https://track.liberty.ioFresh celebrity buzz with instant attention value.en-usSun, 06 Sep 2026 23:29:04 -0400https://track.liberty.io/variance-of-the-mle-for-a-geometric-distribution.htmlhttps://track.liberty.io/variance-of-the-mle-for-a-geometric-distribution.htmlI have derived the MLE for $p$ in the geometric distribution, $p=1/\overline X$. I now need to find its asymptotic variance. Trying to take the variance of $1/\overline X$ directly seems intracta......Mon, 07 Sep 2026 03:29:00 -0400https://track.liberty.io/transpose-of-conjugate-transpose.htmlhttps://track.liberty.io/transpose-of-conjugate-transpose.htmlWhat is the transpose of a conjugate transpose? Let's say we have a matrix $A$ and its conjugate transpose $A^{*}$, would the transpose of the conjugate transpose be the conjugate of $A$, i.e. $\ba......Mon, 07 Sep 2026 03:28:15 -0400https://track.liberty.io/prove-that-the-following-two-definitions-of-the-convex-hull-are-equiva.htmlhttps://track.liberty.io/prove-that-the-following-two-definitions-of-the-convex-hull-are-equiva.htmlI was wondering if a topology expert could help me solve this proof, as I have no idea but want to understand these concepts. This is not for homework. Let X be a point set, not necessarily finit......Mon, 07 Sep 2026 03:24:53 -0400https://track.liberty.io/non-ambiguous-description-of-pasta-property.htmlhttps://track.liberty.io/non-ambiguous-description-of-pasta-property.htmlI find the word "see" as in PASTA (Poisson Arrivals See Time Averages) to be a bit ambiguous and thus wanted some clarification in what it means (see http://pages.cs.wisc.edu/~dsmyers/cs547/...Mon, 07 Sep 2026 03:19:50 -0400https://track.liberty.io/learning-trigonometry-on-my-own.htmlhttps://track.liberty.io/learning-trigonometry-on-my-own.htmlI have been self teaching myself math beginning with a grade 10 level for a while now and need learn trigonometry from near scratch. I am seeking both books and perhaps lectures on trigonometry and ...Mon, 07 Sep 2026 03:19:20 -0400https://track.liberty.io/solving-a-logarithmic-polynomial.htmlhttps://track.liberty.io/solving-a-logarithmic-polynomial.htmlI want to solve this equation for $x$: $${\frac{1}{\sqrt{2 \pi x}} \left(\frac{e z}{2x}\right)^x} = \epsilon$$ Is there a closed form for it, or does it have to be solved numerically? I can turn......Mon, 07 Sep 2026 03:18:16 -0400https://track.liberty.io/derivative-of-sec-1-x.htmlhttps://track.liberty.io/derivative-of-sec-1-x.htmlI'm struggling with problem below which I eagerly want to solve. Let me know from where this problem is, if possible (the origin source of textbook). your answers might really helpful to get throug......Mon, 07 Sep 2026 03:14:18 -0400https://track.liberty.io/how-to-demolish-skyscraper.htmlhttps://track.liberty.io/how-to-demolish-skyscraper.htmlIn Battlefield 4 on map "Siege of Shanghai" there is skyscraper in center. By the end of "Conquest" round it usually destroyed. But every time I miss that moment. What trigger destruction of this ...Mon, 07 Sep 2026 03:06:45 -0400https://track.liberty.io/how-do-you-hide-command-blocks.htmlhttps://track.liberty.io/how-do-you-hide-command-blocks.htmlI am making a server, and I want to teleport the closest player to a certain area. But I don't know how to make the command blocks invisible, so the players don't see them. Does anyone know how?...Mon, 07 Sep 2026 02:55:37 -0400https://track.liberty.io/further-factorisation-of-a-difference-of-cubes.htmlhttps://track.liberty.io/further-factorisation-of-a-difference-of-cubes.htmlWe know that a difference of cubes can be factored using $a^3-b^3=(a-b)(a^2+ab+b^2)$. How do we know that the quadratic can't be factored further. For example, $$ 27x^3-(x+3)^3=((3x-(x+3))(9x^2+3......Mon, 07 Sep 2026 02:49:14 -0400https://track.liberty.io/what-is-48-div2-9-3-duplicate.htmlhttps://track.liberty.io/what-is-48-div2-9-3-duplicate.htmlThere is a huge debate on the internet on the value of $48\div2(9+3)$. I believe the answer $2$ as I believe it is part of the bracket operation in BEDMAS. Mathway yields the same answer. I also be......Mon, 07 Sep 2026 02:27:45 -0400https://track.liberty.io/convolution-of-a-step-function-with-itself.htmlhttps://track.liberty.io/convolution-of-a-step-function-with-itself.htmlI have a step function that states $f(x) = 1$ for $|x| < 1$ and 0 everywhere else. So far I've found this to be $$\int_{-\infty}^{\infty}\, dy\, f(y)\,f(x-y)$$ and from boundaries, $$-1 < y &......Mon, 07 Sep 2026 02:27:20 -0400https://track.liberty.io/how-to-get-the-i-like-it-hot-achievement-in-tsw-l-how-to-make-a-flamin.htmlhttps://track.liberty.io/how-to-get-the-i-like-it-hot-achievement-in-tsw-l-how-to-make-a-flamin.htmlThere is an Regional Achievement (hubs) called "I like it hot" in Secret World: Legends. To get it, you need to eat a "Flamin' Hot Taco". How do you get one?...Mon, 07 Sep 2026 02:13:05 -0400https://track.liberty.io/how-to-find-the-multiplicity-of-eigenvalues.htmlhttps://track.liberty.io/how-to-find-the-multiplicity-of-eigenvalues.htmlI don't understand how to find the multiplicity for an eigenvalue. To be honest, I am not sure what the books means by multiplicity. For instance, finding the multiplicty of each eigenvalue for the ...Mon, 07 Sep 2026 02:08:37 -0400https://track.liberty.io/can-039-t-seem-to-get-a-minecart-to-move.htmlhttps://track.liberty.io/can-039-t-seem-to-get-a-minecart-to-move.htmlI have Minecraft 1.8. I placed a powered rail on the floor of a room in my basement and a button to activate it. I place the minecart, press the button and the cart doesn't move. If I do the same ...Mon, 07 Sep 2026 02:06:53 -0400https://track.liberty.io/cauchy-integral-formula.htmlhttps://track.liberty.io/cauchy-integral-formula.htmlCan someone please help me answer this question as I cannot seem to get to the answer. Please note that the Cauchy integral formula must be used in order to solve it. Many thanks in advance! \begin{...Mon, 07 Sep 2026 02:06:17 -0400https://track.liberty.io/i-can-039-t-craft-a-silver-watch-terarria.htmlhttps://track.liberty.io/i-can-039-t-craft-a-silver-watch-terarria.htmlI have all the materials needed for a Silver Watch. I have 9 chains and 22 Silver. (I know you need 10 silver and 1 chain) But I can't craft it? I try a Crafting Table, Furnace, Anvil, Advanced Cra......Mon, 07 Sep 2026 02:05:20 -0400https://track.liberty.io/questions-tagged-world-of-warcraft.htmlhttps://track.liberty.io/questions-tagged-world-of-warcraft.htmlQ&A for passionate videogamers on all platforms...Mon, 07 Sep 2026 01:59:00 -0400https://track.liberty.io/fallout-4-do-i-really-need-the-institute-to-continue-and-complete-fall.htmlhttps://track.liberty.io/fallout-4-do-i-really-need-the-institute-to-continue-and-complete-fall.htmlI don't think I'm going to move forward anymore with the Institute, as a glitch has occurred in my previous question! But is it worth actually trying to fix this? Do I really need the Institute or ......Mon, 07 Sep 2026 01:51:17 -0400https://track.liberty.io/where-can-i-get-something-better-than-a-bronze-edge-for-frog.htmlhttps://track.liberty.io/where-can-i-get-something-better-than-a-bronze-edge-for-frog.htmlI've finally managed to rescue Frog from his sulking with a really expensive gift. However, he can't use that gift. For that matter, I'm having trouble finding something he can use. I travelled all......Mon, 07 Sep 2026 01:50:34 -0400https://track.liberty.io/can-you-easily-find-friends-039-ghosts.htmlhttps://track.liberty.io/can-you-easily-find-friends-039-ghosts.htmlI don't currently have Mario Kart 8 Deluxe, only the Wii U version. I'm trying to assess whether the deluxe version has some functionality I've been really wanting. On the Wii U, my friends and I ......Mon, 07 Sep 2026 01:30:24 -0400https://track.liberty.io/what-do-you-get-as-a-member-in-runescape.htmlhttps://track.liberty.io/what-do-you-get-as-a-member-in-runescape.htmlI thought of becoming a paid Runescape member, but I'm not sure the expense is worth it. Therefore I'm wondering what are the advantages of becoming a member? What more could I do? I'm already a l......Mon, 07 Sep 2026 01:28:06 -0400https://track.liberty.io/the-nth-order-linear-differential-equation-with-the-constant-coefficie.htmlhttps://track.liberty.io/the-nth-order-linear-differential-equation-with-the-constant-coefficie.htmlWe have the nth order linear differential equation with the constant coefficients such that $a_ny^n + a_{n-1}y^{n-1}+...+ a_1y' + a_0y= 0$. What would be the properties of the solution if the time ......Mon, 07 Sep 2026 01:23:56 -0400https://track.liberty.io/square-root-of-2-is-irrational.htmlhttps://track.liberty.io/square-root-of-2-is-irrational.htmlI am studying the proof that $\sqrt 2$ is an irrational number. Now I understand most of the proof, but I lack an understanding of the main idea which is: We assume $\frac{m^2}{n^2} = 2$. Then ......Mon, 07 Sep 2026 01:23:56 -0400https://track.liberty.io/what-is-a-dual-space.htmlhttps://track.liberty.io/what-is-a-dual-space.htmlI've started studying differential geometry by myself and I ran into dual spaces in a section on 1-forms. I'm not very well versed in linear algebra so any help is much appreciated....Mon, 07 Sep 2026 01:21:38 -0400https://track.liberty.io/macro-to-interact-with-target.htmlhttps://track.liberty.io/macro-to-interact-with-target.htmlIn World of Warcraft, is there an equivalent that you can use in a macro to 'Interact With Target' so that I can create a macro that interacts with specific NPCs with a single keypress?...Mon, 07 Sep 2026 01:11:55 -0400https://track.liberty.io/how-do-i-downgrade-xbox-360-dashboard.htmlhttps://track.liberty.io/how-do-i-downgrade-xbox-360-dashboard.htmlI have an Xbox 360 Elite and I want to downgrade the dashboard version from 2.x to 9199. I tried putting $SystemUpdate (from the official $SystemUpdate9199.zip) folder into a FAT32 formatted USB and ...Mon, 07 Sep 2026 01:07:44 -0400https://track.liberty.io/quadratic-formula-with-negative-square-root.htmlhttps://track.liberty.io/quadratic-formula-with-negative-square-root.htmlI encountered this while doing a very basic physics question (so I hope you don't mind if I post it, the problem lies within my math anyways): A stunt vehicle leaves an inclined slope of 28 degrees......Mon, 07 Sep 2026 00:57:57 -0400https://track.liberty.io/how-do-you-find-a-multiplicative-inverse-in-modulo-arithmetic.htmlhttps://track.liberty.io/how-do-you-find-a-multiplicative-inverse-in-modulo-arithmetic.htmlIn one of my lectures I have been given this example: When Googling 'multiplicative inverse' most of the tutorials seem to indicate it's as easy as just multiplying a number by the number divided ......Mon, 07 Sep 2026 00:27:53 -0400https://track.liberty.io/how-many-players-can-play-splitscreen-in-cod-wwii.htmlhttps://track.liberty.io/how-many-players-can-play-splitscreen-in-cod-wwii.htmlHow many players can play split-screen, and what game modes are (or aren't) available? How many players can play local co-op? How many players can play local multiplayer? How many players can play ...Mon, 07 Sep 2026 00:16:21 -0400https://track.liberty.io/executing-a-command-for-an-entity-in-front-of-me.htmlhttps://track.liberty.io/executing-a-command-for-an-entity-in-front-of-me.htmlI'd like to know if there is a command for detecting an entity in front of me, that upon detection can execute a command for that entity. The command would be like: There's a cow in front the p......Mon, 07 Sep 2026 00:11:20 -0400https://track.liberty.io/subgroups-of-the-klein-4-group.htmlhttps://track.liberty.io/subgroups-of-the-klein-4-group.htmlCan anyone explain to me the subgroups of the Klein-4 group? I'm trying to view it this way: I want some groups that are not empty and $ab^{-1} \in H$, where $H$ denotes the subgroups I am looking......Mon, 07 Sep 2026 00:09:48 -0400https://track.liberty.io/do-normal-region-restrictions-apply-when-playing-ps1-games-on-ps3.htmlhttps://track.liberty.io/do-normal-region-restrictions-apply-when-playing-ps1-games-on-ps3.htmlI got my PS1 while living in the US and I live in EU now. Will my US PS1 games work on a PAL PS3 or do normal region restrictions apply?...Mon, 07 Sep 2026 00:03:57 -0400https://track.liberty.io/derivation-of-chi-squared-pdf-with-one-degree-of-freedom-from-normal-d.htmlhttps://track.liberty.io/derivation-of-chi-squared-pdf-with-one-degree-of-freedom-from-normal-d.htmlHow can we derive the chi-squared probability density function (pdf) using the pdf of normal distribution? I mean, I need to show that $$f(x)=\frac{1}{2^{r/2}\Gamma(r/2)}x^{r/2-1}e^{-x/2} \>, \...Sun, 06 Sep 2026 23:59:01 -0400https://track.liberty.io/what-is-complex-analysis-why-is-it-accompanied-by-linear-algebra.htmlhttps://track.liberty.io/what-is-complex-analysis-why-is-it-accompanied-by-linear-algebra.htmlI hope this doesn't extend to a lengthy question. I studied Linear Algebra recently in my first term at university. I came to the realization however that some institutions would teach that course ......Sun, 06 Sep 2026 23:58:36 -0400https://track.liberty.io/is-there-a-strategy-to-win-this-move-color-game.htmlhttps://track.liberty.io/is-there-a-strategy-to-win-this-move-color-game.htmlMy girlfriend has been playing a game on her smartphone for a while. The rules are: there are $n+2$ tubes (with $n\ge 1$) and $n$ colored liquids. Two tubes are left empty and the others are fille......Sun, 06 Sep 2026 23:41:40 -0400https://track.liberty.io/plot-y-range-on-mathematica.htmlhttps://track.liberty.io/plot-y-range-on-mathematica.htmlI have a plot that I would like to slightly manipulate in Mathematica. Here is the code I am entering: Plot[{x, 2^x, log_2(x)}, {x, -1, 3}] As you can see $x$, $2^x$, and $log_2(x)$ are all plo......Sun, 06 Sep 2026 23:40:49 -0400https://track.liberty.io/fubini-study-metric-on-the-complex-projective-space-mathbb-pc-m.htmlhttps://track.liberty.io/fubini-study-metric-on-the-complex-projective-space-mathbb-pc-m.htmlI study Andrei Moroianu's Lectures on Kähler Geometry and have some troubles to understand the proof of Lemma 7.1. The claim is that the tensor $h$ that gives some called Fubini–Study metric on the ...Sun, 06 Sep 2026 23:30:14 -0400https://track.liberty.io/is-equilateral-trapezium-possible.htmlhttps://track.liberty.io/is-equilateral-trapezium-possible.htmlQ.Is equilateral Trapezium possible? The defination of Trapezium :A quadrilateral with one pair of parallel sides. If we have a trapezium that have equal sides,it becomes a rhombus (& contrad......Sun, 06 Sep 2026 23:28:19 -0400https://track.liberty.io/intuitive-understanding-of-the-constant-e.htmlhttps://track.liberty.io/intuitive-understanding-of-the-constant-e.htmlPotentially related-questions, shown before posting, didn't have anything like this, so I apologize in advance if this is a duplicate. I know there are many ways of calculating (or should I say "e......Sun, 06 Sep 2026 23:26:22 -0400https://track.liberty.io/average-length-of-chord-of-circle-definite-integral.htmlhttps://track.liberty.io/average-length-of-chord-of-circle-definite-integral.htmlGiven a circle C of radius a, and a diameter AB of C. Chords are drawn perpendicular to AB, intercepting equal arcs along the circumference of C. Find the limit of the average of the lengths of t......Sun, 06 Sep 2026 23:19:28 -0400https://track.liberty.io/pairs-of-trig-functions-pythagorean-identities-and-derivatives.htmlhttps://track.liberty.io/pairs-of-trig-functions-pythagorean-identities-and-derivatives.htmlOr, "Why do the six trig functions split up into three pairs of best friends?" There are three Pythagorean identities: $$ \begin{align*} \sin^2 x + \cos^2 x &= 1\\ \tan^2 x + 1 &= \sec^2 ......Sun, 06 Sep 2026 23:05:40 -0400https://track.liberty.io/finding-the-angular-size-of-a-minor-arc.htmlhttps://track.liberty.io/finding-the-angular-size-of-a-minor-arc.htmlThe question is as follows: A 20-inch chord is drawn in a circle with a 12-inch radius. What is the angular size of the minor arc of the chord? What is the length of the arc, to the nearest tent......Sun, 06 Sep 2026 23:01:26 -0400https://track.liberty.io/do-79-42-in-binary-using-2-039-s-complement.htmlhttps://track.liberty.io/do-79-42-in-binary-using-2-039-s-complement.htmlI did: 79_10 = 1001111 42_2 = 101010 I took 2's Complement of 42 and got 010110. Then I did 1001111 + 010110 and got 1100101 but this is 101. If I remove the leftmost one it becomes 37 which is ...Sun, 06 Sep 2026 23:00:54 -0400https://track.liberty.io/when-is-the-frobenius-endomorphism-an-automorphism.htmlhttps://track.liberty.io/when-is-the-frobenius-endomorphism-an-automorphism.htmlI was thinking about some problem: Lef $F$ be a finite field with the characteristic $p$. Then a map $x\mapsto x^p$ is an automorphism. I figured out that this map is called the Frobenius ...Sun, 06 Sep 2026 23:00:18 -0400https://track.liberty.io/showing-whether-a-sequence-is-bounded-above-or-not.htmlhttps://track.liberty.io/showing-whether-a-sequence-is-bounded-above-or-not.htmlI am trying to solve the following problem about a sequence: Consider the sequence ${a_n}$ where $a_n = 1 + \frac{1}{1 \cdot 3} + \frac {1}{1 \cdot 3 \cdot 5} + \frac {1}{1 \cdot 3 \cdot 5 \cdot 7}......Sun, 06 Sep 2026 22:49:01 -0400https://track.liberty.io/what-does-a-mod-2-pi-function-mean.htmlhttps://track.liberty.io/what-does-a-mod-2-pi-function-mean.htmlI read somewhere that cosine is a "$\bmod 2 \pi$ function". I think it means that it repeats every $2\pi$, but what is this "mod"?...Sun, 06 Sep 2026 22:44:08 -0400https://track.liberty.io/what-is-a-series.htmlhttps://track.liberty.io/what-is-a-series.htmlThis question is rather pedantic, but it is something that has been bothering me for some time. Summing up infinitely many terms of a sequence is something that is done in pretty much every subfie......Sun, 06 Sep 2026 22:37:13 -0400https://track.liberty.io/next-book-in-learning-differential-geometry.htmlhttps://track.liberty.io/next-book-in-learning-differential-geometry.htmlI have just finished the book "Manfredo P. do Carmo - Differential Geometry of Curves and Surfaces". My aim is to reach to graduate level to do research, but articles are not only too advanced to ......Sun, 06 Sep 2026 22:31:44 -0400https://track.liberty.io/find-the-solution-to-the-initial-value-problem.htmlhttps://track.liberty.io/find-the-solution-to-the-initial-value-problem.htmlFind the solution of the initial value problem $$ \frac{dy}{dt}=1-y^2,\;y(0)=y_0 $$ (i) in the case $y_0=1$; (ii) $y_0=-1$; (iii) in all other cases. Are there any $y_0$ such that the sol......Sun, 06 Sep 2026 22:22:57 -0400https://track.liberty.io/proof-c-n-r-c-n-n-r.htmlhttps://track.liberty.io/proof-c-n-r-c-n-n-r.htmlHello just want to see if my proof is right, and if not could someone please guide me because I am not clearly seeing the steps to this proof. I don't know if I correctly solve the proof in the sec......Sun, 06 Sep 2026 22:15:22 -0400https://track.liberty.io/how-to-factor-x-5-x-1.htmlhttps://track.liberty.io/how-to-factor-x-5-x-1.htmlI put $x^5 + x + 1$ into Wolfram and my TI-89 to factor it and they both arrived at the same factored answer: $$(x^2+x+1)(x^3-x^2+1)$$ I've tried to expand this out to reverse engineer it: $$\be......Sun, 06 Sep 2026 21:37:36 -0400https://track.liberty.io/why-is-it-wrong-to-take-the-cube-root-of-both-sides-z-1-3-z-3.htmlhttps://track.liberty.io/why-is-it-wrong-to-take-the-cube-root-of-both-sides-z-1-3-z-3.htmlif i take square root of both sides of the equation then the z cancel each other: $ \sqrt[3] {(z+1)^3 }=\sqrt[3]{z^3} $ $ z+1 =z $ $ z-z+1=0$ $ 1=0 $ I thought this complex equation had no solution......Sun, 06 Sep 2026 21:35:12 -0400https://track.liberty.io/how-many-numbers-between-1-10000-can-be-divided-by-2-and-5-but-are-not.htmlhttps://track.liberty.io/how-many-numbers-between-1-10000-can-be-divided-by-2-and-5-but-are-not.htmlFirst i calculated how many numbers are divisible by 2 and 5 $\frac{10000}{2}+\frac{10000}{5}-\frac{10000}{2*5}=6000$ Then i calculated how many numbers are not divisible by 3 or 4 $10000-(\frac{10......Sun, 06 Sep 2026 21:34:01 -0400https://track.liberty.io/finding-center-of-tessellating-hexagons.htmlhttps://track.liberty.io/finding-center-of-tessellating-hexagons.htmlGiven a hexagon such that opposite angles and radii are equal, how can I find the center point of any number of other hexagons (of the same dimensions) that form a tessellation? In this (very ug......Sun, 06 Sep 2026 21:32:07 -0400https://track.liberty.io/what-does-adding-sin-theta-cos-theta-make-my-graph-a-linear-relationsh.htmlhttps://track.liberty.io/what-does-adding-sin-theta-cos-theta-make-my-graph-a-linear-relationsh.htmlWhat is the point of adding sin n cos of theta when graphing range? e.g. I see on hyperphysics a graph of range vs sin n cos of theta and it makes the experimental data embody a linear relationshi......Sun, 06 Sep 2026 21:30:33 -0400https://track.liberty.io/what-is-the-point-of-countable-vs-uncountable-sets.htmlhttps://track.liberty.io/what-is-the-point-of-countable-vs-uncountable-sets.htmlI understand how to use these concepts and how to prove certain sets are countable or uncountable. However I don't get the point of it. What difference does it make whether a set is countable? Peop......Sun, 06 Sep 2026 21:22:40 -0400https://track.liberty.io/cross-product-equation-with-sine-i-don-039-t-understand-the-unit-vecto.htmlhttps://track.liberty.io/cross-product-equation-with-sine-i-don-039-t-understand-the-unit-vecto.html$\vec{a} \times \vec{b}=\mid \vec{a} \mid \mid \vec{b} \mid \sin(\theta)\, \vec{n}$ I know this is probably an incredibly simple question, and I know that "n" is the unit vector at right angles to......Sun, 06 Sep 2026 21:22:22 -0400https://track.liberty.io/rules-of-inference-is-my-application-of-simplification-in-this-proof-c.htmlhttps://track.liberty.io/rules-of-inference-is-my-application-of-simplification-in-this-proof-c.htmlCould someone verify that my proof is valid as the question did not have a solution? $(\lnot R \lor \lnot F) \to (S \land L)$ $(S \to T)$ $(\lnot T)$ $(\lnot S)$ 2,3 Modus tollens $\lnot (\l......Sun, 06 Sep 2026 21:09:44 -0400https://track.liberty.io/is-there-any-commonality-between-math-induction-and-logic-induction.htmlhttps://track.liberty.io/is-there-any-commonality-between-math-induction-and-logic-induction.htmlLogic induction is reasoning by probability. Math induction seems to be related to just Natural numbers and used to prove a statement for every natural number. From these definitions there is no ...Sun, 06 Sep 2026 20:45:09 -0400https://track.liberty.io/expand-function-in-legendre-polynomials-on-the-interval-1-1.htmlhttps://track.liberty.io/expand-function-in-legendre-polynomials-on-the-interval-1-1.htmlExpand the following function in Legendre polynomials on the interval [-1,1] : $$f(x) = |x|$$ The Legendre polynomials $p_n (x)$ are defined by the formula : $$p_n (x) = \frac {1}{2^n n!} \frac{......Sun, 06 Sep 2026 20:42:56 -0400https://track.liberty.io/simplifying-large-exponents-in-modular-arithmetic-like-1007-in-4-1007-.htmlhttps://track.liberty.io/simplifying-large-exponents-in-modular-arithmetic-like-1007-in-4-1007-.htmlHow would I rigorously prove that $4^{1007} \pmod{5} = 4$ and $4^{1008} \pmod{5} = 1$? I was simplifying a larger modular arithmetic problem ($2013^{2014} \pmod{5}$) and got it down to $4^{1007} \......Sun, 06 Sep 2026 20:33:47 -0400https://track.liberty.io/intersection-of-normal-extensions-is-a-normal-extension.htmlhttps://track.liberty.io/intersection-of-normal-extensions-is-a-normal-extension.htmlLet's assume that we have two normal extensions $L_1$ and $L_2$ which means that $L_1=K(A_1)$ and $L_2=K(A_2)$, where $A_1$ and $A_2$ are sets of roots of proper polynomials. How can I show that ...Sun, 06 Sep 2026 20:16:56 -0400https://track.liberty.io/formula-for-infinite-sum-of-a-geometric-series-with-increasing-term.htmlhttps://track.liberty.io/formula-for-infinite-sum-of-a-geometric-series-with-increasing-term.htmlI'm looking for the Expectation of the discrete random variable X, E[X], with pmf: $$p(x)=(\frac 16)^{x+1}, x=0,1,2,3...$$ so what I tried is as follows... $$E[X]= \sum_{0}^\infty xp(x) =$$ so t......Sun, 06 Sep 2026 20:13:01 -0400https://track.liberty.io/how-do-i-find-the-minimum-force-done-by-a-person-pushing-a-box-when-tw.htmlhttps://track.liberty.io/how-do-i-find-the-minimum-force-done-by-a-person-pushing-a-box-when-tw.htmlThe problem is as follows: A box of $320\,N$ is at rest over a horizontal terrain. The coefficients of friction between the box and the terrain are $0.36$ and $0.48$. A girl is pushing the box ...Sun, 06 Sep 2026 20:08:10 -0400https://track.liberty.io/is-the-metric-tensor-relative-to-a-reference-coordinate-system.htmlhttps://track.liberty.io/is-the-metric-tensor-relative-to-a-reference-coordinate-system.htmlI am fairly new to this topic, and I don't know anything about tensors, but I have to know what a metric tensor is and this is how I have been thinking about it: 1)A metric tensor is a mathematical ...Sun, 06 Sep 2026 19:56:29 -0400https://track.liberty.io/what-is-the-fastest-way-to-find-factors-of-a-number-that-add-up-to-a-s.htmlhttps://track.liberty.io/what-is-the-fastest-way-to-find-factors-of-a-number-that-add-up-to-a-s.htmlSo I am studying for the SAT and am re-learning the grouping method for factoring. The current problem I'm on is $$7m^2 +59m+24$$ With grouping, the traditional way is to find two numbers that are ...Sun, 06 Sep 2026 19:37:22 -0400https://track.liberty.io/what-does-cayley-table-for-a-group-mathbb-z-5-cdot-5-tell-us.htmlhttps://track.liberty.io/what-does-cayley-table-for-a-group-mathbb-z-5-cdot-5-tell-us.htmlFirsly, I would like you to explain me what $\mathbb{Z}_5^*$ means. My teacher told me it is a group of units of $\mathbb{Z}_5$, but I'm not sure what a group of units is. If $\mathbb{Z}_5^*=\{0,1,......Sun, 06 Sep 2026 19:34:38 -0400https://track.liberty.io/how-do-i-find-a-unit-vector-orthogonal-to-a-line.htmlhttps://track.liberty.io/how-do-i-find-a-unit-vector-orthogonal-to-a-line.htmlI am looking at an academic paper. In one section of it I need to find a line orthogonal to the between two points. The paper says: Given a point p ∈ δΩ the normal direction is computed as f......Sun, 06 Sep 2026 19:28:56 -0400https://track.liberty.io/if-l-is-context-free-is-l-context-free-as-well.htmlhttps://track.liberty.io/if-l-is-context-free-is-l-context-free-as-well.htmlI know CFL in closed under Closure, but don't know how to prove it back. I try to construct a CFG for L* such as S->ε|AS and say since L* in a context-free language, there must exist a method to......Sun, 06 Sep 2026 19:25:32 -0400https://track.liberty.io/which-one-is-larger-8-factorial-to-the-power-of-1-8-or-9-factorial-to-.htmlhttps://track.liberty.io/which-one-is-larger-8-factorial-to-the-power-of-1-8-or-9-factorial-to-.htmlWhich one is larger? 8 factorial to the power of 1/8 or 9 factorial to the power of 1/9. Show your work without calculator....Sun, 06 Sep 2026 19:25:04 -0400https://track.liberty.io/the-subring-test.htmlhttps://track.liberty.io/the-subring-test.htmlThis is how the Wikipedia article on subring defines the subring test The subring test states that for any ring $R$, a nonempty subset of $R$ is a subring if it is closed under addition and ...Sun, 06 Sep 2026 19:12:34 -0400https://track.liberty.io/what-is-a-standard-curve.htmlhttps://track.liberty.io/what-is-a-standard-curve.htmlI know you may find definitions out there for standard curve. But, how can we define it in a way that makes it more understandable and clear of what it does?...Sun, 06 Sep 2026 19:08:50 -0400https://track.liberty.io/derivative-of-quaternions.htmlhttps://track.liberty.io/derivative-of-quaternions.htmlI am trying to calculate the Jacobian of a function that has quaternions and 3D points in it. I refer to quaternions as $q$ and 3D points as $p$ $$h_1(q)=A C(q)p $$ $$h_2(q)=q_1\otimes q \otimes q_......Sun, 06 Sep 2026 18:56:42 -0400https://track.liberty.io/do-counit-unit-adjunctions-have-the-expected-universal-properties.htmlhttps://track.liberty.io/do-counit-unit-adjunctions-have-the-expected-universal-properties.htmlLet a unit-counit adjunction between the categories $C$ and $D$ be defined as a pair of functors $F: D\rightarrow C$ and $G: C\rightarrow D$ with natural transformations $\varepsilon : FG \rightarr......Sun, 06 Sep 2026 18:52:33 -0400https://track.liberty.io/a-theorem-about-three-lines-any-two-of-them-being-coplanar.htmlhttps://track.liberty.io/a-theorem-about-three-lines-any-two-of-them-being-coplanar.htmlI am pretty convinced the following result holds in dimension 3: If three non-coplanar lines are such that any two of them are coplanar, then either all three lines are parallel, or they all ...Sun, 06 Sep 2026 18:47:29 -0400https://track.liberty.io/examples-of-some-pointwise-convergent-sequences-of-functions.htmlhttps://track.liberty.io/examples-of-some-pointwise-convergent-sequences-of-functions.htmlI have recently come across pointwise/uniformly convergent sequences of functions, and I am hoping if someone could give some examples of certain sequences of functions so that I could understand the ...Sun, 06 Sep 2026 18:36:20 -0400https://track.liberty.io/limit-question-with-bernstein-polynomial-approximating-f-x-x-2.htmlhttps://track.liberty.io/limit-question-with-bernstein-polynomial-approximating-f-x-x-2.htmlI am reading Atkinson's "An Introduction to Numerical Analysis". I am trying to verify a limit on page 198 involving the Bernstein polynomial approximating $f(x) = x^2$. The statement in the book ......Sun, 06 Sep 2026 18:35:15 -0400https://track.liberty.io/what-is-the-proper-way-to-study-more-advanced-math.htmlhttps://track.liberty.io/what-is-the-proper-way-to-study-more-advanced-math.htmlHere's what happens. I get stuck on some proof or some mathematical construction and I end up staring at the problem for hours, sometimes not making any progress. I do this because sometimes I thin......Sun, 06 Sep 2026 18:34:29 -0400https://track.liberty.io/computational-complexity-of-the-matlab-039-s-solver-for-solving-nonlin.htmlhttps://track.liberty.io/computational-complexity-of-the-matlab-039-s-solver-for-solving-nonlin.htmlI have to solve a nonlinear (trigonometric) equation (that always has only one solution) $$f(x)=0$$ where function $f: D \subset \mathbb{R} \to \mathbb{R}$ is strictly increasing. I used the MATLAB ...Sun, 06 Sep 2026 18:20:48 -0400https://track.liberty.io/finding-volume-of-a-frustum-of-a-pyramid.htmlhttps://track.liberty.io/finding-volume-of-a-frustum-of-a-pyramid.htmlI need to find the volume of a frustum of a pyramid with square base of side $b$, square top of side $a$, and height $h$(using integrals). I have no idea how to do questions like these, I only know......Sun, 06 Sep 2026 18:12:40 -0400https://track.liberty.io/proof-by-contradiction-negation-of-conjunction.htmlhttps://track.liberty.io/proof-by-contradiction-negation-of-conjunction.htmlConsider we have the following statement to prove: $P \implies Q \wedge R$. For a proof by contradiction, we assume $P \wedge (\neg Q \lor \neg R)$. How would one go about this? Typically to prov......Sun, 06 Sep 2026 18:02:24 -0400https://track.liberty.io/parallel-algorithms-for-svd.htmlhttps://track.liberty.io/parallel-algorithms-for-svd.htmlI just have completed a preliminar theoretical study of the important SVD decomposition. Now, I'm moving to numerical calculation of SVD. I would like to learn directly a parallel algorithm to ...Sun, 06 Sep 2026 17:34:42 -0400https://track.liberty.io/using-semantic-arguments-in-proving-validity.htmlhttps://track.liberty.io/using-semantic-arguments-in-proving-validity.htmlI have to show that $\forall x (\phi \to \psi) \to (\forall x \phi \to \forall x \psi)$ is valid using semantic arguments. However, what does "semantic arguments" mean? Do I use rules of inference?...Sun, 06 Sep 2026 17:26:45 -0400https://track.liberty.io/laplace-transform-involving-heaviside-step-function.htmlhttps://track.liberty.io/laplace-transform-involving-heaviside-step-function.htmlI'm trying to find the Laplace transform of $7 e^{-3t} u(t-3)$, where $u$ is the heaviside step function. However, we've never really gone through what the Laplace transform of the heaviside step ...Sun, 06 Sep 2026 17:11:44 -0400https://track.liberty.io/solution-for-the-inequality-x-1-x-2-x-3-0.htmlhttps://track.liberty.io/solution-for-the-inequality-x-1-x-2-x-3-0.htmlRecently I came across an inequality like this: $$(x-1)(x-2)(x-3)>0$$ The question was: Which solution for this in inequality is correct? $x>3$ or $x<1$ $x>3$ or $1<x<2$ $x&l......Sun, 06 Sep 2026 17:10:18 -0400https://track.liberty.io/positive-definite-and-transpose.htmlhttps://track.liberty.io/positive-definite-and-transpose.htmlWhen a matrix A has m rows and n columns (m>n), explain why $AA^{T}$ can't be positive definite. For the same matrix A, is $A^{T}A$ always positive definite? If so, explain. If not, what is the ...Sun, 06 Sep 2026 17:09:26 -0400https://track.liberty.io/graph-theory-what-is-the-definition-of-the-sorted-edge-algorithm.htmlhttps://track.liberty.io/graph-theory-what-is-the-definition-of-the-sorted-edge-algorithm.htmlI've been googling for a while and can't find a clear definition of the "sorted edge" algorithm--can anyone provide it please? A description would be helpful, but a simple statement of the algorit......Sun, 06 Sep 2026 17:09:25 -0400https://track.liberty.io/definition-of-very-ample-divisor.htmlhttps://track.liberty.io/definition-of-very-ample-divisor.htmlSorry for my bad English. Let $X$ be scheme over $Y$, and $\mathscr{L}$ be invertible sheaf on $X$. By definition of Hartshorne's algebraic geometry (p120), $\mathscr{L}$ is very ample relative to......Sun, 06 Sep 2026 17:04:36 -0400https://track.liberty.io/limit-approaching-0-of-a-rational-defined-function.htmlhttps://track.liberty.io/limit-approaching-0-of-a-rational-defined-function.htmlFor $f(x)=\begin{cases} e^{x^2}-1, & x \in \mathbb Q\\[2ex] 0, & x \not\in \mathbb Q \end{cases} $ Evaluate $\lim_{x\to 0}f(x)$. Can someone point me in the right direction? I have no ......Sun, 06 Sep 2026 16:58:45 -0400https://track.liberty.io/calculate-in-cm-the-length-of-segment-kl.htmlhttps://track.liberty.io/calculate-in-cm-the-length-of-segment-kl.htmlLet ABCD be a rectangle such that AB = $ \sqrt {2} BC $. Let E be a point on the semicircle with diameter AB, as shown in the following figure. Let K and L be the intersections of AB with ED and EC, ...Sun, 06 Sep 2026 16:53:46 -0400https://track.liberty.io/limit-comparison-test-for-checking-the-convergence-of-an-infinite-seri.htmlhttps://track.liberty.io/limit-comparison-test-for-checking-the-convergence-of-an-infinite-seri.htmlHere's the limit comparison test from an online source: $$\lim_\limits{n \to \infty} \frac{a_{n}}{b_{n}}=c$$ Where $a_{n}$ and $b_{n}$ are the general terms of two different infinite series. If $c$......Sun, 06 Sep 2026 16:41:38 -0400https://track.liberty.io/in-triangle-abc-with-ab-ac-and-angle-bac-20-circ-d-is-on-ac-with-bc-ad.htmlhttps://track.liberty.io/in-triangle-abc-with-ab-ac-and-angle-bac-20-circ-d-is-on-ac-with-bc-ad.htmlIn $\triangle ABC$ with $AB=AC$ and $\angle BAC=20^\circ$, point $D$ is on $AC$, with $BC=AD$. Find $\angle DBC$. I know the correct solution, but I'm more interested in where is the problem in my ...Sun, 06 Sep 2026 16:37:46 -0400https://track.liberty.io/use-calculus-to-maximize-the-area-of-a-triangle.htmlhttps://track.liberty.io/use-calculus-to-maximize-the-area-of-a-triangle.htmlWhat is the maximum area of a triangle that can be inscribed in a circle with radius of $5$ units. Solve using calculus. Is there a way to use $A = 0.5 \times b \times h$ and take the derivative?...Sun, 06 Sep 2026 16:30:20 -0400https://track.liberty.io/how-can-i-solve-a-triangle-knowing-its-area-one-side-and-an-opposite-a.htmlhttps://track.liberty.io/how-can-i-solve-a-triangle-knowing-its-area-one-side-and-an-opposite-a.htmlI need to calculate the missing elements of a triangle, knowing its area one side and the angle opposite the given side. The triangle is not a right angle triangle, nor is it equilateral or isoscel......Sun, 06 Sep 2026 16:24:59 -0400https://track.liberty.io/find-depth-of-a-half-filled-parabolic-cross-section.htmlhttps://track.liberty.io/find-depth-of-a-half-filled-parabolic-cross-section.htmlGiven a cross-section of an object that is parabolic in shape, how do you find the depth of the object when it is "half full". A full example given in an exam: A long trough whose cross-section......Sun, 06 Sep 2026 16:24:24 -0400https://track.liberty.io/definition-of-an-algebraic-singularity.htmlhttps://track.liberty.io/definition-of-an-algebraic-singularity.htmlI'm reading text about generating functions and how to reveal their asymptotic behaviour by means of analysing their singularities. In this context the term "algebraic singularity" or in German "...Sun, 06 Sep 2026 16:14:26 -0400https://track.liberty.io/how-to-make-a-semicircle-graph.htmlhttps://track.liberty.io/how-to-make-a-semicircle-graph.htmlWhat is the formula to make a semicircle graph that is continuous? By continuous I mean like a sine or cos graph but shaped like semicircles one after the other. Thanks...Sun, 06 Sep 2026 16:13:08 -0400https://track.liberty.io/how-to-prove-a-graph-asymmetric.htmlhttps://track.liberty.io/how-to-prove-a-graph-asymmetric.htmlFind a 3-regular graph that is has no other automorphism other than the identity. I searched and found that this means the graph is asymmetric and there is an example: the Frucht graph. But can so......Sun, 06 Sep 2026 16:13:00 -0400https://track.liberty.io/what-is-the-derivative-of-the-folium-of-descartes.htmlhttps://track.liberty.io/what-is-the-derivative-of-the-folium-of-descartes.html$$x^3+y^3=6xy\tag{1}\label{eq1}$$ $$\frac{dy}{dx}=\frac{\left(2y-x^{2}\right)}{y^{2}-2x}\tag{2}\label{eq2}$$ I embed the derivative $(2)$ into a 3D space by including variable $z$: $$z=\frac{\left(......Sun, 06 Sep 2026 16:11:35 -0400