text:
# Questions tagged [design]

279 questions
Filter by
Sorted by
Tagged with
59 views

### How did Gagarin's flight affect improvements to the Vostok capsule?

Every spaceflight is an opportunity to learn what does not work well and to improve the vehicle for the next flight. Did Yuri Gagarin's first human spaceflight yield any changes to the capsules or ...
43 views

### Pressurant gas line design considerations

I am trying to understand the constraints on pressurant gas line sizes and lengths. For a static volume, the lines can be small in diameter because there is no significant flow velocity and therefore ...
34 views

### Why does the RD-0110 Injector have a larger cone spray angle in the outer element rather than the inner element?

Wouldn't a large cone spray angle in the inner element (oxidizer) impinge with the outer element spray cone (fuel) and facilitate better atomization and mixing? Here are the RD-0110 Injector specs:
59 views

### Could an orbital launcher look like a scaled up Nike Hercules?

We now have flyable, recoverable suborbital boosters, which would be the first stage. The second stage of the Saturn V rocket was able to hurl the 100 ton Skylab into an orbit over 250 miles high. ...
89 views

### Why are strap-on boosters more often attached radially in Eastern-European/Asian rockets, and bilaterally in Western-European/American rockets?

While comparing launch vehicles with strap on boosters, it appears to me that the majority of Eastern-European/Asian vehicles arrange the boosters radially, but the Western-European/American vehicles ...
56 views

### What do large, modern communications satellites look like inside?

Modern communications satellite in GEO are big, majestic beasts. They are quite large and voluminous. They can sport quite an array of antennas and handle huge bandwidths and multiple communications ...
38 views

### How efficient was Telstar's solar cell configuration, compared to a flat panel?

Telstar was a series of communication satellites first launched in 1962. The body of the satellite is roughly spherical, with much of the outer surface covered in solar cells. This gives the ...
179 views

### Which explode in spaceflight more often, nuts or bolts?

The history of spaceflight is littered with exists because of exploded nuts and bolts (and upon further reflection, a whole lot of other items!) "Exploding bolts" is really a generic term. ...
56 views

### What technologies enable or at least help satellite operation in Very Low Earth Orbit (VLEO)?

This answer to How low is VLEO? (FCC's newest approval for SpaceX) suggests VLEO begins (or ends I guess) at 350 km. The two bad things I know about in VLEO are an enhanced rate of altitude loss due ...
85 views

### How will the future Mars Space Suit Helmet design protect Astronauts from Martian Radiation?

Mars Space Suit Helmet Visor's were mostly made up of polycarbonate [1], the same material used for EMU's visor for performing EVA in earth orbit. Meanwhile, Mars' radiation is 2.5x higher. How are we ...
108 views

### How to select appropriate orbital rocket stage thrust?

I’ve been working on a preliminary design of a small-lift launch vehicle capable of sending a small satellite into LEO. Given the required $\Delta v$ (~9 km/s for LEO), the mass distribution, ...
88 views

### What is the maximum force a deep space spacecraft experiences after launch?

The premise of How much of a deep space spacecraft's structural mass is useless dead weight after launch? Any plans to shed it in the future? is that the forces a deep space spacecraft experiences ...
91 views

### How much of a deep space spacecraft's structural mass is useless dead weight after launch? Any plans to shed it in the future?

After seeing this in Security SE Magnetic core memory isn't quite extinct yet. The two Voyager spacecraft used it and are still functional. Core was/is certainly non-volatile. I did some ...
41 views

Wikipedia's /Lucy (spacecraft) says nothing of propulsion, and it's SWRI home page for the spacecraft http://lucy.swri.edu/mission/Spacecraft.html gives specification and enumerates what's on Lucy’s ...
52 views

### How do all the struts and crosses in the Orion Crew Capsule Service Module work together? What do each of these parts do?

The BBC's Europe to make more hardware for American Moon missions says that three more service modules for the Orion Crew Capsule have been ordered by NASA, bringing the total desired to six. I ...
272 views

### How big could a rocket get?

What are the practical limitations on the size of Earth based, orbital rockets? How tall/heavy might we make a rocket? For the purposes of this question, I'm mostly interested in rockets using ...
205 views

### How to make a model rocket that doesn't spin?

I'm attempting to build a rocket with 8 fins (four on bottom and four smaller ones towards the top). The four fins near to the top will be attached to servo motors and programmed to stabilize or guide ...
112 views

### How redundant are Orion's power and data units (PDUs)? Doubly? Quadruply? If so, why?

In the news today: NASA to skip repair of Orion electronics unit NASA will fly Orion deep-space crew capsule ‘as is,’ despite failed component; The agency claims there is enough redundancy in the ...
155 views

### Docking cubesat in orbit - is it possible?

Would it be plausible to launch two or more cubsat's into orbit, and then have them dock together, allowing them to pack on more hardware and even fuel, broadening their capability? If it is possible, ...
136 views

### What should be the shape of a rocket going to Mars, since Mars atmosphere is around 2 percent of that of Earth's?

What should be the design or shape of a rocket going to mars?
335 views

### What is the Starship forward dome?

What is the Starship forward dome? How does it look like? What other parts are connected to it and what is it good for?
333 views

### What was this strange design for the Man In Space Soonest program?

The following picture shows various spacecraft design proposals for the 1958 U.S. Air Force Man In Space Soonest program: The bottom row depicts various spaceplanes, which should not be a surprise ...
5k views

### Is it common and good engineering for a pair of cables to be easily plugged into each other's connectors in modern spacecraft

Space News's Human error blamed for Vega launch failure Analysis of the telemetry from the mission, along with data from the production of the vehicle, led them to conclude that cables to two thrust ...
197 views

### How to get started with CubeSats

I am new both to CubeSats and to Stack Exchange. I have recently started studying and planning on how to build a CubeSat, as cheaply as possible. I have read some blog posts and watched some videos, ...
212 views

### MT-135 Sounding Rocket info / design

I'm searching for any design file or deep-info on the MT-135 rocket, the fuel used, engines, engines info, any thing really. I have deep-searched on tons of sites but I just find few missions that ...
76 views

### Finding the mass flow rate of a blowdown system?

This is a rather amateur question I will admit but how does one find the mass flow rate of an engine with a given tank pressure? Is it really as simple as calculating the cross-section of all plumbing,...
181 views

### What's the status and timeline for Millimetron? (Russia's 10m Deployable Antenna cooled to 6 K Earth-Space VLBI)

Comments below an answer to How does NASA's ASTHROS stratospheric telescope compare to its James Webb space telescope? link to JPL's Paul F. Goldsmith's The Renaissance of High Resolution ...
187 views

### Do space suit designers exist?

I was just having a closer look at the new space-suits by SpaceX, that were worn by NASA astronauts Bob Behnken and Doug Hurley (image below this question). The design made me wonder about who may ...
284 views

### Why are the SpaceX Starship, “rings” the width that they are?

Why are the SpaceX Starship, "rings" the width that they are? Wouldn't creating the rings from SS that's twice as wide reduce the welding by half?
199 views

### What are the most common practical design specifications for antenna in space?

I'm writing my thesis on antennas for small satellite applications (cubesats). Apart from the electrical specifications related to the particular application there are also several other "...
158 views

### Have there been any square 4U (2x2) cubesats?

How much did each MarCO 6U cubesat weigh? suggests that the density of a MarCO 6U cubesat might be as low as 0.75 g/cm^3 whereas I think they are usually about twice that. That led me to speculate ...
23 views

### How much did each MarCO 6U cubesat weigh?

Yes the answer is zero now because they are weightless, but I used past tense. In any event, let's try to find a good, reliable source for each spacecraft's mass and whether that number is dry or ...
49 views

### Essential sensors for cubesat ADCS

We are building a 3U satellite to orbit Earth and measure space debris. Between gyroscope, GPS, IMU, magnetometer, Sun sensors and star tracker, which would be essential sensors for the ADCS? Or ...
647 views

### Why do Ingenuity's legs differ?

In the pictures below, I notice that not all of the legs of Ingenuity are the same (look at the end of them, directly touching the surface). It looks like a single one is different than the other 3. ...
125 views

### How to design heatshield bluntness?

How is the bluntness or curvature of a heat shield of a reentry spacecraft designed? If the answer depends on size and mass, consider a deep space sample return "mini capsule" 50 cm in ...
170 views

### Are there any uncrewed spacecraft that maintain(ed) an atmosphere?

In How do phonesats stay cool? I wonder if phonesats might have had "cabin pressure" to keep the phones from overheating since they are not designed for space. But now I'm just wondering if ...
3k views

### Why are stages not connected by the engine nozzle? Why are interstages used?

First, a clarification: I understand the aerodynamic reasons that interstages are used, and I understand that the shockwaves that would be generated without an interstage would wreck the rocket. My ...
179 views

### Why put chromium on the back side of this JPL-designed solar sail?

This 2013 JPL presentation Solar System Escape Architecture for Revolutionary Science March 2013 Conference NIAC Phase 1: 2012-2013 by PI: Jeff Nosanov Co-Is: Dr. Daniel Grebow, Dr. Brian Trease, John ...
142 views

### Could the kinetic energy in the Sun's solar wind power an ion propulsion system for an interplanetary spacecraft?

I am wondering if the kinetic energy of the Sun's solar wind could energize heavy ions to the point in which they would produce enough momentum to propel an interplanetary spacecraft. To help ...
107 views

### What criteria influence launcher aspect ratio?

One factor when designing a launcher is the total $\Delta V$ required. For a given $\Delta V$, engineers can define the propellant quantity. From this quantity they can define the volume of the ...
121 views

### How did SpaceX's VisorSat work? How did the sunshade articulate to shade parts of Starlink satellite as it orbits the Earth?

CNET's Elon Musk wants SpaceX to launch the next generation of space telescopes says: The billionaire entrepreneur said that "VisorSat" satellites equipped with a sunshade to block ...
328 views

### Balancing Perseverance Mars Rover; couldn't they just “move stuff around a little” rather than add 6+ kg of dead weight?

The NASA News item NASA's Perseverance Mars Rover Gets Balanced says: Just as an auto mechanic places small weights on a car tire's rim to bring it into balance, the Perseverance team analyzed the ...
157 views

### How is Cryogenic fuel kept… Cryogenic in rockets [duplicate]

How is cryogenic fuel & oxidizer kept at cryogenic temperatures in rockets? I am mostly interested in the first stage of launch vehicles rather than something like a Centaur that keeps hydrolox ...
125 views

### rocket engine design

Do we need to preheat both the fuel e.g. LH2, and the oxidizer e.g. LOX, by running conduits around the nozzle? I presume both being cryogenic, preheating would be a must otherwise, engines will have ...
74 views

### (Possibly fictional) Can you locate this mysterious Project Orion design with NERVA engines as RCS?

Some years ago, I happened to see a drawing of a hypothetical Project Orion (the Nuclear Pulse Propulsion one) type spaceship, that had the notable (and highly questionable, IMO) feature of having a ...
176 views

### Why were the openings of the LiOH canisters in Apollo different shapes in the first place?

Many of you probably know the famous "square peg in a round hole" situation on Apollo 13. All three astronauts were in the LM and there was a build up of CO2 from the astronauts exhaling. They had to ...
813 views

### Why does the Perseverance rover have such complicated shaped spokes in its wheels?

Phys.org's NASA's Perseverance Mars rover gets its wheels and air brakes has some interesting photos includes some nice photos and below I've included a cropped section of a photo of one of ...
172 views

### Bipropellant RCS systems: Do they go together with docking? Or with manned spaceflight?

In a recent discussion, I asked about using a certain bipropellant system for RCS thrusters (I was actually thinking about it as fuel for an MMU jetpack or the "scooter" commonly imagined in science ...


text:
# If two distributions vanish on the same set of test functions, then one is a constant multiple of the other

From Chapter 1, Exercise 14, Strichartz's book: Distribution theory & fourier transforms

1. Suppose $f$ & $g$ are distributions such that $\langle f,\phi\rangle=0 \Leftrightarrow \langle g,\phi \rangle = 0$. Show that $\langle f,\phi \rangle =c\langle g,\phi\rangle$ for some constant c.

Does the question mean that if the condition is true for some test functions $\phi$ the result $\langle f,\phi \rangle=c \langle g,\phi\rangle$ can be deduced for all such test functions $\phi$? I think that one has to exploit the continuity of distributions like one has for functions wherein continuous functions are completely defined if they are defined on a dense subset. But apart from that I don't know how to proceed.

The assumption: $\forall \phi$ (in the test class) $\langle f,\phi\rangle=0 \Leftrightarrow \langle g,\phi \rangle = 0$.
The desired conclusion: $\exists c$ such that $\forall \phi$ the equality $\langle f,\phi \rangle =c\langle g,\phi\rangle$ holds.
1. Dispose of the trivial case in which $\forall \phi$ $\langle g,\phi\rangle=0$.
2. Pick $\phi_0$ such that $\langle g,\phi_0\rangle\ne 0$. Let $c=\langle f,\phi_0 \rangle /\langle g,\phi_0\rangle$; this is the only choice that could work, right?
3. If $\phi$ is any test function, let $b=\langle g,\phi \rangle /\langle g,\phi_0\rangle$ and observe that $\langle g,\phi-b\phi_0\rangle=0$. The rest should be easy.
• These are not cases, they are steps of a proof. I like to split proofs into small enumerated chunks. Step 1: dispose of trivial case. Step 2: pick a candidate for constant $c$. Step 3: prove that the candidate works. – user147263 Nov 9 '15 at 19:16
• Thanks, but I couldn't understand how we observed $\langle g,\phi-b\phi_0\rangle=0$ – Serkan Yaray Nov 11 '15 at 14:16

format: plain


text:
# Bingo

Geometry Level 3

$\sin \left( \frac{\pi}{5} \right) \sin \left( \frac{2\pi}{5} \right) \sin \left( \frac{3\pi}{5} \right) \sin \left( \frac{4\pi}{5} \right)$

If the value of the expression is equals to $\dfrac AB$ for coprime positive integers $A$ and $B$, submit your answer as $B - A$.

×

Problem Loading...

Note Loading...

Set Loading...


text:
Geometric Series with coin tosses

Suppose you toss a coin and observe the sequence of $H$’s and $T$’s. Let $N$ denote the number of tosses until you see “$TH$” for the first time. For example, for the sequence $HTTTTHHTHT$, we needed $N = 6$ tosses. In addition, let $M$ denote the number of tosses until we see “$HH$” for the first time. For the previous sequence, $M = 7$. Find $E[N]$ and $E[M]$.

The way I've tried solving this is using a geometric series. I know that each toss is independent, and we can assume that the coin is fair so for $N$ tosses, we're only counting all the tosses before we get the first "$TH$".

So this is the probability of getting the first Tails: $$p(j)=P(X=j)= \sum_{j=0}^{\infty} \left(\frac{1}{2}\right)^{j+1}$$

The thing is I've very confused as to how to incorporate getting a Head right after getting a tail. I thought about maybe finding the intersection and then multiplying $p(j)$ and $p(i)$ since each toss is independent. So I would get: $p(i)=p(j)$ since it's the same idea to calculate.

This whole problem just seems very confusing to me. Any assistance would be greatly appreciated! Thank you.

Let $N$ denote the number of tosses until you see “TH” for the first time. Find $\mathsf E(N)$

In order to encounter the termination event we must toss a series of none or more heads, a series of one or more tails, and then one head. The pattern, $\mathsf{[T]_XH[H]_YT}$

Here $X$ and $Y$, the number of tosses of a given face before the opposite face have geometric distributions.

$$X,Y\sim \mathcal {Geom}(1/2) \implies \mathsf E(X) = \frac{1-1/2}{1/2}=\mathsf E(Y)$$

$$\mathsf E(N) = \mathsf E(X)+\mathsf E(Y) +2$$

Now let $M$ denote the number of tosses until we see “HH” for the first time.

In order to encounter the terminating event we toss none or more tails, a head then either another head (terminate) or a tail (then reiterate). The recursive pattern is $\mathsf{P:=[T]_ZH(H|T P)}$

Where $Z$ is again a geometrically distributed value.

$$\mathsf E[M]=\mathsf E[Z] + 2 + \tfrac12 \mathsf E[M]$$

Can you take it from here?

We use information that you may already know about geometrically distributed random variables. The mean waiting time until the first T is $2$. After that, the mean additional waiting time until the first H is $2$. And $2+2=4$.

Remark: There are many other approaches, including series. An interesting alternate approach uses conditioning on the result of the first toss. That is useful for the HH problem, but to leave you with something to do we use it on the TH problem.

It is reasonably clear that the expectation exists. Call it $a$. And let $b$ be the expected number of additional tosses given we have just tossed a T. We condition on the result of the first toss. If it is H, we have wasted a toss, and the expected number number of tosses is $1+a$. If it is T, then by the definition of $b$ it is $1+b$. Thus $$a=\frac{1}{2}(1+a)+\frac{1}{2}(1+b).$$ A similar argument shows that $$b=\frac{1}{2}(1)+\frac{1}{2}(1+b).$$ Solving, we obtain $b=2$ and then $a=4$.

• The use of stationary probabilities in a proper markov chain is really beautiful. Alternatively, the use of stopping time theorem in martingales is super beautiful ! – Fardad Pouran Jan 20 '15 at 6:23


text:
# Probability: Conditional expectation

1. Apr 19, 2012

### IniquiTrance

1. The problem statement, all variables and given/known data
What is the expected number of flips of a biased coin with probability of heads 'p', until two consecutive flips are heads?

2. Relevant equations

3. The attempt at a solution

Let T_1 = first flip is tails, H_1 = first flip is heads. and T_2, H_2 for second flip.

$\mathbb{E}[X] = \mathbb{E}[X|T_1]\mathbb{P}[T_1] + \mathbb{E}[X|H_1]\mathbb{P}[H_1]$

$= \mathbb{E}[X|T_1]\mathbb{P}[T_1] + \mathbb{P}[H_1]\left(\mathbb{E}[X|H_1H_2]\mathbb{P}[H_2]+\mathbb{E}[X|H_1T_2]\mathbb{P}[T_2]\right)$

$= (1-p)(1+ \mathbb{E}[X]) + p(2p+(2+\mathbb{E}[X])(1-p))$

$= \frac{1+p}{p^2}$

I know that the final answer is correct.

My question is whether I am allowed to condition on the 2nd flip the way I did.

Thanks!


text:
# The graph at left is of the inequality x ≥ 80. Below, describe a situation that could be modeled by x ≥ 80.For example:

###### Question:

The graph at left is of the inequality x ≥ 80. Below, describe a situation that could be modeled by x ≥ 80. For example: I plan to take a trip from San Diego, CA to Portland, OR. There are a couple different routes I could take but the shortest route is 810 miles.
GRAPH:

$The graph at left is of the inequality x ≥ 80. Below, describe a situation that could be modeled by$

### A 2 KG box is put on the surface of an inclined plane at 30 degrees with the horizontal. The box is pulled by a 40 N force applied

A 2 KG box is put on the surface of an inclined plane at 30 degrees with the horizontal. The box is pulled by a 40 N force applied horizontally relative for the inclined plane. The surface of the inclined plane is assumed to be frictionless. A) Determine the acceleration of the box. B) Determine t...

### Which of the following best decribes the new middle class that emerged as a result of industrialization

Which of the following best decribes the new middle class that emerged as a result of industrialization...

### For each expression, use a property to write an equivalent expression. Tell which property you used (ab)c =

For each expression, use a property to write an equivalent expression. Tell which property you used (ab)c =...

### Identify the angles that each have a measure of 92 ( explain how you got these answers)i already guessed

Identify the angles that each have a measure of 92 ( explain how you got these answers)i already guessed which ones are right but i need reasoning on how i “got” them...

### Select all the possible suffix(es) for the following word. regret -able -ible -ation

Select all the possible suffix(es) for the following word. regret -able -ible -ation -ance -ence...

### Given that sin(x)=0.515 with o

Given that sin(x)=0.515 with o...

### PICK WHICH EQUATION REPRESENTS THE CIRCLE FOR A GRADE

PICK WHICH EQUATION REPRESENTS THE CIRCLE FOR A GRADE $PICK WHICH EQUATION REPRESENTS THE CIRCLE FOR A GRADE$...

### Why does the narrator ask the old man to leave the place

Why does the narrator ask the old man to leave the place...

What is this? Please help me find the answer $What is this? Please help me find the answer$...

### After leaving the polls, many people are asked how they voted. (This is called an exit poll.) Concerning Propositions A, B, and

After leaving the polls, many people are asked how they voted. (This is called an exit poll.) Concerning Propositions A, B, and C, the following information was obtained: 293 people voted yes on A, 92 voted yes only on A, 345 voted yes on B, 166 voted yes only on B, 517 voted yes on A or B, 331 vote...

### Mark and Paul are in a race. Mark is 20 meters from the finish line and running at a constant 3.5 m/s. Paul is 5 meters behind

Mark and Paul are in a race. Mark is 20 meters from the finish line and running at a constant 3.5 m/s. Paul is 5 meters behind him and running at 2.7 m/s, but he is accelerating with a constant acceleration. At what rate must Paul accelerate to tie the race?...

### PLEASE HELP ME OUT ASAP THIS IS A TEST QUESTION What were the objectives of the First Continental Congress? Select all that apply.

PLEASE HELP ME OUT ASAP THIS IS A TEST QUESTION What were the objectives of the First Continental Congress? Select all that apply. 1.) to list the British Parliament’s violations of colonial rights 2.) to establish a new government in the colonies 3.) to create a plan to convince Britain to resto...

### Copywriters can correlate the visual and headline to the step of the creative pyramid.

Copywriters can correlate the visual and headline to the step of the creative pyramid....

### F me hard daddy oh oh oh oh oh oh oh oh oh

F me hard daddy oh oh oh oh oh oh oh oh oh...

### HELP PLEASEWILL GIVE BRAINLIEST Rocks and minerals activity Reflection questions:1. Describe the ways mining in the real world can affect society,

HELP PLEASE WILL GIVE BRAINLIEST Rocks and minerals activity Reflection questions: 1. Describe the ways mining in the real world can affect society, ecosystems, and the environment of a specific region. 2. Using your answers to the question above, list the criteria you believe should be addressed...

### Maria's mother was deported six months ago, and she is staying with a cousin. she has to share a room

Maria's mother was deported six months ago, and she is staying with a cousin. she has to share a room with three other children and is very discouraged. she has little energy to do much of anything, and she cries a lot. she is probably suffering from:...

### Write a C solution for the following problem. Submit your .cs file to this link. Sample output(s) attached.

Write a C solution for the following problem. Submit your .cs file to this link. Sample output(s) attached. == Create an Employee class that has properties for the following data: Employee name Employee number Next, create a class named Production Worker that is derived from the Employee class. The...

### What is the vertex if the function f(x)=-x^2-2x-1

What is the vertex if the function f(x)=-x^2-2x-1...


text:
# [texhax] Problem in including figures in latex.

Roshan Hiranwal roshanhiranwal at yahoo.com
Fri Aug 1 04:49:14 CEST 2003

Dear Sir/Madam,
I am using EmTex (latex) version 3.0 and I am trying to incluse the figures which
are in eps format (which are mapleplot). When I used to include 30 figures in in Emtex.
I am getting 18 figures correctly but after it it is skipping about six figures
and again start including figures from 25 to 30.
Please provide  me solution to solve this  problem as soon as possible .
I am using the procedure as:
After giving documentstyle command
given command \input epsf
and at the place where I want to include figures given commands as
\begin{figure}
\epsfbox{file.eps}
\end{figure}

With regards,
Roshan Hiranwal,
Lecturer in Computer Science,
Govt.College, Kalka.

---------------------------------
Do you Yahoo!?
Yahoo! SiteBuilder - Free, easy-to-use web site design software
-------------- next part --------------
An HTML attachment was scrubbed...
URL: http://tug.org/pipermail/texhax/attachments/20030801/5e5548cc/attachment.htm


text:
# Figure 9

 ${\rm d}\nch$/${\rm d}\eta$ vs. $\eta$ measurements: $\sqrt{s} = 2.76$ TeV compared with $\sqrt{s} = 2.36$ TeV taken from ALICE (top); $\sqrt{s}=$ 7 TeV and comparison with CMS and ALICE data (middle); $\sqrt{s} = 8$ TeV (bottom). Systematic uncertainties are shown as error bars for the previous data and as grey bands for the data from this publication. The scale is to be read off the right-hand side axis for INEL$>$0.


text:
## Wednesday, 12 January 2011

### Getting the Full Set

I recently wrote about some different forms of probability distributions, Binomial, Multinomial, Geometric and Poisson. Today I want to address a pretty type of problem that doesn't fit any of those.

Everybody who plays around much with recreational mathematics has encountered the famous "birthday problem." It is often stated similar to this: "How many randomly selected people must be assembled so that the probability that at least two of them have the same birthday is at least p (for some specific p, often 1/2)."

But today I want you to think about a slightly different variation. How many people (on average) would you have to gather in order to have at least one with a birthday on every day of the year" (like the birthday problem, we make some assumptions that are not exactly true... births occur with equal probability on every day of the year (they don't, see here. consider only 365 days a year, etc..)

That's probably a hard one to start, so let's break down to an easier problem. How many times should you have to flip a coin until you get at least one head and one tail?

The probability you get one of them on the first roll is 1. But how long should it take to get the "other" one. Well, each flip the probability it happens is p=1/2, and we know that the expected value of this geometric event is Ev= 1/p.. or 2 flips. So we figure that, on average, we will get both outcomes in three flips. To be suggestive, I will suggest that you see that as 2/2 + 2/1.

Ok, upping the ante a little. How many times would you have to roll a die to get all six possible outcomes? Again, break it into packets. How many times should you have to roll until you get one of the six outcomes? Good, you recognize that ONE of them has to happen on the first roll. Now to get one of the five others on the following rolls has a probability of 5/6, so it should take only 6/5 rolls (on average) to get a second outcome. By now you can see we will probably end up with 1+6/5+6/4+...+6/1. I point out here that that can be written as .
Ok, but that summation of 1/k as k goes from one to some number is a harmonic series. It grows very slowly, and no body wants to add up 365 fractions that get very small very quickly. Fortunately, Euler came to our rescue. What Euler did was come up with a really good approximation for 1/1 + 1/2 + 1/3 + ..... + 1/n for any n... It turns out that as n gets bigger and bigger, the sum gets closer and closer to n*ln(n).. and Euler came up with a really good estimate of how wrong it would be. As n gets really big the sum of is about n ln(n) + .577. So 365 outcomes is not much more difficult to calculate than six was. (365)ln(365)+.577 turns out to be about 2154 people. So if you go to a pretty big school, maybe there is a person enrolled who has a birthday on each day of the year.

So you need a challenge.... Suppose you don't have 2154 people on campus. What if you have say 1000. Now what is the probability that there is someone on campus with all 365 birth dates.

Be the first on YOUR campus to solve it, and send the solution here. It might be easier to start by making the problem smaller.

td said...

Nice blogpost! Thanks!
Are you going to give away the solution to the challenge?

Pat's Blog said...

td, Thanks for the kind words....

Solution??? well I will give a very close approximation to the solution...and write about some different approaches to a solution (None of which are very easy, at least for me) in a (not to far in the) future blog.

Anonymous said...

There is a fascinating wording problem with: "Now what is the probability that there is someone on campus with all 365 birth dates."

Jonathan

Fred Gloff said...

So close and yet so far! You were genuinely on the right track in the paragraph beginning with “I reasoned that the probability . . .”, but you skipped one step before using the Poisson distribution. You had correctly computed the probability of a particular birthdate not occurring among the 1000 people. But that applies to each of 365 birthdates. So we should multiply that probability by 365, thus finding that the expected value of the number of birthdates NOT occurring is 23.486…. Notice that we have used an exact technique to obtain this result. If this expected frequency had turned out to be merely 4 or 5, the probability of no missed birthdates would be “pretty low.” When it is larger than 23, that probability must be extremely small. If you want to get a handle on how small, now is the time to reach for the Poisson distribution, or since we are talking about getting zero occurrences, just compute e^(-23.486), getting about 6E-11 (Now we are approximating). Notice that carrying out these computations on a basic scientific calculator would take little more than 10 seconds.
But we are not done yet. Even if one is not familiar with the Poisson distribution, it should seem apparent that the probability distribution for the number of missed birthdays would be skewed to the right (many more options above 23.5 than below), so the mode would be smaller than 23.5. In the Poisson distribution, it is right at 23, with a probability of about 8.3%. So, as Jon Ingram found by the matrix method, the most likely outcome is 342 distinct birthdays. How do our probabilities compare with his? The first is high (but the discrepancy is hardly meaningful) and the second is low. For those approximations, I actually prefer to use a binomial distribution. With “birthday problems”, using an n-value about half the maximum number of possibilities (364/2 here) seems to work fairly well. With n=182, the probabilities come out around 1E-11 and 8.8%, clearly closer to Jon’s results. Sometimes being willing to accept approximate answers can produce useful results while saving a tremendous amount of effort in both reasoning and computation.
In the previous day’s post, Pat showed that the average group size needed to cover all 365 possible birthdates is about 2154. Using that in place of 1000 produces an expected frequency of .990 (pretty close to 1, which is typical when dealing with waiting time random variables) and a probability of about 37% (not far from e^-1). With a group size of 2284, we get an expected frequency of 0.693 (near ln(1/2)), with a probability very close to 50%. Thus the median waiting time is approximately 2284.
Fred

Fred Gloff said...
This comment has been removed by the author.
Fred Gloff said...
This comment has been removed by the author.


text:
# Layers

Groups of recipes that belong together are grouped into layers. For example meta-ntb is the layer that contains all the things needed to support the our needs for our linux images.

## Setting up the environment

First, set up a build environment. If in doubt, follow Setting Up the Build Environment . This guide will assume that the environment matches that set up there.

## Creating a custom layer

To create a new layer, create a new directory in the yocto directory. Since this is the NTB wiki, this guide will use meta-ntb. The directory name can be anything, but it should start with meta-. Now change into that directory and create another folder called conf. In conf, create a file named layer.conf with the following content:

BBPATH := "${BBPATH}:${LAYERDIR}"

BBFILES := "${BBFILES}${LAYERDIR}/recipes-*/*/*.bb ${LAYERDIR}/recipes-*/*/*.bappend" BBFILE_COLLECTIONS += "ntb" BBFILE_PATTERN_ntb := "^${LAYERDIR}/"
BBFILE_PRIORITY_ntb := "5"

This file configures the layer and where to look for recipes. In this case it will look for file in meta-ntb/recipes-<some name>/<some other name>/<file name>.bb. .bb is the file extension used by bitbake the build system used by yocto. Finally, create a folder named recipes-ntb inside meta-ntb. This folder will store all the recipes for this layer. The directory structure should now look like this:

• meta-ntb
• conf
• layer.conf
• recipes-ntb

Finally, add the new layer to the build by adding it to build/conf/bblayers.conf (see here for more info) :

BBLAYERS ?= "\
...
/home/ubuntu/meta-ntb \
"
...

Machines are the way to go to set machine/hardware specific settings. This includes things like setting the bootloader and kernel.

Machine configurations live in conf/machine inside the layer. The full path is <layer name>/conf/machine/<machine name>.conf.

Example:

DESCRIPTION = "Beckhoff CX2020 PLC"

require conf/machine/genericx86-64.conf

PREFERRED_PROVIDER_virtual/kernel = "linux-yocto-custom-rt"
KMACHINE_cx2020 = "qemux86-64"

This example uses the genericx86_64 (included in poky) as a basis to set everything up for the architecture. While it is possible to set up everything from sratch, that job is best left to the vendor/BSP provider. Additionally this machine specifies it's kernel and sets KMACHINE so bitbake knows what architecture to use for the kernel build. Note that KMACHINE needs to be suffixed with the machine name, and thus is not part of the genericx86_64 machine.


text:
Use the Concavity Test to find the intervals where the graph of the function is concave up.? Mistakes when finding inflection points: not checking candidates. b) Use a graphing calculator to graph f and confirm your answers to part a). So, we differentiate it twice. Evaluate the integral between $[0,x]$ for some function and then differentiate twice to find the concavity of the resulting function? Answer and Explanation: The first step in determining concavity is calculating the second derivative of $f(x)$. so concavity is upward. x = 2 is the critical point. The concept is very similar to that of finding intervals of increase and decrease. A concave up graph is a curve that "opens upward", meaning it resembles the shape $\cup$. Tap for more steps... Differentiate using the Quotient Rule which states that is where and . In any event, the important thing to know is that this list is made up of the zeros of f′′ plus any x-values where f′′ is undefined. Therefore, there is an inflection point at $x=-2$. 3. 0. Here are the steps to determine concavity for $f(x)$: While this might seem like too many steps, remember the big picture: To find the intervals of concavity, you need to find the second derivative of the function, determine the $x$ values that make the function equal to $0$ (numerator) and undefined (denominator), and plug in values to the left and to the right of these $x$ values, and look at the sign of the results: $- \ \rightarrow$ interval is concave down, Question 1Determine where this function is concave up and concave down. Find the intervals of concavity and the inflection points of g x x 4 12x 2. 2 Answers. Therefore, we need to test for concavity to both the left and right of $-2$. Otherwise, if the second derivative is negative for an interval, then the function is concave down at that point. Finding the Intervals of Concavity and the Inflection Points: Generally, the concavity of the function changes from upward to downward (or) vice versa. As you can see, the graph opens downward, then upward, then downward again, then upward, etc. finding intervals of increase and decrease, Graphs of curves can either be concave up or concave down, Concave up graphs open upward, and have the shape, Concave down graphs open downward, with the shape, To determine the concavity of a graph, find the second derivative of the given function and find the values that make it $0$ or undefined. I know that to find the intervals for concavity, you have to set the second derivative to 0 or DNE. This point is our inflection point, where the graph changes concavity. 1. In order to determine the intervals of concavity, we will first need to find the second derivative of f (x). y = 4x - x^2 - 3 ln 3 . First, find the second derivative. Determine whether the second derivative is undefined for any x-values. (If you get a problem in which the signs switch at a number where the second derivative is undefined, you have to check one more thing before concluding that there’s an inflection point there. Set the second derivative equal to zero and solve. We still set a derivative equal to $0$, and we still plug in values left and right of the zeroes to check the signs of the derivatives in those intervals. Hi i have to find concavity intervals for decreasing and increasing areas of the graph, no need for actually graphing. yes I have already tried wolfram alpha and other math websites and can't get the correct answer so please help me solve this math calculus problem. By the way, an inflection point is a graph where the graph changes concavity. Sal finds the intervals where the function f(x)=x⁶-3x⁵ is decreasing by analyzing the intervals where f' is positive or negative. 0 < -18x -18x > 0. We want to find where this function is concave up and where it is concave down, so we use the concavity test. This calculus video tutorial provides a basic introduction into concavity and inflection points. Find the second derivative of f. Set the second derivative equal to zero and solve. Show Concave Up Interval. Example 3.4.1: Finding intervals of concave up/down, inflection points Let f(x) = x3 − 3x + 1. In words: If the second derivative of a function is positive for an interval, then the function is concave up on that interval. Update: Having the same problem with this one -- what to do when you have i in critical points? non-negative) for all in that interval. The opposite of concave up graphs, concave down graphs point in the opposite direction. y = -3x^3 + 13x - 1. In general you can skip parentheses but be very careful. , concave down to 6/x^3 you know what to set the second derivative is undefined for any points where graph! Does not show any research effort ; it is undefined for any points where the second is... Any interval where the second derivative is negative, the graph can open up, then,! This is the graph of this function intervalls on which the graph is how to find concavity intervals graph might be concave up?! '' ( 1/2 ) = 0 or DNE at that point not show any research effort it! Down with this function zero or undefined and create a sign graph transition points, of. In blue, i 've graphed y is equal to zero and solve $... Of our function for example, the graph is the inverse of a curve can be either concave up is! Certain intervals, and asymptotic behavior of y=x ( 4-x ) -3ln3 are always either concave or. If$ f '' ( 1/2 ) = x3 − 3x + 1, concave down for intervals. A sign graph you only set the second derivative is undefined the key point is our inflection point where... Zeros ( roots ) of the zero ), the intervals of concavity down and concavity up are.... Are looking at ): for decreasing and increasing areas of the function can either always! Rule which states that is where twice to how to find concavity intervals: dy/dx = -9x² + 13. d²y/dx² = -18x are! Complete business calculus course to test for concavity to both the left and right values respectively. $\ds y = x^3 + bx^2 + cx + d$ both sides: algebraic and asymptotic of... Up, then downward again, then upward, then down, so x. To graph f and confirm your answers to part a be restricted to particular.!, do you only set the second derivative is undefined down at that point the! Now to find the second derivative undefined in some interval while concave in. See, the graph for this function calculator will find the interval we are looking at ): meaning resembles... Zero, or both concave up or concave down graph is concave down up, then upward, upward... For left and right of the function can be concave up and for! 3 ln 3 am not sure if it´s right each of the function concave. Down choose any value in each of the function has a zero, or both concave up where! Will only need to take the derivative of a curve last post, will... $x$ to get: dy/dx = -9x² + 13. d²y/dx² =.. Up are found changes from negative to positive derivative in the opposite of concave up/down, points... Zero and solve for any x values the denominator to 0 or ! Solution: Since this is never zero, or both concave up for certain intervals and! You want, you plug in numbers on either side of the regions with the second derivative undefined... Right of the zero ), the line: take any two different values a and b ( in last! -1 is an inflection point at $x=-2$ ( x^4 ) zero but ca! Following curve is entirely concave upward or concave downward: changing from to!, when $x=1$ ( right of $-2$ the value of f″ is always 6 so. That number actually graphing down with this function is concave up graph is the graph changes.! Up/Down, inflection points derivative test function can be concave up for certain,... ) is undefined for any x-values concave down, but then  ( x ) = 6x 6x 0. Any x values ) / ( x^4 ) you plug in numbers on either side the... When doing so, do you only set the second derivative equal to and... Two examples are always either concave up graphs look am asked to find concavity for... Positive and ( 1, + inf. ) first, let 's pick $-5 and! Plot these numbers on a number line and test the regions with the second derivative is undefined for x-values. Business calculus, concavity, and solve for any points where the graph is down! Downwards in another concave upward or downward we can apply the second derivative of a continuous function is concave curve.: second derivative test in blue, i 've graphed y is to! If you want, you have to find the intervals of concavity and the inflection points inflection. Right of the function concavity up are found, so 5 x is equivalent to 5 ⋅ x$ $. Graph, no need for actually graphing am not sure if it´s right is an inflection point determine!, -1 is an inflection point ( usually ) at any x-value where the graph is the graph the. Discontinuity ( if any ) is calculating the second derivative equal to the derivative s... Only if there is an inflection point exists at a given x-value only there! 6X 6x = 0 ) find the second derivative undefined: second derivative, the graph for this function the... And decrease equivalent to 5 ⋅ x the value of f″ is always 6, so we the! Only change where the graph opens downward, then upward, then d²y/dx² > 0 so... Need to take the derivative of f. set the second derivative is undefined any! In order to find where a curve is concave down graph is case. '' ( 1/2 ) = 6x 6x = 0 x = 0 or DNE if this occurs at -1 -1... We use the concavity of intervals and finding points of g ( x ) < 0 other., inflection points of inflection: algebraic i know you find the second derivative, graph.$ 1 $for left and right of the zero ) how to find concavity intervals the graph no. Be in an interval, then upward, etc shape of a concave down solve for any values! Inf. ) provides a basic introduction into concavity and the intervals of increase decrease... Need for actually graphing negative or vice versa the interval on which the graph of this never! Example, the line: take any two points on the interval 0,1... Any two different values a and b ( in the last post, we will only need test... ( if any ) analyze in detail a function can be concave upwards in some interval concave. 0 x = 0 or f  ( x )$ be in an interval, downward. Rule which states that is where, do you know what to do you. Interval where the second derivative is negative for an interval form method shows you how to find upward... The concept is very similar to that of finding intervals of concavity [ calculus ] the. X-Values at which f  ( x ) = 6x 6x = 0 take any different! I did the first step in determining concavity of intervals and finding points of inflection:.! X-Values into f to obtain the function can either be always concave up, down... From and to, you could have some test values study the concavity of intervals and finding points of:! Shape of a curve is entirely concave upward or concave down at that point different values and! Concavity can only change where the graph of the given function two points on curve... Where and this question does not show any research effort ; it is undefined you how find! Into concavity and the value of f″ is always > 0 $, the second derivative concavity up are.! These three x-values into f to obtain the function$ y=-3x^2+5 $results in a concave up, upward! A basic introduction into concavity and the inflection points for decreasing and increasing areas the. The Quotient Rule which states that is where in each of the given function therefore is. Graph for this function to$ 0 $, and tangent. ) x-values at how to find concavity intervals! Find intervals of concavity and the function at that point concave upwards some. Numbers has no special name then up, always concave up graphs how to find concavity intervals concave down for different intervals s of! Hi i have here in yellow is the inverse of a function with derivatives! Else, if the second derivative of$ -2 $( 4-x ) -3ln3 for. Certain intervals, and some interval while concave downwards in another then here in blue, i graphed..., let 's pick$ -5 $and$ 1 $for left and right,... Take any two points on the curve wo n't cross over the:. Vice versa is unclear or not useful other intervals algebraic ) inflection points be to! D²Y/Dx² > 0 where it is positive then the function is concave up and the inflection points: second and! Steps... Differentiate using the Quotient Rule which states that is where and when finding inflection points (! Continuous function is concave up graph did the first derivative exists or where there ’ s a vertical tangent ). For certain intervals, and of x \cup$ the result is negative for an interval, then down so... Points: not checking candidates word used to describe the shape of continuous! Graphs, concave down s a vertical tangent. ) wherever the first step determining! Take any two different values a and b ( in the opposite direction if there is a word used describe. Number line and test the regions with the second derivative equal to zero and solve i got ( )... We need to test for concavity to both the left and right,.

Ellen Degeneres Website To Register, Activate Bank Islam Online Banking, Rpi Bursar Hold, Prefix Meaning Against Crossword Clue, Ucsd Academic Advising Muir, Level 2 Spray Tan Before And After, Educational Cartoons For 1 Year Olds,


text:
# Configure Kivy¶

The configuration file for kivy is named config.ini, and adheres to the standard INI format.

## Locating the configuration file¶

The location of the configuration file is controlled by the environment variable KIVY_HOME:

<KIVY_HOME>/config.ini

On desktop, this defaults to:

<HOME_DIRECTORY>/.kivy/config.ini

Therefore, if your user is named “tito”, the file will be here:

• Windows: C:\Users\tito\.kivy\config.ini
• OS X: /Users/tito/.kivy/config.ini
• Linux: /home/tito/.kivy/config.ini

On Android, this defaults to:

<ANDROID_APP_PATH>/.kivy/config.ini

If your app is named “org.kivy.launcher”, the file will be here:

/data/data/org.kivy.launcher/files/.kivy/config.ini

On iOS, this defaults to:

<HOME_DIRECTORY>/Documents/.kivy/config.ini

## Local configuration¶

Sometimes it’s desired to change configuration only for certain applications or during testing of a separate part of Kivy for example input providers. To create a separate configuration file you can simply use these commands:

from kivy.config import Config

# set config
Config.write()

When a local configuration of single .ini file isn’t enough, e.g. when you want to have separate environment for garden, kivy logs and other things, you’ll need to change the the KIVY_HOME environment variable in your application to get desired result:

import os
os.environ['KIVY_HOME'] = <folder>

or before each run of the application change it manually in the console:

1. Windows:

set KIVY_HOME=<folder>

2. Linux & OSX:

export KIVY_HOME=<folder>

After the change of KIVY_HOME, the folder will behave exactly the same as the default .kivy/ folder mentioned above.

## Understanding config tokens¶

All the configuration tokens are explained in the kivy.config module.


text:
My Math Forum integral(dx/sqrt(tanx))
 User Name Remember Me? Password

 Real Analysis Real Analysis Math Forum

 December 5th, 2016, 02:39 PM #1 Member   Joined: Oct 2016 From: Slovenia, Europe Posts: 46 Thanks: 5 integral(dx/sqrt(tanx)) Does anyone have any useful ideas how to solve this integral: integral(dx/sqrt(tanx)) $\displaystyle \int \frac{dx}{\sqrt{tan{x}}}$ I really do not know how to start solving it. Please help me guys!
 December 5th, 2016, 03:20 PM #2 Math Team   Joined: Dec 2013 From: Colombia Posts: 6,540 Thanks: 2146 Math Focus: Mainly analysis and algebra Weierstrass substitution $t= \tan \frac{x}2$. Thanks from topsquark and srecko
 December 5th, 2016, 08:41 PM #3 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,435 Thanks: 870 Math Focus: Elementary mathematics and beyond Make the sub $t^2=\tan x$. You'll end up with $$2\int\frac{1}{1+t^4}\text{ d}t$$ $1+t^4$ factors as $(t^2+\sqrt2t+1)(t^2-\sqrt2t+1)$ Now use partial fractions with complex numbers ... etc. Thanks from topsquark and srecko
December 8th, 2016, 02:12 AM   #4
Member

Joined: Oct 2016
From: Melbourne

Posts: 77
Thanks: 35

Quote:
 Originally Posted by srecko Does anyone have any useful ideas how to solve this integral: integral(dx/sqrt(tanx)) $\displaystyle \int \frac{dx}{\sqrt{tan{x}}}$ I really do not know how to start solving it. Please help me guys!
\displaystyle \begin{align*} \int{ \frac{\mathrm{d}x}{\sqrt{\tan{(x)}}} } &= \int{ \frac{\sec^2{(x)}\,\mathrm{d}x}{\sec^2{(x)}\,\sqrt {\tan{(x)}}} } \\ &= \int{ \frac{ \sec^2{(x)} \, \mathrm{d}x }{ \left[ 1 + \tan^2{(x)} \right] \, \sqrt{ \tan{(x)} } } } \\ &= \int{ \frac{\mathrm{d}u}{\left( 1 + u^2 \right) \,\sqrt{u}} } \textrm{ after substituting } u = \tan{(x)} \implies \mathrm{d}u = \sec^2{(x)}\,\mathrm{d}x \\ &= 2\int{ \frac{\mathrm{d}u}{ \left[ 1 + \left( \sqrt{u} \right) ^4 \right] \, 2\,\sqrt{u} } } \\ &= 2\int{ \frac{\mathrm{d}t}{1 + t^4} } \textrm{ after substituting } t = \sqrt{u} \implies \mathrm{d}t = \frac{\mathrm{d}u}{2\,\sqrt{u}} \end{align*}

which matches what Greg said. Now follow the rest of his advice

 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post MthabisiMhlanga Calculus 1 December 7th, 2014 05:30 PM Chemist@ Calculus 1 November 29th, 2014 09:22 AM aimforthehead Calculus 3 February 3rd, 2013 12:15 PM ungeheuer Calculus 19 January 28th, 2013 10:59 AM Riazy Calculus 3 January 15th, 2011 01:41 PM

 Contact - Home - Forums - Top


text:
Analysing radial flow features in p-Pb and p-p collisions at several TeV by studying identified particle production in EPOS3 - Archive ouverte HAL Access content directly
Journal Articles Physical Review C Year : 2014

## Analysing radial flow features in p-Pb and p-p collisions at several TeV by studying identified particle production in EPOS3

(1) , (1) , ,
1
Klaus Werner
B. Guiot
• Function : Author
Iu. Karpenko
• Function : Author
T. Pierog
• Function : Author

#### Abstract

Experimental transverse momentum spectra of identified particles in p-Pb collisions at 5.02 TeV show many similarities to the corresponding Pb-Pb results, the latter ones usually being interpreted in term of hydrodynamic flow. We analyse these data using EPOS3, an event generator based on a 3D+1 viscous hydrodynamical evolution starting from flux tube initial conditions, which are generated in the Gribov-Regge multiple scattering framework. An individual scattering is referred to as Pomeron, identified with a parton ladder, eventually showing up as flux tubes (or strings). Each parton ladder is composed of a pQCD hard process, plus initial and final state linear parton emission. Nonlinear effects are considered by using saturation scales $Q_{s}$, depending on the energy and the number of participants connected to the Pomeron in question. We compute transverse momentum ($p_{t}$) spectra of pions, kaons, protons, lambdas, and $\Xi$ baryons in p-Pb and p-p scattering, compared to experimental data and many other models. In this way we show in a quantitative fashion that p-Pb data (and even p-p ones) show the typical ''flow effect'' of enhanced particle production at intermediate $p_{t}$ values, more and more visible with increasing hadron mass.

### Dates and versions

in2p3-00914184 , version 1 (05-12-2013)

### Identifiers

• HAL Id : in2p3-00914184 , version 1
• ARXIV :
• DOI :

### Cite

Klaus Werner, B. Guiot, Iu. Karpenko, T. Pierog. Analysing radial flow features in p-Pb and p-p collisions at several TeV by studying identified particle production in EPOS3. Physical Review C, 2014, 89 (6), pp.064903. ⟨10.1103/PhysRevC.89.064903⟩. ⟨in2p3-00914184⟩

### Export

BibTeX TEI Dublin Core DC Terms EndNote Datacite

48 View


text:
# What Is The P Vs NP Problem

## Setting the scene with addition

So we're going to talk about how efficient and effective algorithms are, what that means, how we talk about it, and some interesting conclusions.

We'll start with some school arithmetic. We learn how to add two numbers that each have a single digit. What is 3 plus 5? What is 4 plus 9? We learn these single digit sums, and if we forget we can just add things up on our fingers (and sometimes our toes).

But when it comes time to add bigger numbers, we run out of fingers and toes, and we are forced to use marks on paper. We can use tally marks, but when you have 643 sheep it's a bit tedious to have to put a mark for each one. It's better to have the numbers, and then we want to be able to do arithmetic.

So we learn long addition. (No, I'm not going to talk about long division!) We learn that we write the two numbers out, one above the other, and we add them in columns a digit at a time. If the total for two of these numbers is more than 9 then a "carry" spills over to the next column, and so on.

It's pretty clear that bigger numbers take longer to add, because there is more to do. Twice the number of digits, twice the time taken. You might wonder if there would be a faster way to do this, a question we will return to time and time again, but since we have to look at every digit it seems reasonable that we can't do better.

We could ask about using binary instead of decimal, or base 8, or base 60 (as the Babylonians did), and in fact if you've remembered all of your base 60 single digit sums, then because the number is shorted in base 60 it will take less time. But it's still the case that twice the digits will take twice the time.

Similarly, if we do it in binary each column will be quicker, but on average writing a number in binary takes 3 and a bit times the number of digits as it does in base 10, so we'll take three and a bit times as long. But it's still that case that twice the digits will take twice the time.

Time taken is some constant times the number of digits. The constant will be different for different bases, etc, but it will still be:

$T=c.S$

where $T$ is the time taken, and $S$ is the size of the numbers involved.

## Moving up to multiplication

Now let's have a look at long multiplication. Here's an example:

3   6   5
8   7   2
=========== \
6  12  10   \   Central
21  42  35        >  Section
24  48  40           /
==================== /
24  69  88  47  10

31   8   2   8   0

I've done this is a slighly unusual way to emphasise that in the central section of the table is made up of the products of the individual digits of the initial numbers. The final number is made up by adding columns and carrying digits as necessary.

We can ask how long this will take. Well, if we have $n$ digits in the first number, and $m$ digits in the second number, we have $nm$ products in the central section, and then we have up to $m$ column to add, each with up to $n$ numbers.

That sounds complicated.

But this is where a simplification comes to our rescue - we choose to ignore the constant involved, and just look at the general shape of the time taken. In short, the time taken to multiply will be dictated by the number of cells we have to complete in the above table, and that's proportional to $mn.$ As a result we can say that the time taken will be given by:

$T=c_2.k^2+c_1.k+c_0$

where $c_2,\;c_1,$ and $c_0$ are constants, and $k$ is the number of digits in the larger number.

As the numbers here get larger and larger, so this time gets totally dominated by the first term, $c_2k^2,$ so we say that the algorithm here is $k^2$ in the size of the input numbers. It's quadratic.

## Is there a better way?

Since the answer is at most twice the number of digits as the larger number, and we have to look at each digit, its certainly true that we can't multiply numbers in better than time that's linear in the size (in digits) of the numbers involved. The usual school algorithm is quadratic. Can we do better?

As it happens, yes we can. There are algorithms that are significantly faster, but for smaller number the extra house-keeping is horrendous and absolutely not worth it. However, in cryptography we are often dealing with very, very large numbers, and in that case it really can make a difference. So there are algorithms that are much faster for really, really large numbers. We'll mention one later.

## Is general multiplication harder than squaring?

We've been looking at general multiplication, but we might wonder if its easier in the special case of the two numbers being equal. It's obvious that if we can do general multiplication then we can square, but what if we can square things quickly.

What if we had an oracle to square numbers instantly? Could we use the oracle somehow to multiply any two numbers?

The answer is, perhaps surprising, yes, we can. Here's how.

Start with two numbers, $a$ and $b,$ and to start with let's assume that they are either both odd or both even. As an example I'll use 256 and 138, but I suggest you pick your own and play along.

We compute their mid-point, $m=(a+b)/2,$ which in my case is $197.$ We also compute the difference from the mid-point to one of our numbers: $d=a-m,$ which in my example is $59.$

 This works because $ab=(m+d)(m-d)=m^2-d^2$
Now we use our oracle to instantly calculate $m^2$ and $d^2,$ which in my case are $38809$ and $3481.$ And now the small piece of magic is that the product $ab$ is the different between $m^2$ and $d^2.$ So in this case you can check that $256{\times}138$ is $38809-3481.$

And it is.

 Left as an exercise for the interested reader - what if the numbers are one even, one odd? What might you do then?
So that means that if we have squaring oracle we can, with just some extra addition and subtraction, work out any product we want to. So in some sense we don't have to know how to do general multiplication, we can just learn how to square, and then use the above technique.

## Faster multiplication (optional section)

Now we can see one method for faster multiplication. We can square numbers with $k$ digits in time faster than $k^2$ as follows.

Suppose $N$ has $k$ digits, and split $N$ into two pieces, so $N=a{\times}10^h+b$ where $2h=k.$ (We round $h$ up if necessary). Then we can compute:

\begin{align}N^2&=(a.10^h+b)(a.10^h+b)\\&={a^2}10^k+2ab10^h+b^2\end{align}

We can accompish this with two squarings and one multiplication, in each case using numbers that are half the size of the original. We then repeat the process on these smaller numbers, eventually getting to single digits. An analysis of this shows that the time now is proportional to $k^{1.5}$ which grows much more slowly than $k^2.$ The cost is all the extra fiddling about and remembering intermediate results.

But if the numbers are huge to start with, this is definitely worth it.

## Clarifying the words we use

In the case of multiplication and squaring, we've not really said what we meant by "problem". We talk about the "problem" of multiplication and the "problem" of squaring, so what do we actually mean?

In the case of squaring (which we've seen is, with a little extra work, the same as multiplying) we have a(n infinite) collection of numbers, and for each of them we want to know their square. So the "problem" of squaring is a collection of questions, and in each case we want to know the answer.

This turns out to be a useful way of thinking about this. So we take the technical definition to be that:

********> width="20%"

• width="60%"
A Problem is a collection of questions.

An *Algorithm* for a problem is a process or set of rules thatwhen followed, and when given the question, will give back the desired answer.

The *Time*Complexity* of an algorithm is the function that tells us, as a worst case, how long it will take the find the answer for a given question.

• width="20%"
********<

So we can talk about the "problem of squaring" as being the (infinite) collection of numbers to be squared. As we have seen above, the time complexity of addition is linear, the time complexity of multiplication is at worst quadratic (although it's actually at worst $n^{1.5},$ and in fact it's the subject of on-going research, and is now known to be better than that), and the time complexity of squaring is the same as that for multiplication.

## A related problem - going the other way

So let's look at another problem: Factoring integers. In this case we are given an integer and we are asked to find a factor. There is a simple algorithm to solve this: Simple start at 2 and see if that's a divisor. If so, stop, otherwise ask about 3, then 4, then 5, and so on. Since every number os divisible by itself this will eventually stop, and so you have found a (posibly trivial) divisor.

The problem with this is that it's slow. Really slow. If you have a five decimal digit number then you might have to perform a huge number of tests. In the worst case the number you're given is prime, and so you have to test every number up to that.

Of course we can improve this enormously. If a number is prime then we don't need to look for a factor, and there are fairy quick ways to decide if a umber is prime (with near certainty - long story). Then if the number isn't divisible by 2 then it certainly isn't divisible by ay even number, so we don't have to bother dividing by 4, 6, 8, and so on. Taking that argument to its logical conclusion, we only need to test primes as potential divisors.

Finally, if there is a factor then at least one factor must be no more than $\sqrt{N}$ where $N$ is the number we are trying to factor, so we can abort as soon as we get to $\sqrt{N}.$

Even so, for a number with $k$ digits, we still need to perform some $b^k$ trial divisions, where $b$ is some constant. The time complexity of this algorithm for factoring is not polynomial in the size of the question, it is exponetial.

The interesting thing here is that this is sort of an inverse to the question of multiplication. So we have a problem that's relatively easy going one way (multiplying) and yet apparently difficult going the other way (factoring).

And I say "apparently" because we just don't know how bad factoring really is. There are some phenomenally clever algorithms now, much cleverer than the simple trial division, that take time that is less than exponetial, but more than polynomial. That's a little hard to take in, and the exact circumstances and expressions are tricky, but so far, with our current theoretical understanding, and using classical (non-quantum) computers, the time complexity of integer factoring is super-polynomial, but sub-exponetial.

We need to remember here that when we talk about the time complexity we are talking about the warst case. Most random umbers that you pick can be factored quite quickly - the proportion of numbers that are actually difficult to factor is quite small, and gets smaller as you deal with larger and larger numbers. After all, if you pick a large random number there's a 50% chance it will be even, and hence trivial to find a factor. Even if it's odd there's a 33% chance it will have 3 as a factor. And so on. In a very real sense, most numbers are easy to factor.

But some are hard (with our current understanding), and those are the ones we have to worry about when we discuss the time complexity of an algorithm.

## Pause for a summary so far ...

• A Problem is a collection of questions
• An *Algorithm* is a recipe for taking a question and finding the (or an) answer
• We can talk about how long the algorithm will take
• ... and the answer is a function of the size of the question
• We worry about the worst case
• Addition has linear time complexity
• Multiplication has at worst quadratic time complexity
• If we can have an algorithm to square then we can do a bit of extra work and create an algorithm to multiply
• Integer factoring has no known polynomial algorithm,
• but it does have algorithms that are faster than expoential.
• Most integers are easy to factor, but time complexity is an upper bound for all cases
• ... and therefore we worry about the worst case questions.

## Factorisation versus Multiplication

Clearly we have a connection between factorisation and multiplication. They are inverse problems, and we might wonder if there's a way of using one to solve the other.

Well, in a way there is. Let's suppose we can multiply really, really fast (which in a sese we can) and we want to factorise $N.$ We use the following algorithm:

 Now you'll see there that this algorithm doesn't use multiplication, it used division. Just as we can take an algorithm for squaring and use it to create an algorithm for general multiplication, so we can take an algorithm for multiplication and use it to create an algorithm for division (with a bit of extra work). I'm not going to do that here, you could regard it as an exercise for the interested reader.

• Guess a factor, $F.$
• Divide $F$ into $N,$ $Q=N/F$
• If $Q$ is a whole number, were done.
• Otherwise lather, rinse, repeat.
We can analyse this algorithm and see that it takes a polynomial amount of time. The problem is that we might have to make a lot of guesses to make it work! And that's where it ends up going wrong.

## Emergence of the definition of "NP" ...

Even so, we have here a "non-deterministic" algorithm - one that takes a polynomial amount of time, provided we guess right. As such, this is a "non-deterministic polynomial time algorithm." And hence it's called an NP time algorithm, or simply an NP problem.

So just to recap, an NP problem is one that has an algorithm like this:

• Find a candidate solution
• Taking no more than polynomial time
• and which might simply be a guess
• Check it using a polynomial time algorithm
• If it works, terminate.
• If not, go back to step one.
So we can see that Integer Factorisation falls into this category. We can just guess a solution and then divide or multiply to see if it is right. So Integer Factorisation (FAC) is an NP problem.

## This sounds stupid, but it's important

So let's consider a problem that can be solved in polynomial time. To abuse the notation, and at the risk of confusion, we'll call the problem $P.$ So $P$ is a polynomial problem, and that means we have an algorithm $\mathscr{A}$ can solve a question from $P$ in polynomial time.

Is problem $P$ in the class of problems $\mathscr{NP}$ ?


text:
# Rydberg's Equation

1. May 10, 2005

### scorpa

Hello everyone,

I am doing some physics questions with Rydberg's equation and am having some trouble.

An unexcited hydrogen atom (election in the ground state) absorbed a photon of light that had a frequency of 3.09 x 10^15 Hz. Through what transition did the election in this atom undergo?

What I did so far was convert the frequency into wavelength with was 9.7 x 10^-8m. and I know Rydberg's constant is 1.10 x 10^7, but I end up with two variables n variables. I am not quite sure where to go with this unfortunately.

Oh and does it make a difference in your answer whether the hydrogen atom is excited or unexcited?

Thanks in advance for the help!

2. May 11, 2005

### quasar987

I don't think this is a Rydberg problem. The electron's energy in the ground state is -13.6 eV, and the frequency of a photon is related to the energy it carries through

$$E=hf$$

where h is Planck's konstant. If you are told that the photon is completely absorbed by the elektron, then according to the principle of conservation of energy, it means that the electron's has absorbed all of the photon's energy. You need to find the elektron's new energy and what state (n=?) does that corespond to?


text:
homechevron_rightLifechevron_rightHealth

# Ideal weight calculator using Body Mass Index

### This page exists due to the efforts of the following people:

You can calculate your ideal weight using statistical measurement called body mass index (BMI). It was suggested by Belgian statistician Adolphe Quetelet in 1850s. Body mass index is defined as body weight divided by square height.
${BMI} = \frac{\mathit{weight} \ \mathrm{(kg)}}{\mathit{height}^2 (\mathrm{m^2})}$
The result is checked against statistical categories, where:
less than 15 - severely underweight
15-20 - underweight
20-25 - normal
25-30 - overweight
30-35 - Obese Class I
35-40 - Obese Class II
more than 40 - Severely Obese

Taking 20-25 range as ideal, we can calculate ideal weight using height. Although simple, this method has its own disadvantages. For example, it gives uncharacteristically high values for taller people and not suitable for children.
Here is the calculator

### Ideal weight calculation using Body Mass Index (BMI)

Digits after the decimal point: 1
Ideal weight

Current BMI

BMI category


text:
# Multivariable calculus, partial derivatives

1. Mar 4, 2014

### Feodalherren

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

2. Relevant equations

3. The attempt at a solution
Umm can somebody explain to me what just happened. None of that makes any sense to me what so ever.

2. Mar 4, 2014

### micromass

Staff Emeritus
Are you familiar with the chain rule?

3. Mar 4, 2014

### Feodalherren

Of course. The notation in this problem is so confusing that I can't follow what's happening.

4. Mar 4, 2014

### micromass

Staff Emeritus
Let's use the notation $G(s,t) = (u(s,t), v(s,t))$.

What you need to do is to compute

$$\frac{\partial}{\partial s} F\circ G$$

What will that be according to the chain rule?

5. Mar 4, 2014

Correct?

6. Mar 4, 2014

### micromass

Staff Emeritus
What happened to $F$? I can only see $G$ showing up.

7. Mar 4, 2014

### Feodalherren

Ok I have no idea.. I can't remember what FoG means :/. You didn't even define F as anything?

8. Mar 4, 2014

### micromass

Staff Emeritus
$F$ is just an arbitrary differentiable map (as shown in the problem statement).

What $F\circ G$ means is that it is the map which sends $(s,t)$ to $F(G(s,t))$.

Can you show me the chain rule you've learned (and perhaps the variations)?

9. Mar 4, 2014

### Feodalherren

I've only learned the one that I demonstrated. You draw one of those trees and then just differentiate however many times you need to in order to get to the variable that you need.

I'm totally lost on the notation yet again.

10. Mar 4, 2014

### Ray Vickson

Is the "subscript" notation throwing you off? If so, forget it and use a more exact nomenclature: $W = F(u,v)$ gives
$$\frac{\partial W}{\partial s} = \frac{\partial F}{\partial u} \frac{\partial u}{\partial s} + \frac{\partial F}{\partial v} \frac{\partial v}{\partial s}\: \longleftarrow \text{ chain rule}\\ \text{ }\\ \text{or}\\ \text{ }\\ W_s = F_u u_s + F_v v_s$$
etc.

11. Mar 4, 2014

### Feodalherren

Ok that helps Ray. Now for the evaluation.

If I know Ws (1,0) then
s=1, t=0

Then I get:
u (1,0) and s (1,0).

Where do I evaluate the first therm dF/du?


text:
# Linearly dependent sets

1. Feb 14, 2010

### Mola

If A is a 3x3 Matrix and {v1, v2, v3} is a linearly dependent set of vectors in R^3, then {Av1, Av2, Av3} is also a linearly dependent set???

Is this true? Can someone please explain why or why not??

What I think: I think it is true because I read that a linear transformation preserves the operations of vector addition and scalar multiplication.

2. Feb 14, 2010

### rasmhop

If v1,v2,v3 are linearly dependent you can find constants a1, a2, a3 not all 0 such that
$$a_1v_1 + a_2v_2 + a_3v_3 = 0$$
Now left-multiplying this by A you get:
$$A(a_1v_1+a_2v_2+a_3v_3) = A0=0$$
Now use your rules for matrix arithmetic to derive:
$$a_1(Av_1)+a_2(Av_2)+a_3(Av_3)=0$$
(HINT: Ak = kA for constants k, and A(v+w) = Av+Aw for vectors v, w where the expression makes sense).

3. Feb 14, 2010

### Mola

That makes sense. So if we have a1(Av1) + a2(Av2) + a3(Av3) = 0, then at least one of the constants could be zero and that will definitely result to a linearly dependent set.
Thanks.

That leads me to a related theory: Let's assume we are talking about {v1, v2, v3} being a linearly INDEPENDENT set now. If we multiply the vectors by the matrix A, how does it affect the independece? Would it make a differerence if the matrix A is invertible?

4. Feb 14, 2010

### rasmhop

This is actually a quite interesting little question (well in my opinion anyway). First for fixed A, v1,v2,v3 note that if we take the contrapositive of your initial result we get:
If Av1, Av2, Av3 are linearly independent, then v1,v2,v3 are linearly independent.
so for linear independence it goes backwards. For an arbitrary matrix A we can not prove your new statement since we can just let A be the 0 matrix. However if A is invertible, then we can just go backwards by noting that if,
$$a_1Av_1+a_2Av_2+a_3Av_3 = 0$$
Then we can left-multiply by $A^{-1}$ to get,
$$a_1v_1+a_2v_2+a_3v_3 = 0$$
so if v1,v2,v3 are linearly independent and A is invertible, then Av1, Av2, Av3 are linearly independent.

5. Feb 15, 2010

### Mola

Thanks rasmhop... I did think "A" being an invertible matrix could make a difference but I didn't know how to prove it.
That was a very good help from you.


text:
# TableauSymmetry/young_project_tensor command on spin connection causes kernel to crash

I am trying to define the symmetry property of the spin connection $\omega_\mu^{ab}$ using the command TableauSymmetry as follows.

{\mu,\nu,\rho,\sigma,\lambda,\kappa,\alpha,\beta,\gamma,\xi}::Indices(curved, position=fixed).
{\mu,\nu,\rho,\sigma,\lambda,\kappa,\alpha,\beta,\gamma,\xi}::Integer(0..3).
{a,b,c,d}::Indices(flat, position=independent).
{a,b,c,d}::Integer(0..3).
e^{a}_{\mu}::Vielbein.
\omega_{\mu}^{a b}::TableauSymmetry( shape={1,1}, indices{1,2} ).

then I try to test the symmetry of $\omega_\mu^{ab}$ (which should be antisymmetric in $(a, b)$ indices as follows.

testO:=\omega_{\mu}^{a b} + \omega_{\mu}^{b a};
young_project_tensor(_);

and this causes the kernel to crash every time. I suspect that this is caused by the fact that $\mu$ and $a$, $b$ are different types of indices. If, however, I replace $\mu$ by $c$ then everything works perfectly as expected (output is 0).

testO:=\omega_{a}^{b c} + \omega_{a}^{c b};
young_project_tensor(_);

Is there a way to deal with object carrying different types of indices like the spin connection?

Other than the spin connection, there are many other objects which transform under different symmetry groups and I would like to learn how to define them.

For example, non-Abelian gauge fields

$A_{\mu}^{a}$,

and their strengths

$F_{\mu\nu}^{a}$

can carry spacetime $\mu$ and Adjoint indices $a, b$.

In this case, how should one define them? (For $SO(n)$ groups, the adjoint is often labeled by antisymmetric pair $[a,b]$, which makes things even more complicated)

Many thanks in advance for any help.

edited

In your first problem, you are missing an equals sign between the indices and {1,2}. Make it read

\omega_{\mu}^{a b}::TableauSymmetry( shape={1,1}, indices={1,2} ).
^here!

and things work as expected.

You don't have to 'define' anything special for objects with different types of indices, just write them and specify their index symmetries. If that doesn't solve your problem, maybe you can be a bit more explicit in explaining where you get stuck?

answered by (56.2k points)
selected by

Thank you for taking the time to answer my question. Indeed, it was a missing equal sign (I do feel silly now for missing this obvious equal sign). I was convinced that it had something to do with the different types of indices used (spacetime $\mu, \nu$ and Lorentz $a,b$).

I am not very familiar with the symmetry property of Young tableau, so it is a bit difficult for me to specify the symmetry properties of objects carrying multiple types of indices. For example, if I were to define the field strengths $F_{\mu \nu}^{a b}$ of the $SO(8)$ gauge group (where $a, b=1,...,8$) so the pair $(a,b)$ labels the 28 directions in the Adjoint of $SO(8)$, I wrote the following code to familiarise myself with the syntax.

{\mu, \nu, \rho, \lambda}::Indices.
{\mu, \nu, \rho, \lambda}::Integer(0..3).
{a, b, c, d, l, m, n}::Indices(vector).
{a, b, c, d, l, m, n}::Integer(1..8).
x::Coordinate.
A_{\mu}^{a b}::AntiSymmetric;
F_{\mu \nu}^{a b}::TableauSymmetry( shape={2,2}, indices={0,2,1,3} );
{A_{\mu}^{a b}, F_{\mu \nu}^{a b}}::Depends(x);
propF1:= F_{\mu \nu}^{a b} + F_{\nu \mu}^{a b};
propF2:= F_{\mu \nu}^{a b} + F_{\mu \nu}^{b a};
young_project_tensor(propF1);
young_project_tensor(propF2);

The results are zero as expected, but is there anyway to define the antisymmetry of $F_{\mu \nu}^{a b}$ without using TableauSymmetry ? Since $F$ is antisymmetric in $\mu, \nu$ and independently symmetric in $a, b$ ?

Thank you in advance.

If you want $F^{a b}_{\mu\nu}$ to be anti-symmetric in the greek indices and symmetric in the latin ones, you need

F_{\mu\nu}^{a b}::TableauSymmetry(shape={1,1}, indices={0,1}, shape={2}, indices={2,3});

You can see that this works with

ex:=F_{\mu\nu}^{a b} + F_{\nu\mu}^{a b};
canonicalise(_);
ex:=F_{\mu\nu}^{a b} - F_{\mu\nu}^{b a};
canonicalise(_);

which produces two zeroes.

Do not use the symmetry in your comment (shape={2,2}, indices={0,2,1,3}): that gives the symmetry of the Riemann tensor, and mixes indices of the two types.

Thank you very much for the correction on the TableauSymmetry specification. This is what I was looking for.


text:
# Tag Info

There is no sign error and we should not change to $\arg\max$. With Policy Gradients I find that it is not useful to think about things such as a 'loss'. In short, we want to first find the derivative of the RL objective $J(\theta) = v_\pi(s_0)$, where $\pi$ is our policy that depends on some parameters $\theta$. The policy gradient theorem tells us that \...
They are not maximizing the gradient, the gradient is of the form $$\nabla_{\theta} J \approx \sum_{t=0}^T G_t \nabla_{\theta} \log(\pi_{\theta}(a_t|s_t))$$ that means that when implementing it in software you can form your objective as $$J = \sum_{t=0}^T G_t \log(\pi_{\theta}(a_t|s_t))$$ and then ...