Let $X:=S^1 \times S^1$ be the torus.   Is the zero and second singular cohomology group $\mathbb{Z}$, the first $\mathbb{Z}^2$ and otherwise vanishing?   To show this: One first computes singular homology which gives that the zero and second homology group equals $\mathbb{Z}$ and first homology group equals $\mathbb{Z}^2$. Using universal coefficient theorem and that homology groups are free over $\mathbb{Z}$, we get the desired.  I was just wondering, because I wanted to compute cohomology groups of the sheaf of locally constant function with values in $\mathbb{Z}$, which should coincide with the singular cohomology group, but in this case I get that the first cohomology group equals $\mathbb{Z}^4$.
I computed the homology groups of the torus, can someone tell me if this is correct? The calculation, not the result that is. Thanks!  The cells of $T^2$ are $e^0, e^1_a, e^1_b, e^2$  The chain groups are  $$ C_0(T^2) = \{ k e^0 |&#160;k \in \mathbb{Z} \} = \mathbb{Z}$$  $$ C_1(T^2) = \{&#160;k_1 e^1_a + k_2 e^1_b |&#160;k_1 , k_2 \in \mathbb{Z} \} = \mathbb{Z} \oplus \mathbb{Z}$$  $$ C_2(T^2) = \mathbb{Z}$$  $$ C_k(T^2) = \{ 0 \} , k &gt; 2$$  Now the homology groups:  $$ H_0(T^2) = \ker \partial_0 / im \partial_1 = \mathbb{Z} / 0 = \mathbb{Z}$$  where $im \partial_1 = 0$ because there is no chain in $C_1(T^2)$ whose boundary is a zero-chain in $C_0(T^2)$. (Is this reasoning correct?)  $$ H_1(T^2) = \ker \partial_1 / im \partial_2 = \mathbb{Z} \oplus \mathbb{Z}$$  where $\ker \partial_1 =  \mathbb{Z} \oplus \mathbb{Z} $, i.e. again everything maps to zero because there is no element in $C_1$ whose boundary maps to an element in $C_0$.  $im \partial_2 = 0$ again because there is no element in $C_2$ whose boundary is an element of $C_1$.  I don't want to use Hurewicz to do $H_1$.  $$ H_2(T^2) = \mathbb{Z}$$ using similar arguments as above.  Thanks for your help.  Edit  I posted the computations as an answer below. I got two up votes but I don't know by who so I'm not yet sure I can trust them...
Is there a way to set app-specific volume? call function declared below Is it possible to call a function which is declared below in bash?  Example  if [ "$input" = "yes" ]; then     YES_FUNCTION elif [ "$input" = "no" ]; then     NO_FUNCTION else     exit 0; fi  YES_FUNCTION() {   .....   ..... }  NO_FUNCTION() {   .....   ..... } I'm a bit overwhelmed by the available options in regards to the three major photo editing apps from Adobe.  Does anyone know of an article or site that lists the features of Lightroom, PS Elements, and PS, side by side?  I know that Lightroom revolves around edits and workflow, and that PS is the big kahuna of image editing.  Where does Elements fit in to this understanding? How could I execute a script in Shell when a shortcut key is pressed.   Essentially what I need is when a shortcut key is pressed the script should read from a file and display that content in the terminal. I have to deal with a large number of files nested inside of directories (one file per directory), resembling something like this:  fred/result.txt john/result.txt mary/result.txt ...   I am currently using the following command to process each of those files:  find . -maxdepth 1 -type d \( ! -name . \) -exec bash -c \"cd '{}' &amp;&amp; processResult result.txt\" \;   I am looking for something I can add into the end of this command that will change the filenames to fred.txt, etc. and then later move the file into the parent directory to eliminate the extra layer of directories.  What would be the best way to do this? Checking the way to stop duplicate names - validation rules or triggers Is it possible to fail an ability check on purpose? Here are the general guidelines regarding ability checks:     The GM calls for an ability check when a character or monster attempts an action (other than an attack) that has a chance of failure. When the outcome is uncertain, the dice determine the results.   In some cases, you could want to fail an ability check. If you are climbing a wall an want to drop off, you fail simply because you stop attempting to climb, but in other cases, success and failure are both tied to the actual action and may both have desirable outcomes.  Example 1: Spell scrolls  An evil adversary wants you to cast a spell scroll for him, but you have him figured out and want to destroy his spell scroll instead, seemingly by accident. The rules on spell scrolls read:     If the spell is on your class’s spell list but of a higher level than you can normally cast, you must make an ability check using your spellcasting ability to determine whether you cast it successfully. The DC equals 10 + the spell’s level. On a failed check, the spell disappears from the scroll with no other effect.   A failure means that you successfully destroy the spell scroll.  Example 2: Amulet of the Planes  You are in possesion of the Amulet of the Planes. Your party is being overrun by terrible monsters, and the only way to save your party is to sacrifice yourself by sending yourself and the monsters to another plane of existence.     While wearing this Amulet, you can use an action to name a location that you are familiar with on another plane of existence. Then make a DC 15 Intelligence check. On a successful check, you cast the Plane Shift spell. On a failure, you and each creature and object within 15 feet of you travel to a random destination. Roll a d100. On a 1-60, you travel to a random location on the plane you named. On a 61-100, you travel to a randomly determined plane of existence.   By walking up to the monsters and failing the check, you and the monsters would be sent to another plane of existence.  The question  The rules seem to assume that you always want to succeed when you perform an ability check. Sometimes, however, you could want to fail on purpose. A sensible DM would rule that if you are skilled enough to succeed the check, then you would undoubtly be able to fail it on purpose as well. What does the rules as written say -- if anything -- about this circumstance?  According to RAW, is it possible to attempt an ability check and intentionally fail? Is there a way to see who voted on your posts? Why does cancellation of dots $\frac{\partial \dot{\mathbf{r}}_i}{\partial \dot{q}_j} = \frac{\partial \mathbf{r}_i}{\partial q_j}$ work? I have some questions regarding suspended users:   Why do suspended users have a reputation of one? According to the default reputation of a new user, he/she will have reputation of one and he/she has the privilege of posting questions. When comparing this, this is getting violated when compared to suspended users' reputation. Even though they have one reputation point, they cannot post questions. As suspended users do not have privilege of posting questions, there is no use of making "Ask Question" visible. This is an unnecessary server hit. Am I right? As suspended users do not have the privilege of answering questions, there is no use of making "Answer Question" visible. Why does $k/m=\omega^2$ for harmonic motion? Can anyone please give me a proof for $k/m=w^2$ in simple harmonic motion?  I have tried energy conservation and Newton's laws as follows :  In the case of a mass-spring system,  $$F=ma =-kx\\ F=ma = mr\omega^2 $$  hence   $$\frac km  = \omega^2$$  Or   $$\frac12 mv^2 = \frac12kx^2$$  $$\frac km  =  \frac{V^2}{x^2}   =   \omega^2$$  therefore $k/m  =  \omega^2$  Are these valid and correct? I have been stuck in this for the  entire day. The strange thing is that opened scripts or newly opened ones are working fine. It's the new created C# scripts that are not working.  With not working, I mean the MonoBehaviour is not in light blue color it's in black color. In this screenshot I took, I marked it with a red circle.    What have I tried so far? I have closed Visual Studio and started over again.  Since it didn't solve it, I closed the unity editor and started it over again, to no avail.  I pressed 'clean solution', I pressed 'rebuild solution', I pressed 'build solution'.  What else can I do ? And why are the other scripts working fine? If I type gameo...it will auto complete it to GameObject but in this specific script it's just not working.  This is a screenshot of the editor. I have in the Hierarchy a Ladder object.  In the Assets I created a folder name Ladder and a script name Ladder. But this Ladder script is not working.    This is the Ladder script and this script was working before but once I created another script inside the ladder folder name Raise empty script the Raise script didn't work and now the Ladder is not working either. But other scripts are working fine.  using System.Collections; using System.Collections.Generic; using UnityEngine;  public class Ladder : MonoBehaviour {     public Transform charcontroller;      private bool inside = false;     private float heightFactor = 3.2f;      private void OnTriggerEnter(Collider other)     {             inside = true;     }      private void OnTriggerExit(Collider other)     {             inside = false;     }      private void Update()     {         if (inside == true &amp;&amp; Input.GetKey("w"))             charcontroller.transform.position += Vector3.up / heightFactor;     } } I am trying to find the name of a direct to TV movie, possibly a 2-3 part miniseries which I believe aired on the SciFi channel in the 90s. I don't remember who starred in it, but the movie opened with an astronaut attempting reentry whose shuttle ends up exploding. He awakes in a large bubble of air in a river and after reaching the surface and swimming to shore, meets several other people, from seemingly many different times throughout history (including one alien, who died on Earth).  Over the course of the movie, we learn the world is the afterlife, where everyone goes when they die. The major plot involves launching a steamboat (built by Mark Twain, no less) to travel up a river to the mountains, where the beings responsible for bringing everyone to this world are.  Does anyone know what the title of this movie/miniseries is? Does there exist a set L of pairwise disjoint lines, each line in an axis direction, in $\mathbb{R}^3$ such that their union is equal equal to $\{ x : x \text{ has at least one irrational coordinate}\}$? I have a trained logistic regression model that I am applying to a testing data set. The dependent variable is binary (boolean). For each sample in the testing data set, I apply the logistic regression model to generates a % probability that the dependent variable will be true. Then I record whether the acutal value was true or false. I'm trying to calculate an $R^2$ or Adjusted $R^2$ figure as in a linear regression model.  This gives me a record for each sample in the testing set like:  prob_value_is_true         acutal_value    .34                          0    .45                          1    .11                          0    .84                          0     ....                        ....             I am wondering how to test the accuracy of the model. My first attempt was to use a contingency table and say "if prob_value_is_true > 0.80, guess that the actual value is true" and then measure the ratio of correct to incorrect classifications. But I don't like that, because it feels more like I'm just evaluating the 0.80 as a boundary, not the accuracy of the model as a whole and at all prob_value_is_true values.  Then I tried to just look at each prob_value_is_true discrete value, as an example, looking at all samples where prob_value_is_true=0.34 and measuring the % of those samples where the acutal value is true (in this case, perfect accuracy would be if the % of samples that was true = 34%). I might create a model accuracy score by summing the difference at each discrete value of prob_value_is_true. But sample sizes are a huge concern here, especially for the extremes (nearing 0% or 100%), such that the averages of the acutal values are not accurate, so using them to measure the model accuracy doesn't seem right.  I even tried creating huge ranges to ensure sufficient sample sizes (0-.25, .25-.50, .50-.75, .75-1.0), but how to measure "goodness" of that % of actual value stumps me. Say all samples where prob_value_is_true is between 0.25 and 0.50 have an average acutal_value of 0.45. Is that good since its in the range? Bad since its not near 37.5% (the center of the range)?  So I'm stuck at what seems like should be an easy question, and hoping someone can point me to a resource or method to calculate an accuracy stastic for a logistic regression model. After what speed air friction starts to heat up an object? I understand that air friction cools off an object at low speeds. For example, if you blow on a spoon of hot soup, it cools off. Or if you swing a hot frying pan in the air, it cools off faster.  But at higher speeds, the situation changes in the opposite. For example, consider a meteor falling on Earth. It is so fast that it heats up to such a high temperature that it burns into ashes.  What is the critical speed in which an object starts to heat up?  For example, consider a spherical object with radius of 1 meter. Let its density be 1 g/cm3 if needed. And let the air pressure be 1 atm. Assume that the temperature of the object is 400 K and the temperature of the air is 300 K. Also assume that the specific heat constant of the material is 1 cal/(gr.K). What is the critical speed for this object after which its temperature starts to rise above 500 K? Rising edge pulse detector from logic gates The circuits I describe are entirely made of 7400 series logic gates (7402, 7404 and 7408 ic). I'm trying to build a rising (positive) edge pulse detector using logic gates.  The following circuit should work in theory:   (see )  I do not expect the short output to be seen through a LED so, to test it, I make it trigger an SR latch to its up state:    However, it doesn't have effect on the latch. So I added an inductor to delay the input to the NOT gates in the pulse detector circuit and it worked:   But now I can see the short flashing output through a LED which I should not because it would mean it is too long to work with a circuit like that:   Which intends to toggle the D Latch output on each clock rising edge (Note that this is a D Latch not a D Flip-flop)  And anyway there is no place for inductors in integrated circuits so there must be a way to do this only with logic gates. Can someone solve this mystery?  BTW It does not show in my schematics but I did put 10K pull-down resistors where there might be floating pins. It is possible to press ⎇ Backspace on macOS in order to delete the previous word (left of the cursor), and ⌘ Backspace to delete the previous line (left of the cursor). If you have an external keyboard with a "delete" key (or the "forward delete key"), or are able to emulate the delete key via software, it is possible to press ⎇ Delete to delete the word to the right of the cursor. However, perhaps surprisingly, it is not possible to press ⌘ Delete in order to delete the line to the right of the cursor.  Now, I have only been able to make this possible by emulating a ⌘⇧→ and then a Backspace with BetterTouchTool. However, I want to avoid doing something like this because BTT does not work at a "low enough level" for me – that is to say, the action is not entirely reliable.  Is there any official macOS way to delete the line to the right of the cursor? What is the !! (not not) operator in JavaScript? Proving that either $2^n-1 $ or $ 2^n+1$ is not prime