https://track.liberty.ioFresh celebrity buzz with instant attention value.en-usMon, 07 Sep 2026 16:39:59 -0400https://track.liberty.io/how-to-maximize-the-profit-for-the-given-equation.htmlhttps://track.liberty.io/how-to-maximize-the-profit-for-the-given-equation.htmlIt is a maximize profit question. Basically: There is a play which costs $180$ Each attendee costs $0.4$ Ticket price affects the overall attendance. When the ticket price is $5$, then there are ......Mon, 07 Sep 2026 20:37:18 -0400https://track.liberty.io/1-minus-percentage-what-is-this-called.htmlhttps://track.liberty.io/1-minus-percentage-what-is-this-called.htmlI'm trying to convert a percentage: The original percentage represents how bad the condition of a asset is (in other words, a high number is bad). I would like to convert the percentage to represen......Mon, 07 Sep 2026 20:30:44 -0400https://track.liberty.io/incapacitating-dahl-quest.htmlhttps://track.liberty.io/incapacitating-dahl-quest.htmlTowards the end of the game, you can incapacitate Dahl instead of outright killing him. This opens up an option to leave the station, as well as allow for any crew members you've saved to leave. I......Mon, 07 Sep 2026 20:25:37 -0400https://track.liberty.io/i-dismissed-jenassa-now-i-can-039-t-find-her-at-the-drunken-huntsman-w.htmlhttps://track.liberty.io/i-dismissed-jenassa-now-i-can-039-t-find-her-at-the-drunken-huntsman-w.htmlPossible Duplicate: How do I find my follower if and when they leave me? I paid Jenassa 500 gold and hired her as a companion. This was before I became the thane of Dragonsreach. I was really ......Mon, 07 Sep 2026 20:23:59 -0400https://track.liberty.io/circle-least-squares-fit.htmlhttps://track.liberty.io/circle-least-squares-fit.htmlSo my question is this: Find the equation of the circle that gives the best least squares circle fit to the points $(-1,-2), (0,2.4), (1.1,-4),$ and $(2.4,-1.6).$ So far I have this general equa......Mon, 07 Sep 2026 20:20:28 -0400https://track.liberty.io/do-ps1-or-ps2-memory-cards-die-over-time-like-cartridges.htmlhttps://track.liberty.io/do-ps1-or-ps2-memory-cards-die-over-time-like-cartridges.htmlI have a PS1, PS2, PSP and a PS3. With an adapter (official, made by Sony), I was able to copy my old file saves from my old games into my PS3, so I can keep them without being erased for long time......Mon, 07 Sep 2026 20:19:33 -0400https://track.liberty.io/is-it-worth-it-to-get-better-at-contest-math.htmlhttps://track.liberty.io/is-it-worth-it-to-get-better-at-contest-math.htmlI have never done well in math competitions, and am now past the point at which I can participate in them. I am asking if it is worth it to go back and practice such types of problems until I gain ......Mon, 07 Sep 2026 20:14:54 -0400https://track.liberty.io/how-to-find-the-domain-of-this-function.htmlhttps://track.liberty.io/how-to-find-the-domain-of-this-function.htmlf(x)= $\frac{(\sqrt{x}-\sqrt{x-1} )}{( \sqrt{x}+\sqrt{x-1} )}\;$ first off $\sqrt{x}$ is defined for: $$x > 0 \tag{1}$$ and $\sqrt{x-1}$ is defined for: $$x \ge 1 ......Mon, 07 Sep 2026 20:13:34 -0400https://track.liberty.io/magnitude-of-cross-product-given-2-vector-magnitudes-and-angle-between.htmlhttps://track.liberty.io/magnitude-of-cross-product-given-2-vector-magnitudes-and-angle-between.htmlif the magnitude of vector U is 6 and the magnitude of vector V is 9, and the angle between these vectors is 60 degrees, what is ||U x V||? Why is the answer not 54Sin(60)?...Mon, 07 Sep 2026 20:05:04 -0400https://track.liberty.io/how-can-i-soft-reset.htmlhttps://track.liberty.io/how-can-i-soft-reset.htmlI've already completed the game 3 times and I only recently found out about Soft Resetting. I wanted to try it but after 17 on Heatran, I got really Impatient. If anyone has an easily explained tu......Mon, 07 Sep 2026 20:03:01 -0400https://track.liberty.io/what-is-a-homomorphism.htmlhttps://track.liberty.io/what-is-a-homomorphism.htmlI am starting to see the term everywhere I look, but every time I do, I get confused and can't get past it. I've seen various definitions: ("linear" homomorphism, i think) $$f:S\rightarrow T $$ $$\...Mon, 07 Sep 2026 19:57:13 -0400https://track.liberty.io/prerequisites-for-atiyah-macdonald.htmlhttps://track.liberty.io/prerequisites-for-atiyah-macdonald.htmlI am currently doing a one semester course on groups and rings where we have learned about (so far): Definitions of groups, subgroups, cyclic and normal subgroups, the symmetric group, homomorphis......Mon, 07 Sep 2026 19:56:34 -0400https://track.liberty.io/difficulty-in-understanding-find-e-n-where-n-min-n-0-x-n-x-0.htmlhttps://track.liberty.io/difficulty-in-understanding-find-e-n-where-n-min-n-0-x-n-x-0.htmlIt is Question 30 on page 168 in Ross's book (Introduction to Probability Models-11th edition) Let $X_i, i\ge 0$ be independent and identically distributed random variables with probability mass ...Mon, 07 Sep 2026 19:49:12 -0400https://track.liberty.io/which-buildings-have-lots-of-ingredients.htmlhttps://track.liberty.io/which-buildings-have-lots-of-ingredients.htmlI'm prioritizing improvement of the Alchemy and Enchanting skills because they can enhance practically everything else, including each other. I'm focusing on creating potions with a Paralyze effect......Mon, 07 Sep 2026 19:46:21 -0400https://track.liberty.io/classification-of-finite-abelian-groups.htmlhttps://track.liberty.io/classification-of-finite-abelian-groups.htmlI'm new to group theory and struggling with understanding what is required in an example problem I'm working through. I need to classify the following using the classification og finite abelian gr......Mon, 07 Sep 2026 19:42:19 -0400https://track.liberty.io/how-can-i-talk-to-neloth-without-him-attacking-me.htmlhttps://track.liberty.io/how-can-i-talk-to-neloth-without-him-attacking-me.htmlEvery time I walk into the mushroom area, Neloth and his assistant try to kill me. I need to talk to them for the quest Path of Knowledge, but I'm not able to....Mon, 07 Sep 2026 19:41:38 -0400https://track.liberty.io/do-diamonds-spawn-less-in-the-mountains-and-extreme-hills-biomes.htmlhttps://track.liberty.io/do-diamonds-spawn-less-in-the-mountains-and-extreme-hills-biomes.htmlI don't know if diamonds spawn less in mountains and extreme hills biomes or not, but I always seem to get extreme-hilly unlucky (sorry) with finding diamonds in those biomes. The other day I spent......Mon, 07 Sep 2026 19:40:58 -0400https://track.liberty.io/deducing-pa-039-s-axioms-in-zfc.htmlhttps://track.liberty.io/deducing-pa-039-s-axioms-in-zfc.htmlRecently I've stumbled across this claim: Peano axioms can be deduced in ZFC I found a lot of info regarding this claim (e.g. what would (one version of) the natural numbers look like within the ...Mon, 07 Sep 2026 19:09:21 -0400https://track.liberty.io/proof-of-a-determinant-identity.htmlhttps://track.liberty.io/proof-of-a-determinant-identity.htmlI have a matrix identity that I wish to prove, which relates the determinant of a matrix to determinants of sub-matrices (essentially, cofactors of the larger matrix). In general terms, consider the ...Mon, 07 Sep 2026 19:07:57 -0400https://track.liberty.io/find-the-maximum-and-minimum-value-of-variance-duplicate.htmlhttps://track.liberty.io/find-the-maximum-and-minimum-value-of-variance-duplicate.htmlLet $X$ be an arbitrary random variable takes values in $\{0,1,2,...,10\}$. Then the minimum and maximum values of variance of random variable $X$ are $0$ and $30$ $1$ and $30$ $0$ and $25$ $1$......Mon, 07 Sep 2026 19:06:11 -0400https://track.liberty.io/where-can-i-find-iron-gold-and-diamond-in-survival-mode.htmlhttps://track.liberty.io/where-can-i-find-iron-gold-and-diamond-in-survival-mode.htmlWhere can I find Iron, Gold, and Diamond in Minecraft Pocket Edition Survival Mode? I can only find gravel, dirt, wood, sand, and cobblestone....Mon, 07 Sep 2026 18:57:56 -0400https://track.liberty.io/matchmaking-claiming-i-have-to-reconnect-to-a-finished-game.htmlhttps://track.liberty.io/matchmaking-claiming-i-have-to-reconnect-to-a-finished-game.htmlToday I played 2 competitive matches in CS:GO. After the first competitive match ended, and tried to start the second game, it told me I still had a game going, and should reconnect from the main m......Mon, 07 Sep 2026 18:50:05 -0400https://track.liberty.io/prove-the-converse-of-wilson-039-s-theorem-duplicate.htmlhttps://track.liberty.io/prove-the-converse-of-wilson-039-s-theorem-duplicate.html... namely that if $n > 1$ and $(n − 1)!\equiv−1\pmod{n}$, then $n$ is prime. This is for a number theory class I'm in at Penn State. My idea is to follow accordingly, but I can't get it concre......Mon, 07 Sep 2026 18:42:15 -0400https://track.liberty.io/using-the-arnoldi-iteration-to-find-the-k-largest-eigenvalues-of-a-mat.htmlhttps://track.liberty.io/using-the-arnoldi-iteration-to-find-the-k-largest-eigenvalues-of-a-mat.htmlI'm trying to obtain a general understanding of this algorithm which determines the k-largest eigenvalues of a matrix $A\in \mathbb{R}^{n\times n}$. How I see it: power iteration: take random ...Mon, 07 Sep 2026 18:31:05 -0400https://track.liberty.io/does-collecting-keys-and-dogtags-have-any-purpose-other-than-trophies.htmlhttps://track.liberty.io/does-collecting-keys-and-dogtags-have-any-purpose-other-than-trophies.htmlIs there a specific reason to collect all of these car keys, or all of the dog tags? To put it simply, are these only in the game for those try-hard gamers that think they have to get everything in a ...Mon, 07 Sep 2026 18:29:29 -0400https://track.liberty.io/droprate-of-prideful-gear-from-celestials.htmlhttps://track.liberty.io/droprate-of-prideful-gear-from-celestials.htmlAs PvP season 15 is dawning upon us, Blizzard has decided to give us a chance at next seasons gear by letting us do Celestials and getting a chance for the gear. What are the drop rates of the items ...Mon, 07 Sep 2026 18:23:27 -0400https://track.liberty.io/which-npcs-can-point-me-to-san-francisco.htmlhttps://track.liberty.io/which-npcs-can-point-me-to-san-francisco.htmlI want to travel to San Francisco. Unfortunately it's been a long time and I forgot where it is and people in other towns don't want to tell me. I have asked for directions and got plenty, but none......Mon, 07 Sep 2026 18:21:40 -0400https://track.liberty.io/how-many-different-eulerian-circuits-are-there-in-this-graph.htmlhttps://track.liberty.io/how-many-different-eulerian-circuits-are-there-in-this-graph.htmlI'm trying to figure out how many different Eulerian circuits there are in the following graph. I have found one Eulerian circuit so far (see photo below, start- and end point to the far left). How ...Mon, 07 Sep 2026 17:59:37 -0400https://track.liberty.io/orthogonal-art-gallery-problem.htmlhttps://track.liberty.io/orthogonal-art-gallery-problem.htmlHere is a problem: One should position the least amount of cameras in an art gallery. The walls in the gallery are represented only by horizontal and vertical line segments. I am trying to find the ...Mon, 07 Sep 2026 17:54:04 -0400https://track.liberty.io/find-the-projection-of-b-onto-the-column-space-of-a.htmlhttps://track.liberty.io/find-the-projection-of-b-onto-the-column-space-of-a.html$A= \left[ {\begin{array}{ccccc} 1 & 1 \\ 1 & -1 \\ -2 & 4 \end{array} } \right] $ and $b = \left[ {\begin{array}{cccc} 1 \\ 2 \\ 7 \end{array} } \right] $ ......Mon, 07 Sep 2026 17:53:50 -0400https://track.liberty.io/minecraft-sound-effects.htmlhttps://track.liberty.io/minecraft-sound-effects.htmlI am doing a Minecraft animation and I need Minecraft sound effects does anyone know any safe website where java Minecraft sounds can be downloaded?...Mon, 07 Sep 2026 17:40:18 -0400https://track.liberty.io/markov-chains-and-stochastic-matrix.htmlhttps://track.liberty.io/markov-chains-and-stochastic-matrix.htmlGive an example of a $2\times 2$ regular transition matrix $T$ and an initial state probability vector $x$ such that all the entries in $T^{1000}$x is still different from the steady state vector w......Mon, 07 Sep 2026 17:36:34 -0400https://track.liberty.io/the-case-of-being-vacuously-true-trivially-true-and-the-empty-set.htmlhttps://track.liberty.io/the-case-of-being-vacuously-true-trivially-true-and-the-empty-set.htmlLet's suppose that we need to prove a statement of the form $\forall x(x\in A\longrightarrow\phi)$, where $\phi$ is any property (for example when we need to prove that $A\subseteq A\cup B$). Is it ...Mon, 07 Sep 2026 17:28:27 -0400https://track.liberty.io/is-it-correct-to-say-that-square-root-of-0-9-is-0-9-itself.htmlhttps://track.liberty.io/is-it-correct-to-say-that-square-root-of-0-9-is-0-9-itself.htmlWhen I calculate Square Root of 0.9, it comes around 0.9486832980505138. Though I have heard people occasionally saying that the square root of 0.9 is 0.9 itself. Would it be correct to make a stat......Mon, 07 Sep 2026 17:15:56 -0400https://track.liberty.io/find-the-remainder-when-123412341234-written-1234-times-is-divided-by-.htmlhttps://track.liberty.io/find-the-remainder-when-123412341234-written-1234-times-is-divided-by-.htmlFind the remainder when $123412341234$...(written $1234$ times) is divided by $13$ I divided $1234$ by $13$ and I got the answer $12$ which is the correct answer but what is the approach behind so......Mon, 07 Sep 2026 17:05:39 -0400https://track.liberty.io/intuitive-explanation-of-double-expectation.htmlhttps://track.liberty.io/intuitive-explanation-of-double-expectation.htmlThis has been bugging me for some time. The famous result in probability is like $E[Y] = E[E[Y|X]]$ Can someone write an intuitive explanation of the above?...Mon, 07 Sep 2026 17:04:22 -0400https://track.liberty.io/the-proper-symbol-to-denote-proper-subset.htmlhttps://track.liberty.io/the-proper-symbol-to-denote-proper-subset.html⊂ and ⊃ symbols Some authors use the symbols ⊂ and ⊃ to indicate subset and superset respectively; that is, with the same meaning and instead of the symbols, ⊆ and ⊇ Other authors prefer to use the ...Mon, 07 Sep 2026 17:01:57 -0400https://track.liberty.io/how-to-avoid-introducing-a-spurious-solution.htmlhttps://track.liberty.io/how-to-avoid-introducing-a-spurious-solution.htmlSuppose I have $\sqrt{r-x}-\frac{x}{\sqrt{r-x}}-o=0$ with $r,x$ and $o$ real. To solve I multiply by $\sqrt{r-x}$. I assume therefore $r>x$ $r-x-x-o\sqrt{r-x}=0$ $r-2x=o\sqrt{r-x}$ Squaring......Mon, 07 Sep 2026 16:56:17 -0400https://track.liberty.io/show-that-the-inverse-of-a-strictly-diagonally-dominant-matrix-is-mono.htmlhttps://track.liberty.io/show-that-the-inverse-of-a-strictly-diagonally-dominant-matrix-is-mono.htmlI have been struggling with this problem for awhile. Given that $A$ is a strictly diagonally dominant matrix with positive diagonal entries and non-positive off-diagonal entries, show that $A$ is ...Mon, 07 Sep 2026 16:56:12 -0400https://track.liberty.io/why-do-the-ivs-change-based-on-level.htmlhttps://track.liberty.io/why-do-the-ivs-change-based-on-level.htmlI know how IVs in Pokemon Go work, but I don't understand why when I check my Pokemon's IVs on an IV calculator, I get multiple IV combinations back. For instance, say I check IVs on a Pikachu. T......Mon, 07 Sep 2026 16:56:06 -0400https://track.liberty.io/can-a-system-of-linear-equations-be-inconsistent-even-if-the-last-colu.htmlhttps://track.liberty.io/can-a-system-of-linear-equations-be-inconsistent-even-if-the-last-colu.htmlI have a simple question which is troubling me. I have seen this theorem in Linear Algebra which I quote here : The Row Echelon Form of an Inconsistent System: An augmented matrix corresponds t......Mon, 07 Sep 2026 16:55:39 -0400https://track.liberty.io/how-can-one-prove-that-pi-4-pi-5-e-6.htmlhttps://track.liberty.io/how-can-one-prove-that-pi-4-pi-5-e-6.htmlA proof of the inequality using properties of $\pi$ and $e$, for example, is what I'm looking for. Not calculator approximations showing the inequality holds....Mon, 07 Sep 2026 16:48:24 -0400https://track.liberty.io/meaning-of-limit-as-h-approaches-0.htmlhttps://track.liberty.io/meaning-of-limit-as-h-approaches-0.htmlI am learning differentiation, and am having some trouble understanding what is meant by h approaching zero, from a positive or negative direction. If I understand correctly, h is one point on the ......Mon, 07 Sep 2026 16:40:34 -0400https://track.liberty.io/missing-orcalculum-for-ultima-weapon.htmlhttps://track.liberty.io/missing-orcalculum-for-ultima-weapon.htmlI’m literally missing a orcalculum+! I just started gathering them, and gathered the one from Pirates in the chest and the one from final world in the chest. It said I obtained one both times but my ...Mon, 07 Sep 2026 16:40:18 -0400https://track.liberty.io/how-many-rows-and-columns-are-in-an-m-x-n-matrix.htmlhttps://track.liberty.io/how-many-rows-and-columns-are-in-an-m-x-n-matrix.htmlA simple question: By definition, does an m x n matrix have m rows and n columns, or is it vice versa?...Mon, 07 Sep 2026 16:27:46 -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 -0400