text:
# ThoughtWorks 的一道笔试题

PROBLEM ONE: TRAINS

Problem: The local commuter railroad services a number of towns in Kiwiland. Because of monetary concerns, all of the tracks are 'one-way.' That is, a route from Kaitaia to Invercargill does not imply the existence of a route from Invercargill to Kaitaia. In fact, even if both of these routes do happen to exist, they are distinct and are not necessarily the same distance!

The purpose of this problem is to help the railroad provide its customers with information about the routes. In particular, you will compute the distance along a certain route, the number of different routes between two towns, and the shortest route between two towns.

Input: A directed graph where a node represents a town and an edge represents a route between two towns. The weighting of the edge represents the distance between the two towns. A given route will never appear more than once, and for a given route, the starting and ending town will not be the same town.

Output: For test input 1 through 5, if no such route exists, output 'NO SUCH ROUTE'. Otherwise, follow the route as given; do not make any extra stops! For example, the first problem means to start at city A, then travel directly to city B (a distance of 5), then directly to city C (a distance of 4).

1. The distance of the route A-B-C.
2. The distance of the route A-D.
3. The distance of the route A-D-C.
4. The distance of the route A-E-B-C-D.
5. The distance of the route A-E-D.
6. The number of trips starting at C and ending at C with a maximum of 3 stops. In the sample data below, there are two such trips: C-D-C (2 stops). and C-E-B-C (3 stops).
7. The number of trips starting at A and ending at C with exactly 4 stops. In the sample data below, there are three such trips: A to C (via B,C,D); A to C (via D,C,D); and A to C (via D,E,B).
8. The length of the shortest route (in terms of distance to travel) from A to C.
9. The length of the shortest route (in terms of distance to travel) from B to B.
10. The number of different routes from C to C with a distance of less than 30. In the sample data, the trips are: CDC, CEBC, CEBCDC, CDCEBC, CDEBC, CEBCEBC, CEBCEBCEBC.

Test Input:

For the test input, the towns are named using the first few letters of the alphabet from A to D. A route between two towns (A to B) with a distance of 5 is represented as AB5.

Graph: AB5, BC4, CD8, DC8, DE6, AD5, CE2, EB3, AE7

Expected Output:

Output #1: 9
Output #2: 5
Output #3: 13
Output #4: 22
Output #5: NO SUCH ROUTE
Output #6: 2
Output #7: 3
Output #8: 9
Output #9: 9
Output #10: 7
==========

1-5:

package com.log4think.code;

public class TrainMap {

public int[][] map = {
//    A   B   C   D   E
{-1,  5, -1,  5,  7},
{-1, -1,  4, -1, -1},
{-1, -1, -1,  8,  2},
{-1, -1,  8, -1,  6},
{-1,  3, -1, -1, -1},
};

public int getDistance(String path)
{
int distance = 0;

for (int i=0; i 0)
System.out.println(distance);
else
System.out.println("NO SUCH ROUTE");

//		g.findDistance("AEBCD");
//		g.findDistance("AED");
}
}

6:

public class TrainMap {

public int[][] map = {
{-1, 5, -1, 5, 7},
{-1, -1, 4, -1, -1},
{-1, -1, -1, 8, 2},
{-1, -1, 8, -1, 6},
{-1, 3, -1, -1, -1},
};

// used to record the trip round count
public int tripCount = 0;

/***
*
* @param end , the end node. if we reach it we found a path
* @param path , the current path we already have
* @param maxLength, the maximum stops
*/
public void dfs(String end, String path, int maxLength)
{
// this is for debug and trace
// System.out.println(";; " + path);

// if the path reach the maximum stops, then cancel search
if (path.length() - 1 > maxLength) return;

// check if we have reach the "end" node
if ( path.length() > 1 && path.endsWith(end) ) {
System.out.println(path + ", " + tripCount);
tripCount ++;
}

// caculate the lastest node index in map
char lastChar = path.charAt(path.length()-1);
int lastNodeIndex = lastChar - 'A';

// loop all nodes in map which connected to lastNode, and try it
for ( int i=0; i 0) {
dfs(end, path + newChar, maxLength);
}
}
}

public static void main(String[] args) {
TrainMap g = new TrainMap();

g.dfs("C", "C", 3);
}
}

7:

public class TrainMap {

public int[][] map = {
{ -1, 5, -1, 5, 7 },
{ -1, -1, 4, -1, -1 },
{ -1, -1, -1, 8, 2 },
{ -1, -1, 8, -1, 6 },
{ -1, 3, -1, -1, -1 }
};

public void bfs(String start, String end, int hops) {
String lastRoute = start;

for (int hop = 0; hop < hops; hop++) {
String route = "";
for (int i = 0; i < lastRoute.length(); i++) {
char c = lastRoute.charAt(i);
int id = c - 'A';

for (int j = 0; j < map[id].length; j++) {
if (map[id][j] > 0)
route = route + (char) (j + 'A');
}
}
//          System.out.println(lastRoute);
lastRoute = route;
}

//      System.out.println(lastRoute);
System.out.println(lastRoute.split(end).length - 1);
}

public static void main(String[] args) {
TrainMap g = new TrainMap();

g.bfs("A", "C", 4);
}
}

8/9:

public class TrainMap {

public int[][] map = {
{-1, 5, -1, 5, 7},
{-1, -1, 4, -1, -1},
{-1, -1, -1, 8, 2},
{-1, -1, 8, -1, 6},
{-1, 3, -1, -1, -1},
};

public void dfs(String end, String path, int cost) {
if (path.endsWith(end) && cost < bestCost && cost > 0) {
bestPath = path;
bestCost = cost;
return;
}
char lastChar = path.charAt(path.length() - 1);
int lastNodeIndex = lastChar - 'A';

for (int i = 0; i < map[lastNodeIndex].length; i++) {
char newChar = (char) (i + 'A');
int newCost = map[lastNodeIndex][i];
if (newCost > 0) {
if (path.indexOf(newChar) > 0)
continue;
dfs(end, path + newChar, cost + newCost);
}
}
}

public String bestPath = "";
public int bestCost = Integer.MAX_VALUE;

public static void main(String[] args) {
TrainMap g = new TrainMap();

g.dfs("C", "A", 0); // 8
//      g.dfs("B", "B", 0); // 9

System.out.println("Best Path: " + g.bestPath + "\nCost: " + g.bestCost);
}
}

10:

public class TrainMap {

public int[][] map = {
{-1, 5, -1, 5, 7},
{-1, -1, 4, -1, -1},
{-1, -1, -1, 8, 2},
{-1, -1, 8, -1, 6},
{-1, 3, -1, -1, -1},
};

public void dfs(String end, String path, int cost) {
if (cost >= 30)
return;

if (cost > 0 && path.endsWith(end)) {
System.out.println(path + ", " + cost);
}

char lastChar = path.charAt(path.length() - 1);
int lastNodeIndex = lastChar - 'A';

for (int i = 0; i < map[lastNodeIndex].length; i++) {
char newChar = (char) (i + 'A');
int newCost = map[lastNodeIndex][i];
if (newCost > 0) {
dfs(end, path + newChar, cost + newCost);
}
}
}

public static void main(String[] args) {
TrainMap g = new TrainMap();

g.dfs("C", "C", 0);
}
}


text:
New interface

# Motivic, logarithmic, and topological Milnor fibrations

Abstract : We compare the topological Milnor fibration and the motivic Milnor fibre of a regular complex function with only normal crossing singularities by introducing their common extension: the complete Milnor fibration. We give two equivalent constructions: the first one extending the classical Kato-Nakayama log-space and the second one, more geometric, based on the real oriented multigraph construction, a version of the real oriented deformation to the normal cone. In particular we recover A'Campo's model of the topological Milnor fibration, by quotienting the motivic Milnor fibration with suitable powers of $\mathbb R_{>0}$, and show that it determines the classical motivic Milnor fibre. We give precise formulae expressing how the introduced objects change under blowings-up thus showing, in particular, that the motivic Milnor fibre is well-defined as an element in a Grothendieck ring and not merely in its localisation.
Document type :
Preprints, Working Papers, ...

https://hal.archives-ouvertes.fr/hal-03460891
Contributor : Jean-Baptiste Campesato Connect in order to contact the contributor
Submitted on : Wednesday, December 1, 2021 - 10:45:19 AM
Last modification on : Monday, October 10, 2022 - 4:06:07 PM

### Identifiers

• HAL Id : hal-03460891, version 1
• ARXIV : 2111.14881

### Citation

Jean-Baptiste Campesato, Goulwen Fichou, Adam Parusiński. Motivic, logarithmic, and topological Milnor fibrations. {date}. ⟨hal-03460891⟩

Record views

format: plain


text:
Math Help - L'Hospitals Rule ---- easy quick help

1. L'Hospitals Rule ---- easy quick help

Find lim as n--> infinity of 2n/(1+3sqrt(n))

since its infinity over infinity use L'Hospitals rule

2 / (3/2 n^-1/2)

2(3/2 sqrt(n))

OK so i know this is wrong and i know it should be

2sqrt(n) / (3/2)

but i dont get why only the n^(-1/2) comes up to the top and the 3/2 stays in the denominator cause i thought the 3/2 and n^-1/2 should go up top since they are connected (multiplying).

2. Originally Posted by mybrohshi5
Find lim as n--> infinity of 2n/(1+3sqrt(n))

since its infinity over infinity use L'Hospitals rule

2 / (3/2 n^-1/2)

2(3/2 sqrt(n))

OK so i know this is wrong and i know it should be

2sqrt(n) / (3/2)

but i dont get why only the n^(-1/2) comes up to the top and the 3/2 stays in the denominator cause i thought the 3/2 and n^-1/2 should go up top since they are connected (multiplying).
2/(3/2)*n^-1/2
= 2*(n^1/2)/(3/2)

3. I understand that it is 2*(n^1/2)/(3/2) but i dont get why it is this. why doesnt the (3/2) go to the numerator along with the n^-1/2 since they are "attatched to one another"

4. Originally Posted by mybrohshi5
I understand that it is 2*(n^1/2)/(3/2) but i dont get why it is this. why doesnt the (3/2) go to the numerator along with the n^-1/2 since they are "attatched to one another"
When 3/2 goes to numerator it becomes (3/2)^-1 = 2/3


text:
# How many ways can you get a straight or a flush in poker?

Continuing with the work we’ve done in Counting Balls and Counting, Permutations and Combinations we will look at how to count the number of flushes or straight that can be dealt in a poker hand.

## Poker Hands

There are wealth of different types of poker. In each of the cases, we have different number of cards, ways these are dealt, and rules for constructing hands. Additionally, there are different types of card decks that can be used, so we will begin by defining our rules.

For these examples, we will assume that we will be working with a 5 card poker hand where all 5 cards are dealt at once. You can then rearrange these cards as you like in order to make the hand you want. While you would generally have to declare your hand according to your best outcome, we won’t require that you choose one type of outcome. That is, we will count a hand that has four of the same number card as a four of a kind, three of a kind, two pairs and a pair.

Furthermore, we will work with the standard 52 card deck. In this deck, we will have 4 suits; hearts, diamonds, clubs and spades. Each of these suits will have 13 distinct numbered cards; ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, king. When ordering, the cards follow the above ordering with the exception that the ace can also be used as the card following the king.

In general, two pokers hands will be considered the same if the same 5 cards are in the hand, in any order. You cannot have 2 of the same card, because each card appears 1 in the deck. Furthermore, in order to define a hand, we would have to define both the suit and the number for each card. Therefore, if we wanted to count all possible poker hands, we would note that when choosing a hand, repetition of results is not allowed and order does not matter. We saw in the previous posts that this will mean that we are finding the number of combinations of 52 cards, 5 at a time, so there will be $$\binom{52}{5} \approx 2.6$$ million different hands.

### Flushes

A flush in poker is when all 5 cards in your hand are from the same suit. If we want to count the number of ways that we can get a flush, we want to determine the number of ways that we can have distinct hands that result in a flush. In order to do this, we will construct a hand, determining all of the options that are available to us.

In order to construct a flush, the first choice we will have is to choose a suit. Since there are 4 suits in our card deck, there will be four ways we can choose a suit. Once we have chosen a suit, all cards must be this suit. After this, we will need to choose the number for each of the cards. Since there are 13 numbers available and we need 5 of them, we will be picking 5 from 13. Since the order in which we choose them doesn’t matter, there will be exactly $$\binom{13}{5}$$ ways to do this.

Combining this information, we find that the total number of flushes will be $$4 * \binom{13}{5} =5148$$.

### Straights

A straight in poker is when all 5 cards can be arranged so that the numbers of the cards are consecutive. For example, the hand consisting of $$5 \heartsuit, 2 \clubsuit, 6 \spadesuit, 3 \clubsuit, 4 \heartsuit$$ would be a straight because we can rearrange the cards to read $$2,3,4,5,6$$. We will again find all the ways we can construct such a hand.

We begin by choosing the suit of each card. Since a straight is not dependent on the suit of the cards, we can choose any suit we like for each card. We note here that we can also choose the same suit for all the cards since we would count this as straight as well as a flush. Therefore, we have 4 choices for each card. Since there are 5 cards, we have a total of $$4^{5}=1024$$ choices for suit.

Now that we have chosen the suits of the cards, we now have to choose the number for each card. In this case, we note that once we have chosen the lowest numbered card, the remaining cards must follow consecutively from the previous card. Now, note that we can have a straight starting with ace through 10. However, if it starts at a Jack, we will run out of cards to place in our straight, so there are only 10 options for the lowest card.

If we combine this together, we note that we have $$1024$$ choices for suit and 10 choices for numbers of cards. We, therefore, have a total of 10240 ways to construct a straight.

### Straights or Flushes

As we count the number of hands that are a straight or a flush, we will do so by noticing that we can do so by adding the number of straights to the number of flushes and subtracting the number of hands that are both. Note we have to subtract off the hands that are both because we are counting them twice by adding the number of straights to number of flushes.

We will therefore first need to find the number of hands that are both flushes and straight. Such hands would be called a straight flush and would be a hand in which you can rearrange the numbers so that they are consecutive and all the cards will be the same suit. If we follow the process we had above for choosing number and suit, we can find the number of such hands. Since this hand will be a flush, we will have exactly 4 choices for suits. Also, since the hand is a straight, we will have 10 choices for the number. We, therefore, get that there are 40 total straight flushes.

Combining all this, we find that the total number of hands that are a straight or a flush would be $$5148+10240-40=15,348$$.

## Conclusion

We were able to count the number of different ways to get a flush or straight in poker by combining our different counting techniques. We could use these same counting techniques, with some modification in order to count the other types of hands as well. As we do so, make sure to determine how to construct each handing, noting whether or not order matter in each case.

I hope you enjoyed and learned something from this post. If you did, make sure to like it and share it on Social Media with anyone else that may find it useful.

This site uses Akismet to reduce spam. Learn how your comment data is processed.


text:
# Moon Earth binar system by math

In Science - Posted on Jul 02, 2015 17:30 - From: physicsforums.com

Moon Earth binar system by math

Hello,
I'm interesting in Earth Moon evolution history. I understand it is very complex problem, and I can find many articles about that. But I would like to start from beginning and add real condition on my model step by step (I hope good way to do something).

Moon Earth binar system by math

Top similar posts to Moon Earth binar system by math

## Possibility of multiple moons in Earth-Moon system

I am interested in theoretical (what if) possibility of our Earth having more than one moon, I wonder if they can be arranged at the same orbital resonance as the Galilean moons 1:2:4 If yes, would such system be stable? If yes, should our Moon be the first, second or third satellite out of the... Possibility of multiple moons in Earth-Moon system...

## Question: Gravitational Potential Energy in a Ball-Earth-Moon system?

Hello, here's a questions I was wondering if any of you could solve. I don't have the exact numbers, but the scenario is this: a guy standing on the Earth throws a ball upwards and catches it a few seconds later. How does would affect the potential energy and/or mechanical energy of the... Question: Gravitational Potential Energy in a Ball-Earth-Moon system?...

## Middle-earth: Shadow of Mordor - 6 Tips for Mastering the Nemesis System

Get a leg up in your struggle for power with tips straight from the game's lead designer. More... ...

## 'Lightning Storms: Exploring Nature's Awesome Fireworks' 'The Moon: Exploring the Earth's Night light' 'Voices of Earth' 'Dinosaurs Trilogy' + [Kindle Edns] (Kids & Adults alike)!

Lightning Storms: Exploring Nature's Awesome Fireworks (A Window Into Our Weird, Wacky, Wonderful World Book 3) [Kindle Edition] Link 117p http://www.amazon.com/dp/B00MJYRV...&tag=false See it all for yourself with over 85 Stunning Photos, 15 Quality Diagrams, and 4 amazingly easy to read Charts. If you are like most people you are fascinated with the beauty of lightning, but also a little fearful of it as well. Watching a good lightning storm can make for a fantastic light show and be a lo...

## Could the moon provide all of Earth’s power?

Next time you look up at the night sky, focus on the moon. One day It could be a global power station, and the end of energy shortfalls....

### Universal Gravitation w/ Earth & Moon

1. The problem statement, all variables and given/known data The acceleration of gravity on the Moon is 1/6 what it is on Earth. The radius of the Moon is 1/4 that of the Earth. What is the Moon's mass compared to the Earth's? 2. Relevant equations $F_g = \frac{GMm} {r^2}$... Universal Gravitation w/ Earth & Moon...

### How on earth do you westernize with the new rebel system?

Playing a game as the Incas, and I've conquered all my neighbours and am colonizing tons. It's about 1520 and I'm in the Caribbean and Westernizing off of a Portugal core in Haiti. Harsh treatment no longer works (except to delay a rebellion by about 5 months for bloody 375 MP!!!!) so I have nothing I can do stop the rebellions, except for increase autonmy. Bad luck, I decreased autonomy 19 years ago in all these provinces so I basically am buggered. The fact that you need to plan WAAAAY in ad...

### Gravity on Earth vs The Moon

1. The problem statement, all variables and given/known data On Earth, an average person's vertical jump is 0.40m. What is it on the Moon? 2. Relevant equations Fg 1 on 2 = G(m1m2/r^2) G= 6.67 x 10^-11 3. The attempt at a solution r= 0.40m F earth on person on Earth's surface =... Gravity on Earth vs The Moon...

### Satellite orbiting the Earth vs the moon?

I know that the the satellite will lose speed(due to perhaps small air resistance), and therefore lose centripetal force? I am not sure how losing centripetal force and speed causes the satellite in orbit to lose radius. If force is the same, and velocity is lowered, than radius would... Satellite orbiting the Earth vs the moon?...

### E: 19/12 Win Moon Indigo, Under The Skin & The Man Who Fell To Earth On DVD

Email: [email protected] Subject/Answer: Michel Gondry Include your full contact details. Clicky ...

### In which point between Earth and Moon is gravity field zero

1. The problem statement, all variables and given/known data I already searched for solution and i found similar one but i'm still confused. -In which point between Earth and Moon gravity field equals 0 if distance between moon and earth is 60.3 * radius Earth and mass of the earth is 81 *... In which point between Earth and Moon is gravity field zero...

### SERIOUSLY IF BLACK HOLE EXIST HOW COME EARTH AND SOLAR SYSTEM STILL SURVIVE?

we should have been sucked in and end of the world liao how come bo daiji until now? ...

### Around the Earth to the Moon

My first video tracing a path around the earth and out ot the moon. Using Blender 2.6 running on debian linux wheezy. There are a couple of things that need to be changed, such as stars appear too close and change in tracking lock from earth onto the moon around frame 285 could be smoother. Thanks to Andrew Price at Blenderguru for providing the inspiration and a great tutorial. https://www.youtube.com/watch?v=pbYSjc3O7nk Attached Thumbnails   ...

### Gravity force between Earth and Moon from this angle

1. The problem statement, all variables and given/known data Hello the text is in swedish but ill try my best to translate, What it says is: Determine the resultant force on the moon with regard to thr gravitation force from the sun and the earth when it's in this position shown in the... Gravity force between Earth and Moon from this angle...

### This week’s near-Earth asteroid had its own moon

An asteroid called 2004 BL86 just passed right by Earth, and astronomers spied an exciting surprise: this asteroid has a moon to call its very own....


text:
## The Bell Series of an Arithmetical Function

Given an arithmetical functionand a primewe denote bythe power seriesand call this the Bell series ofmodulo

Bell series are very useful whenis multiplicative. We need the following theorem.

Theorem

The Bell series for an arithmetical function is unique. Supposeandare multiplicative functions, thenif and only iffor all primes

Proof: Ifthenfor alland allsoConversely iffor allthenfor allSinceandare multiplicative and take the same values for all prime powers they agree for all the positive integers so

Examples

Bell series for the Mobius function. Sinceandforwe have

Bell series for Euler's totient functionSinceforwe have

Ifis a completely multiplicative function so thatfor allthen the Bell series is a geometric series and

as long as

Hence we have the following


text:
# If $B(t)$ is Brownian motion then prove $W(t)$ defined as follows is also Brownian motion

Let $B(t)$ be standard Brownian motion on $[0.1]$

Define $W(t)$ as follows

$W(t) = B(t) - \int_0^t \frac{B(1)-B(s)}{1-s} \, ds$

Prove $W(t)$ is also Brownian motion

So I'm not sure how to deal with the integral here. In order to show it, too, is Brownian motion I think I would need to

1. Make an argument that the transformation is linear and hence also Gaussian
2. Show that $E[W(t_1) - W(t_2)] = 0$ for any $t_1>t_2>0$
3. Show the variance of $W(t_1) - W(t_2) = t_1 - t_2$ or an equivalent covariance function
4. Finally make an argument that separate increments are independent. Though this would follow trivially I think from the fact $B(t)$ is brownian motion with independent increments.

But yeah, how do I deal with the integral?

• Hint: Integration is a linear functional. – A.S. Oct 31 '15 at 6:59
• Is it valid to say $E[\int[B(1)-B(s)]/(1-s)ds$ =$$\int (EB(1) - EB(s))/(1-s)ds since the integral is Linear? As that would make arguing the expectation is 0 trivial I guess, and when calculating the variance it would just give me a usual integral in s. By the way I appreciate the help A.S. – Patrick Oct 31 '15 at 7:00 • Yes. Expectation is just another integration and you can swap order of well-behaved integrals. – A.S. Oct 31 '15 at 7:04 • When you say "well-behaved" would that just mean the expectation doesn't depend on the bounds of the integral, or would it be a variety of things? – Patrick Oct 31 '15 at 7:07 • Look up Fubini's theorem – user223391 Oct 31 '15 at 7:10 ## 1 Answer Recall the following characterization of (one-dimensional) Brownian motion A stochastic process (W_t)_{t \geq 0} is a Brownian motion, if and only if, 1. (W_t)_t has continuous sample paths. 2. (W_t)_t is a Gaussian process with mean 0 and covariance \mathbb{E}(W_s W_t) = \min\{s,t\} for all s,t \geq 0. As (W_t)_t has obviously continuous sample paths, we just have to check the second property. Since (B_t)_{t \geq 0} is a Brownian motion, it is in particular a Gaussian process and so$$B_t - \sum_{j=0}^{n-1} (B_1-B(t_j)) \frac{1}{1-t_j} (t_{j+1}-t_j)$$is Gaussian for each n \in \mathbb{N} where t_j := \frac{t}{n} j. If we let n \to \infty, then we get$$W_t = \lim_{n \to \infty} \left( B_t - \sum_{j=0}^{n-1} (B_1-B(t_j)) \frac{1}{1-t_j} (t_{j+1}-t_j) \right)$$is Gaussian as a limit of Gaussian random variables. Since this argumentation applies in exactly the same way to the joint distributions (W_{s_1},\ldots,W_{s_m}) where s_j \geq 0, we get that (W_t)_{t \geq 0} is a Gaussian process. It remains to check mean and covariance. By Fubini's theorem, we have$$\begin{align*} \mathbb{E}(W_t) &= \underbrace{\mathbb{E}(B_t)}_{0} - \mathbb{E} \left( \int_0^t\frac{B_1-B_s}{1-s} \, ds \right) = - \int_0^t \underbrace{(\mathbb{E}(B_1-B_s)}_{0} \frac{1}{1-s} \, ds = 0. \end{align*}$$Now fix r \leq t.$$\begin{align*} \mathbb{E}(W_r W_t) &= \mathbb{E}(B_t B_r)- \mathbb{E} \left( B_t \int_0^r \frac{B_1-B_s}{1-s} \, ds \right) - \mathbb{E} \left( B_r \int_0^t \frac{B_1-B_s}{1-s} \, ds \right) \\ &\quad + \mathbb{E} \left( \int_0^t \int_0^r \frac{B_1-B_u}{1-u} \frac{B_1-B_v}{1-v} \, du \, dv \right) \\ &=: \mathbb{E}(B_r B_t) +I_2+I_3+I_4 \end{align*}$$If we can show that$$I_2+I_3+I_4 = 0$$we are done. Using \mathbb{E}(B_u B_v) = \min\{u,v\} for any u,v \in [0,1] and Fubini's theorem, we find$$ \begin{align*} I_2 &= \int_0^r \frac{\mathbb{E}(B_1 B_t-B_tB_s)}{1-s} \, ds = \int_0^r \frac{t-s}{1-s} \, ds \\ &= - \log (1-r) t + r + \log(1-r) \end{align*}$$as r \leq t. Similarly,$$\begin{align*} I_3 &= \int_0^t \frac{r- \min\{r,s\}}{1-s} \, ds = \int_0^r \frac{r-s}{1-s} \, ds + \int_r^t \underbrace{\frac{r-r}{1-s}}_{0} \, ds = \int_0^r \frac{r-s}{1-s} \, ds \\ &= (1-\log(1-r)) r + \log(1-r) \end{align*}$$and, finally,$$\begin{align*} I_4 &= \int_0^t \int_0^r \frac{1-v-u+ \min\{u,v\}}{(1-u)(1-v)} \, du \, dv \\ &= \int_r^t \int_0^r \frac{1-v-u+ u}{(1-u)(1-v)} \, du \, dv + \int_0^r \int_0^r \frac{1-v-u+ \min\{u,v\}}{(1-u)(1-v)} \, du \, dv \\ &= (t-r) \int_0^r \frac{1}{1-u} \, du + 2 \int_0^r \int_v^r \frac{1}{1-v} \, du \, dv\\ &= -(t-r) \log(1-r) + 2 ((1-\log(1-r))r + \log(1-r)) \end{align*}$$where we have used in the penultimate equation that$$\begin{align*} \int_0^r \int_0^r \frac{1-v-u+ \min\{u,v\}}{(1-u)(1-v)} \, du \, dv &= \int_0^r \int_0^v \frac{1}{1-u} \, du \, dv + \int_0^r \int_v^r \frac{1}{1-v} \, du \, dv \\ &= \int_0^r \int_v^r \frac{1}{1-u} \, dv \, du + \int_0^r \int_v^r \frac{1}{1-u} \, dv \, du \\ &= 2 \int_0^r \int_v^r \frac{1}{1-v} \, du \, dv. \end{align*}$Adding all up, we get$I_2+I_3+I_4 = 0\$ and this finishes the proof.

• Thanks so much, your answer also helped me solve two similar proofs. I can't upvote because of my rep, but your help is very much appreciated. – Patrick Oct 31 '15 at 22:17
• @Patrick You are welcome. Thanks for letting me know that the answer helped you. – saz Nov 1 '15 at 6:16


text:
# GUI Documentation¶

## Fitting options¶

Global Fitting Specify the scope of the global variables.

• Pixel-wise the decays are fitted on a pixel-by-pixel basis
• Image-wise global fitting is performed across each image individually, i.e. lifetimes are constant across each image
• Global global fitting is performed across the entire dataset, i.e. lifetimes are constant across all images.

Global Variable combo-box to confine the scope of the global fit to a particular multi-well plate metadata parameter (e.g. ‘Row’ or ‘Column’). If a parameter is selected all images with the same parameter value will be globally analysed together.

Global Mode combo-box to set up the method of global analysis. The following choices are available:

• Global binning All pixels in the global fit are binned into a single decay. This decay is then fitted to determine the global parameters. Each pixel is then fitted individually with the global parameters fixed (using the method of linear least square minimisation) to determine the local variables.
• Global analysis All the pixels in the global fit are simultaneously analysed using Variable Projection algorithm (see in Advanced tab)

No. Exp combo-box to set up the number of exponential decay components $$n_\tau$$

No. Fixed The number of fixed fluorescence lifetimes; the first No. Fixed values in the Initial Tau list are fixed.

Fit contributions combo-box to set up the fitting mode for the amplitudes Ak (formula (3) and (4)). The following choices are available:

• Fixed the fractions $$\beta_i$$ are fixed for corresponding amplitudes (set up by the textboxes appearing to the right)
• Fit locally amplitudes are fitted locally, i.e. on a pixel-by-pixel basis
• Fit globally amplitudes are fitted globally, i.e. they are invariant across the image. Use this option when fitting a complex donor FRET model or polarisation resolved data.
• Fit globally (Grouped): use this option to fit a model where we have a mixture of several fluorophores with complex decays. The lifetimes and contributions are specified in groups, indexed from zero. Within each group the lifetime and fractional contributions are invariant across the image but the fractional population of each group varies locally. The groups are specified using the Initial tau list.

Fit Reference combo-box to set up the fitting mode for the variable $$\tau_R$$ (formula (4)), as Fixed or Fitted. The reference lifetime is fitted globally.

Initial Tau combo-box to choose the initial values of lifetimes. These are estimated automatically unless ‘Automatically estimate initial guesses’ is deselected.

### Stray Light tab¶

You can select the fitting scope of the “stray light” variables entering the model expressions (3-5), by choosing either Fixed, or Fitted Locally, or Fitted Globally. The scope is set for the following variables:

• Offset A constant background value, Z in the formula (5)
• Scatter A fast-light scatter contribution, S in the formulas (3) and (4)
• TV Background A time-varying background contribution that is scaled by the coefficient V, see formula (5). This option can only be used if a time varying background has been loaded. The V coefficient can be fitted globally or fixed by setting the TVB Background to ‘Fixed’ and TVB to 1.

### Anisotropy tab¶

These options are used to set up an anisotropy decay model if polarisation resolved data has been loaded.

No Decays combo-box to specify the number of rotational correlation times to fit.

No Fixed combo-box to specify the number of rotational correlation times that are fixed. The first No Fixed are fixed to the value specified in the Phi list.

Use Phi textbox to specify the initial guesses and fixed values for the anisotropy correlation times.

### FRET tab¶

These options are used to fit a complex-donor FRET decay model. In this model we assume that the donor fluorophore has two or more conformations with associated lifetimes whose FRET efficiencies are linked by their relative quantum yields. To use this mode Fit Contributions should be set to Fixed or Globally in the Lifetime tab, as we assume that the fractional contribution of the various donor species are invariant across the image.

No. FRET Species combo-box to specify the number of different FRET conformations present. For example, for an intra-molecular FRET probe where there are likely to be two different conformations with different (but non-zero) FRET efficiencies, set this to two and set Include Donor Only to No. For an inter-molecular FRET probe where the unbound state will have a zero FRET efficiency, set this to one and set Include Donor Only to Yes.

No. Fixed combo-box to set the number of FRET conformations which have fixed efficiencies. The first No. Fixed FRET efficiencies in the E list will be fixed.

Include Donor Only combo-box to specify whether a donor-only contribution should be included, that is a contribution from a species which is not FRETing.

No. Threads Set up the number of threads used in multi-threading processing. This is automatically set to the number of cores present on the local machine which in general is optimal.

Algorithm combo-box to choose the method and the type of error function minimization. The following methods are available:

• Variable Projection: Use separable fitting to minimise the mean-square error function.
• Maximum Likelihood: Use the maximum likelihood fitting metric, only applicable to pixel-wise fitting. Use this option for fitting data with low photon count numbers, particularly TCSPC data.
Weighting Mode combo-box to select the method of weighting the residuals entering the mean-square error function when using Variable Projection.
Error function =

where ti are the bin (delay) times, σ(ti) is weighting coefficient, and Dmodel(ti) and y(ti) are the model and measured intensity values, respectively.

• Average Data Weight according to the average measured decay across the global scope. This is the default, and advised option.
• Pixelwise Data Weight according the measured decay in each pixel $$chi^2(ti)=y(ti)$$. Can display significant bias at low photon counts.
• Model Weight according to the model decay $$\chi^2(ti)=D_{model}(ti)$$. May lead to problems with convergence if initial estimates are significantly different from the true values.

Pulse Train Correction combo-box to specify whether incomplete decays should be included in the fit. Unless you have a very low repetition rate laser source or are fitting very fast decays, this option should only be disabled for testing and when fitting data simulated without incomplete decays.

Auto Resampling combo-box to dynamically combine bins to ensure that there are at least 10 counts in each bin. This option is superseded by Maximum Likelihood fitting.

IRF combo-box to set up the type of IRF. The following choices are available:

• Single Point Use the same IRF for all pixels
• SV IRF Use a spatially varying IRF that has been loaded from the IRF menu. Each pixel is fitted using the IRF from the corresponding pixel in the spatially varying IRF.

IRF shift map Use an IRF shift map that has been loaded from the IRF menu. The same IRF is used in each pixel, time shifted according to the shift map loaded.

Live Fit checkbox to dynamically refit the currently selected decay as fitted parameters are changed or as the selected region is changed. This option does not automatically fit the whole dataset.

Calculate Errors checkbox to estimate confidence intervals on the global parameters based on the F statistic. This option should currently be considered experimental and significantly increases the fitting time.

## Instrument and preprocessing options¶

### Data tab¶

These settings are used to specify data acquisition parameters and set any required pre-processing.

Smoothing The kernel size used for spatially smoothing the data to reduce the noise. The equivalent binning setting in Becker and Hickl software is provided for reference.

Note

To ensure that the smoothing does not affect the choice of background or threshold values, data is averaged rather than summed over the kernel. The $$\chi^2$$ value correctly accounts for the level of smoothing applied.

Integrated Min. Pixels which have an integrated intensity of less than this value will be excluded from the fit. This may be used to exclude pixels which are too dim to give a good fit.

Note

The integrated intensity is calculated after accounting for the specified background and is not smoothed.

Time Min. and Time Max. The limits of the valid FLIM signal. Time bins or gates outside of these values will be excluded from the fit. Use this to remove, e.g. TCSPC data which falls outside of the linear range of the TAC.

Counts/Photon The average number of digital counts recorded per photon. This is used to correct the $$\chi^2$$ value to account for the gain provided by the intensifier and camera.

Tip

For TCSPC data this should be left at 1.

Rep. Rate The repetition rate of the laser used to acquire the data, in MHz. This is used to correct for incomplete decays.

Warning

This value must be set correctly; if it is set incorrectly the fitting may fail or produce a very poor fit.

Gate Max. Saturation limit of the data. Pixels which have a time bin or gate above this limit will be excluded from the fit. This can be used to exclude saturated pixels which will bias the fit.

Tip

For 16-bit TCSPC data (e.g. B&H, LaVision) use 65,536.
For 12-bit time gated data use 4,096.

### Background tab¶

The settings in this tab are used to set the background subtraction. These options subtract a background from the data before fitting.

Tip

A background due to detector dark noise or stray room light should be added to the model rather than being subtracted, otherwise the fit will be weighted incorrectly. See Dealing with backgrounds for more information.

Background The type of background to subtract:

 None No background value Single Value A constant value, e.g. camera offset Image A background image, subtracted uniformly from every time bin or gate. Use to correct for a non-uniform background due to, e.g. camera non-uniformities TV Image A time varying background (TVB) image. Use to account for time and spatially varying background fluorescence, e.g. due to plate fluorescence with non-uniform illumination

Background Value The background value subtracted if Single Value is selected in Background.

### IRF tab¶

The settings in this tab relate to the Instrument Response Function (IRF) and how it is processed. See Instrument Response Function for more information.

IRF Type The type of IRF loaded

 Scatter An IRF recorded with a scattering sample or raman signal. Reference A mono-exponential decay recorded with a reference dye.

Reference Lifetime The lifetime of the reference decay. This value will only be used if IRF type is set to Reference.

Background The level background present in the IRF. The background signal level is estimated automatically after loading, see Estimate IRF Background option in the IRF menu.

BG is Afterpulsing. Whether the background in the IRF is due to afterpulsing. If so, the background value is used to compensate for the afterpulsing, otherwise the background value is subtracted from the IRF.

Time Min. and Time Max.` The limits of the valid FLIM signal. Time bins or gates outside of these values will be removed from the IRF.

IRF Shift Shift in picoseconds to apply to the IRF.

G Factor The relative sensitivities of the parallel and perpendicular detector channels for polarisation resolved data.


text:
# What is the domain and range of y= 1 / (x-3) ?

Mar 20, 2017

Domain: $\mathbb{R} - \left\{3\right\}$, or $\left(- \infty , 3\right) \cup \left(3 , \infty\right)$
Range: $\mathbb{R} - \left\{0\right\}$, or $\left(- \infty , 0\right) \cup \left(0 , \infty\right)$

#### Explanation:

You can't divide by zero, meaning the denominator of the fraction cannot be zero, so
$x - 3 \ne 0$
$x \ne 3$
Thus, the domain of the equation is $\mathbb{R} - \left\{3\right\}$, or $\left(- \infty , 3\right) \cup \left(3 , \infty\right)$
Alternately, to find the domain and range, look at a graph:
graph{1/(x-3) [-10, 10, -5, 5]}
As you can see, the x never equals 3, there is a gap at that point, so the domain doesn't include 3 - and there is a vertical gap in the range of the graph at y=0, so the range doesn't include 0.
So, again, the domain is $\mathbb{R} - \left\{3\right\}$, or $\left(- \infty , 3\right) \cup \left(3 , \infty\right)$
And the range is $\mathbb{R} - \left\{0\right\}$, or $\left(- \infty , 0\right) \cup \left(0 , \infty\right)$.

NOTE: Another way to find y that may or may not be allowed (solving for x):
Multiply both sides by x:
$y \left(x - 3\right) = 1$
Divide by y:
$x - 3 = \frac{1}{y}$
$x = \frac{1}{y} + 3$
Since you can't divide by zero, $y \ne 0$, and the range of y is $\mathbb{R} - \left\{0\right\}$ or $\left(- \infty , 0\right) \cup \left(0 , \infty\right)$.


text:
# My quadratics

Algebra Level 4

If $$a+b+c=0$$ and $$a$$, $$b$$ and $$c$$ are rational, the roots of the equation $$(b+c-a)x^{2}+(c+a-b)x+(a+b-c)=0$$ are

×

Problem Loading...

Note Loading...

Set Loading...


text:
logarithm (1 synonym found)

### Sort logarithm by:    Alphabetically | Length | Random | Definition

Definition of logarithm:

(n) : for a number $x$, the power to which a given base number must be raised in order to obtain $x$. Written $\log_b x$. For example, $\log_\left\{10\right\} 1000 = 3$ because $10^3 = 1000$
History
This word has been
viewed 1,790 times.

Definition of logarithm:

(n) : for a number $x$, the power to which a given base number must be raised in order to obtain $x$. Written $\log_b x$. For example, $\log_\left\{10\right\} 1000 = 3$ because $10^3 = 1000$


text:
This is part 8 of an ongoing Tuesday Sanity Check series on Getting Things Done (GTD) for Academics. You can find the first seven posts here: Setting the Stage, Engaging the System, Acquiring the Habits, Collect, Process, Planning, and Doing.

So far in this series, I have stressed that Getting Things Done, GTD, is a system for making intelligent choices about what you should be doing and what you should not be doing at any given point in time. It is predicated on some basic habits: Collecting, Processing, Planning, and Doing. These four habits form the core of what Leo Babauta calls "Mimimal ZTD" in his Zen To Done system. These habits, when done consistently, add up to a dramatic improvement in productivity and peace of mind. Although it can be hard work to build those habits, it's not rocket science, and anybody --- even college professors --- can do it.

One mistake a lot of new GTDers make is in thinking that GTD is not about these habits or about behavioral change, but about tools --- apps, fancy notebooks, and so on. This is like saying that getting healthy is about having good exercise equipment. You can buy an expensive treadmills, but without the behavioral change and the consistent practice of basic habits, that treadmill will just collect dust and good health will seem elusive and complicated. GTD isn't complicated. But if you try to practice it without the habits in place, it's not likely to stick.

That said --- if you are making an effort to practice the basic habits, there's a lot to be said about having good tools. This post in the series is going to focus on the fifth habit in GTD and ZTD, the habit of using a simple trusted system for housing and managing information about tasks and projects. Having a good "tool stack" for using GTD is important for making the practices of collecting, processing, planning, and doing more fluid and frictionless, more a part of the way you think.

And by "good", we mean simple and trusted. Use only the simplest possible tools needed to implement the basic habits, and use only tools that reduce the amount of time and energy you spend on those habits rather than the ones that add to that expenditure. The rule is that the best GTD tools are the ones you actually use.

## What a good system looks like

At the absolute minimum, a GTD implementation should include:

• A calendar. As you collect and process your stuff, a lot of it will have a time and/or date constraint. Those items should go on the calendar, along with the times that you set up in the Planning stage every week to work on your Big Rocks and MIT's.
• A collection of lists. Not a single, monolithic to-do list (which is not a good system) but lists, plural, that contain action items, "Someday/Maybe" lists, "Waiting For" lists, and other lists for the stuff that you process.
• A place to store non-actionable but useful information. Remember that in processing, we separate the actionable stuff from the non-actionable stuff. Actionable stuff goes on a list (above). Non-actionable stuff that isn't to be deleted needs a home too, a place where you'll always look and where you can search it up easily when needed.

Notice that none of these elements strictly requires the use of a computer. You can use plain paper calendars, spiral notebooks, and filing cabinets and have a highly functioning GTD system as long as the basic four habits are in place. However, in my view, using computer software is the way to go, because the computer tools tick off all the boxes above, and also because the idea of having all my tasks and projects in a single notebook that could be easily lost or destroyed totally freaks me out.

Whatever form your tools take, they must be

• Simple: The act of using the tool should be a nearly frictionless experience, and it shouldn't take a long time to learn how to use the tool. For example, a hammer is the epitome of a simple tool. Hammers don't come with instructions. Everybody knows how to use one intuitively. However, the flip side of this analogy also holds: Hammers don't really do much besides pound things. While a creative person might find multiple uses for a hammer, at some point the job might be complex enough that they need to move up to a more complex tool. But even then, simplicity is important. For example, my power drill does stuff that a hammer doesn't. It's more complicated than a hammer, but it's still pretty simple to use. The point here is that while simplicity is paramount, it's also a moving target based on your needs and experiences. As you gain more experience with GTD, you can consider using more complex tools. But start simple and always keep "simplest possible tool" in mind as the main idea.
• Trusted: You should spend little to no energy thinking about the tool itself after you have it set up. That seems ironic when we're in the midst of a long post about tools, but the whole purpose of a tool it to allow you to do more on the job for which you bought the tool in the first place. Example: Around here in Michigan, it snows a lot. I have a tool, a snowblower, that helps me get the snow off my driveway. I need to be able to trust my snowblower. That means that when I'm using it, I want to focus on getting snow removed, not on whether the snowblower is functioning properly. And I do not ever want to think about the snowblower when I am not using it, unless absolutely necessary. Similarly, once you get comfortable with GTD tools, whatever they look like, they should fade into the background and stay there, occupying close to zero of your mental space.

A point to go along with that last sentence: Don't spend a lot of time jumping around from one tool to another. Experiment, but eventually pick a tool and stick with it. This is hard for me, because I am constantly trying out new apps and services and dealing with an ongoing sense of digital FOMO. But in the end, I find I'm a lot more productive if I just stick with what works for me rather than expending insane amounts of time setting up my stuff in one app after the next.

## My system: The basics

So now, let me try to explain the "tool stack" that I've set up to implement GTD.

Disclaimer What I'm about to describe might be way more complex than a new GTD person might be ready to handle. That's OK and nobody should feel obligated to do things the way I do them. If you're new to this, keep reading, but you'll want to think about What subset or mutation of this guy's setup would work for me? My setup isn't perfect, and it's undergoing constant evolution. But it's the result of a long period of trial and error and it works well enough for me now --- no more, no less.

The main components of my GTD setup are:

• Google Calendar, which I use as, well, my calendar. There's nothing earth-shattering about how I use it. When stuff gets processed that is more of an event than a task, it goes on the calendar.
• ToDoist, which I use for managing my lists of actions. This is the heart of my GTD setup, and I'm going to go into mind-numbing detail on this in a second.
• Evernote, which I use as my digital filing cabinet. Anything that is not actionable but which is not to be deleted or delegated, goes into Evernote. That includes emails; stuff I clip off the web; scans of paper documents; files (sometimes); drafts of notes; and a lot of other stuff. I have quite a bit to say about Evernote and so I am going to make a separate post about it later, rather than make this one any longer.

I have a few ancillary tools that I use to help out, like Google Keep which I mentioned in an earlier post for capturing quick notes. But 95% of my operations use some combination of the three tools above.

## ToDoist

ToDoist is the glue that holds my GTD together. At its core, ToDoist is nothing more than an app for making lists. It does one thing; but it does it so well, and has so many useful features, that it stands out in a crowded field of similar task management apps. I don't think ToDoist is perfect, but I've tried many different to-do list apps and I keep coming back to this one because it really fits the way I want to work.

ToDoist is available on Windows, Mac, Android, and iOS and can be accessed through a web browser if all else fails. (This is one of the main reasons I use it.) It's free to download, but some of the features I'm going to describe require the Premium version of ToDoist, which costs \$28.99 per year. (Which I consider one of the better deals in all of paid software.)

Lists in ToDoist are referred to as Projects which is how we think of them in GTD. There's no one right way to set up a Project list in ToDoist. You could have one monolithic list if you wanted, as a single Project. But in GTD it's better practice to have one list per project. Within each project are the items in the list. We'll refer to these as "tasks", but in reality you could use ToDoist for any list at all regardless of what's in the list. In ToDoist, there is one default project, the Inbox. But you can create as many of your own as you need in addition to the Inbox.

When you enter a task in ToDoist --- which can be done in several ways, including a Quick Add feature that's triggered by a keyboard shortcut --- you type out the task itself, and then you can do several different things to it:

• Set a due date and time, using either a dropdown calendar menu --- or by just typing out the due date and time, and ToDoist automatically reads it and converts it into a due date.
• Add a label, which is a piece of metadata attached to the task, by typing the @ symbol followed by the label text (for example @email). In David Allen's GTD language, these labels are contexts that will help you filter tasks according to certain rules (see below).
• Add a priority level, defined to be "level 1" (highest priority) to "level 4" (lowest), by typing p1, p2, p3, or p4 or clicking on a flag icon colored red (level 1), orange (level 2), or yellow (level 3).
• Add a reminder, which will send you an email or a system notification or both, at either a certain time or when you enter a certain location.
• Add a recurrence so that the tasks repeats on a fixed cycle. Do this just by typing in the period of the cycle, for example "daily" or "every Monday, Wednesday, and Friday". (ToDoist's capabilities of understanding English are very impressive.)

You can do all of these to each task, or none of them.

Every project in ToDoist comes with a unique email address, so when you are processing your email inbox, if an email contains actionable information, you can just forward it from your inbox straight into ToDoist. I only ever use the email for the Inbox project, but theoretically any project at all can have emails forwarded into it. This is an essential feature of to do list apps for GTD because it makes it totally easy to keep your email inbox at zero. A large portion of my processing habit consists of forwarding emails into ToDoist.

For me, what makes ToDoist so useful for GTD is its ability to search and filter the tasks that I've entered. There's a search box on the ToDoist screen that can accept Boolean search queries. For example if I want to find all the tasks that have the string "repo" in them, I just type in "repo", which will return not only tasks but also projects that have this string in them:

You can search using almost any data you've entered, including priority level, due date, labels, and project name. For example, I can search for all the tasks that are due before June 1 by typing due before: June 1:

I can search for all tasks that are either priority 1 or priority 2 and which have no due date by typing (p1 | p2) & no date:

I'll say a little more about searching below. Suffice to say that the search syntax, described in full here, is simple and powerful for slicing and dicing your way through a big list of hundreds of things to do. Every search can be saved into what's called a filter which are housed in anarea on the main ToDoist screen so you can implement common searches with one click.

ToDoist has lots of other features as well, like the ability to set up shared lists with other users and the ability to export tasks to Google Calendar. It's well worth exploring once you've read this post.

## My ToDoist setup

Now that I've described ToDoist in general, let me describe the particular way I have it set up.

First, I have five main project areas: NEXT, PROJECTS, Recurring, Future, and Someday/Maybe:

The last three are easiest to explain. Recurring is a "project" that's just a list that holds any task that repeats. I find it convenient to have these in a separate space from non-repeating tasks because when I cross off a task from a project, I like to see it gone. Future holds tasks that I need to get done in the future but which don't need attention yet; when I collect and process such a task, I enter it into Future and stick a reminder on it that gives me a notification at the point in the future when I need to start thinking about it. (This is the analogue of the tickler file, a staple of paper-based GTD implementations.) And Someday/Maybe is a place where I put stuff that I "maybe" want to do "someday" but which isn't actionable right now.

The NEXT and PROJECTS areas hold the vast majority of my tasks. Both of these are containers for a bunch of projects housed inside them, and in each of those project are tasks that I decide are actionable when I do my processing. First, here's NEXT:

Tasks inside NEXT are "stand-alone" tasks like "Add editorial board information to the CV" or "Schedule kids' passport appointment". Although these don't belong to a particular project, they do fall into distinct categories in terms of my life commitments. Those are the categories you see listed: Teaching, Scholarship, Service, Professional, Speaking, Writing, Personal, and Reading. Adding info to my CV is a Professional task, for instance; scheduling my kids' passport appointments is a Personal task.

The reason I have those categories set up is because it forces me to process my actions in terms of the bigger picture. When I collect something and then process it, and decide it's actionable, I have to stop and think about where it fits in my life. By extension I have to think about whether I should be doing that thing at all if there's no clear answer to the question of where it fits. This extra step adds friction to the system, but it's a big help in keeping my day-to-day work connected to the larger context of work-life balance.

Below NEXT is PROJECTS, which is a container for all my actual projects:

And each of these projects contains a list of tasks associated with those projects, like this one:

A person getting started using ToDoist could just leave it at that -- projects that contain tasks. I go a few steps further by adding extra data to those tasks.

First there labels. The labels I use the most are:

Label What it's for
@next Designates the "next action" that needs to be taken on each project
@el ("energy low") Tasks that can be done with low energy
@eh ("energy high") Tasks that require high energy
@ts ("time short") Tasks that can be done in 2--5 minutes
@tl ("time long") Tasks that take more than 45 minutes
@email Tasks that involve sending email
@campus Tasks that can only be done on campus
@office Tasks that can only be done at my office
@home Tasks that can only be done at home

I have a handful of lesser-used labels like @phone for tasks that involve phone calls. But I use these so infrequently I'm gradually starting to phase them out to keep the labels as few as necessary. In fact the vast majority of my tasks only use a combination of the first five labels above -- denoting only whether an action is a "next action" and what sort of time and energy requirements it has. And I only take the time and energy requirements that are on the extreme ends; I used to have labels for "medium-time" and "medium-energy" but now I just assume that anything that isn't on the high or low end is medium. Many tasks have no labels at all because they have no special requirements for time, energy, location, or tools.

Many of my tasks use priority levels as well. Priority 1 (red) indicates an MIT. Priority 2 (orange) is for tasks that are not MIT's but which are high priority. Priority 3 (yellow) is for stuff that needs my attention. Priority 4 (white) is everything else.

All of these task items and the metadata they carry are entered into ToDoist when I process. The workflow goes like this: Once I decide an item in my inbox is actionable:

1. Enter it into ToDoist (unless it's really a project and not a single task)
2. Add labels for time, energy, next action, etc. if it needs any
3. Add priority level if it needs one
4. Add a due date if it has one
5. Move the task to an appropriate project or one of the life-areas in the NEXT project.

I just loop through this process one task at a time until the inbox is clear. ToDoist makes this very easy through an ingenious implementation of natural-language processing. Here's how that works.

For example, this morning I remembered that I need to book a flight for an upcoming speaking engagement in Spain. Like a good GTDer, I captured that thought immediately rather than try to remember it. I know what the task is ("Book flight to Spain"). I figure this will take about 15 minutes, but I need to have my full attention on the task so I consider it high-energy. It's not urgent, but needs attention (to prevent it from becoming urgent) so I estimate that's a priority level 3. Moreover, the host institution told me they need to have that by March 12. Also there's no particular place I need to be and no particular tool that I need (other than my computer, which is always with me). And it belongs in the project "UBC talk" (UBC = initials of the university that's hosting me). I also happen to know this is not the next action to take on this project.

All of this mental processing takes me about 5 seconds. Entering the task into ToDoist with all that info doesn't take much longer:

And that's that. The task is now in the system, in its appropriate place with all the metadata I need, and out of my brain and inbox. (This was using the Quick Add feature; it's just about as fast and frictionless if you enter it in the app.)

If you have an email inbox that's sitting at 300+ emails right now, like a lot of professors, do the math for a second. Assume that half of those emails do not contain actionable tasks, so they'll be either deleted or filed away --- at any rate, they are easy to remove from the inbox. If you spend 1 minute on each of the remaining tasks doing mental processing and then sending them into ToDoist like I just described --- and that's probably an overestimate of how much time it takes --- you can get your entire inbox zeroed out in less than 3 hours. One dedicated Saturday, or a day during spring or summer break, can be enough to get you out of email hell once and for all. It's totally doable for anybody with a simple, trusted system.

## Using the setup

I've added structure to my ToDoist setup that isn't strictly necessary, with all the labels and priority levels and so on. Beginners probably shouldn't use that level of structure when they are getting started. But where all of this structure pays off, once you're comfortable with it, is when you start using ToDoist's search capabilities.

Remember in this post we posed a bunch of scenarios professors encounter and asked, In each case, what could I be doing, what should I be doing, and what should I not be doing? Using ToDoist's search capabilities, we can query our tasks lists and get a custom-made menu of next actions that are perfectly suited to the situation.

For example: It’s 8:45am on a Monday. My three-hour block of teaching starts in 15 minutes (and the classroom is across the hall). I have plenty of energy but not much time. What could I be doing in this situation? To find out, I just open ToDoist and ask it to show me all the tasks that are short in time duration. I can do that by typing @ts into the search field and I get:

This happens to be the only short-time task I have right now. So this is is a perfect fit; I need to get it done anyway, and it's the right thing to be working on given the moment I'm in.

Or, suppose that I'm entering into a block of time I set aside for working on my MTH 325 class I'm teaching right now. I'm feeling high-energy and I want to focus on the highest-priority MTH 325-related tasks (say, level 1 or level 2), or those tasks that are due within the next week. What should I be doing? I can ask ToDoist by entering the search: (p1 | p2 | 7 days) & #325 (which in English would read: "priority 1, or priority 2, or due within 7 days; and the project name contains the string 325"):

And just like that, out of hundreds of tasks that I need to get done, I have a short menu of four tasks that I should be working on because they fit the moment. I can just pick whichever one off of this menu that I want to do the most.

I cannot overstate how awesome and crucial it is that I can slice my task lists according to priority, context, time required, energy level, project group, due date, tool availability and so on. It's like having a superpower. You can cut a massive list of tasks down to a very short list, all of which are equally perfectly suited to the moment --- then just pick the one you want the most.

I also have a few filters set up, which are just saved searches that I can do with one click:

• next actions ((p:NEXT) | @next) which gives me a master list of all next actions, both the ones that belong to projects and the ones in the disaggregated NEXT area. When I need to get something done but don't have a search criterion, I just pull up this list and pick one thing off of it to do.
• MIT's (p1). This returns a list of all priority 1 tasks.
• Quick ((@next | p:NEXT) & @ts). Returns a list of all next actions that take less than 5 minutes.
• Brainless (@el & (@next | p:NEXT)). Returns a list of all low-energy next actions, perfect for those times when I'm tired out and can't think straight. Even if I'm wiped out, I can still find something to do, like moving a stack of papers from one place to another or deleting a GitHub repository. The capability of GTD to allow me to be productive even when I am in a zombie-like state was what originally sold me on this philosophy, back around 2004 when my wife and I started having kids and sleep was, to put it mildly, elusive.

## Conclusion

GTD is not about tools but about behavioral change. There are habits that have to be developed before any productivity system or philosophy has a chance of gaining traction. But, good tools do support good habits, and help them to become more, well, habitual. "Good" tools are the ones you actually use, and with GTD it's important to pick simple tools that work like the air conditioning and heating in your house --- they make life better and you don't think about them unless they break, which shouldn't be often.

A companion post I'll put up later will talk about Evernote which is the other part of my system. Stay tuned for that in the next few days.

My personal challenge to you: This week, keep working on those first four habits, and scout out tools that you like and that you could use to support them. Tell us about those in the comments and how you plan to use them.


text:
# Skills Handbook - Exponents - Exercises - Page 806: 10

252

#### Work Step by Step

In order to write a number in standard form, we have to take it out of exponential form. 6 is raised to the second power and 7 is raised to the first power, so we know that this is the same as 6 multiplied by itself and then multiplied by 7 once: $6 \times 6\times 7= 252$

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


text:
# Implementing an Adjacency List in C

I haven't yet implemented any sort of graph thus far in C and decided to give it a go by trying to implement an adjacency list in C. Is there anything in my code that you see that I can improve and is there any other sort of basic functionality that is missing in my adjacency list and should be added?

Graph.h

#ifndef GRAPH_H_INCLUDED
#define GRAPH_H_INCLUDED

typedef enum {UNDIRECTED = 0, DIRECTED} Edge_t;

typedef struct Graph
{
Edge_t typeOfGraph;
int totalVertices;
int totalEdges;
}Graph_t;

Graph_t *createGraph(int, Edge_t);
void deleteGraph(Graph_t *);
int doesEdgeExist(Graph_t *, int, int);
int calculateNumOfEdges(int, Edge_t);
void displayGraph(Graph_t *);

#endif

Graph.c

#include <stdio.h>
#include <stdlib.h>
#include "Graph.h"

{
int vertex;
};

{
int totalMembers;
};

{

if(newNode == NULL)
{
return NULL;
}

newNode->vertex = vertex;
newNode->next = NULL;

return newNode;
}

Graph_t *createGraph(int totalVertices, Edge_t typeOfGraph)
{
int i;

Graph_t *graph = (Graph_t *)malloc(sizeof(Graph_t));

if(graph == NULL)
{
return NULL;
}

graph->totalVertices = totalVertices;
graph->totalEdges = 0;
graph->typeOfGraph = typeOfGraph;

{
free(graph);
return NULL;
}

for(i = 0; i < totalVertices; i++)
{
}

return graph;
}

void deleteGraph(Graph_t *graph)
{
if(graph != NULL)
{
{
int vertex;

for(vertex = 0; vertex < graph->totalVertices; vertex++)
{

while(listIterator != NULL)
{
listIterator = listIterator->next;
free(temp);
}
}

}

free(graph);
}
}

void addEdge(Graph_t *graph, int src, int dest)
{
if((src >= graph->totalVertices || src  < 0) || (dest >= graph->totalVertices || dest < 0))
return;

if(doesEdgeExist(graph, src, dest))
return;

if(newNode != NULL)
{
graph->totalEdges++;

if(graph->typeOfGraph == UNDIRECTED)
{

if(newNode != NULL)
{
graph->totalEdges++;
}
}
}
}

int doesEdgeExist(Graph_t *graph, int src, int dest)
{

while(srcVertexPtr != NULL)
{
if(srcVertexPtr->vertex == dest)
{
return 1;
}
else
srcVertexPtr = srcVertexPtr->next;
}

return 0;
}

int calculateNumOfEdges(int totalNumberOfEdges, Edge_t typeOfGraph)
{
/*
I'm assuming the graph has no self loops or multi-edges.
*/

if(typeOfGraph == UNDIRECTED)
{
}
else
}

void displayGraph(Graph_t *graph)
{
int vertex;

for(vertex = 0; vertex < graph->totalVertices; vertex++)
{

printf("Vertex %d is adjacent to ", vertex);

while(listIterator != NULL)
{
printf("%d->", listIterator->vertex);
listIterator = listIterator->next;
}

printf("NULL\n");
}
}

Main.c

#include <stdio.h>
#include <stdlib.h>
#include "Graph.h"

int main()
{
Graph_t *uGraph = createGraph(5, UNDIRECTED);

if(uGraph == NULL)
{
printf("Couldn't not allocate any memory. Terminating Program. ");
exit(EXIT_FAILURE);
}

printf("Undirected Graph\n\n");
displayGraph(uGraph);

printf("\nThe total number edges in this graph is %d\n",
calculateNumOfEdges(uGraph->totalEdges, uGraph->typeOfGraph));

deleteGraph(uGraph);

return 0;
}

Output

Your code is pretty clean, but it's unclear what you imagine it being used for, except possibly just to illustrate adjacency lists. If you're looking to build something actually useful, then I suggest coming up with a few model applications, and thinking about what you need to provide to serve those applications efficiently and easily. For example, how would you model a shortest-path problem with your graph, and how would you implement Dijkstra's algorithm to solve that problem? Or how would you compute (and represent) a minimal spanning tree?

### graph.h

• Graph_t.totalVertices and Graph_t.totalEdges cannot meaningfully take negative values, so you should consider declaring them to have unsigned types. If you do change these, however, then you'll need to propagate corresponding changes throughout.

• Your function prototypes do not include parameter names. Although these are optional, including them makes the header easier to understand, and it may help the compiler produce more meaningful diagnostic messages.

• You declare function createAdjacentNode(int), returning AdjListNode_t *. There's nothing inherently wrong with that, but AdjListNode_t is an opaque type, and you've no other function or other data type declared that uses AdjListNode_t, so what use does anyone who cannot see the definition of AdjListNode_t have for createAdjacentNode(int)? It looks like that function wants instead to have internal linkage in graph.c (and be omitted from the header).

• public function calculateNumOfEdges() sticks out as an oddball on account of accepting as its arguments members (presumably) of a Graph_t, instead of accepting a Graph_t *. Your API is mixing different levels of abstraction.

• As a matter of style (only), I recommend against spelling names in camel case in C code.

• The suffix _t for type names is reserved by POSIX. That perhaps can be ignored if you don't intend your code to be used in a POSIX environment. In practice, your use of that suffix will probably not present a problem even in a POSIX environment, but as a matter of form, I recommend against using it.

### graph.c

Your implementation code is fairly straightforward, but it does present a few issues:

• do not cast the return value of malloc() in C
• addEdge() fails silently in the event of an allocation error. It should somehow alert the caller if it is unable to add the requested edge.
• When operating on an undirected graph, an allocation failure in the wrong place can make addEdge() add just one (directed) edge instead of the two it normally would add. This failure mode is also silent.
• doesEdgeExist() is a public function, so it ought to perform proper argument checking. As it is, if its first argument is a null pointer, the function will attempt to dereference it, and if its second argument is not a valid vertex number for the provided graph then an out-of-bounds array access will be performed.
• the (documented) assumption by calculateNumOfEdges() that the graph has no multi-edges is reasonable because addEdge() enforces it. If you want to also assume (as you do) that there are no self-edges, then you should enforce that constraint, too.

### Missing features

You asked about missing features. I can imagine a lot of things that you could provide, but here is a fairly minimal list of additional things I think you should provide:

• Many graph-based algorithms rely on or work with data associated with edges and / or vertices, such as tags and weights. You have no built-in mechanism for associating any such data with either vertices or edges.

• On the other hand, all the adjacency information is opaque to users anyway, so there are not many interesting things one can actually do with the graphs you provide for.

• There is no mechanism for adding or removing vertices.

• There is no mechanism for deleting edges.


text:
• RAMAKRISHNAIAH C R

Articles written in Journal of Earth System Science

• Spatial variability of ground water quality: a case study of Udupi district, Karnataka State, India

Groundwater is a reliable source of fresh water for domestic and agricultural water users. It supports subsurface ecosystem by balancing the geo-biological and bio-geochemical processes at micro- and macro-scales of the ecosystem. Overexploitation, anthropogenic activities and improper agricultural practices have contributed to the pollution of groundwater sources all around the globe. The water quality index (WQI) is the most extensively used indicator which transforms the water quality information derived from several parameters into a single value/rating to categorize and provide a general perception of water quality standard. Groundwater quality analysis and mapping via geographical information system (GIS) proves to be beneficial in identifying the locations where the groundwater quality is deteriorating. In the current study, the WQI of groundwater was determined for the samples collected from open and tube wells located within the Udupi district of Karnataka state, India. The groundwater quality parameters such as pH, hardness, calcium, chlorides, nitrates, iron, fluoride, sulfates, manganese, sodium, magnesium, potassium, turbidity, and phosphate were analyzed for water samples collected from 112 randomly chosen open/tube wells in order to determine the WQI. Interpolation approaches such as inverse distance weighting (IDW) and kriging were adopted in the GIS environment to quantify the spatial variability of groundwater quality over the entire geographical area. The groundwater quality maps were generated using the best fit models. The results portray that, the accuracy of interpolation using IDW and kriging methods was dependent on the measures of central tendency and variability of water quality data of different parameters. The kriging interpolation was much accurate for most of the groundwater quality parameters compared to IDW maps. The WQI maps, perhaps signposted the poor quality of groundwater quality in about 1.88% of the geographical area of Udupi district. Further, about 21.69% of the area was affected by poor quality of groundwater where suitable strategies for replenishment of groundwater resources should be taken up by the concerned authorities. The spatial distribution maps of groundwater quality aid to locate vulnerable places where immediate action is required.

$\bf{Highlights}$

$\bullet$ Water quality index mapping clearly depicts the critical areas that need policy measures for the groundwater sustainability.

$\bullet$ Current study acts as a decision support system for taking up water quality management activities for groundwater remediation in the study area.

$\bullet$ Geostatistical methods prove to be ideal for the evaluation of spatial groundwater quality assessment and distribution.

• # Journal of Earth System Science

Volume 130, 2021
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019


text:
### Search for supersymmetry in events with a $\tau$ lepton pair and missing transverse momentum in proton-proton collisions at $\sqrt{s} =$ 13 TeVSearch for supersymmetry in events with a $\tau$ lepton pair and missing transverse momentum in proton-proton collisions at $\sqrt{s} =$ 13 TeV

Access Restriction
Open

 Source arXiv.org Content type Text File Format PDF Language English
 Learning Resource Type Article


text:
Examveda

# If 80% of A = 50% of B and B = x% of A, then the value of x is :

A. 400

B. 300

C. 160

D. 150

### Solution(By Examveda Team)

80% of A = 50% of B
⇒ $$\frac{80}{100}$$A = $$\frac{50}{100}\,$$B
⇒ 8A = 5B
⇒ A = $$\frac{5}{8}$$B
Put value of A in given equation,
B = x% of A
⇒ B = $$\frac{x}{100}$$ × $$\frac{5}{8}\,$$B
⇒ x = $$\frac{100 × 8}{5}$$
⇒ x = 160
Related Questions on Percentage


text:
# Are questions of the form "has this ever been studied?" appropriate?

Are requests of the form "has this ever been studied, and if so, may I please have a reference?" considered appropriate?

But let me be more specific.

So firstly, there's the point-set or "classical" approach to topology, which concerns itself with ordered pairs $(X,\tau)$ called topological spaces.

Then there's the pointless approach to topology, which concerns itself with lattices $(\tau,\wedge,\vee)$ called frames (in which finite meets distribute over arbitrary joins.)

I'm interested in a concept halfway between the two. We might call it "the classical approach, but with lattices." Rather than $(X,\tau)$, we concern ourselves $(P,\tau),$ where $P$ is a lattice that is isomorphic to a powerset lattice, and $\tau$ is a subset of $P$ that is closed with respect to arbitrary joins etc.

The motivation for this idea is as follows: we may be able to weaken the requirement that $P$ needs to be isomorphic to a powerset, and still be able to develop classical topology just fine.

So my question is, has this idea been studied before, and if so, may I please have reference recommended?

• One problem with this type of question is that they are semi-answerable: if the object has been studied, one can point to a reference, but there is no way to show that the subject has not been studied. It may have been studied, for example, and found to be useless so nothing was written as a result. Jan 28 '13 at 17:41
• @Mariano This comes to mind... ;-) Jan 28 '13 at 20:07
• @MichaelGreinecker, well, that is somewhat different. There can't be a standard notion of something for which one cannot give references! :-) Jan 28 '13 at 20:38
• Ask away, but now, ask it at math.stackexchange, the main site! Feb 1 '13 at 18:05


text:
If LCM of and is and the HCF of and is , then the value of is? - Padhle.Online
Students' favourite free learning app with LIVE online classes, instant doubt resolution, unlimited practice for classes 6-12, personalized study app for Maths, Science, Social Studies, video e-learning, online tutorial, and more. Download Now!

# If LCM of and is and the HCF of and is , then the value of is?

Solution

## Calculate the value of $p$:We know that the relation between the HCF and the LCM of any two numbers $a$ and $b$ is given by,It is given that, And, So, the numbers are $p$ and $12$.Now, using the above formula,$⇒$  $⇒$ $⇒$ $⇒$ Or, Hence, the required value of $p$ is $8$.

brings to you the BEST FACULTY for students of Class 9th - 12th. I (Balkishan Agrawal) aim at providing complete preparation for CBSE Board Exams (Maths) along with several other competitive examinations like NTSE, NSO, NSEJS, PRMO, etc. & Maths…


text:
1. ## Basis

I have no idea how to start solving this problem. My professor gives really bad notes. I know you guys dont like to work problems without the asker doing some work, but i dont know where to start. If someone could walk me through this i would greatly appreciate it.

I dont know how to format matrices so i just copied the problem from the website. Each set of numbers in parentheses should be in a 1 column matrix (top, bottom).

Any help is greatly appreciated.

Consider the basis of consisting of vectors
(-2, 5) and (-5, 4) .
Find in whose coordinate vector relative to the basis is
(-1, 6) .

2. You want to solve $
a\left[ \begin{gathered}
- 2 \hfill \\
3 \hfill \\
\end{gathered} \right] + b\left[ \begin{gathered}
- 5 \hfill \\
4 \hfill \\
\end{gathered} \right] = \left[ \begin{gathered}
- 1 \hfill \\
6 \hfill \\
\end{gathered} \right]$

or $\left\{ \begin{gathered}
- 2a - 5b = - 1 \hfill \\
5a + 4b = 6 \hfill \\
\end{gathered} \right.$

3. I used substition and got a=26/17 and b=-7/17.

Is that my answer? Cause the website is telling me it's wrong.

4. Well the way you have formated the question makes it very hard to read.
I may not have understood what is being asked.

Do you do correctly?

5. Here's a screenshot of the question. It looked like you had understood it correctly.

6. See if this is what is meant by that wording.

7. Awesome! That worked. Could you explain how that works, cause i was certain that i was solving for an a and b like you had originally said.

8. Originally Posted by manofsteele888
Could you explain how that works, cause i was certain that i was solving for an a and b like you had originally said.
Well I am not sure I can.
The way that question is worded is missleading I think.
But I am not someone who regularly works with linear algebra wording.
Perhaps someone else here can explain that wording.

9. Ok, thanks man. Much appreciated.

10. Originally Posted by manofsteele888
I have no idea how to start solving this problem. My professor gives really bad notes. I know you guys dont like to work problems without the asker doing some work, but i dont know where to start. If someone could walk me through this i would greatly appreciate it.

I dont know how to format matrices so i just copied the problem from the website. Each set of numbers in parentheses should be in a 1 column matrix (top, bottom).

Any help is greatly appreciated.

Consider the basis of consisting of vectors
(-2, 5) and (-5, 4) .
Find in whose coordinate vector relative to the basis is
(-1, 6) .

.


text:
Feeds:
Posts

## HANS ENGLER’S TRAVELING WAVE SOLUTIONS FOR TIME-FRACTIONAL DIFFUSIONS

The idea is to replace Engler’s stable density with the Wright function and systematically replace $1/\alpha$ (where $\alpha$ is a parameter for the stable distribution with $\beta/\alpha$).

The fundamental solution of $D^\beta_t u + A u = 0$ is

$W(x,t) = t^{-\beta/\alpha} f( |x| t^{-\beta/\alpha})$

Following Engler () consider $U_{\tau}(\xi) = F(\xi \tau^{-\beta/\alpha})$ where F is the cumulative distribution function, and set $u_{c\tau}(x,t) = U_{\tau}(x+ct)$.  Then verify that

$D^{\beta}_t u_{c\tau}(x,t) + A u_{c\tau}(x,t) = c\tau^{-\beta/\alpha} g_0(u_{c\tau}(x,t) + \beta/\alpha \tau g_1(u_{c\tau}(x,t))$

with $g_0(\zeta) = f(F^{-1}(\zeta)$ and $g_1(\zeta) = F^{-1}(\zeta) f(F^{-1}(\zeta))$.

These traveling wave solutions provide unbounded speed for spread, i.e., $lim_{t\rightarrow \infty} u(t,x) = 1$