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Asymptotics of uniformly continuous functions over $\mathbb R$ Weird thing in references/bibliography page Sounds of the letter a How can I know, precisely, when to differentiate the sounds of the letter a, like in: apple and vault? Compute Taylor error of $10^{-5}$ for $\cos(\pi/20)$ Evaluate the integral $\int_{-1}^1\frac{dx}{(e^x+1)(x^2+1)}$. Evaluate:      $$\int_{-1}^1\frac{dx}{(e^x+1)(x^2+1)}$$ Question on showing a bijection between $\pi_1(X,x_0)$ and $[S^1, X]$ when X is path connected. I am trying to do this question taken from Hatchers algebraic topology and I am struggling to understand the notation and the concepts.  As far as I know $\pi_1(X,x_0)$ is the set of end point preserving homotopy classes of loops in X based at $x_0$ and a loop is just a path    $f:I \rightarrow X$ with $f(0)=f(1)$. The question says we can regard $\pi_1(X, x_0)$ as the set of basepoint preserving homotopy classes of maps $(S^1, s_0) \rightarrow (X, x_0)$.   This confuses me, does it mean the set of basepoint preserving homotopy class on maps $g:S^1 \rightarrow X$ which map loops in $S^1$ based at $s_0$ to loops in X based at $x_0$?  Then how can $\pi_1(X,x_0)$ be regarded as this set?  If someone could explain this it would be really useful How would I write the equivalent of C#'s  in JavaScript?  var haystack = 'hello world'; var needle = 'he';  haystack.startsWith(needle) == true   Note: This is an old question, and as pointed out in the comments ECMAScript 2015 (ES6) introduced the  method. However, at the time of writing this update (2015) . I'm running Windows 7. It seems that svchost is eating up a lot of memory.   The user name for the particular svchost instance is "Network Service". The description is "Host Process for Windows Services".   Sometimes it eats up 50% of the CPU usage. Other times, it's just under 5%. I can't figure out when it's 50% and when it's not. But it fluctuates often -- even when I've just booted up and no programs were started.  How do I diagnose this? Formula field is too big to execute (5,480 characters) I got an error when i am trying to add fomrula in formula field.  Error: Compiled formula is too big to execute (5,480 characters). Maximum size is 5,000 characters  My formula is   (ED__c+Service_Tax__c+Total_Site_Work_Price1__c+Total_Selling_Price_Based_on1__c )*WCTper__c   In this formula ED__c, Total_Site_Work_Price1__c, Total_Selling_Price_Based_on1__c are also Formula fields. And these formulas are also contain fomrula fields.  How can i calculate the formula..? How do I prove the following statement by induction?  $$n^2 \lt 2^n$$  $P(n)$ is the statement $n^2 \lt 2^n$  Claim: For all $n \gt k$, where $k$ is any integer, $P(n)$  (since $k$ is any integer, I assume I have to prove this for positive and negative integers)  So let base case be $P(1)$, and I have to prove $P(n)$ for $n \ge 1$ and $n \lt 1$  $P(1)$ is $1^2 \lt 2^1$ which is clearly true.  Induction hypothesis: $k^2 \lt 2^k$  Inductive step $(k+1)^2 \lt 2^{(k+1)}$  $(k+1)^2 = k^2 + 2k + 1$  $k^2 + 2k + 1 \lt 2^k + 2k + 1$ by inductive hypothesis  Not sure how to proceed. Is my previous intuition that I have to prove this for $n \ge 1$ and $n \lt 1$ correct? Derive Fourier transform of sinc function We know that the Fourier transform of the sinc function is the rectangular function (or top hat). However, I'm at a loss as to how to prove it. Most textbooks and online sources start with the rectangular function, show that   $$\int_{-\infty}^\infty \text{rect}(x)e^{i\omega x}dx=\int_{-1/2}^{1/2}e^{i\omega x}dx=\left.\frac{e^{i\omega x}}{i\omega}\right\vert_{-1/2}^{1/2}=\text{sinc}(\omega/2)$$  and then just invoke duality and claim that the Fourier transform of the sinc function is the rectangular function. Is there any way of deriving this directly? i.e., starting with the sinc function?   I've tried, but I'm not sure as to how to proceed. I know that the sinc is not Lebesgue integrable and only improper Riemann integrable. Some vague recollection of these being important in the Fourier transform hinders my thought process. Can someone clear things up for me? Find $x$ such that $(x^2 + 4x + 3)^x + (2x + 4)^x = (x^2 + 4x + 5)^x$ Is the universe fundamentally deterministic? I'm not sure if this is the right place to ask this question. I realise that this maybe a borderline philosophical question at this point in time, therefore feel free to close this question if you think that this is a duplicate or inappropriate for this forum. Anyway, I'm an electrical engineer and I have some basic knowledge of quantum mechanics. I know that Schrödinger's equation is deterministic. However, quantum mechanics is much deeper than that and I would like to learn more. If this question is not clearly answerable at this point than can anyone point out some recognized sources that try to answer this question. I would appreciate it if the source is scientific and more specifically, is related to quantum theory. Suppose we have a magnetic field directed into the computer screen. Assume that there is no gravitational field. Now suppose we have a stationary positively charged particle in the magnetic field. The magnetic force is given by $$\mathbf{F}=q\mathbf{v}\times\mathbf{B}$$  Since $\mathbf{v}=0, \mathbf{F}=0$.  Now pick a reference frame moving to the left with constant velocity. In this frame, the particle is moving to the right with constant velocity. So, the magnetic force is directed upwards.  From what I've learnt in physics, forces are independent of choice of reference frames as long as the frames are inertial. But in the 2 inertial reference frames I have chosen, I get 2 different magnetic forces. How do we reconcile this? I am trying to prove the following.  Proposition Let $g$ be a Lie algebra such that the first derived algebra $Dg$ is a proper ideal of $g$. Consider the quotient $g/D^1 g$, and show that the preimage of any codimension 1 subspace is   an ideal, and has codimension 1.   I could show that the preimage is indeed an ideal, but I need some help with the part about the dimensions:  So suppose that $\phi: x \mapsto [x]$ is the quotient projection. And write $[h] + Dg$  for a codimension 1 subspace of $g$ (here $h\subset g$). Then the preimage of this subspace is just the linear span $h+Dg\leq g$. Let $y+z \in h+Dg$ with $y\in h$ and $z\in Dg$. Let $x\in g$. We want to show that $[x, y+z]\in h$, ie that $h + Dg \lhd g$. We can write $x$ as $x_0+x_1$ for $x_0 \in Dg$, $x_1\not\in Dg$. Then  $$[x,y+z] = [x_0, y+z] + [x_1, y] + [x_1,z] \in h + Dg$$  The inclusion follows since $Dg$ is an ideal so the first and third terms belong $Dg$; and $[x_1, y] \in D_g$, as $g/Dg$ is abelian. Should I capitalize the word "that" in a title? Since the version of the word "that" might matter to the correct answer, I'll provide the actual title in question:  Title:  Existing Solutions that Didn't Work  If someone could explain the why of the answer, I would be appreciative.  EDIT:  It has been suggested that this is a duplicate, and that may be, in a sense.  However, the after reading the suggested prior question, I'm no closer to my answer.  I think it may have to do with my use of the word "that."  Perhaps it is incorrect?  Using these definitions of the word "that:" I'm having trouble figuring out which definition of that matches my title.  Am I even using it correctly?  Perhaps I should use which instead:  Possible Title:  Existing Solutions which Didn't Work  If my original usage is incorrect, and I should use which, then the answer is simple since which is always a pronoun or adjective, and those are both capitalized in a title.  However, I'm having difficulty understanding which part of speech that is in my original title. How to Store Design Data in Unity including Formulas? My team is switching to Unity 3D for an upcoming project. I'm tasked with converting our game data system into Unity.  Our existing games use spreadsheet documents (Google Docs) for designers to set values and formulas. This is then exported into an XML doc (and obfuscated) and read in by our game client. I'm trying to figure out a way to accomplish this same thing but solely through Unity. Storing, exposing, and reading data through Unity is pretty straightforward but I'm not sure on supporting formulas.  I realize I can write a custom parser and interpreter which can handle formulas but I'm wondering if there is already a way to do this in Unity.  For example, let's say I have a unit with various stats: Level, Max HP, HP, Damage, Victory Reward, etc. That unit may receive a reward after combat which is based on its Level and remaining HP. A designer would be able to enter into the Victory Reward field either a flat value, like 3, or a formula such as "Floor({Level} * ({HP} / {Max HP}))", something like that.  My question is basically, is there an existing solution for handling data with formulas in Unity, via a downloadable asset or something in Unity itself, or is there a simple framework or technology which already can do something like this which can be easily integrated (example: some kind of JSON extension which supports self-referencing other values and basic mathematical/logical operations)? For holomorphic functions, if $\{f_n\}\to f$ uniformly on compact sets, then the same is true for the derivatives. Let $\Omega$ be an open subset in $\mathbb{C}$. Let $\{f_n\}$ be a sequence of holomorphic functions on $\Omega$ such that $f_n\to f$ pointwise and converges uniformly on any compact subset $K\subseteq \Omega$. Then by Cauchy Theorem and Morera Theorem, $f$ is holomorphic. Let $f_n$ and $f'$ be the derivatives of $f_n$ and $f$ respectively. Prove that $f_n'\to f'$ uniformly on any compact subset $K\subseteq \Omega$.   How to prove? Detect if a method was overridden using Reflection (C#) What does the [Flags] Enum Attribute mean in C#?