r

Dec 2008
you know...around...

23×3×52 Posts

Quote:
 Originally Posted by NBtarheel_33 Floor(p#*y) is the largest integer less than or equal to the product of p# and y, correct?
That is most definitely correct.

2014-09-02, 23:01   #139
NBtarheel_33

"Nathan"
Jul 2008
Maryland, USA

100010110012 Posts

Quote:
 Originally Posted by mart_r That is most definitely correct.
Then how are you claiming that quantity is prime?

2014-09-02, 23:08   #140
NBtarheel_33

"Nathan"
Jul 2008
Maryland, USA

3·7·53 Posts

Quote:
 Originally Posted by mart_r Floor(p#*y) must be prime by definition.
Definition? Or do you mean *by assumption*? In other words, are you restricting your choices of p and y such that floor(p#y) must be prime, and ignoring all other candidates for p and y?

Quote:
 And apart from 2, all other primes are odd. If y>=2 then 2#*y is already >= 4, thus an odd prime.
This claim is false. Counterexample: Let y = 6. Then floor(2#y) = floor(2*6) = floor(12) = 12, neither odd nor prime.

2014-09-03, 03:36   #141
axn

Jun 2003

127616 Posts

Quote:
 Originally Posted by NBtarheel_33 Definition? Or do you mean *by assumption*? In other words, are you restricting your choices of p and y such that floor(p#y) must be prime, and ignoring all other candidates for p and y?
He is trying to prove that there are no y >= 1.5 that can satisfy the conjecture using Reductio. Assume there exists y >= 1.5 that satisfies the conjecture. By definition (from the conjecture), floor(p#*y) must be prime for all primes. That leads to the conclusion that there exists an m such that m*p#-1 is prime for all primes, which is then considered as absurd, and hence y (if it exists) must be < 1.5. QED.

2014-09-04, 20:15   #142
mart_r

Dec 2008
you know...around...

23×3×52 Posts

Quote:
 Originally Posted by NBtarheel_33 Definition? Or do you mean *by assumption*? In other words, are you restricting your choices of p and y such that floor(p#y) must be prime, and ignoring all other candidates for p and y?
Restricting the choices for y, that is.

Quote:
 Originally Posted by NBtarheel_33 This claim is false. Counterexample: Let y = 6. Then floor(2#y) = floor(2*6) = floor(12) = 12, neither odd nor prime.
That simply means y=6 is not a solution.

While I'm here, may I add that
$\lim_{p\to\infty} \int_0^p \frac{dx}{\log x}-\sum_{x=2}^p \frac{1}{\log x}=0.24323834289098075541505913546...$

Or, equivalently,
$\lim_{p\to\infty} \int_a^p \frac{dx}{\log x}=\sum_{x=2}^p \frac{1}{\log x}$
for a=1.54983177364507780342792218629...

There is yet another constant arising in the computation which I can make neither head nor tail of: The value above, 0.243..., can be approximated quite well for integer p with
$\int_0^p \frac{dx}{\log x}-\sum_{x=2}^p \frac{1}{\log x}+\frac{1}{2\log x}-\frac{0.1734474150838006...}{x(\log_2 x)^2}$

Do these constants belong to someone here?

 2014-12-10, 02:54 #143 Trilo     "W. Byerly" Aug 2013 1423*2^2179023-1 1378 Posts Have you made any progress on calculating a y value that floor(p#*y) is prime? I am interested in your work. Last fiddled with by Trilo on 2014-12-10 at 02:55

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Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

Chapter Contents
Chapter Introduction
NAG Toolbox

## Purpose

nag_ode_bvp_ps_lin_quad_weights (d02uy) obtains the weights for Clenshaw–Curtis quadrature at Chebyshev points. This allows for fast approximations of integrals for functions specified on Chebyshev Gauss–Lobatto points on $\left[-1,1\right]$.

## Syntax

[w, ifail] = d02uy(n)

## Description

Given the (Clenshaw–Curtis) weights ${w}_{\mathit{i}}$, for $\mathit{i}=0,1,\dots ,n$, and function values ${f}_{\mathit{i}}=f\left({t}_{\mathit{i}}\right)$ (where ${t}_{\mathit{i}}=-\mathrm{cos}\left(\mathit{i}×\pi /n\right)$, for $\mathit{i}=0,1,\dots ,n$, are the Chebyshev Gauss–Lobatto points), then $\underset{-1}{\overset{1}{\int }}f\left(x\right)dx\approx \sum _{\mathit{i}=0}^{n}{w}_{i}{f}_{i}$.
For a function discretized on a Chebyshev Gauss–Lobatto grid on $\left[a,b\right]$ the resultant summation must be multiplied by the factor $\left(b-a\right)/2$.

## References

Trefethen L N (2000) Spectral Methods in MATLAB SIAM

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
$n$, where the number of grid points is $n+1$.
Constraint: ${\mathbf{n}}>0$ and n is even.

None.

### Output Parameters

1:     $\mathrm{w}\left({\mathbf{n}}+1\right)$ – double array
The Clenshaw–Curtis quadrature weights, ${w}_{\mathit{i}}$, for $\mathit{i}=0,1,\dots ,n$.
2:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Errors or warnings detected by the function:
${\mathbf{ifail}}=1$
Constraint: ${\mathbf{n}}>0$.
Constraint: n is even.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

The accuracy should be close to machine precision.

A real array of length $2n$ is internally allocated.

## Example

This example approximates the integral $\underset{-1}{\overset{3}{\int }}3{x}^{2}dx$ using $65$ Clenshaw–Curtis weights and a $\mathrm{65}$-point Chebyshev Gauss–Lobatto grid on $\left[-1,3\right]$.
```function d02uy_example

fprintf('d02uy example results\n\n');

n = int64(64);
a = -1;
b =  3;

% Set up solution grid
[x, ifail] = d02uc(n, a, b);

% Get integrand values on grid
f = 3*x.^2;
scale = 0.5*(b-a);

[w, ifail] = d02uy(n);

% Evaluate apprimation to definite integral
integ = dot(w, f)*scale;

% Print solution
fprintf('Integral of f(x) from %4.1f to %4.1f = %7.4f\n', a, b, integ);
fprintf('\nError in approximation = %12.2e\n', integ-28);

```
```d02uy example results

Integral of f(x) from -1.0 to  3.0 = 28.0000

Error in approximation =     3.55e-15
```


text:
# 6.3: Maxwell's Derivation of the Gas-velocity Probability-density Function

To this point, we have been developing our ability to characterize the gas-velocity distribution functions. We now want to use Maxwell’s argument to find them. We have already introduced the first step, which is the recognition that three-dimensional probability-density functions can be expressed as products of independent one-dimensional functions, and that $${\rho }_{\theta }\left(\theta \right)$$, and $${\rho }_{\varphi }\left(\varphi \right)$$ are the constants $${1}/{2}$$ and $${1}/{2\pi }$$. Now, because the probability density associated with any given velocity is just a number that is independent of the coordinate system, we can equate the three-dimensional probability-density functions for Cartesian and spherical coordinates: $$\rho \left(v_x,v_y,v_z\right)=\rho \left(v,\ \theta ,\varphi \right)$$ so that

${\rho }_x\left(v_x\right){\rho }_y\left(v_y\right){\rho }_z\left(v_z\right)=\frac{{\rho }_v\left(v\right)}{4\pi }$

We take the partial derivative of this last equation with respect to $$v_x$$. The probability densities $${\rho }_y\left(v_y\right)$$ and $${\rho }_z\left(v_z\right)$$ are independent of $$v_x$$. However, $$v$$ is a function of $$v_x$$, because $$v^2=v^2_x+v^2_y+v^2_z$$. We find

$\frac{{d\rho }_x\left(v_x\right)}{dv_x}{\rho }_y\left(v_y\right){\rho }_z\left(v_z\right)=\frac{1}{4\pi }{\left(\frac{{\partial \rho }_v\left(v\right)}{\partial v_x}\right)}_{v_yv_v}=\frac{1}{4\pi }\left(\frac{d{\rho }_v\left(v\right)}{dv}\right){\left(\frac{\partial v}{\partial v_x}\right)}_{v_yv_z}$ Since $$v^2=v^2_x+v^2_y+v^2_z$$, $$2v{\left({\partial v}/{\partial v_x}\right)}_{v_yv_z}=2v_x$$ and ${\left(\frac{\partial v}{\partial v_x}\right)}_{v_yv_z}=\frac{v_x}{v}$

Making this substitution and dividing by the original equation gives

$\frac{{d\rho }_x\left(v_x\right)}{dv_x}\frac{{\rho }_y\left(v_y\right){\rho }_z\left(v_z\right)}{{\rho }_x\left(v_x\right){\rho }_y\left(v_y\right){\rho }_z\left(v_z\right)}=\frac{v_x}{v}\frac{1}{{\rho }_v\left(v\right)}\frac{d{\rho }_v\left(v\right)}{dv}$

Cancellation and rearrangement of the result leads to an equation in which the independent variables $$v_x$$ and $$v$$ are separated. This means that each term must be equal to a constant, which we take to be $$-\lambda$$. We find

$\left(\frac{1}{v_x \rho_x\left(v_x\right)}\right) \frac{d \rho_x \left(v_x\right)}{dv_x}=\left(\frac{1}{v\rho_v \left(v\right)}\right)\frac{d \rho_v\left(v\right)}{dv}=-\lambda$

so that

$\frac{d \rho_x\left(v_x\right)}{ \rho_x\left(v_x\right)}=-\lambda v_x dv_x$

and

$\frac{d\rho_v\left(v\right)}{\rho_v\left(v\right)}=-\lambda vdv$

From the first of these equations, we obtain the probability density function for the distributions of one-dimensional velocities. (See Section 4.4.) The three-dimensional probability density function can be deduced from the one-dimensional function. (See Section 4.5.)

From the second equation, we obtain the three-dimensional probability-density function directly. Integrating from $$v=0$$, where $${\rho }_v\left(0\right)$$ has a fixed value, to an arbitrary scalar velocity, $$v$$, where the scalar-velocity function is $${\rho }_v\left(v\right)$$, we have

$\int^{\rho_v\left(v\right)}_{\rho_v\left(0\right)} \frac{d \rho_v\left(v\right)}{ \rho_v\left(v\right)}=-\lambda \int^v_0 vdv$

or

${\rho }_v\left(v\right)={\rho }_v\left(0\right)exp\left(\frac{-\lambda v^2}{2}\right)$

The probability-density function for the scalar velocity becomes

$\frac{df_v\left(v\right)}{dv}=v^2{\rho }_v\left(v\right)={\rho }_v\left(0\right)v^2exp\left(\frac{-\lambda v^2}{2}\right)$

This is the result we want, except that it contains the unknown parameters $${\rho }_v\left(0\right)$$ and $$\lambda$$. The value of $${\ \rho }_v\left(0\right)$$ must be such as to make the integral over all velocities equal to unity. We require

\begin{aligned} 1 & =\int^{\infty }_0 \left(\frac{df_v\left(v\right)}{dv}\right) dv \\ ~ & = \rho_v\left(0\right)\int^{\infty }_0 v^2\mathrm{exp}\left(\frac{-\lambda v^2}{2}\right)dv \\ ~ & =\frac{\rho_v\left(0\right)}{4\pi } \left(\frac{2\pi }{\lambda }\right)^{3/2} \end{aligned}

so that

$\rho_v\left(0\right)=4\pi \left(\frac{\lambda }{2\pi }\right)^{3/2}$

where we use the definite integral $$\int^{\infty }_0 x^2 \mathrm{exp}\left(-ax^2\right)dx=\left(1/4\right)\sqrt{\pi /a^3}$$. (See Appendix D.) The scalar-velocity function in the three-dimensional probability-density function becomes

$\rho_v\left(v\right)=4\pi \left(\frac{\lambda }{2\pi }\right)^{3/2}\mathrm{exp}\left(\frac{-\lambda v^2}{2}\right)$

The probability-density function for the scalar velocity becomes

\begin{aligned} \frac{df_v\left(v\right)}{dv} & =v^2 \rho_v \left(v\right) \\ ~ & =4\pi \left(\frac{\lambda }{2\pi }\right)^{3/2}v^2\mathrm{exp}\left(\frac{-\lambda v^2}{2}\right) \end{aligned}

The three-dimensional probability density in spherical coordinates becomes

\begin{aligned} \rho \left(v,\ \theta ,\varphi \right) & = \rho_v\left(v\right)\rho_{\theta}\left(\theta \right) \rho_{\varphi }\left(\varphi \right) \\ ~ & =\left(\frac{\lambda }{2\pi }\right)^{3/2}\mathrm{exp}\left(\frac{-\lambda v^2}{2}\right) \end{aligned}

The probability that an arbitrarily selected molecule has a velocity vector whose magnitude lies between $$v$$ and $$v+dv$$, while its $$\theta$$-component lies between $$\theta$$ and$$\ \theta +d\theta$$, and its $$\varphi$$-component lies between $$\varphi$$ and $$\varphi +d\varphi$$ becomes

\begin{aligned} dP\left(\textrm{ʋ}\prime \right) & =\left(\frac{df_v\left(v\right)}{dv}\right)\left(\frac{df_{\theta }\left(\theta \right)}{d\theta }\right)\left(\frac{df_{\varphi }\left(\varphi \right)}{d\varphi }\right)dvd\theta d\varphi \\ ~ & =\rho \left(v,\theta ,\varphi \right)v^2 \mathrm{sin} \theta dvd\theta d\varphi \\ ~ & =\left(\frac{1}{4\pi }\right) \rho_v \left(v\right)v^2 \mathrm{sin} \theta dvd\theta d\varphi \\ ~ & = \left(\frac{\lambda }{2\pi }\right)^{3/2}v^2exp\left(\frac{-\lambda v^2}{2}\right) \mathrm{sin} \theta dvd\theta d\varphi \end{aligned}

In Section 4.6, we again derive Boyle’s law and use the ideal gas equation to show that $$\lambda ={m}/{kT}$$.