Is there an explanation for the contradiction between "Voyager" and "The Undiscovered Country"?
The curious case of Dmitri Valtane
Mental estimate for tangent of an angle (from $0$ to $90$ degrees) Does anyone know of a way to estimate the tangent of an angle in their head? Accuracy is not critically important, but within $5%$ percent would probably be good, 10% may be acceptable.  I can estimate sines and cosines quite well, but I consider division of/by arbitrary values to be too complex for this task. Multiplication of a few values is generally acceptable, and addition and subtraction are fine.  My angles are in degrees, and I prefer not have to mentally convert to radians, though I can if necessary. Also, all angles I'm concerned with are in the range of [0, 90 degrees].  I am also interested in estimating arc tangent under the same conditions, to within about 5-degrees would be good.  Backstory  I'm working on estimating the path of the sun across the sky. I can estimate the declination pretty easily, but now I want to estimate the amount of daylight on any given day and latitude. I've got it down to the arc cosine of the product of two tangents, but resolving the two tangents is now my sticking point. I also want to calculate the altitude of the sun for any time of day, day of the year, and latitude, which I have down to just an arc tangent. Plot Option Precedence while combining Plots with Show[] When I installed the latest update of Ubuntu, I was asked for the username but I do not remember what my username is. How do I find out what my username is? Suppose $x&gt;0$ and we have function $f(x)=\frac{\sin x}{x}$ how can we show $$\forall n \in \mathbb{N} :|f^{(n)}(x)|\leq \frac{1}{n+1}$$ I need a hint to show this property .Thanks in advance . I tried for $n=1 ,2$  by finding maximum of $|f'| ,|f''|$ but I get stuck to show for $n$ I am generating a fairly large index of .txt, .tex, and .pdf files. This process has worked for a long time, and the size of the index is not the issue. I have added logic to insert additional links in the index and now am running into:  TeX capacity exceeded, sorry [save size=80000]  Questions:  What are the typical causes of this kind of problem, and any suggestions on how to determine the cause of this. Can the save size be increased?  Of course any solution that just allows me to increase the save size would be preferable if it works, the first question is only important if the size can't be increased or I run into this problem again. Notes:  Generating smaller indices seems to work just fine, only when I attempt to generate the entire index do I run into problems, so coming up with a MWE will be difficult. Here is my player code.  Rigidbody rb;  Vector3 currMovement; public float jumpSpeed = 10; public float moveSpeed = 10; public float rotSpeed = 180;  float distToGround;  public float posSmoothTime = 0.5f; float currPos; float targetPos; float posVel;  public float rotSmoothTime = 0.5f; float currRot; float targetRot; float rotVel;  void Start() {     rb = GetComponent&lt;Rigidbody&gt;();     distToGround = GetComponent&lt;Collider&gt;().bounds.extents.y; }  bool isGrounded() { return Physics.Raycast(transform.position, Vector3.down, distToGround + 0.1f); }  void Update() {     Move(); }  void Move() {     // Rotation smoothing.     targetRot = Input.GetAxisRaw("Horizontal") * rotSpeed * Time.smoothDeltaTime;      if (targetRot &gt; 360)         targetRot -= 360;     if (targetRot &lt; 0)         targetRot += 360;     currRot = Mathf.SmoothDampAngle(currRot, targetRot, ref rotVel, rotSmoothTime * Time.smoothDeltaTime);      transform.eulerAngles += new Vector3(0, currRot, 0);      // Movement smoothing.     targetPos = Input.GetAxisRaw("Vertical") * moveSpeed;      currPos = Mathf.SmoothDamp(currPos, targetPos, ref posVel, posSmoothTime * Time.smoothDeltaTime);     currMovement = new Vector3(0, 0, currPos);     currMovement = transform.rotation * currMovement;      if (isGrounded())     {         if (Input.GetButtonDown("Jump"))             rb.velocity += Vector3.up * jumpSpeed;     }      rb.position += currMovement * Time.smoothDeltaTime; }   I have a Rigidbody attached to my player. I think the problem is with my camera script. Here is my camera script.  public Transform player; Quaternion targetLook; Vector3 targetMove; public float smoothLook = 0.5f; public float smoothMove = 0.5f; public float distFromPlayer = 5, heightFromPlayer = 3; Vector3 moveVel;  void LateUpdate() {     CameraMove(); }  void CameraMove() {     targetMove = player.position + (player.rotation * new Vector3(0, heightFromPlayer, -distFromPlayer));     transform.position = Vector3.SmoothDamp(transform.position, targetMove, ref moveVel, smoothMove);      targetLook = Quaternion.LookRotation(player.position - transform.position);     transform.rotation = Quaternion.Slerp(transform.rotation, targetLook, smoothLook); }   }  The player is not an parent of my camera. When I parent the player to my camera the shake stops. But I want a custom smooth camera movement with my custom scirpt, so I can't make the player a parent of the camera. Zombies are always after human beings. How do Zombies differentiate between a zombie and a human being? Why don't zombies eat each other? Android Nougat .  It adds features such as multi-window support.  You can review the high-level changes at .  Of course, not all devices will get this update right away, even at release; others won't get it at all. Manufacturers and carriers often choose to add their own custom modifications, which tend to delay the official release.  When will each device get Nougat, either officially or unofficially, if ever?  See also: I got a brand new Windows 7 machine, installed the operating system, created one account and forgot its password. What can I do?  There is no external CD, the operating system is loaded from somewhere inside the machine.  I already tried to remember passwords and tried all candidates with all possible combinations of caps lock, num lock etc. TikZ or cryptocode package for cryptographic protocol diagrams Is it possible to use the Lagrangian formalism involving physical terms to answer the question of why all planets are approximately spherical?  Let's assume that a planet is 'born' when lots of particles of uniform density are piled together, and constrained only by the Newtonian gravity of the $N$ particles with no external gravity.    My question is: Is it possible to use the variational calculus starting from this situation to prove that planets should be spherical? How did nobody at Hogwarts (especially Dumbledore) figure out what killed Myrtle? The information is all clearly there, since Harry and Ron, two 12-year-olds, were able to figure it out (with a bit of help from a petrified Hermione), thought at the time Aragog was much younger and smaller and wouldn't have been as easy to find, so maybe the "spiders flee before it" clue would have been lost, although to be honest it's not exactly as vital a piece of the puzzle as the others.  Dumbledore himself is obviously a very capable and intelligent Wizard, and after Myrtle died she came back as a ghost and was able to provide (admittedly limited) details of her own death.  I don't know about you, but I'd have thought at least someone would have tried to piece it together at the time, and it wouldn't have taken them long to figure out "Oh, she saw a pair of big yellow eyes after hearing a boy speak a funny language in the bathroom and dropped dead? Sounds like she might have heard somebody order a Basilisk to look her in the eyes in parseltongue", especially if the investigator in question was someone like Dumbledore.  Do we have any info on why nobody seemed to have found out anything in 50 years prior to Harry Potter (and, probably more importantly, Hermione Granger) turning up at Hogwarts? At the end of the game, both remaining players have a Baron.   The sum of all cards discarded by both players is equal.  Who wins? My question is in the title. I'm a little lost in how to solve this problem. There is a hint associated with the problem that states the following:   Use a line integral to show that $ |f(z)| \leq A + |z| $ where A = $\max\{1, |f(0)|\}$  Thanks for the help. Prove that the sequence with $T(0)=1$ and $T(n) = 1 + \sum_{j=0}^{n-1}T(j)$ is given by $T(n)=2^n$ $T(0)=1 \\ T(n) = 1 + \sum_{j=0}^{n-1}T(j) \\ $  Show that $T(n) = 2^n$.  I know how to prove this by induction, but I would like to know how to show this using first principles.  Edit: The way I want to solve this problem is manipulate $T(n)$ in such a way that it ends up as $2^n$. Including Bitmap (.bmp) I can't include bitmaps with pdflatex. Here is my mwe:  \documentclass[12pt,a4paper]{article} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage[turkish]{babel} \addto\extrasturkish{\uccode`i=\string"9D \lccode`I=\string"19 } \addto\noextrasturkish{\uccode`i=`I \lccode`I=`i }   \usepackage{graphicx}  \begin{document}  \shorthandoff{=}  \includegraphics[width=5cm,height=5cm]{000.bmp} \end{document}   Gives the error Unknown graphics extension: .bmp. Canonical URL for a home page and trailing slashes Does it give higher quality to shoot in low-res mode in-camera, or to downsize high-res photos later? Suppose you want to store your photos at some lower resolution, let's say about ~1 MB large.  You own a camera able to make photos at 5 MPix and more.  And, you want to have the maximum quality at the smaller size.  Would it be better to make a photo at highest resolution and then resize it with PC software (with a good filter of course), or to directly set the camera to make photos at 2 MPix only? What is the behavior of "delete" with stack objects? How do you give a C# auto-property an initial value?  I either use the constructor, or revert to the old syntax.    Using the Constructor:  class Person  {     public Person()     {         Name = "Initial Name";     }     public string Name { get; set; } }   Using normal property syntax  (with an initial value)  private string name = "Initial Name"; public string Name  {     get      {         return name;     }     set     {         name = value;     } }   Is there a better way?