Trouble understanding internal and external forces
Do internal forces cause change in velocity?
What is the value of lim$_{n\to \infty} a_n$ if $\lim_{n \to \infty}\left|a_n+ 3\left(\frac{n-2}{n}\right)^n \right|^{\frac{1}{n}}=\frac{3}{5}$? Let $\{a_n\}$ be a sequence of real numbers such that $$\lim_{n \to \infty}\left|a_n+ 3\left(\frac{n-2}{n}\right)^n \right|^{\frac{1}{n}}=\frac{3}{5}.$$  What is the value of $\lim_{n\to \infty} a_n$? $\operatorname{arsinh}$ vs $\operatorname{arcsinh}$ I note that some people like to write the inverse hyperbolic functions not with the prefix "arc" (like regular inverse trigonometric functions), but rather "ar".  This is because the prefix "arc" (for arcus) is misleading, because unlike regular trigonometric functions, they are not used to find lengths (inverse trigonometric functions can be used to find arc length of ellipses like $x^2+y^2=1$) but rather find area of a sector of the unit hyperbola.  Which version should be preferred?  $\operatorname{arsinh}$ initially confused me as I did not know why "ar" was used.  However, some seem to prefer this notation as it is more of an accurate description of the function. Do we need a global test before post hoc tests? I often hear that post hoc tests after an ANOVA can only be used if the ANOVA itself was significant.    However, post hoc tests adjust $p$-values to keep the global type I error rate at 5%, don't they?  So why do we need the global test first?  If we don't need a global test is the terminology "post hoc" correct? Or are there multiple kinds of post hoc tests, some assuming a significant global test result and others without that assumption? Ball $X^*$ is weak-star metrizable iff X is separable in Conway's book. This is a Theorem in Conway's book     I don't understand in the third paragraph why there are open sets $\{U_n:n\geq1\}$ in $(ball X^*,wk^*)$ such that $0\in U_n$ and $\bigcap^{\infty}_{n=1}U_n=\{0\}$. By the definition of the relative weak-star topology on ball $X^*$, for each $n$, there exists a finite set $F_n$ contained in $X$ such that $$\{x^*\in\text{ball }X^*:|\langle x,x^*\rangle|&lt;1 \text{ for all }x\in F_n\}\subseteq U_n.$$  Can someone explain this for me？ Thank you so much!! Do we say "-1 thing" or "-1 things"?  I am interested in both      two things minus one thing(s)    and      minus/negative one thing(s) Types of books or Types of book? can anybody tell me which one of these is the correct one? I have been searching all over the internet but I have still got no answer.  Types of books Types of book How do I redirect to another webpage? Ok, so PHP isn't the best language to be dealing with arbitrarily large integers in, considering that it only natively supports 32-bit signed integers. What I'm trying to do though is create a class that could represent an arbitrarily large binary number and be able to perform simple arithmetic operations on two of them (add/subtract/multiply/divide).  My target is dealing with 128-bit integers.  There's a couple of approaches I'm looking at, and problems I see with them. Any input or commentary on what you would choose and how you might go about it would be greatly appreciated.  Approach #1: Create a 128-bit integer class that stores its integer internally as four 32-bit integers. The only problem with this approach is that I'm not sure how to go about handling overflow/underflow issues when manipulating individual chunks of the two operands.  Approach #2: Use the bcmath extension, as this looks like something it was designed to tackle. My only worry in taking this approach is the scale setting of the bcmath extension, because there can't be any rounding errors in my 128-bit integers; they must be precise. I'm also worried about being able to eventually convert the result of the bcmath functions into a binary string (which I'll later need to shove into some mcrypt encryption functions).  Approach #3: Store the numbers as binary strings (probably LSB first). Theoretically I should be able to store integers of any arbitrary size this way. All I would have to do is write the four basic arithmetic functions to perform add/sub/mult/div on two binary strings and produce a binary string result. This is exactly the format I need to hand over to mcrypt as well, so that's an added plus. This is the approach I think has the most promise at the moment, but the one sticking point I've got is that PHP doesn't offer me any way to manipulate the individual bits (that I know of). I believe I'd have to break it up into byte-sized chunks (no pun intended), at which point my questions about handling overflow/underflow from Approach #1 apply. MCP4131: How to know what value resistors to use on the multiplexed SPI IO? There are two aspects here. The first question is related to R1 below:    The datasheet says: "R1 must be sized to ensure VIL and VIH of the devices are met" but gives no formula. So first question:  (1) How does one determine value of R1?  The datasheet gives maximum SPI SCK frequency for this device as 10MHz for 5V supply as given in Tables 1-1 and 1-2. However, for the read operation on the device where the SDI/SDO are multiplexed, the idea is different because:    The point 2 under the Table above says that the max SCK can be increased by using a pull up resistor. About these things I am wondering the following:  (2) Why is max SCK frequency lower for multiplexed SDI/SDO read operation only? (3) If I connect a pull-up resistor it will certainly form a voltage divider with the R1 in the first picture. How do I determine the value of both of these resistors then?  Finally, since the Table 1-1 and 1-2 give SCK Input Frequency for voltage 1.8V to 2.7V as 1MHz rather than 10MHz which applies to 2.7V to 5.5V.   (4) Why is the max SCK lowered at lower supply voltage? I'd like to play Dragon Age Inquision in english instead of the language of my operating system, but it seems to default to that and I can't find a way to change it. I already tried setting the Origin client language to english, but that didn't change anything.  Where can I change the language for Dragon Age: Inquisition? Sci-Fi Movie/Show Where a Soul is Captured in a Glass Vessel I remember very little of this, I was quite young -  It was on television in the mid 90's, but seemed older. My wife was telling me about the 21 grams experiment and the memory clicked into place. Setting Inside of a mad scientist sort of lab, with the plasma balls and everything else, a character (Who in my mind resembles Gene Wilder in Young Frankenstein) is conducting an experiment. The room is hazy, lots of hues of purple and red. He has an assistant present. They were conducting some sort of experiment to try to figure out the secrets of immortality (or possibly reanimation). I remember them &quot;trapping&quot; a human soul in a glass vessel to see what it weighed or what kind of volume it took up - I believe they shot someone in this vessel, and because their soul could not &quot;escape&quot;, they did not die. Dirac Delta function inverse Fourier transform We know that the Fourier transform of the Dirac Delta function is defined as $$\int_{-\infty}^{\infty} \delta(t) e^{-i\omega t} dt = 1,$$  and if I were to reconstruct the function back in time domain, the inverse Fourier transform is defined as  $$\delta(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty} e^{i\omega t} d\omega.$$  How do I compute this integral analytically? Using fflush(stdin) Help with proving that $\pi$ is irrational Number agreement between subject and object Show that $\text{C}/c \cong (c/(\text{C}^{\text{op}}))^{\text{op}}$ I'm self studying from Category Theory in Context by Emily Riehl and have encountered the following question regarding slice categories:     Let $\text{C}$ be a category and $c$ be an object in $\text{C}$. Show that $\text{C}/c \cong (c/(\text{C}^{\text{op}}))^{\text{op}}$    The exercise appears in an chapter that introduces the notion of opposite categories and duality proofs.     My intuition of why the statement is true comes from the following observation:  Let $x,y$ be objects in $\mathbf{C}$ (or equivalently $\mathbf{C}^{\text{op}}$). Let $g^{\text{op}} : c \to y$ and $f^{\text{op}} : c \to x$ be objects in $c/(\mathbf{C}^{\text{op}})$. By the definition of slice category, there exists a morphism $h^{\text{op}} : y \to x$ in $\mathbf{C}^{\text{op}}$ such that the diagram below commutes:  $\hskip2in$   Now the 'corresponding diagram' in the opposite category $(c/(\mathbf{C}^{\text{op}}))^{\text{op}}$ would be:  $\hskip2in$    which, as I understand it, would also be commutative. Now, to me, this indicates:   that the objects in $(c/(\mathbf{C}^{\text{op}}))^{\text{op}}$ are the same as the objects in $\mathbf{C}/c$, namely, a morphism in $\mathbf{C}$ from an object $z \in \mathbf{C}$ to $c$. the commutativity of the seconds diagram describes morphisms $h$ in both $\mathbf{C}/c$ and $(c/(\mathbf{C}^{\text{op}}))^{\text{op}}$ if you reverse the arrows again you return to the same diagram. This makes me think you could construct a map between morphisms by first making a map from $ \mathbf{C}/c \to (\mathbf{C}/c)^{\text{op}} ( \cong c/(\mathbf{C}^{\text{op}}) )$ by $h \mapsto h^{op}$     Now, what I am really looking for is a way to structure a proof. I feel like I have some intuition but am very unsure of how to write it down in a concise way. I am new to category theory so I'm still very much learning how to construct these types of proofs using duality; I feel like what I have written is maybe overkill but I'm trying to be as explicit as possible to make sure I understand everything.  In particular, I don't know how I would get a concrete correspondence between morphisms in the two categories that respects associativity of composition etc.  Any help or pointers would be appreciated! Find count of all combination of numbers whose sum is x I want to find the sum of all combination of numbers whose sum is x, for e.g.  when x = 3 f(x) = countOf(111,12,21,3) = 4 How to find out the variant of my Galaxy Nexus? Let $\Gamma$ be a set of formulas and $\phi$ be a formula. Show that $\Gamma \cup \{\neg \phi\}$ is satisfiable if and only if $\Gamma\not \models \phi$.   This seemed pretty obvious but I wanted to see if my proof made sense:  Proof: $(\Rightarrow)$  To derive for a contradiction, suppose that: $\Gamma \models \phi$. That means for all truth assignments $v$, for $\gamma \in \Gamma$, if $v(\gamma) = T$, then $v(\phi) = T$.  But this contradicts our assumption that $\Gamma \cup \{\neg \phi \}$ is satisfiable by the fact that $v(\phi) = T$ cannot happen so: $\Gamma \not \models \phi$ .  $(\Leftarrow)$   So by the definition of $\Gamma \not \models \phi$, we have that there is some truth assignment $v$ which satisfies $\Gamma$ but does not satisfy $\phi$. So  that means $v(\phi) = F$ since $v(\phi) \not = T$ which implies that $v(\neg \phi) = T$ which means $v$ satisfies $\Gamma \cup \{\neg \phi\}$.    I feel like I'm missing something in the forward direction, but at the same time... It looks pretty trivial as well. Am I missing anything crucial?  Thank you! I am trying to count the number of bit-strings of length 8 with 3 consecutive zeros or 4 consecutive ones. I was able to calculate it, but I am overcounting. The correct answer is $147$, I got $148$.  I calculated it as follows:  Number of strings with 3 consecutive zeros = $2^5+5\times2^4 = 112$, because the 3 zeros can start at bit number 1, 2, 3, .., 6  Number of strings with 4 consecutive ones = $2^4+4\times2^3 = 48$, I used the same reasoning.  Now I am trying to count the number of bit-strings that contain both 3 consecutive zeros and 4 consecutive 1s. I reasoned as follows:   the strings can be of the following forms: 0001111x, 000x1111, x0001111..thus there are $2+2+2 = 6$ possibilities for bit-strings where the 3 consecutive zeros come first. Symmetrically there are $6$ bit-strings where the 4 consecutive ones come first.  Thus the answer should be = $112+48-12 = 148$.  clearly there's something wrong with my reasoning, if someone could point it out, that would be awesome. Thanks