https://track.liberty.ioFresh celebrity buzz with instant attention value.en-usMon, 07 Sep 2026 12:25:09 -0400https://track.liberty.io/proof-of-euler-039-s-theorem-without-abstract-algebra.htmlhttps://track.liberty.io/proof-of-euler-039-s-theorem-without-abstract-algebra.htmlEvery proof I've seen of Euler's Theorem (that $\gcd(a,m) = 1 \implies a^{\phi(m)} \equiv 1 \pmod m$) involves the fact that the units of $\mathbb{Z}/m\mathbb{Z}$ form a group of order $\phi(m)$. ......Mon, 07 Sep 2026 16:15:38 -0400https://track.liberty.io/max-number-of-players-in-multiplayer.htmlhttps://track.liberty.io/max-number-of-players-in-multiplayer.htmlHow many players can play simultaneously in a multiplayer game ? Is it the same for both internet and LAN games ?...Mon, 07 Sep 2026 16:14:33 -0400https://track.liberty.io/find-closed-form-of-a-sequence-2-5-11-23.htmlhttps://track.liberty.io/find-closed-form-of-a-sequence-2-5-11-23.htmlFind closed form of a sequence $2,5,11,23,\dots$ How to get generating function for this sequence (closed form)? Explicit form is $f(x)=2+5x+11x^2+23x^3+\cdots$ Is it possible to get to geometric ...Mon, 07 Sep 2026 16:13:53 -0400https://track.liberty.io/does-the-nintendo-switch-not-adjust-to-your-tv-039-s-aspect-ratio.htmlhttps://track.liberty.io/does-the-nintendo-switch-not-adjust-to-your-tv-039-s-aspect-ratio.htmlJust bought my Nintendo Switch, but when I tried docking it to the TV and the image was much larger than the TV screen. When I tried adjusting it, I had to make the screen so small that there were ......Mon, 07 Sep 2026 15:51:00 -0400https://track.liberty.io/finding-equilibrium-points-of-a-differential-equation.htmlhttps://track.liberty.io/finding-equilibrium-points-of-a-differential-equation.htmlOK, this is a weak question, but I'm a bit rusty on differential equations, so here it is: I need to find the general solution of x'=ax+3 and then find the equilibrium points as well as which poin......Mon, 07 Sep 2026 15:49:46 -0400https://track.liberty.io/riemann-explicit-formula-for-pi-x-and-its-evaluation.htmlhttps://track.liberty.io/riemann-explicit-formula-for-pi-x-and-its-evaluation.htmlWe have for $x>1$: $$\tag{1}\pi^*(x)=R(x)-\sum_{\rho} R(x^{\rho})$$ $\rho$-s are zeros of Riemann zeta function (trivial and nontrivial). I know how to derive $(1)$ and also know how to deri......Mon, 07 Sep 2026 15:42:29 -0400https://track.liberty.io/clarifications-about-the-definition-of-algebraic-systems-and-algebraic.htmlhttps://track.liberty.io/clarifications-about-the-definition-of-algebraic-systems-and-algebraic.htmlI am learning about Groups from a discrete mathematics textbook for a computer science course by Grimaldi. An algebraic system is defined as a set along with operations on elements of the set. An ...Mon, 07 Sep 2026 15:41:38 -0400https://track.liberty.io/expectation-of-standard-brownian-motion.htmlhttps://track.liberty.io/expectation-of-standard-brownian-motion.htmlLet Let $\{W_t\}_{t\ge 0}$ be a standard Brownian motion on some filtered probability space $(\Omega , \mathcal{F}_{t}, \{\mathcal{F}_{t}\}_{t\ge 0}, \mathbb{P}).$ How can we show that the expect......Mon, 07 Sep 2026 15:40:02 -0400https://track.liberty.io/excellent-charge-moves-in-pvp.htmlhttps://track.liberty.io/excellent-charge-moves-in-pvp.htmlIn Trainer Battles (Arena, Rocket fights, etc), you can "charge" a move by swiping all of the icons that appear. Depending on how many you hit, you can get a caption showing how well you ......Mon, 07 Sep 2026 15:39:47 -0400https://track.liberty.io/harvest-or-rescue-little-sisters-in-bioshock-2.htmlhttps://track.liberty.io/harvest-or-rescue-little-sisters-in-bioshock-2.htmlI just tested it out and it seems that if you harvest a little sister you get 160 adam out of her. If you use her to harvest 2 bodies you get 40-40 each plus you get 80 more when you rescue her at ......Mon, 07 Sep 2026 15:39:25 -0400https://track.liberty.io/how-to-solve-and-find-the-value-of-log-k-equation.htmlhttps://track.liberty.io/how-to-solve-and-find-the-value-of-log-k-equation.htmlI am doing a mathematics problem where the problem equation led me to below conclusion: $$\log k = 52.79$$ Now I am not sure how to solve and get value of $k$? What way we can find the value of $k$?...Mon, 07 Sep 2026 15:39:21 -0400https://track.liberty.io/calculating-ray-cylinder-intersection-points.htmlhttps://track.liberty.io/calculating-ray-cylinder-intersection-points.htmlHow is it done? I couldn't find a general formula/algorithm for this problem. I've read this article, I've managed to calculate the intersection points around the $z$-axis like they explained, but ......Mon, 07 Sep 2026 15:34:01 -0400https://track.liberty.io/functions-and-vertical-line-proof.htmlhttps://track.liberty.io/functions-and-vertical-line-proof.htmlA curve is a function if satisfy the vertical line test. (a) What happens with this?: (b) do not exist a function for graphing this curve? Really? P.S. I have a TI graph calculator and I once ......Mon, 07 Sep 2026 15:32:26 -0400https://track.liberty.io/monomorphisms-epimorphisms-and-isomorphisms-of-groups-category.htmlhttps://track.liberty.io/monomorphisms-epimorphisms-and-isomorphisms-of-groups-category.htmlI would like to solve the Problem 2.12 in Szekeres, A Course in Modern Mathematical Physics: Show that the class of groups as objects with homomorphisms between groups as morphisms forms a cat......Mon, 07 Sep 2026 15:21:03 -0400https://track.liberty.io/rules-for-moving-over-equality-sign.htmlhttps://track.liberty.io/rules-for-moving-over-equality-sign.htmlWhat are the rules for moving over the other side of equal sign in boolean equations ? For example lets take XOR, just speculating : x ^ y = z (x ^ y) & z OR (x ^ y) & z = 0 OR wh......Mon, 07 Sep 2026 15:04:15 -0400https://track.liberty.io/volume-of-triangular-pyramid-given-4-vertices.htmlhttps://track.liberty.io/volume-of-triangular-pyramid-given-4-vertices.htmlI want to calculate the colume of the polyhedra with vertices $(a_1,0,0),(0,a_2,0),(0,0,a_3),(a_1,a_2,a_3)$. My solution is: Using translation and a reflection I am considering the polyhedra $(a_1......Mon, 07 Sep 2026 15:02:00 -0400https://track.liberty.io/how-exactly-does-daruk-039-s-protection-work-deplete-its-charges-and-r.htmlhttps://track.liberty.io/how-exactly-does-daruk-039-s-protection-work-deplete-its-charges-and-r.htmlSo I recently earned Daruk's protection, and am excited to use it, but I'm a little confused about exactly what it does, and how often I can use it. The explanation I got when it was granted basica......Mon, 07 Sep 2026 14:54:00 -0400https://track.liberty.io/stokes-theorem-for-a-circle-in-the-plane-x-y-z-5.htmlhttps://track.liberty.io/stokes-theorem-for-a-circle-in-the-plane-x-y-z-5.htmlFind $\int_C \vec F \cdot d \vec r $ where $C$ is a circle of radius $2$ in the plane $x + y + z = 5$, centered at $(4,4,-3)$ and oriented clockwise when viewed from the origin, if $\vec F = 4y\ve......Mon, 07 Sep 2026 14:51:34 -0400https://track.liberty.io/how-do-i-add-minecraft-as-a-firewall-exception.htmlhttps://track.liberty.io/how-do-i-add-minecraft-as-a-firewall-exception.htmlI cannot play LAN on Minecraft as the game I am trying to join does not appear on the list. The other person can see my game, but is unable to join as they receive a timed out error. If I turn ......Mon, 07 Sep 2026 14:39:45 -0400https://track.liberty.io/basic-and-non-basic-variables-in-linear-programming.htmlhttps://track.liberty.io/basic-and-non-basic-variables-in-linear-programming.htmlI dont understand what are Basic and non basic variables,why we are talking them specially, what they have got to do with the rank of the coefficient matrix and augmented matrix ,and some deal with......Mon, 07 Sep 2026 14:36:08 -0400https://track.liberty.io/calculating-equal-playing-time-in-a-soccer-game-with-minimum-number-of.htmlhttps://track.liberty.io/calculating-equal-playing-time-in-a-soccer-game-with-minimum-number-of.htmlI need to produce a formula that takes the following parameters: T = time of game in minutes p = number of players on field at one time s = number of substitute players Each of these is variable ......Mon, 07 Sep 2026 14:33:17 -0400https://track.liberty.io/integral-of-sec-x-tan-x.htmlhttps://track.liberty.io/integral-of-sec-x-tan-x.htmlmethod 1:Substitution $\int \sec x\tan x dx=\int \frac{\sin x}{\cos ^2x}dx$ Let $u=\cos x \implies -du=\sin x dx$ $-\int \frac{1}{u^2}du=\frac{1}{u}+c$ So $\int \sec x\tan x dx=\frac{1}{\cos x}......Mon, 07 Sep 2026 14:26:59 -0400https://track.liberty.io/concept-of-linearity.htmlhttps://track.liberty.io/concept-of-linearity.htmlI hear so many terms involving the word "linear". Linear function, linear equation, linear system, linear operator, linear transformation, linear mapping, linear space, linear algebra, linear elect......Mon, 07 Sep 2026 14:12:16 -0400https://track.liberty.io/do-i-have-any-disadvantages-playing-cs-go-on-linux.htmlhttps://track.liberty.io/do-i-have-any-disadvantages-playing-cs-go-on-linux.htmlI started to play CS:GO once a week with my friends. Since I have a powerful developer laptop running Linux, I use it for these weekly meetings. Assuming we are on the exact same system, do I hav......Mon, 07 Sep 2026 13:54:15 -0400https://track.liberty.io/installation-error-80029564-on-ps3.htmlhttps://track.liberty.io/installation-error-80029564-on-ps3.htmlI have a problem when installing the game Midnight Club Los Angeles: Complete Edition, which is available for a test of 60 minutes via the PlayStationPlus membership. All went well during the downl......Mon, 07 Sep 2026 13:54:06 -0400https://track.liberty.io/how-to-make-mobs-spawn-where-the-player-is-at-random-intervals.htmlhttps://track.liberty.io/how-to-make-mobs-spawn-where-the-player-is-at-random-intervals.htmlHello I'm trying to make a plugin/mod where mobs(1 - 31) randomly spawn at the players coords, but I'm very new to coding in Minecraft so any help will be appreciated. I'm playing on 1.18...Mon, 07 Sep 2026 13:52:39 -0400https://track.liberty.io/minimum-value-of-quadratic-equation.htmlhttps://track.liberty.io/minimum-value-of-quadratic-equation.htmlI know that $ax^2+bx+c$ if $a < 0$ it has maximum value and we find $m = -b/2a$ but if $a>0$ it should have minimum value how can find it? Thanks in advance....Mon, 07 Sep 2026 13:39:00 -0400https://track.liberty.io/ender-pearl-textures-not-showing.htmlhttps://track.liberty.io/ender-pearl-textures-not-showing.htmlI edited my resource packs ender pearl texture by replacing the original texture with another ender pearl texture from a different pack and now whenever I see an ender pearl it is a pink and black ...Mon, 07 Sep 2026 13:37:27 -0400https://track.liberty.io/can-3-planes-intersect-at-a-single-point-closed.htmlhttps://track.liberty.io/can-3-planes-intersect-at-a-single-point-closed.htmlI know that three planes can intersect having a common straight line as intersection. But I have seen in some references that three planes intersect at single point.The three planes were represente......Mon, 07 Sep 2026 13:32:41 -0400https://track.liberty.io/can-i-read-diary-entries-i-missed.htmlhttps://track.liberty.io/can-i-read-diary-entries-i-missed.htmlHat Kid has a secret diary you can read if you It looks like you can only see one entry at a time, usually something relevant to plot points you just cleared. Is there any way to see entries you m......Mon, 07 Sep 2026 13:27:45 -0400https://track.liberty.io/how-can-i-get-a-blaster-launcher.htmlhttps://track.liberty.io/how-can-i-get-a-blaster-launcher.htmlFrom what I understand it requires a Fusion Core, but where can I get one? Is there a certain type of UFO of special mission it comes from? Is any specific research required?...Mon, 07 Sep 2026 13:24:34 -0400https://track.liberty.io/path-around-rectangular-fountain-closed.htmlhttps://track.liberty.io/path-around-rectangular-fountain-closed.htmlA landscaper is designing a rectangular fountain with a 4-foot-wide path around it. The equation $A = 4p + 64$ will relate the area $A$, in square feet, of the path to the perimeter $p$, in feet, o......Mon, 07 Sep 2026 13:15:45 -0400https://track.liberty.io/is-there-any-reward-for-completing-rdr-2-missions-with-gold-medal.htmlhttps://track.liberty.io/is-there-any-reward-for-completing-rdr-2-missions-with-gold-medal.htmlI have noticed after the first chapter of RDR 2, there are medals awarded for completing the side objectives in the missions. From what I have read you earn an achievement if you get at least 70 g......Mon, 07 Sep 2026 13:13:41 -0400https://track.liberty.io/what-is-the-assassin-039-s-creed.htmlhttps://track.liberty.io/what-is-the-assassin-039-s-creed.htmlAt some point in every Assassin's Creed game, the creed itself comes up, but I can never remember what it is. I know it has something to do with "everything is permitted" and there's another part ......Mon, 07 Sep 2026 13:04:26 -0400https://track.liberty.io/average-value-of-e-x-from-0-to-infty.htmlhttps://track.liberty.io/average-value-of-e-x-from-0-to-infty.htmlI just signed up for this site so I hope I'm doing this right. I've been having trouble wrapping my head around the average value of the function $e^{-x}$ from 0 to $\infty$. I know that the average ...Mon, 07 Sep 2026 12:51:49 -0400https://track.liberty.io/fourth-derivative-of-f-x-e-4-x.htmlhttps://track.liberty.io/fourth-derivative-of-f-x-e-4-x.htmlFourth derivative of $f(x)=e^{4/x}$ I keep trying to calculate this but I can't get it right... Can somebody walk me through this? I get it's just the product rule and chain rule, but dang!! The co......Mon, 07 Sep 2026 12:42:04 -0400https://track.liberty.io/smallest-square-that-can-be-fitted-outside-the-regular-hexagon.htmlhttps://track.liberty.io/smallest-square-that-can-be-fitted-outside-the-regular-hexagon.htmlThey have the derivation here http://www.drking.org.uk/hexagons/misc/deriv4.html The figure is In this derivation the second line is $ a^2 = b^2 + b^2 $ .How have they assumed $ AB = AC $ ? . I......Mon, 07 Sep 2026 12:30:36 -0400https://track.liberty.io/what-does-the-notation-x-and-x-mean.htmlhttps://track.liberty.io/what-does-the-notation-x-and-x-mean.htmlI have been reading through this document, and on the bottom of page 7 it uses the notation $x(A) = x^+(A) - x^-(A)$ where $x$ is a function. What could this notation mean?...Mon, 07 Sep 2026 12:30:30 -0400https://track.liberty.io/simple-geometric-proof-for-snell-039-s-law-of-refraction.htmlhttps://track.liberty.io/simple-geometric-proof-for-snell-039-s-law-of-refraction.htmlSnell's law of refraction can be derived from Fermat's principle that light travels paths that minimize the time using simple calculus. Since Snell's law only involves sines I wonder whether this m......Mon, 07 Sep 2026 12:29:18 -0400https://track.liberty.io/do-double-damage-rune-stack.htmlhttps://track.liberty.io/do-double-damage-rune-stack.htmlIf I was able to bottle a rune of Double Damage at around 1:55 and after I waited in the same rune spot, another Double Damage rune appears and I took it and use also the rune in my bottle, will it......Mon, 07 Sep 2026 12:28:39 -0400https://track.liberty.io/what-is-the-giant-coin-slot-before-the-boss-area-in-the-sand-kingdom.htmlhttps://track.liberty.io/what-is-the-giant-coin-slot-before-the-boss-area-in-the-sand-kingdom.htmlIn the Sand Kingdom, after deepest underground and before the big arena, there's an oversized coin slot in a big circle. You can jump into it; I can't find out what to do with it. What is the gian......Mon, 07 Sep 2026 12:21:17 -0400https://track.liberty.io/how-do-you-find-your-port-for-minecraft-lan-worlds.htmlhttps://track.liberty.io/how-do-you-find-your-port-for-minecraft-lan-worlds.htmlI've looked at a lot of websites and the ports are either outdated or don't show how to find it. I know how to get into the cmd thing and all that, but I cannot find my port for the life of me. Is it ...Mon, 07 Sep 2026 12:16:54 -0400https://track.liberty.io/how-long-is-vayne-039-s-tumble-animation.htmlhttps://track.liberty.io/how-long-is-vayne-039-s-tumble-animation.htmlVayne's Q; Tumble, is a short CD, short distance dash, that increases her damage for her next shot. How long is the animation, in seconds? The sub-question is, will Vayne hit a point when tumbling ...Mon, 07 Sep 2026 12:13:48 -0400https://track.liberty.io/do-variables-without-any-number-beside-them-have-a-coefficient.htmlhttps://track.liberty.io/do-variables-without-any-number-beside-them-have-a-coefficient.htmlI've seen variables like $x, r, z, q, t$ that don't have any numbers beside them. Do they even have coefficients, likewise for negative variables without any numbers in front of the negative signs......Mon, 07 Sep 2026 12:04:40 -0400https://track.liberty.io/how-can-i-glide-longer.htmlhttps://track.liberty.io/how-can-i-glide-longer.htmlI want to glide longer but I don't know if I need the grappel accelerator and where I can get it? Any hints?...Mon, 07 Sep 2026 11:55:02 -0400https://track.liberty.io/which-3d-movie-file-format-is-automatically-recognized-as-3d-by-the-ps.htmlhttps://track.liberty.io/which-3d-movie-file-format-is-automatically-recognized-as-3d-by-the-ps.htmlI just bought a 3D TV. With 3D blu-rays and 3D games, the PS3 tells the TV that the following thing is 3D and the TV automatically switches to 3D mode. I have a few 3D shorts in mp4 files (side-by......Mon, 07 Sep 2026 11:54:39 -0400https://track.liberty.io/which-gods-and-game-mode-are-most-suitable-for-the-20-to-1-achievement.htmlhttps://track.liberty.io/which-gods-and-game-mode-are-most-suitable-for-the-20-to-1-achievement.htmlIn Smite there is an achievement for getting 20 kills and 1 death in a single game. Which gods, general strategy and game mode should I use to maximise my chances of obtaining this achievement?...Mon, 07 Sep 2026 11:44:23 -0400https://track.liberty.io/what-are-the-differences-between-counter-strike-global-offensive-and-c.htmlhttps://track.liberty.io/what-are-the-differences-between-counter-strike-global-offensive-and-c.htmlAs Counter Strike: Source is also a graphics improved version of original CS. But what are new features in Global Offensive and how they differs from Counter Strike: Source?...Mon, 07 Sep 2026 11:43:23 -0400https://track.liberty.io/how-do-you-unlock-each-party-member.htmlhttps://track.liberty.io/how-do-you-unlock-each-party-member.htmlThere are a lot of characters to unlock. What are the pre-requisites and what do you have to do to get them?...Mon, 07 Sep 2026 11:41:40 -0400https://track.liberty.io/show-that-the-gram-matrix-g-b-is-positive-definite.htmlhttps://track.liberty.io/show-that-the-gram-matrix-g-b-is-positive-definite.htmlSuppose we have $\boldsymbol P_{\leqslant 1}=\operatorname{span}\{1,x\}$ that is an inner product space with respect to $\int^1_0p(x)q(x)dx$. Consider the basis $B=\left\{b_1 = 1, b_2 = x\righ......Mon, 07 Sep 2026 11:28:26 -0400https://track.liberty.io/what-is-the-use-of-delta-symbol-in-set-theory.htmlhttps://track.liberty.io/what-is-the-use-of-delta-symbol-in-set-theory.htmlWhat is the use of $ \Delta $ in set theory?...Mon, 07 Sep 2026 11:23:09 -0400https://track.liberty.io/why-symmetric-relation-doesn-039-t-have-loops.htmlhttps://track.liberty.io/why-symmetric-relation-doesn-039-t-have-loops.htmlI have been reading Velleman's How to prove book and have come across following section while reading "Relations" chapter: Suppose R is a relation on A. R is said to be reflexive on A (o......Mon, 07 Sep 2026 10:57:05 -0400https://track.liberty.io/how-can-we-take-the-laplace-transform-of-a-piecewise-function.htmlhttps://track.liberty.io/how-can-we-take-the-laplace-transform-of-a-piecewise-function.htmlHow can we take the LaPlace transform of a function, given piece-wise function notation? For example, $f(t)=\begin{cases} 0 &\mbox{for } 0<t<2\\ t&\mbox{ for } 2<t \end{cases}$ ...Mon, 07 Sep 2026 10:45:50 -0400https://track.liberty.io/8-fold-symmetry-wrt-one-plane-implies-8-fold-symmetry-wrt-all-planes.htmlhttps://track.liberty.io/8-fold-symmetry-wrt-one-plane-implies-8-fold-symmetry-wrt-all-planes.htmlLet a bounded, open, connected set $A \subset \mathbb{R}^N$ (with $N \geq 3$) satisfy the following symmetry assumptions: $A$ is 4-fold rotational symmetric with respect to any coordinate plane $(......Mon, 07 Sep 2026 10:45:08 -0400https://track.liberty.io/funny-integral-meme-from-facebook.htmlhttps://track.liberty.io/funny-integral-meme-from-facebook.htmlThe following is a meme from Facebook. It's supposed to be a joke, but is there any serious mathematical way to evaluate this? (Tbh I just want to impress my friends by giving a smartass answer)...Mon, 07 Sep 2026 10:22:14 -0400https://track.liberty.io/what-039-s-the-difference-between-f-x-and-f-x-closed.htmlhttps://track.liberty.io/what-039-s-the-difference-between-f-x-and-f-x-closed.html(There is a question similar to mine, but it talks about f[x] and F[x].) I've seen some people use F(x) when denoting functions, but what is the difference between using a capital F and a lower ca......Mon, 07 Sep 2026 10:06:59 -0400https://track.liberty.io/is-frac-x-2-a-2-frac-y-2-b-2-frac-y-b-the-equation-of-an-ellipse-shoul.htmlhttps://track.liberty.io/is-frac-x-2-a-2-frac-y-2-b-2-frac-y-b-the-equation-of-an-ellipse-shoul.htmlWhile solving a question today I came across a locus of the form $$\frac{x^2}{a^2}+\frac{y^2}{b^2} = \frac{y}{b}$$ It was told in my book that it is the equation of an ellipse. How is that so? For......Mon, 07 Sep 2026 10:06:30 -0400https://track.liberty.io/how-do-i-allow-unit-stacking.htmlhttps://track.liberty.io/how-do-i-allow-unit-stacking.htmlI am browsing Steam Community Mods and I can't find a mod that does this....Mon, 07 Sep 2026 10:03:23 -0400https://track.liberty.io/how-to-calculate-fermat-point-in-a-triangle-most-efficiently.htmlhttps://track.liberty.io/how-to-calculate-fermat-point-in-a-triangle-most-efficiently.htmlI am aware of this question, but mine is a bit more specific. I want to find the coordinates of the Fermat point for a given triangle. Assuming that no angle in the triangle is larger than 120 degr......Mon, 07 Sep 2026 09:48:18 -0400https://track.liberty.io/trig-sum-tan-21-circ-tan-22-circ-cdots-tan-2-89-circ-text.htmlhttps://track.liberty.io/trig-sum-tan-21-circ-tan-22-circ-cdots-tan-2-89-circ-text.htmlAs the title suggests, I'm trying to find the sum $$\tan^21^\circ+\tan^2 2^\circ+\cdots+\tan^2 89^\circ$$ I'm looking for a solution that doesn't involve complex numbers, or any other advanced bran......Mon, 07 Sep 2026 09:45:56 -0400https://track.liberty.io/by-first-expanding-cos-2x-sin-2x-3-otherwise-show-that-cos-6x-sin-6x-1.htmlhttps://track.liberty.io/by-first-expanding-cos-2x-sin-2x-3-otherwise-show-that-cos-6x-sin-6x-1.htmlBy first expanding $(\cos^2x + \sin^2x)^3$, otherwise, show that $$ \cos^6x + \sin^6x = 1 - (3/4)\sin^2(2x)$$ Here's what I've done so far (starting from after expansion): $\cos^6x + (3\cos^4x\sin^......Mon, 07 Sep 2026 09:40:40 -0400https://track.liberty.io/which-armour-gives-a-larger-boost-to-ancient-weapons.htmlhttps://track.liberty.io/which-armour-gives-a-larger-boost-to-ancient-weapons.htmlAssuming I'm using ancient weapons, and have the barbarian armour set and ancient armour set upgraded twice, which set will I do more damage with with regular attacks, and which with charge attacks (...Mon, 07 Sep 2026 09:33:43 -0400https://track.liberty.io/laplace-transform-of-impulse-function.htmlhttps://track.liberty.io/laplace-transform-of-impulse-function.htmlThe laplace tranform of the following function impulse function is $$\int_{0}^{\infty}\delta(t).e^{-st}\,dt$$ $$=\int_{0}^{\infty}\delta(t)\,dt=1$$ (area under unit impulse is always 1) $$=1.\in......Mon, 07 Sep 2026 09:32:28 -0400https://track.liberty.io/quiz-on-android-ios-game.htmlhttps://track.liberty.io/quiz-on-android-ios-game.htmlI have played this game but I´ve lost my phone and forgot the name. What I know This game is available for iOS and Android This game is a quiz battle This game has levesl. This game has a lot of ......Mon, 07 Sep 2026 09:29:33 -0400https://track.liberty.io/understanding-identity-theorem-proof-in-complex-analysis.htmlhttps://track.liberty.io/understanding-identity-theorem-proof-in-complex-analysis.htmlIdentity theorem Let $f$ be holomorphic functions on some domain $D\subset\mathbb{C}$, and let $S$ be the set of all zeros of $f$ which has a limit point in $D$. Then $f$ is identically zero in $D$. ...Mon, 07 Sep 2026 09:29:08 -0400https://track.liberty.io/parametrization-of-a-line-segment-using-angle-as-parameter.htmlhttps://track.liberty.io/parametrization-of-a-line-segment-using-angle-as-parameter.htmlI know this is probably elementary level for most people here, but I've been stuck on this problem for no less than 4 hours and I am completely clueless as to how to figure this out. Is it possible......Mon, 07 Sep 2026 09:27:58 -0400https://track.liberty.io/proving-that-0-cdot-x-0-using-field-axioms.htmlhttps://track.liberty.io/proving-that-0-cdot-x-0-using-field-axioms.htmlConsider the following axiomatic definition of a field: A field is a set $F$ together with two binary operations $+$ and $\cdot$ on $F$ such that $(F,+)$ is an Abelian group with identity $0$ an......Mon, 07 Sep 2026 09:12:19 -0400https://track.liberty.io/how-to-prove-that-the-derivative-of-heaviside-039-s-unit-step-function.htmlhttps://track.liberty.io/how-to-prove-that-the-derivative-of-heaviside-039-s-unit-step-function.htmlHere is a problem from Griffith's book Introduction to E&M. Let $\theta(x)$ be the step function $$\theta = \begin{cases} 0, & x \le 0, \\ 1, & x \gt 0. \end{cases} $$ The qu......Mon, 07 Sep 2026 09:06:25 -0400https://track.liberty.io/what-should-i-do-with-nuggets.htmlhttps://track.liberty.io/what-should-i-do-with-nuggets.htmlI visited the quarry and acquired some silver & gold nuggets. I'd like to sell them, but only if there's nothing more useful I can do with them. I also am not sure where to sell them since the ...Mon, 07 Sep 2026 09:04:34 -0400https://track.liberty.io/simpsons-tapped-out-finding-maggie.htmlhttps://track.liberty.io/simpsons-tapped-out-finding-maggie.htmlIn the minigame where you have to find Maggie somewhere in Springfield, I always manage to locate her, but when I tap on her on the screen, nothing happens. A few times now I've had her in the cent......Mon, 07 Sep 2026 09:00:21 -0400https://track.liberty.io/cantor-set-and-fractals.htmlhttps://track.liberty.io/cantor-set-and-fractals.htmlI have read that the Cantor set is considered a fractal. I am referring to the Cantor set in which the middle third of a real line is removed recursively. I see that this is recursively defined, bu......Mon, 07 Sep 2026 08:52:02 -0400https://track.liberty.io/how-to-find-where-a-function-is-increasing-at-the-greatest-rate.htmlhttps://track.liberty.io/how-to-find-where-a-function-is-increasing-at-the-greatest-rate.htmlGiven the function $f(x) = \frac{1000x^2}{11+x^2}$ on the interval $[0, 3]$, how would I calculate where the function is increasing at the greatest rate? Differentiating the function will give its ...Mon, 07 Sep 2026 08:48:48 -0400https://track.liberty.io/are-there-any-treasure-vaults-to-break-into.htmlhttps://track.liberty.io/are-there-any-treasure-vaults-to-break-into.htmlWhile on my travels in Skyrim i found one place (Markarth) which reminded me of Vivec from Morrowind, in the sense that there was a secured area with the possibility of gold and loot behind locked ......Mon, 07 Sep 2026 08:45:07 -0400https://track.liberty.io/cash-register-key-locations.htmlhttps://track.liberty.io/cash-register-key-locations.htmlOn the map Interchange, I am looking for the cash register keys to unlock them for roubles, however, I haven't had any luck on finding any of them. What are their spawn locations (picture would be......Mon, 07 Sep 2026 08:44:36 -0400https://track.liberty.io/how-do-i-increase-the-maximum-amount-of-weight-my-character-can-carry-.htmlhttps://track.liberty.io/how-do-i-increase-the-maximum-amount-of-weight-my-character-can-carry-.htmlIn Skyrim, as with most RPGs, I find myself overwhelmed by a compulsive desire to obtain all of the things. All of them. To this end, I wound up picking up five iron warhammers...only to find out t......Mon, 07 Sep 2026 08:42:34 -0400https://track.liberty.io/when-do-multi-parries-happen.htmlhttps://track.liberty.io/when-do-multi-parries-happen.htmlDoing a regular parry is already hard enough, and I can only ever do it by accident, but there are some instances where I was hit by what I think were multi hit projectiles or moves, but somehow pa......Mon, 07 Sep 2026 08:41:37 -0400https://track.liberty.io/has-prof-otelbaev-shown-existence-of-strong-solutions-for-navier-stoke.htmlhttps://track.liberty.io/has-prof-otelbaev-shown-existence-of-strong-solutions-for-navier-stoke.htmlModerator Notice: I am unilaterally closing this question for three reasons. The discussion here has turned too chatty and not suitable for the MSE framework. Given the recent pre-print......Mon, 07 Sep 2026 08:31:26 -0400https://track.liberty.io/probability-of-one-normdist-being-greater-than-another-duplicate.htmlhttps://track.liberty.io/probability-of-one-normdist-being-greater-than-another-duplicate.htmlI have two independant normally distributed random variables. X ~ N(657, 3) Y ~ N(661, 2) P(x > y) = ? How do I calculate the probability of X being greater than Y? Using R for simulating, I am ...Mon, 07 Sep 2026 08:30:41 -0400https://track.liberty.io/backwards-epsilon.htmlhttps://track.liberty.io/backwards-epsilon.htmlWhat does the $\ni$ (backwards element of) symbol mean? It doesn't appear in the Wikipedia list of mathematical symbols, and a Google search for "backwards element of" or "backwards epsilon" turns......Mon, 07 Sep 2026 08:30:38 -0400https://track.liberty.io/deriving-the-formula-for-the-radius-of-the-circle-inscribed-in-an-equi.htmlhttps://track.liberty.io/deriving-the-formula-for-the-radius-of-the-circle-inscribed-in-an-equi.htmlI am trying to derive the formula for the radius of the circle inscribed in an equilateral triangle from scratch. Given $2*n$ = length of a side $H$ = the altitude of the triangle = $h + a......Mon, 07 Sep 2026 08:27:41 -0400https://track.liberty.io/is-there-a-way-to-earn-resources-fast.htmlhttps://track.liberty.io/is-there-a-way-to-earn-resources-fast.htmlIn Metal Gear Solid 5: The Phantom Pain, is there a quick/easy method to getting resources to upgrade Mother Base and unlocking Mission 46? Can I somehow buy resources?...Mon, 07 Sep 2026 08:18:25 -0400https://track.liberty.io/killing-animals-in-dishonored-will-that-ruin-achievements.htmlhttps://track.liberty.io/killing-animals-in-dishonored-will-that-ruin-achievements.htmlCan I kill animals, such as the dogs, and still get the achievements? I'm looking at one achievement that says I can kill "no one". I'm not sure if that includes the animals or not....Mon, 07 Sep 2026 08:12:05 -0400https://track.liberty.io/can-you-turn-weather-off-in-minecraft.htmlhttps://track.liberty.io/can-you-turn-weather-off-in-minecraft.htmlThe arrival of Minecraft Beta 1.5 brought weather conditions. However, it can at times be inconvenient when you're trying to screenshot something and show it off (as it reduces lighting and obstructs ...Mon, 07 Sep 2026 08:11:54 -0400https://track.liberty.io/can-an-autonomous-differential-equation-be-nonhomogeneous.htmlhttps://track.liberty.io/can-an-autonomous-differential-equation-be-nonhomogeneous.htmlWhen a differential equation $\dfrac{dy}{dt}=f(y)$ does not depends on $y$ then it is autonomous, and it can't be non-homogeneous? Can anyone explain this? Thanks!...Mon, 07 Sep 2026 08:07:45 -0400https://track.liberty.io/speed-of-a-particle-given-parametric-equations-of-x-and-y.htmlhttps://track.liberty.io/speed-of-a-particle-given-parametric-equations-of-x-and-y.htmlSo my $x(t)=\pi t +\cos(2\pi t -(\pi/2))$, and my $y(t)=\pi t +\sin(2\pi t -(\pi/2))$. I implicitly derived and got $$\frac{dy}{dt}=\frac{(\pi+\cos(2\pi t -(\pi/2))*2\pi)}{(\pi-\sin(2\pi t -(\pi/2)......Mon, 07 Sep 2026 08:03:09 -0400https://track.liberty.io/what-is-the-trick-to-the-final-boss-fight-in-uncharted-3.htmlhttps://track.liberty.io/what-is-the-trick-to-the-final-boss-fight-in-uncharted-3.htmlThis may seem like a bit of a weird question, especially considering I beat the game on Crushing last night, but here goes... What is the trick to winning the final boss fight against Talbot on the ...Mon, 07 Sep 2026 08:00:14 -0400https://track.liberty.io/notation-for-set-of-constant-functions.htmlhttps://track.liberty.io/notation-for-set-of-constant-functions.htmlI have a constant function that always returns the same integer value. How do I represent a set of functions where each function is a constant function that returns some arbitrary constant? For exa......Mon, 07 Sep 2026 07:55:21 -0400https://track.liberty.io/ivoerth-039-s-path-from-a-bygone-era.htmlhttps://track.liberty.io/ivoerth-039-s-path-from-a-bygone-era.htmlHow do you unlock this quest for Chapter 3, if you are on Ivoerth's path?...Mon, 07 Sep 2026 07:54:15 -0400https://track.liberty.io/are-proofs-by-contradiction-and-counterexample-two-different-technique.htmlhttps://track.liberty.io/are-proofs-by-contradiction-and-counterexample-two-different-technique.htmlNew to proofs. Suppose I want to prove $f: \mathbb Z \to \mathbb Z$ as $f(n) = 2n$ for all $n$ is not onto. Are the two arguments below valid, separate proofs? Are they invalid iterations of a sin......Mon, 07 Sep 2026 07:51:04 -0400https://track.liberty.io/proving-equation-has-only-one-solution.htmlhttps://track.liberty.io/proving-equation-has-only-one-solution.htmlSo i want to prove that $$x^2e^x=1$$ has at least one solution for $$x\in\mathbb{R}$$ I am kinda lost and would appreciate any help. This is suppose to be solved using basic calculus but i am not......Mon, 07 Sep 2026 07:45:05 -0400https://track.liberty.io/completing-the-lost-patrol-bos-side-quest.htmlhttps://track.liberty.io/completing-the-lost-patrol-bos-side-quest.htmlI seem to be having a problem completing the Brotherhood of Steel sidequest Lost Patrol, and I'm wondering if I've hit a bug/if anyone knows how to workaround it/has experienced the same problem/et......Mon, 07 Sep 2026 07:35:31 -0400https://track.liberty.io/how-to-use-a-bank-in-neverwinter-without-being-in-a-guild.htmlhttps://track.liberty.io/how-to-use-a-bank-in-neverwinter-without-being-in-a-guild.htmlI read that it is possible to deposit up to 16 items in bank, but when I approach the bank in Protector's Enclave it denies me access for "not being in a guild" Is it possible to use the bank to s......Mon, 07 Sep 2026 07:26:30 -0400https://track.liberty.io/what-is-the-difference-between-tekkit-and-normal-minecraft.htmlhttps://track.liberty.io/what-is-the-difference-between-tekkit-and-normal-minecraft.htmlI have the normal minecraft loader, and a lot of people tell me about Tekkit, what is the difference?...Mon, 07 Sep 2026 07:24:39 -0400https://track.liberty.io/what-happens-to-the-gem-when-i-salvage-an-item-with-a-gem-in-it.htmlhttps://track.liberty.io/what-happens-to-the-gem-when-i-salvage-an-item-with-a-gem-in-it.htmlI currently have no gold (just went shopping in the auction house... bought me some fancy and expensive weapon), but want to remove a gem from an item since I want to use it in another item. I'm ......Mon, 07 Sep 2026 07:10:03 -0400https://track.liberty.io/if-m-6x-5-what-equation-is-equivalent-to-6x-5-2-10-18x-15-in-terms-of-.htmlhttps://track.liberty.io/if-m-6x-5-what-equation-is-equivalent-to-6x-5-2-10-18x-15-in-terms-of-.htmlThis question I am asking right now comes from the Khan Academy practice task entitled 'Solve quadratic equations by using structure', so the credit goes to them. So, if $m = 6x + 5$, then what ...Mon, 07 Sep 2026 07:06:17 -0400https://track.liberty.io/what-is-the-difference-between-deviation-and-standard-deviation.htmlhttps://track.liberty.io/what-is-the-difference-between-deviation-and-standard-deviation.htmlIn my textbook, it describes the standard deviation "as a number that measures how far data values are from their mean". On the other hand, it says this about deviation: "if x is a number, then the ...Mon, 07 Sep 2026 07:03:19 -0400https://track.liberty.io/how-to-check-this-number-sqrt-47-is-irrational-duplicate.htmlhttps://track.liberty.io/how-to-check-this-number-sqrt-47-is-irrational-duplicate.htmlProve that $\sqrt{47}$ is irrational number. I know that a rational number is written as $\frac{p}{q}$ where $p$ & $q$ are co-prime numbers. But I do not have any idea to prove it irrational ......Mon, 07 Sep 2026 06:49:53 -0400https://track.liberty.io/solving-a-linear-system-of-differential-equations.htmlhttps://track.liberty.io/solving-a-linear-system-of-differential-equations.htmlGiven that $v_1 = \begin{bmatrix}1&1\end{bmatrix}$ and $v_2 = \begin{bmatrix}2 &1\end{bmatrix}$ are eigenvectors of the matrix $$ \begin{bmatrix}-1&-2\\1&-4\end{bmatrix} $$ which is......Mon, 07 Sep 2026 06:49:23 -0400https://track.liberty.io/can-quicksilver-sash-remove-airborne.htmlhttps://track.liberty.io/can-quicksilver-sash-remove-airborne.htmlIs this item able to remove airborne? To be more specific, during Yasuo's ult, can this item or Mercurial Scimitar dismiss the effect of Yasuo's ult?...Mon, 07 Sep 2026 06:44:27 -0400https://track.liberty.io/a-dot-with-custom-position-on-screen.htmlhttps://track.liberty.io/a-dot-with-custom-position-on-screen.htmlSo if you know about 3rd-person shooters, some of them don't have a cross and you have to predict where you shoot. Some people use a transparent tape with a marker dot to aim and I do that too, its......Mon, 07 Sep 2026 06:39:36 -0400