https://liberty.ioCurated star highlights with fresh energy.en-usMon, 07 Sep 2026 02:26:10 -0400https://liberty.io/cdf-of-pareto-distribution.htmlhttps://liberty.io/cdf-of-pareto-distribution.htmlI want to compute the CDF of a Pareto distribution In particular, I have a Pareto distribution with p.d.f. $$f(x) = \lambda x^{-(\lambda + 1)},$$ and I want to find the CDF of $f$. When I perform ...Mon, 07 Sep 2026 06:24:58 -0400https://liberty.io/radians-negative-and-positive-values.htmlhttps://liberty.io/radians-negative-and-positive-values.htmlRecently I have been reading books on DSP where I came across Polar co-ordinates. I understand that on Polar graph (4 quadrants) we have $0, \frac{\pi}{2}, \pi, \frac{3}{2\pi}$ and $2 \pi$ radians ......Mon, 07 Sep 2026 06:23:31 -0400https://liberty.io/parametric-equation-of-a-circle-using-a-line.htmlhttps://liberty.io/parametric-equation-of-a-circle-using-a-line.htmlConsider the unit circle $$ x^2+y^2=1. $$ How can I parametrize it using the line $y=m(x+1)$, where $m$ is its slope?...Mon, 07 Sep 2026 06:08:08 -0400https://liberty.io/number-of-occurrences-in-contiguous-subarrays.htmlhttps://liberty.io/number-of-occurrences-in-contiguous-subarrays.htmlHow to count the number of occurrences of each element in all contiguous subarrays? For an array $[1,2,3]$, the contiguous subarrays are: $[1]$, $[2]$, $[3]$ $[1,2]$, $[2,3]$ $[1,2,3]$ $1$ occu......Mon, 07 Sep 2026 06:07:51 -0400https://liberty.io/what-are-the-domain-s-of-self-inverse-functions.htmlhttps://liberty.io/what-are-the-domain-s-of-self-inverse-functions.htmlA self inverse function is a function $f$, such that $y=f(x)$, with the special property that $ff(x)=x$, or written another way, $f(x) = f^-1(x)$ Example: Imagine there's a function $f$, such that......Mon, 07 Sep 2026 06:06:25 -0400https://liberty.io/give-a-serious-explanation-of-the-difference-between-an-equation-and-a.htmlhttps://liberty.io/give-a-serious-explanation-of-the-difference-between-an-equation-and-a.htmlWhat's the difference between an equation and a function? I mean, I am not seeking for a high school-like answer like "an equation has an equals sign". I want to know what is the fundamental differ......Mon, 07 Sep 2026 06:03:27 -0400https://liberty.io/what-do-you-get-for-tsanna-039-s-sacrificial-offering-in-chapter-7.htmlhttps://liberty.io/what-do-you-get-for-tsanna-039-s-sacrificial-offering-in-chapter-7.htmlIn Chapter 7 - Tsanna offers to sacrifice Evindra to give you the blessings of Lamashtu. If you do it, what benefit do you see for the final battle?...Mon, 07 Sep 2026 06:01:33 -0400https://liberty.io/infinite-sum-and-partial-sum-not-equal-in-wolframalpha.htmlhttps://liberty.io/infinite-sum-and-partial-sum-not-equal-in-wolframalpha.html$$\sum_{n=0}^{\infty} |(-0.8)^n \theta(n)-(-0.8)^{n-1} \theta(n-1)|$$ $\theta(n) = 1$ for $n \geq 0$ and 0 otherwise. My attempt to calculate this summation numerically with Python yielded 10. To ......Mon, 07 Sep 2026 05:57:56 -0400https://liberty.io/how-do-i-use-the-undermounted-shotgun-attachment-for-assault-rifles.htmlhttps://liberty.io/how-do-i-use-the-undermounted-shotgun-attachment-for-assault-rifles.htmlIn call of duty: modern warfare 3 (PC version) when playing multiplayer. You can select at weapon level 20 the shotgun attachment for the assault rifle. How do you fire it? Does it happen automati......Mon, 07 Sep 2026 05:56:05 -0400https://liberty.io/is-it-worth-it-to-upgrade-the-clan-castle.htmlhttps://liberty.io/is-it-worth-it-to-upgrade-the-clan-castle.htmlI was wondering if the clan castle building is worth the resources to upgrade it. I rarely attack getting all my resources from farming and use all them for upgrading defenses. I'm questioning th......Mon, 07 Sep 2026 05:53:21 -0400https://liberty.io/mega-stone-of-chosen-starter-pokemon-buyable.htmlhttps://liberty.io/mega-stone-of-chosen-starter-pokemon-buyable.htmlIf I am choosing Charmander as my starter, will I be able to buy the Charizardite Mega Stone again? So I could buy another one and trade it to someone else. Edit: I just found this site and it sa......Mon, 07 Sep 2026 05:49:33 -0400https://liberty.io/my-world-keeps-not-responding-correctly-how-do-i-fix-that.htmlhttps://liberty.io/my-world-keeps-not-responding-correctly-how-do-i-fix-that.htmlI was using a lan server on my favorite world and when i saved and exited the world everything worked fine. Then later i tried to enter the world and it glitched out a bit at the load screen and wh......Mon, 07 Sep 2026 05:45:41 -0400https://liberty.io/find-max-vertical-distance.htmlhttps://liberty.io/find-max-vertical-distance.htmlWhat is the maximum vertical distance between the line $y = x + 20$ and the parabola $y = x^2$ for $−4 ≤ x ≤ 5?$ What steps do I take to solve this? Do I have to use the distance formula and wh......Mon, 07 Sep 2026 05:37:02 -0400https://liberty.io/quote-from-mathematician-poet.htmlhttps://liberty.io/quote-from-mathematician-poet.htmlI read once a quote by a mathematician (or scientist) who was also a poet and it was something similar to: As a mathematician I try to make complicated things obvious, as a poet I try to make ob......Mon, 07 Sep 2026 05:23:40 -0400https://liberty.io/when-to-play-the-old-hunters-during-the-first-run-or-ng.htmlhttps://liberty.io/when-to-play-the-old-hunters-during-the-first-run-or-ng.htmlI'm doing my first Bloodborne run. I beat the boss of Byrgenwerth. Currently I have the option of playing the new DLC, The Old Hunters. If I waited until I finish my first run, it means I have to......Mon, 07 Sep 2026 05:21:14 -0400https://liberty.io/do-i-need-to-play-the-previous-games.htmlhttps://liberty.io/do-i-need-to-play-the-previous-games.htmlDarksiders II is out for free on XboxLive. Do I need to play Darksiders to understand whats going on? Or is the game completely different to the first?...Mon, 07 Sep 2026 05:14:34 -0400https://liberty.io/quick-way-of-selling-at-the-trading-post.htmlhttps://liberty.io/quick-way-of-selling-at-the-trading-post.htmlIs there a way to quickly sell goods at the trading post in GW2? I know there is the sell at trading post option by right clicking the desired item but that loads up the trading post every time and......Mon, 07 Sep 2026 04:59:15 -0400https://liberty.io/what-039-s-a-good-trophy-army-for-a-town-hall-8-closed.htmlhttps://liberty.io/what-039-s-a-good-trophy-army-for-a-town-hall-8-closed.htmlI've maxed out my troops for town hall 8 and I'm trying to do a big push, any suggestions?...Mon, 07 Sep 2026 04:56:49 -0400https://liberty.io/does-walking-fast-prevent-eggs-hatching.htmlhttps://liberty.io/does-walking-fast-prevent-eggs-hatching.htmlI have a couple of eggs that I've picked up from some Pokestops (I'm lucky enough to live above one). I walk a lot. I get to work through a 2 mile walk / train journey every day, so I should be p......Mon, 07 Sep 2026 04:55:22 -0400https://liberty.io/elementary-proof-of-multiplicative-property-of-determinant.htmlhttps://liberty.io/elementary-proof-of-multiplicative-property-of-determinant.htmlWhy LU decomposition is not used to give elementary proof of $\det (AB) = \det (A) \det (B) $ for square matrices? It seems for any square matrix $ A $, $ A = PLU$ P is permutation matrix, L lo......Mon, 07 Sep 2026 04:45:02 -0400https://liberty.io/tao-real-analysis-question-on-function-definition-should-it-be-an-axio.htmlhttps://liberty.io/tao-real-analysis-question-on-function-definition-should-it-be-an-axio.htmlIn Terence Tao's book "Analysis 1", in definition 3.3.1 (function definition), he says Let $X, Y$ be sets, and let $P(x, y)$ be a property pertaining to an object $x \in X$ and an object......Mon, 07 Sep 2026 04:18:17 -0400https://liberty.io/calculate-the-norm-of-a-polynomial-knowing-inner-product.htmlhttps://liberty.io/calculate-the-norm-of-a-polynomial-knowing-inner-product.htmlThe polynomial $p(x) = 3 - 2x$ is a polynomial in an inner product space with the inner product defined as $$ \langle p, q \rangle = \int_0^1 p(x)q(x) \, dx$$ on the interval $[0, 1]$ in the vec......Mon, 07 Sep 2026 03:52:23 -0400https://liberty.io/what-is-the-frobenius-map-really.htmlhttps://liberty.io/what-is-the-frobenius-map-really.htmlI know what the frobenius map is the naive sense, just the $p$th power map on an $\mathbb{F}_p$ algebra, and that this can be upgraded somewhat to the generator of the monoid of natural transformat......Mon, 07 Sep 2026 03:51:28 -0400https://liberty.io/how-do-you-get-back-from-z8n-tp.htmlhttps://liberty.io/how-do-you-get-back-from-z8n-tp.htmlI can get TO the CL4PTP unit in Tycho's Ribs for it's quest but when I try to use the jumppad to get back, it always tosses me into the rotating spire in the middle and instakills me. I have tried......Mon, 07 Sep 2026 03:51:25 -0400https://liberty.io/where-are-the-fairy-fountains.htmlhttps://liberty.io/where-are-the-fairy-fountains.htmlI am playing Ocarina of Time on the 3DS, and I vaguely recall there being a fairy fountain somewhat close to each dungeon/temple. Is that true? Where are the fairy fountains?...Mon, 07 Sep 2026 03:47:06 -0400https://liberty.io/definite-integration-having-d-dx-operator-inside-the-integral-sign-clo.htmlhttps://liberty.io/definite-integration-having-d-dx-operator-inside-the-integral-sign-clo.html$$\int_{-1}^1 \frac{d}{dx}\ \frac{1}{1 + 2^{1/x}}\ dx$$ How to approach this integral ? Should I differentiate the expression and then integrate ? I tried that but couldn't proceed Thank you....Mon, 07 Sep 2026 03:40:36 -0400https://liberty.io/every-superperfect-number-except-2-is-a-square-number.htmlhttps://liberty.io/every-superperfect-number-except-2-is-a-square-number.htmlI recently read about superperfect numbers: $σ^2(n) = 2n$, where $σ(n)$ is the divisor function. I saw that the first few numbers were: $2, 4, 16, 64, 4096, 65536, 262144$, which are all square-nu......Mon, 07 Sep 2026 03:30:00 -0400https://liberty.io/what-are-the-differences-in-ingress-portal-shield-types.htmlhttps://liberty.io/what-are-the-differences-in-ingress-portal-shield-types.htmlI have collected a few different types of shields, with different colours and of differing rarity. I assume the rarer shield types are more powerful in some way but I can't find any information o......Mon, 07 Sep 2026 03:26:50 -0400https://liberty.io/simplify-abc-a-039-bc-ab-039-c-a-039-b-039-c-039-to-logic-gates.htmlhttps://liberty.io/simplify-abc-a-039-bc-ab-039-c-a-039-b-039-c-039-to-logic-gates.htmlI've simplified $ABC+A'BC+AB'C+A'B'C'$ to $BC+AC+A'B'C'$. However, I want to go further to logic gates for which ICs are readily available. I would like to use at most three such ICs....Mon, 07 Sep 2026 03:20:18 -0400https://liberty.io/can-rotations-in-4d-be-given-an-explicit-matrix-form.htmlhttps://liberty.io/can-rotations-in-4d-be-given-an-explicit-matrix-form.htmlRotation in 2D by an angle $t$ can be performed using $$R=\begin{pmatrix}\cos(t) &-\sin(t) \\ \sin(t) & \cos(t)\end{pmatrix}$$ matrix. But, if I want to rotate a point or vector in 4D, is t......Mon, 07 Sep 2026 03:00:02 -0400https://liberty.io/is-it-possible-to-split-a-division-problem-into-parts-like-in-multipli.htmlhttps://liberty.io/is-it-possible-to-split-a-division-problem-into-parts-like-in-multipli.htmlIn multiplication we can mentally split a problem that is too big into multiple problems. For example: 26 * 40 = (20 * 40) + (6 * 40) = 800 + 240 = 1040 This is a very quick way to multiply other......Mon, 07 Sep 2026 02:56:47 -0400https://liberty.io/i-need-help-finding-the-mean-radius-of-a-cylinder.htmlhttps://liberty.io/i-need-help-finding-the-mean-radius-of-a-cylinder.htmlThe question is: What is the mean radius $\overline{r}$ from the midpoint of a cylinder of radius $a$ and height $h$ to its boundary surface? Evaluate $\overline{r}$ for $a = h/2 = 10~\mathrm{cm}$. ...Mon, 07 Sep 2026 02:51:56 -0400https://liberty.io/what-cosmetic-items-can-you-get-for-free.htmlhttps://liberty.io/what-cosmetic-items-can-you-get-for-free.htmlI recently completed a quest, The Death of Chivalry, and unlocked the cosmetic armors "Dulcin Armour" and "Templar Armour", which got me to wondering... What other cosmetic items can be unlocked th......Mon, 07 Sep 2026 02:49:16 -0400https://liberty.io/questions-tagged-group-theory.htmlhttps://liberty.io/questions-tagged-group-theory.htmlQ&A for people studying math at any level and professionals in related fields...Mon, 07 Sep 2026 02:41:16 -0400https://liberty.io/find-the-locus-of-centroid-of-right-angled-isoceles-triangle-for-the-f.htmlhttps://liberty.io/find-the-locus-of-centroid-of-right-angled-isoceles-triangle-for-the-f.htmlAn isosceles right angled triangle whose sides are $1, 1, \sqrt{2}$ lies entirely in the first quadrant with the ends of the hypotenuse on the coordinate axes. If it slides prove that the locus of......Mon, 07 Sep 2026 02:32:08 -0400https://liberty.io/about-math-notation-the-set-of-the-first-n-natural-numbers.htmlhttps://liberty.io/about-math-notation-the-set-of-the-first-n-natural-numbers.htmlIs it true that I can write the set of the the first $n$ natural numbers as $[ n] $? For example, $[10]= \{1,2,3,4,5,6,7,8,9,10 \}$. And in which math context this is used?...Mon, 07 Sep 2026 02:28:23 -0400https://liberty.io/3-2x-34-15-x-1-5-2x-0.htmlhttps://liberty.io/3-2x-34-15-x-1-5-2x-0.htmlI've never seen anything like this. Do somebody have a way to solve it? I've tried the basci exponential functions techniques but it does not work. Even substitution does not work... I'm really ...Mon, 07 Sep 2026 02:24:32 -0400https://liberty.io/conditions-for-a-quadratic-equation-to-be-positive.htmlhttps://liberty.io/conditions-for-a-quadratic-equation-to-be-positive.htmlI was solving a problem in physics when I encountered this problem. I wanted to find the condition where the value of a quadratic equation is positive. Suppose the function is : $f(x) = Ax^2+Bx+C$ ......Mon, 07 Sep 2026 02:23:05 -0400https://liberty.io/multiplying-3d-matrix.htmlhttps://liberty.io/multiplying-3d-matrix.htmlI was wondering if it is possible to multiply a 3D matrix (say a cube $n\times n\times n$) to a matrix of dimension $n\times 1$? If yes, then how. Maybe you can suggest some resources which I can r......Mon, 07 Sep 2026 02:14:05 -0400https://liberty.io/what-effects-do-the-sword-enchantments-have.htmlhttps://liberty.io/what-effects-do-the-sword-enchantments-have.htmlIn the forge, you can enchant the sword with different scrolls. What are the different enchantments I can get and what is the effect of each?...Mon, 07 Sep 2026 02:09:02 -0400https://liberty.io/is-there-a-difference-between-d-2x-and-dx-2.htmlhttps://liberty.io/is-there-a-difference-between-d-2x-and-dx-2.htmlI've just started reading through Calculus Made Easy by Silvanus Thompson and am trying to solidify the concept of differentials in my mind before progressing too far through the text. In Chapter 1 ...Mon, 07 Sep 2026 02:08:30 -0400https://liberty.io/question-regarding-horizontal-shrink-and-horizontal-stretch.htmlhttps://liberty.io/question-regarding-horizontal-shrink-and-horizontal-stretch.htmlI know the property of Horizontal shrink is $y=f(cx)$ where $ c>1$ and we need to divide the $x$ coordinates by the factor. Horizontal Stretch is $y=f(cx)$ where $0<c<1$ and we also need......Mon, 07 Sep 2026 02:07:09 -0400https://liberty.io/what-is-the-difference-between-frequency-and-density-in-a-histogram.htmlhttps://liberty.io/what-is-the-difference-between-frequency-and-density-in-a-histogram.htmlI know count is simply the amount of times the observations occur in a single bin width, but what is density?...Mon, 07 Sep 2026 02:02:28 -0400https://liberty.io/what-does-x2-mean-is-it-x-cdot-2-or-the-variable-x2.htmlhttps://liberty.io/what-does-x2-mean-is-it-x-cdot-2-or-the-variable-x2.htmlI'm working on a online mathematical learning environment. There is a mathematical editor where students can give their answers using a graphical interface: However, we have had the problem in the......Mon, 07 Sep 2026 01:59:07 -0400https://liberty.io/when-is-the-bipod-used-and-can-this-be-controlled.htmlhttps://liberty.io/when-is-the-bipod-used-and-can-this-be-controlled.htmlSometimes, either prone or crouched, my player will fold down the bipod, forcing the gun to move slightly and blinding my scope with leaves in the process. I've unlocked the bipod on a number of ...Mon, 07 Sep 2026 01:57:12 -0400https://liberty.io/finding-the-t-value-in-order-to-find-the-maximum-curvature-for-a-space.htmlhttps://liberty.io/finding-the-t-value-in-order-to-find-the-maximum-curvature-for-a-space.htmlEditted work. I am attempting to find the maximum curvature for the space curve mentioned below. space curve: $\vec{r}(t) = <3t, 4t, e^{2t}>$ From the space curve, I'm able to find that: ......Mon, 07 Sep 2026 01:44:10 -0400https://liberty.io/how-can-we-derive-the-interpolation-formula-for-higher-order.htmlhttps://liberty.io/how-can-we-derive-the-interpolation-formula-for-higher-order.htmlLet, $$y = mx + c \tag 1$$ is the equation of a straight line. Let, it passes through the point, $$(x_0, y_0)$$. So, from (1) we find, $$c = y_0 + mx_0 \tag 2$$ On the othet hand, from the form......Mon, 07 Sep 2026 01:40:40 -0400https://liberty.io/how-do-i-keep-my-animals-from-eating-my-people-food.htmlhttps://liberty.io/how-do-i-keep-my-animals-from-eating-my-people-food.htmlI've got a reasonably flourishing village with decent population and all the amenities. However, lately I've been having severe issues because my tame animals keep breaking into the food warehouse ......Mon, 07 Sep 2026 01:40:34 -0400https://liberty.io/partial-derivative-max-and-minimum.htmlhttps://liberty.io/partial-derivative-max-and-minimum.htmlI'm faced with the following problem, find the maximum and minimum value of the function: $$f(x,y)= x^{5}y^{7}e^{-2 x-2 y}$$ within the triangle with the corners (0,0), (3 ,0) and (0,3 ). My in......Mon, 07 Sep 2026 01:40:20 -0400https://liberty.io/calculate-which-of-the-four-fundamental-subspaces-a-vector-belongs-to.htmlhttps://liberty.io/calculate-which-of-the-four-fundamental-subspaces-a-vector-belongs-to.htmlAbove you have the question (b) and it's answer underneath. I am out of the blue as how they proved that it belongs to the nullspace of $A^{T}$?...Mon, 07 Sep 2026 01:35:49 -0400https://liberty.io/what-does-game-of-the-year-actually-mean.htmlhttps://liberty.io/what-does-game-of-the-year-actually-mean.htmlOften times, many games have an additional variant named "Game X: Game of the Year edition" or "Game X: GOTY edition". Within the same year, there are multiple games that seemingly have GOTY versio......Mon, 07 Sep 2026 01:35:33 -0400https://liberty.io/how-can-i-fix-pollution.htmlhttps://liberty.io/how-can-i-fix-pollution.htmlMy city has developed a bit of a ground pollution and a water pollution problem. In other similar games, you can mitigate the effects of pollution by planting trees, among other things. What can ......Mon, 07 Sep 2026 01:33:27 -0400https://liberty.io/frequency-of-the-golden-bowser-train.htmlhttps://liberty.io/frequency-of-the-golden-bowser-train.htmlDoes anyone happen to know how often you can go to the Golden Bowser Train? For those that don't yet know, it is a level with nothing but coins, and is a great way to bolster your extra lives....Mon, 07 Sep 2026 01:30:12 -0400https://liberty.io/is-there-a-way-to-get-ds-games-to-use-the-full-top-screen-on-a-3ds.htmlhttps://liberty.io/is-there-a-way-to-get-ds-games-to-use-the-full-top-screen-on-a-3ds.htmlI noticed that when I play DS games on my 3DS, the games don't fill the entirety of the top screen on the 3DS - there are black bars on either side of the image. Is there a way to get the game to ......Mon, 07 Sep 2026 01:23:03 -0400https://liberty.io/using-mass-and-density-of-a-sphere.htmlhttps://liberty.io/using-mass-and-density-of-a-sphere.htmlQuestion: A sphere of radius $R$ has total mass $M$ and density function given by $ρ = kr$, where $r$ is the distance a point lies from the centre of the sphere. Give an expression for the constant......Mon, 07 Sep 2026 01:20:09 -0400https://liberty.io/does-orthogonality-of-vectors-depend-on-the-chosen-basis.htmlhttps://liberty.io/does-orthogonality-of-vectors-depend-on-the-chosen-basis.htmlgeometrically, two vectors are "orthogonal" if there is a right angle between them, which we intuitively understand. Algebraically, the definition of "orthogonal" members of a vector space, is th......Mon, 07 Sep 2026 01:19:20 -0400https://liberty.io/about-zeno-039-s-paradox-and-its-answers.htmlhttps://liberty.io/about-zeno-039-s-paradox-and-its-answers.htmlI read this question How can Zeno's dichotomy paradox be disproved using mathematics? . The first (ie the one on the top) answer uses the fact that $\sum\limits_{n=1}^\infty\frac{1}{2^n}=1$, b......Mon, 07 Sep 2026 01:13:55 -0400https://liberty.io/how-to-evaluate-sqrt-6-sqrt-6-sqrt-cdots-duplicate.htmlhttps://liberty.io/how-to-evaluate-sqrt-6-sqrt-6-sqrt-cdots-duplicate.htmlI was playing mental maths in which we solve problems without pen and pencil. Then I came across a problem $$\sqrt {6 \sqrt{6 \sqrt{6\sqrt\cdots}}}.$$ I asked my teacher and he told me that the a......Mon, 07 Sep 2026 01:08:39 -0400https://liberty.io/how-to-calculate-ln-17-without-calculator.htmlhttps://liberty.io/how-to-calculate-ln-17-without-calculator.htmlHi I've been playing with ln(Natural logarithm) since I first learn it in physic class and my goal is to calculate it without using the calculator. I done $$\ln(12,14,15,16,18,\ldots)$$ just fine ......Mon, 07 Sep 2026 01:06:46 -0400https://liberty.io/how-do-i-build-huge-constructions-in-minecraft.htmlhttps://liberty.io/how-do-i-build-huge-constructions-in-minecraft.htmlIf I want to build a really, really huge structure in Minecraft, say, a castle with towers, walls, several floors and over a thousand of squaremeters/tiles per floor, what is the way to go? How ca......Mon, 07 Sep 2026 00:56:13 -0400https://liberty.io/volume-of-a-solid-defined-by-two-curves-rotating-around-x-axis.htmlhttps://liberty.io/volume-of-a-solid-defined-by-two-curves-rotating-around-x-axis.htmlI'm trying to determine the volume of the solid that is obtained by rotating around the $x$-axis the region limited by $y=2x$ and $y=x^2$. I'm trying to follow James Stewart's "Calculus. Single var......Mon, 07 Sep 2026 00:42:19 -0400https://liberty.io/how-do-the-marriages-that-i-bring-about-through-the-relationship-optio.htmlhttps://liberty.io/how-do-the-marriages-that-i-bring-about-through-the-relationship-optio.htmlI noticed that when I got Owain to join my entourage, he had the same hair color as my character. Are there other effects that come about by different marriages?...Mon, 07 Sep 2026 00:29:08 -0400https://liberty.io/best-intermediate-advanced-computer-science-book.htmlhttps://liberty.io/best-intermediate-advanced-computer-science-book.htmlI'm very interested in Computer Science (computational complexity, etc.). I've already finished a University course in the subject (using Sipser's "Introduction to the Theory of Computation"). I k......Mon, 07 Sep 2026 00:24:01 -0400https://liberty.io/how-do-i-sheathe-my-weapons-in-dishonored.htmlhttps://liberty.io/how-do-i-sheathe-my-weapons-in-dishonored.htmlSo I'm currently in the Boyle Estate party and people keep telling me to put my weapons away, but I've no idea how. I've had a quick look through the controls to see if there is a way to bind a bu......Mon, 07 Sep 2026 00:21:03 -0400https://liberty.io/discontinuous-functions-in-numerical-integration.htmlhttps://liberty.io/discontinuous-functions-in-numerical-integration.htmlI have a system of ODEs whose time courses I'd like to numerically integrate. One of these should have a continuous strictly decreasing term and a discontinuous increasing term. It looks something ......Mon, 07 Sep 2026 00:12:22 -0400https://liberty.io/what-is-the-interval-of-definition-of-a-solution-of-an-ode.htmlhttps://liberty.io/what-is-the-interval-of-definition-of-a-solution-of-an-ode.htmlI don't understand this question. Verify that the indicated function $y=\phi(x)$ is an explicit solution of the given ODE. Consider $\phi$ simply as a function, to give its domain. Then by conside......Mon, 07 Sep 2026 00:10:43 -0400https://liberty.io/how-do-i-calculate-frac-ln-2-ln-3-frac-ln-4-ln-5-frac-ln-2n-ln-2n-1.htmlhttps://liberty.io/how-do-i-calculate-frac-ln-2-ln-3-frac-ln-4-ln-5-frac-ln-2n-ln-2n-1.htmlThe product has only positive factors so it has zero as lower bound. Also the product is decreasing as all its factors are less than one. In conclusion the series must have a limit. I also compute ......Mon, 07 Sep 2026 00:09:26 -0400https://liberty.io/graphing-absolute-value-inside-absolute-value-equation.htmlhttps://liberty.io/graphing-absolute-value-inside-absolute-value-equation.htmlHow would I graph an equation with an absolute value inside an absolute value? For example, $\left|-\left|x\right|+1\right|+\left|y-2\right|=3$ I tried graphing this on Desmos and it gave me some ...Mon, 07 Sep 2026 00:06:12 -0400https://liberty.io/how-to-rotate-big-structures-90-degrees-in-minecraft-pc.htmlhttps://liberty.io/how-to-rotate-big-structures-90-degrees-in-minecraft-pc.htmlI have a big sculpture in Minecraft on my computer and wanted to clone it so I did it cloned but it's the wrong way. Could someone tell me a command to rotate it 90 degrees?...Mon, 07 Sep 2026 00:03:38 -0400https://liberty.io/product-of-two-odd-numbers-is-odd.htmlhttps://liberty.io/product-of-two-odd-numbers-is-odd.htmlHow do I prove that the product of odd integers is odd? I know that I'm supposed to use an algebraic equation....Mon, 07 Sep 2026 00:00:04 -0400https://liberty.io/why-is-n-1-choose-1-0-when-1-is-undefined.htmlhttps://liberty.io/why-is-n-1-choose-1-0-when-1-is-undefined.htmlWhy is ${n-1 \choose -1} = 0$ when (-1)! is undefined? Shouldn't ${n-1 \choose -1}$ also be undefined?...Sun, 06 Sep 2026 23:38:50 -0400https://liberty.io/what-is-a-harmonic-complex-function.htmlhttps://liberty.io/what-is-a-harmonic-complex-function.htmlSo, as far as I have learned, a complex function $f(u)$ is considered harmonic if and only if it satisfies the undermentioned equation: $$ \frac{\partial ^2 u}{\partial x^2} + \frac{\partial ^2 u}{\...Sun, 06 Sep 2026 23:37:45 -0400https://liberty.io/why-are-the-central-angles-of-a-regular-polygon-equal-to-the-exterior-.htmlhttps://liberty.io/why-are-the-central-angles-of-a-regular-polygon-equal-to-the-exterior-.htmlI am currently in Geometry, doing proof, when me, my teacher, and my classmates realized that the exterior angles of a polygon are equal to the central angles of a polygon (for a regular polygon). ......Sun, 06 Sep 2026 23:00:33 -0400https://liberty.io/alternative-quadratic-formula.htmlhttps://liberty.io/alternative-quadratic-formula.htmlWell the formula for solving a Quadratic equation is : $$\text{If }\space ax^2+bx+c=0$$ then $$x=\dfrac{-b \pm \sqrt{b^2 -4ac} }{2a}$$ But looking at this : [Wolfram Mathworld] (And also in other p......Sun, 06 Sep 2026 22:58:03 -0400https://liberty.io/derivative-of-unit-step-function.htmlhttps://liberty.io/derivative-of-unit-step-function.htmlHow would I find the derivative of a unit step function? I understand that the unit impulse function will be used but I'm not sure how to use it. I am trying to find the derivative of this: $v(t)......Sun, 06 Sep 2026 22:41:44 -0400https://liberty.io/sample-mean-is-uniformly-integrable.htmlhttps://liberty.io/sample-mean-is-uniformly-integrable.htmlLet $\{X_n\}$ be a sequence of iid random variables such that $E|X_1| = \mu < \infty$. I would like to show that $\{\bar{X}_n\}$ is uniformly integrable. Thoughts By the strong law of large nu......Sun, 06 Sep 2026 22:38:17 -0400https://liberty.io/definition-of-a-bounded-sequence.htmlhttps://liberty.io/definition-of-a-bounded-sequence.htmlMy professor gave the following definition: A sequence $\{x_n \}$ is said to be bounded if $\exists M > 0$ such that $|x_n| \le M$ for all $n \in \mathbb N^+.$ But then what about the sequenc......Sun, 06 Sep 2026 22:37:51 -0400https://liberty.io/is-there-any-theorem-about-figures-of-equal-area-and-perimeter-being-c.htmlhttps://liberty.io/is-there-any-theorem-about-figures-of-equal-area-and-perimeter-being-c.htmlI had an idea, that all geometric objects, that are different, as they're not a translation, rotation, and a reflection of one another cannot have the same area AND perimeter, as compared to ONE AN......Sun, 06 Sep 2026 22:33:31 -0400https://liberty.io/gcd-of-rationals.htmlhttps://liberty.io/gcd-of-rationals.htmlDisclaimer: I'm an engineer, not a mathematician Somebody claimed that $\gcd$ only is applicable for integers, but it seems I'm perfectly able to apply it to rationals also: $$ \gcd\left(\frac{......Sun, 06 Sep 2026 22:30:01 -0400https://liberty.io/how-to-calculate-maximum-or-minimum-of-two-numbers-without-using-if.htmlhttps://liberty.io/how-to-calculate-maximum-or-minimum-of-two-numbers-without-using-if.htmlHow to to calculate the maximim or minimum of two numbers without using "if" ( or something equivalant to that manner)? The above question is often asked in introductory computer science courses a......Sun, 06 Sep 2026 22:27:52 -0400https://liberty.io/list-of-rules-and-laws-for-calculating-limits.htmlhttps://liberty.io/list-of-rules-and-laws-for-calculating-limits.htmlWhile looking through our Analysis Script, I got suddenly aware that we had already learned quite an amount of rules and laws for calculating limits. We got: $\displaystyle \lim_{x \to a}\;c = c$ $\...Sun, 06 Sep 2026 22:27:16 -0400https://liberty.io/sylvester-039-s-law-of-inertia.htmlhttps://liberty.io/sylvester-039-s-law-of-inertia.htmlConsidering the bilinear form on $R$, $\phi$, defined by the matrix \begin{bmatrix}{-1}&{-1}&{-1}\\{-1}&{1}&{0}\\{-1}&{0}&{1}\end{bmatrix} Classify, according to the Syl......Sun, 06 Sep 2026 22:10:40 -0400https://liberty.io/the-square-of-any-odd-integer-is-odd.htmlhttps://liberty.io/the-square-of-any-odd-integer-is-odd.htmlSuppose that $n$ is an odd integer. Then $n = 2k + 1$ for some integer $k$. Hence $n^2=(2k+1)(2k+1)=4k^2+4k+1=2(2k^2+2k)+1$. Since $k$ is an integer, $2k^2 + 2k$ is an integer. Thus $n^2 = 2k' +......Sun, 06 Sep 2026 22:06:24 -0400https://liberty.io/general-process-to-find-global-extrema-of-a-function.htmlhttps://liberty.io/general-process-to-find-global-extrema-of-a-function.htmlI have been reading and watching videos about local and global extrema, but all of this material covers the topic just graphically, and nobody really explicitly cares on how to find the global maxi......Sun, 06 Sep 2026 21:56:49 -0400https://liberty.io/coin-flipping-likely-to-hit-7-heads-or-tails-in-a-row-answered-and-bet.htmlhttps://liberty.io/coin-flipping-likely-to-hit-7-heads-or-tails-in-a-row-answered-and-bet.htmlI am editing this question as requested so I am more clear in what I am asking. Assuming that a coin flipped has a $50\%$ chance of landing heads and a $50\%$ chance of landing tails, I had wondere......Sun, 06 Sep 2026 21:43:38 -0400https://liberty.io/convex-side-of-a-spherical-mirror.htmlhttps://liberty.io/convex-side-of-a-spherical-mirror.htmlA convex set has the property that if you take any two points in the set and draw the line segment connecting those two points, that line segment lies entirely in the set. My textbook says tha......Sun, 06 Sep 2026 21:39:24 -0400https://liberty.io/bracket-of-8-tournament.htmlhttps://liberty.io/bracket-of-8-tournament.htmlIf I had a bracket style tournament, how many possibilities are there for a 8 team tournament. Once a team gets eliminated, they do not play a game for the 3rd place and below....Sun, 06 Sep 2026 21:35:24 -0400https://liberty.io/product-of-an-m-matrix-and-an-inverse-m-matrix.htmlhttps://liberty.io/product-of-an-m-matrix-and-an-inverse-m-matrix.htmlThis problem comes from the book "Topics in matrix analysis" by Horn and Johnson. Let $A$ and $B$ be two $M$-matrices. Show that $B^{-1}A$ is an $M$-matrix if and only if $B^{-1}A$ is a $Z$-matrix......Sun, 06 Sep 2026 21:31:29 -0400https://liberty.io/what-exactly-is-the-difference-between-gateaux-derivative-and-directio.htmlhttps://liberty.io/what-exactly-is-the-difference-between-gateaux-derivative-and-directio.htmlThe definition of the limit looks very similar between the two derivatives. It seems that directional derivative is the "amount" of the function going in the direction of a vector (arrow), wherea......Sun, 06 Sep 2026 21:26:03 -0400https://liberty.io/transposing-multiple-matrices.htmlhttps://liberty.io/transposing-multiple-matrices.htmlSo I have a question which says the following (a) Given $A= \begin{bmatrix} 3 & 0 \\ -1 & 2 \\ 1 & 1 \end{bmatrix} $, $B= \begin{bmatrix} 4 &-1 \\ 0 & 2 \\ \end{bmat......Sun, 06 Sep 2026 21:18:00 -0400https://liberty.io/why-does-the-taylor-series-of-frac-1-ln-x-have-a-non-infinite-radius-o.htmlhttps://liberty.io/why-does-the-taylor-series-of-frac-1-ln-x-have-a-non-infinite-radius-o.htmlShouldn't the taylor series of a function be equal to that function for any input value? Why does this not work for the taylor series of $\frac {1}{\ln x}$ when $|x| \gt 1$? Edit: I do mean the ......Sun, 06 Sep 2026 21:11:01 -0400https://liberty.io/how-do-you-find-the-probability-of-p-a-and-b.htmlhttps://liberty.io/how-do-you-find-the-probability-of-p-a-and-b.htmlThe probability model has a sample space of {A,B,C} with P(A) = 0.1, P(B) = 0.8, P(C) = 0.1. I found that P(B or C) = 0.9 because the probabilities add, but I cannot figure out P(A and B)....Sun, 06 Sep 2026 21:02:08 -0400https://liberty.io/confusion-about-topologist-039-s-whirlpool.htmlhttps://liberty.io/confusion-about-topologist-039-s-whirlpool.htmlI've found a solution to the title here https://static1.squarespace.com/static/5aff705c5ffd207cc87a512d/t/5c950a447817f75bd2262096/1553271370324/Topology.pdf on page $71$ But why the arc give a ...Sun, 06 Sep 2026 20:57:59 -0400https://liberty.io/what-does-rightarrow-mean-in-p-rightarrow-q.htmlhttps://liberty.io/what-does-rightarrow-mean-in-p-rightarrow-q.htmlI was looking at an exercise where it asked the following: $$\begin{array}{ccc} p&q&p\rightarrow q \\ T&T&T \\ &\ldots \end{array}$$ So, for the third column, I just put $T$ w......Sun, 06 Sep 2026 20:54:25 -0400https://liberty.io/up-to-date-advice-on-the-best-way-to-take-notes-maths.htmlhttps://liberty.io/up-to-date-advice-on-the-best-way-to-take-notes-maths.htmlI have read some old discussions about this topic and would like to get some up-to-date advice if possible. I'm going to start university next year (maths), and I know how important is to have a se......Sun, 06 Sep 2026 20:49:07 -0400https://liberty.io/average-proportion-for-proportions-with-different-denominators.htmlhttps://liberty.io/average-proportion-for-proportions-with-different-denominators.htmlSay I have an experiment in which subjects are asked to respond to some stimulus. Their responses are transcribed and coded, first as "Valid/Invalid", and then for the valid responses, "Correct/Inc......Sun, 06 Sep 2026 20:48:37 -0400https://liberty.io/matlab-vector-assignment-of-arbitrary-length.htmlhttps://liberty.io/matlab-vector-assignment-of-arbitrary-length.htmlGreetings: let's suppose I have a loop that will execute 20 times, and within that loop I'll obtain a vector of information whose length will increase, each iteration, from 2 to 20. Now suppose I h......Sun, 06 Sep 2026 20:48:22 -0400https://liberty.io/why-is-compactness-so-important.htmlhttps://liberty.io/why-is-compactness-so-important.htmlI've read many times that 'compactness' is such an extremely important and useful concept, though it's still not very apparent why. The only theorems I've seen concerning it are the Heine-Borel the......Sun, 06 Sep 2026 20:43:52 -0400https://liberty.io/recursion-tree-t-n-t-n-3-t-2n-3-cn.htmlhttps://liberty.io/recursion-tree-t-n-t-n-3-t-2n-3-cn.htmlI have a task: Explain that by using recursion tree that solution for: $T(n)=T(\frac n3)+T(\frac {2n}{3})+cn$ Where c is constance, is $\Omega(n\lg n)$ My solution: Recursion tree for $T(n)=T(\fra......Sun, 06 Sep 2026 20:26:38 -0400https://liberty.io/what-are-some-measures-of-connectedness-in-graphs.htmlhttps://liberty.io/what-are-some-measures-of-connectedness-in-graphs.htmlI am not a mathematician (I am an engineer who is working on improving his mathematics), so I apologize in advance if my question is trivial. Consider a graph of $N$ nodes, with some defined crite......Sun, 06 Sep 2026 20:20:00 -0400