https://liberty.ioCurated star highlights with fresh energy.en-usWed, 19 Aug 2026 05:16:43 -0400https://liberty.io/user-minus-one-twelfth-mathematics-stack-exchange.htmlhttps://liberty.io/user-minus-one-twelfth-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Wed, 19 Aug 2026 09:12:02 -0400https://liberty.io/maximum-principle-for-harmonic-functions.htmlhttps://liberty.io/maximum-principle-for-harmonic-functions.htmlI know the following classical maximum principle for harmonic functions: If $\Omega \subset \mathbb{C}$ is open and connected and $u \in C^2(\Omega)$ is harmonic, then $u$ has maximum (or ...Wed, 19 Aug 2026 09:08:07 -0400https://liberty.io/what-would-you-see-inside-a-spherical-mirror.htmlhttps://liberty.io/what-would-you-see-inside-a-spherical-mirror.htmlImage to build a huge spherical shell made of semitransparent glass, and to cover the internal part with a reflecting material. In such structure some light can enter, and an observer inside it ......Wed, 19 Aug 2026 09:04:32 -0400https://liberty.io/how-to-find-range-of-a-quadratic-quadratic-function-easily-without-plo.htmlhttps://liberty.io/how-to-find-range-of-a-quadratic-quadratic-function-easily-without-plo.htmlIs there any way to find range of a Quadratic/Quadratic function, without plotting its graph?...Wed, 19 Aug 2026 09:03:48 -0400https://liberty.io/algorithm-to-get-the-maximum-size-of-n-squares-that-fit-into-a-rectang.htmlhttps://liberty.io/algorithm-to-get-the-maximum-size-of-n-squares-that-fit-into-a-rectang.htmlI am looking for an algorithm that can return the number of size of n squares that fit into a a rectangle of a given width and height, maximizing the use of space (thus, leaving the least amount of ...Wed, 19 Aug 2026 09:03:40 -0400https://liberty.io/is-there-a-variant-of-the-dot-product-operation-that-returns-frac-a-1-.htmlhttps://liberty.io/is-there-a-variant-of-the-dot-product-operation-that-returns-frac-a-1-.htmlThe dot product of $[a_1,a_2]$ and $[b_1, b_2]$ is $a_1b_1 + a_2b_2$ (and so on for bigger vectors). What I'm wondering is if there's any definition of a function s.t. the "invdot" produc......Wed, 19 Aug 2026 08:59:02 -0400https://liberty.io/expected-value-of-a-poisson-process.htmlhttps://liberty.io/expected-value-of-a-poisson-process.htmlI've been searching for a while but I can't seem to figure out how to find the expected value of a poisson process up to an arbitrary time. Let {$N(t),t≥0$} be a Poisson process with rate $λ$. H......Wed, 19 Aug 2026 08:51:33 -0400https://liberty.io/coordinate-matrix-of-a-vector-in-terms-of-a-basis.htmlhttps://liberty.io/coordinate-matrix-of-a-vector-in-terms-of-a-basis.htmlI am taking a linear algebra class currently and working through Hoffman's textbook. One of the exercises I am unsure about is, Find the coordinate matrix of the vector $\alpha=(1,0,1)$ in the b......Wed, 19 Aug 2026 08:49:40 -0400https://liberty.io/derivative-of-a-square-root-with-exponential-function.htmlhttps://liberty.io/derivative-of-a-square-root-with-exponential-function.htmlSo I have the following function: $f(x)= \sqrt{e^{2x}}$ After applying the chain rule I sit with: $$\frac{1}{2\sqrt{e^{2x}}}2e^{2x}$$ From there I got: $$\frac{e^{2x}}{\sqrt{e^{2x}}}$$ While ......Wed, 19 Aug 2026 08:36:38 -0400https://liberty.io/creating-truth-tables-variable-columns.htmlhttps://liberty.io/creating-truth-tables-variable-columns.htmlI'm looking for the mathematically equation/algorithm to create the variable columns in a truth table. I really want to understand how this is done, so any directions will be a plus. Ok, so imagin......Wed, 19 Aug 2026 08:36:31 -0400https://liberty.io/combinatorics-formula-reference.htmlhttps://liberty.io/combinatorics-formula-reference.htmlI'm trying to organize the formulas for permutations and combinations. Let's say we sample $r$ objects from an urn which contains $n$ objects, $m$ of which are duplicates. So if the objects in the......Wed, 19 Aug 2026 08:26:40 -0400https://liberty.io/how-to-find-the-point-where-the-curvature-is-maximum.htmlhttps://liberty.io/how-to-find-the-point-where-the-curvature-is-maximum.htmlHow to find the point on the curve $y=e^x$ at which the curvature is maximum. from the graph of the above curve i think at (0,1) the curvature is maximum.Is it true? please help....Wed, 19 Aug 2026 08:24:33 -0400https://liberty.io/user-lucusk-mathematics-stack-exchange.htmlhttps://liberty.io/user-lucusk-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Wed, 19 Aug 2026 08:24:30 -0400https://liberty.io/is-x-1-3-differentiable-at-0.htmlhttps://liberty.io/is-x-1-3-differentiable-at-0.htmlIt occurred to me that functions that are "smooth" but have "infinite slope" may not be considered differentiable at the points where their slope is infinite. An easy example of this is $x^{1/3}$, ......Wed, 19 Aug 2026 08:21:38 -0400https://liberty.io/what-is-a-natural-isomorphism.htmlhttps://liberty.io/what-is-a-natural-isomorphism.htmlI came across the adjective "natural" many times in my reading and I think this has something to do with category theory. Could someone please illustrate the idea behind this adjective to me? Many ...Wed, 19 Aug 2026 08:14:41 -0400https://liberty.io/formal-definition-of-counterexample.htmlhttps://liberty.io/formal-definition-of-counterexample.htmlWhat is the preferred formal definition of “counterexample” as in: zero is a counterexample for "every integer is either positive or negative". Where in the literature is the notion of “counterexam......Wed, 19 Aug 2026 08:12:59 -0400https://liberty.io/projectile-motion-with-air-resistance-proportional-to-velocity-squared.htmlhttps://liberty.io/projectile-motion-with-air-resistance-proportional-to-velocity-squared.htmlA plane flies at a constant altitude of 1000 ft with a constant speed of 300 mph. The plane drops a relief package to a person on the ground. Assume the origin is point where the supply pack is rel......Wed, 19 Aug 2026 08:11:59 -0400https://liberty.io/how-to-find-average-rate-of-change.htmlhttps://liberty.io/how-to-find-average-rate-of-change.htmlHow would I find the average rate of change over $8$ minutes, of a person that runs at a rate of $v(t)=x\sin(x^2-7x)$ ft/min? I missed when this was taught and I have no clue on how to do it. Help......Wed, 19 Aug 2026 08:03:27 -0400https://liberty.io/how-to-estimate-the-range-of-a-normal-distribution-when-the-mean-and-s.htmlhttps://liberty.io/how-to-estimate-the-range-of-a-normal-distribution-when-the-mean-and-s.htmlFor example, how would you respond to this question? The earnings of one-hundred workers in a company are normally distributed. If the mean of this data set is 24 and the standard deviation is 4......Wed, 19 Aug 2026 08:02:10 -0400https://liberty.io/factoring-x-6-y-6-what-is-going-on-here.htmlhttps://liberty.io/factoring-x-6-y-6-what-is-going-on-here.htmlI apologize if this question already exists, but it was quite difficult to word. In my math class today, we learned how to factor a difference of two perfect cubes. One of our practice questions w......Wed, 19 Aug 2026 07:55:44 -0400https://liberty.io/what-does-sin-sin-x-mean.htmlhttps://liberty.io/what-does-sin-sin-x-mean.htmlWhat does an equation like $\sin(\sin(x))$ mean? I know it can be seen as a composite function $f(f(x))$, where $f(x)=\sin(x)$. Is there a way to simplify functions like this, and where will this b......Wed, 19 Aug 2026 07:54:48 -0400https://liberty.io/binomial-expansion-of-1-x-n.htmlhttps://liberty.io/binomial-expansion-of-1-x-n.htmlWe've been given with following binomial expansion $$(x+1)^n= 1+ nx + \frac{n(n-1)}{ 2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3\cdots$$ How can I get the formula of $(1-x)^n$...Wed, 19 Aug 2026 07:45:22 -0400https://liberty.io/which-versions-of-chernoff-bound-are-applied-to-binomial-distribution-.htmlhttps://liberty.io/which-versions-of-chernoff-bound-are-applied-to-binomial-distribution-.htmlThere are several versions of Chernoff bounds. I was wodering which versions are applied to computing the probabilities of a Binomial distribution in the following two examples, but couldn't. I have ...Wed, 19 Aug 2026 07:44:49 -0400https://liberty.io/parabolic-word-problem.htmlhttps://liberty.io/parabolic-word-problem.htmlA rectangular barge is traveling under a bridge with a parabolic archway. The barge is 60 feet tall and 80 feet wide. The bridge is 80 feet tall and 200 feet wide. If the barge must travel down the ...Wed, 19 Aug 2026 07:41:10 -0400https://liberty.io/why-is-the-expected-value-e-x-2-neq-e-x-2.htmlhttps://liberty.io/why-is-the-expected-value-e-x-2-neq-e-x-2.htmlI wish to use the Computational formula of the variance to calculate the variance of a normal-distributed function. For this, I need the expected value of $X$ as well as the one of $X^2$. Intuitive......Wed, 19 Aug 2026 07:40:51 -0400https://liberty.io/how-to-understand-complex-rotation.htmlhttps://liberty.io/how-to-understand-complex-rotation.htmlI come across A Visual, Intuitive Guide to Imaginary Numbers, and find it's difficult for me to understand this section: Let’s take a look. Suppose I’m on a boat, with a heading of 3 units East for ...Wed, 19 Aug 2026 07:38:43 -0400https://liberty.io/finding-the-no-of-ways-to-count-the-letters-in-an-english-alphabet.htmlhttps://liberty.io/finding-the-no-of-ways-to-count-the-letters-in-an-english-alphabet.htmlHow many strings of six lower case letters from the English alphabet contain a) the letter $a$? b) the letters $a$ and $b$? c) the letters $a$ and $b$ in consecutive positions with $a$ preceding......Wed, 19 Aug 2026 07:38:01 -0400https://liberty.io/find-an-equation-for-the-plane-that-contains-the-two-parallel-lines-l-.htmlhttps://liberty.io/find-an-equation-for-the-plane-that-contains-the-two-parallel-lines-l-.htmlI'm don't know where to start on this question and need some assistance....Wed, 19 Aug 2026 07:31:45 -0400https://liberty.io/free-lecture-notes-to-carl-bender-039-s-mathematical-physics-video-lec.htmlhttps://liberty.io/free-lecture-notes-to-carl-bender-039-s-mathematical-physics-video-lec.htmlI am just watching Carl Bender's Mathematical Physics video lecture course (about asymptotics and its application in physics) http://www.perimeterscholars.org/328.html which is great! Are there ......Wed, 19 Aug 2026 07:12:54 -0400https://liberty.io/choose-the-true-statement.htmlhttps://liberty.io/choose-the-true-statement.htmllet f(z) be the meromorphic function fiven by function $f(z)=\frac{z}{(1-e^z).\sin z}.$ then choose the True statement a) For every $ k$ $\in$ Z \ {0} ,$ z= k\pi$ is a simple pole b) $z=......Wed, 19 Aug 2026 07:11:47 -0400https://liberty.io/difference-between-arccos-and-cos-1.htmlhttps://liberty.io/difference-between-arccos-and-cos-1.htmlI'm trying to use inverse cosine, in the cosine rule to find an angle of a triange when you know 3 sides. I know this formula and have it written down. However I've left my calculator and I'm havin......Wed, 19 Aug 2026 07:09:14 -0400https://liberty.io/calculus-does-the-particle-ever-change-direction-at-what-time-interval.htmlhttps://liberty.io/calculus-does-the-particle-ever-change-direction-at-what-time-interval.htmlProblem: A particle moves in a straight line with position relative to the origin given by $s(t)= 2t-\frac{4}{t+1}$ cm, where $t ≥ 0$ is the time in seconds. Here's what I know: a) Find velocity ......Wed, 19 Aug 2026 07:07:37 -0400https://liberty.io/pi-controller-and-finding-poles-of-a-transfer-function.htmlhttps://liberty.io/pi-controller-and-finding-poles-of-a-transfer-function.htmlFor a Wind turbine, the transfer function is $$G(s) = \frac{4.2(s-827)(s^2-5.4s+194)}{(s+0.2)(s^2+0.1s+482)}\;.$$ Furthermore, let $K(s) = b+c/s$ be a $PI$ controller; find $$\frac{1}{1+GK},\qua......Wed, 19 Aug 2026 07:07:07 -0400https://liberty.io/finding-the-centre-of-an-abritary-set-of-points-in-two-dimensions.htmlhttps://liberty.io/finding-the-centre-of-an-abritary-set-of-points-in-two-dimensions.htmlI am currently working on a program that needs to transform one set of coordinates by shifting them to the center of the screen. The points are offset from the middle of the screen - either to the ......Wed, 19 Aug 2026 07:06:34 -0400https://liberty.io/calculate-the-angles-of-a-isosceles-triangle.htmlhttps://liberty.io/calculate-the-angles-of-a-isosceles-triangle.htmlIn the triangle below, is there a way to calculate the $x$ and $y$? To be more specific, $b = 12.8\rm\,cm\ $ and $h = 10\rm\,cm$, hence $a = 11.87\rm\,cm$. I don't know what to do from here....Wed, 19 Aug 2026 06:47:52 -0400https://liberty.io/find-in-radians-the-general-solution-of-cos-3x-sin-5x.htmlhttps://liberty.io/find-in-radians-the-general-solution-of-cos-3x-sin-5x.htmlI am studying maths as a hobby. I have come across this problem: Find a general solution for the equation cos 3x = sin 5x I have said, $\sin 5x = \cos(\frac{\pi}{2} - 5x)$ so $\cos 3x = \sin 5x \im......Wed, 19 Aug 2026 06:40:38 -0400https://liberty.io/definition-of-jointly-normal.htmlhttps://liberty.io/definition-of-jointly-normal.htmlProblem X ~ N(0, 1) Let W be independent of X with P(W = 1) = P(W = −1) = 1 Y = WX It is found that Y ~ N(0,1). Are X and Y jointly normal? Question Answer key said: "Clearly X and Y are not ...Wed, 19 Aug 2026 06:36:59 -0400https://liberty.io/the-dual-transformation-has-the-same-eigenvalues.htmlhttps://liberty.io/the-dual-transformation-has-the-same-eigenvalues.htmlThe Question: Let $T:V \rightarrow W$ be a linear transformation where $V$ is a finite-dimensional vector space over the field $\Bbb F$. (i) Prove that $\text{Im}(T')=(\text{Ker}(T)) ^ \circ$ (ii) ...Wed, 19 Aug 2026 06:33:27 -0400https://liberty.io/use-lagrange-multipliers-to-find-maximum-and-minimum-values.htmlhttps://liberty.io/use-lagrange-multipliers-to-find-maximum-and-minimum-values.htmlI am having trouble understanding how to solve the problem below. Can anyone show me how to solve this? Here is the problem definition: "Use LaGrange multipliers to find the maximum and minimum ...Wed, 19 Aug 2026 06:26:28 -0400https://liberty.io/simple-asymptotic-function.htmlhttps://liberty.io/simple-asymptotic-function.html(I have seen this question but it is too complicated for my needs, and my math skills are not good enough to convert the answer.) I am writing a game and I need a way to increase the armor of the ...Wed, 19 Aug 2026 06:22:51 -0400https://liberty.io/is-a-line-a-convex-set.htmlhttps://liberty.io/is-a-line-a-convex-set.htmlI was thinking about whether a $\textbf{line}$ is a convex set. My intuition is as follows: A set is convex if it includes all convex combinations of points in the set. Or in other words, if it ...Wed, 19 Aug 2026 06:21:12 -0400https://liberty.io/trigonometry-why-we-need-to-relate-to-circles.htmlhttps://liberty.io/trigonometry-why-we-need-to-relate-to-circles.htmlI'm a trigonometry teaching assistant this semester and have a perhaps basic question about the motivation for using the circle in the study of trigonometry. I certainly understand Pythagorean Theo......Wed, 19 Aug 2026 06:20:13 -0400https://liberty.io/intuition-of-the-connection-between-the-graph-laplacian-and-the-laplac.htmlhttps://liberty.io/intuition-of-the-connection-between-the-graph-laplacian-and-the-laplac.htmlIn Physics SO the intuition of the Laplace operator (divergence of the gradient) is explained by resorting to the finite difference version: the Laplace equation is satisfied as long as the value a......Wed, 19 Aug 2026 06:17:59 -0400https://liberty.io/how-do-we-introduce-subtraction-from-these-field-axioms.htmlhttps://liberty.io/how-do-we-introduce-subtraction-from-these-field-axioms.htmlI am familiar with two different sets of field axioms. The first one is from "Mathematical Analysis" by Apostol. It has the first $3$ usual axioms, but the $4^{th}$ one is different: Axiom 1: ...Wed, 19 Aug 2026 06:15:34 -0400https://liberty.io/fraction-of-non-integers.htmlhttps://liberty.io/fraction-of-non-integers.htmlBy definition of a fraction, can the numerator or the denominator be a non integer ? For example, can we call the expression $\frac{\pi}{2}$ a fraction ? The popular answer is no, but since the ca......Wed, 19 Aug 2026 06:12:37 -0400https://liberty.io/find-a-vector-non-orthogonal-to-a-given-set.htmlhttps://liberty.io/find-a-vector-non-orthogonal-to-a-given-set.htmlLet $S = \{\mathbf{v}_1, \dots, \mathbf{v}_n\}$ be a set of vectors in $\mathbb{R}^{n}$. I would like to find a vector $\mathbf{u} \in \mathbb{R}^n$ such that, for all $i \in [1, n]$, $\mathbf{u}$ ......Wed, 19 Aug 2026 06:07:19 -0400https://liberty.io/technique-for-finding-the-nth-term.htmlhttps://liberty.io/technique-for-finding-the-nth-term.htmlI'm a trainee teacher and I need to pass a skills test in maths, but I'm having trouble in this particular area of algebra in finding the nth term of any sequence. I've been given a series of quest......Wed, 19 Aug 2026 06:06:55 -0400https://liberty.io/set-notation-what-does-the-bar-symbol-mean.htmlhttps://liberty.io/set-notation-what-does-the-bar-symbol-mean.htmlIn set notation could somebody explain the meaning of $\mid$ in the equation below please? How does it read? I read it as $s$ and $j$ are an element of $E$ but what does the $j \mid$ mean? Simila......Wed, 19 Aug 2026 06:06:42 -0400https://liberty.io/finding-the-middle-coordintes-of-each-side-of-an-equilateral-triangle.htmlhttps://liberty.io/finding-the-middle-coordintes-of-each-side-of-an-equilateral-triangle.htmlI have a problem with finding the center coordinates of each side in an equilateral triangle. I've linked an image below that shows exactly which coordinates I'm after. I understand that I can wor......Wed, 19 Aug 2026 06:06:08 -0400https://liberty.io/visually-prove-area-1-2-base-height-for-scalene-triangle.htmlhttps://liberty.io/visually-prove-area-1-2-base-height-for-scalene-triangle.htmlPlease refer to image for explanation....Wed, 19 Aug 2026 05:39:14 -0400https://liberty.io/show-that-the-arithmetic-mean-is-less-or-equal-than-the-quadratic-mean.htmlhttps://liberty.io/show-that-the-arithmetic-mean-is-less-or-equal-than-the-quadratic-mean.htmlI tried to solve this for hours but no success. Prove that the arithmetic mean is less or equal than the quadratic mean. I am in front of this form: $$ \left(\frac{a_1 + ... + a_n} { n}\right)^......Wed, 19 Aug 2026 05:37:41 -0400https://liberty.io/right-angles-in-circles.htmlhttps://liberty.io/right-angles-in-circles.htmlI have been posed the following conjecture: Let $AB$ be the diameter of a circle. Let $C$ be the mid point of the arc $AB$. Let $D$ be somewhere on the arc $AC$ and $E$ be on the chord $DB$ such......Wed, 19 Aug 2026 05:31:54 -0400https://liberty.io/how-would-you-explain-to-a-9th-grader-the-negative-exponent-rule.htmlhttps://liberty.io/how-would-you-explain-to-a-9th-grader-the-negative-exponent-rule.htmlLet us assume that the students haven't been exposed to these two rules: $a^{x+y} = a^{x}a^{y}$ and $\frac{a^x}{a^y} = a^{x-y}$. They have just been introduced to the generalization: $a^{-x} = \fra......Wed, 19 Aug 2026 05:30:20 -0400https://liberty.io/definition-of-a-convex-cone.htmlhttps://liberty.io/definition-of-a-convex-cone.htmlIn the definition of a convex cone, given that $x,y$ belong to the convex cone $C$,then $\theta_1x+\theta_2y$ must also belong to $C$, where $\theta_1,\theta_2 > 0$. What I don't understand is why ...Wed, 19 Aug 2026 05:23:23 -0400https://liberty.io/how-to-find-the-fixed-field.htmlhttps://liberty.io/how-to-find-the-fixed-field.htmlLet $\mathbb{Q}$ be the field of rational number, then the splitting field of $x^{3}-2$ over $\mathbb{Q}$ is $\mathbb{Q}[\sqrt[3]{2}, \zeta]$ where $\zeta\neq 1$ be the third root of unity. The e......Wed, 19 Aug 2026 05:22:03 -0400https://liberty.io/what-vector-field-property-means-is-the-curl-of-another-vector-field.htmlhttps://liberty.io/what-vector-field-property-means-is-the-curl-of-another-vector-field.htmlI know that a vector field $\mathbf{F}$ is called irrotational if $\nabla \times \mathbf{F} = \mathbf{0}$ and conservative if there exists a function $g$ such that $\nabla g = \mathbf{F}$. Under ...Wed, 19 Aug 2026 05:13:49 -0400https://liberty.io/chinese-remainder-theorem-backwards-duplicate.htmlhttps://liberty.io/chinese-remainder-theorem-backwards-duplicate.htmlI was reading a description of the CRT which went like this $x = a \mod p$ $x = b \mod q$ has a unique solution for $x \mod pq$ It also said - The reverse direction is trivial. Give $x \mod pq$ we ......Wed, 19 Aug 2026 05:12:17 -0400https://liberty.io/area-of-a-five-pointed-star.htmlhttps://liberty.io/area-of-a-five-pointed-star.htmlA 5 pointed star is inscribed in a circle of radius $r$. Prove that the area of the star is $$ \frac{10 \tan\left(\tfrac{\pi}{10}\right)}{3-\tan^2\left(\tfrac{\pi}{10}\right)} r^2 $$...Wed, 19 Aug 2026 04:57:36 -0400https://liberty.io/how-to-use-double-angle-identities-to-find-sin-x-and-cos-x-from-sin-2x.htmlhttps://liberty.io/how-to-use-double-angle-identities-to-find-sin-x-and-cos-x-from-sin-2x.htmlIf $\sin 2x =\frac{5}{13}$ and $0^\circ < x < 45^\circ$, find $\sin x$ and $\cos x$. The answers should be $\frac{\sqrt{26}}{26}$ and $\frac{5\sqrt{26}}{26}$ Ideas The idea is to use double ......Wed, 19 Aug 2026 04:56:57 -0400https://liberty.io/graphing-multivariate-functions-by-hand-impossible.htmlhttps://liberty.io/graphing-multivariate-functions-by-hand-impossible.htmlMy question here is derivative of my last one but for increased brevity of both questions I've separated them. I'm exploring the following function for learning:$$f(x,\ y) = (x-2)^2 + (y-2)^2$$ An......Wed, 19 Aug 2026 04:41:56 -0400https://liberty.io/learning-functional-analysis-and-measure-theory.htmlhttps://liberty.io/learning-functional-analysis-and-measure-theory.htmlI have taken a first course in real analysis and I'm currently studying analysis in $\mathbb{R}^N$ on my own. I want to start functional analysis after this, and I also want to learn measure theory......Wed, 19 Aug 2026 04:41:44 -0400https://liberty.io/proof-of-the-right-and-left-cancellation-laws-for-groups-duplicate.htmlhttps://liberty.io/proof-of-the-right-and-left-cancellation-laws-for-groups-duplicate.htmlI was asked to proof the right and left cancellation laws for groups, i.e. If $a,b,c \in G$ where $G$ is a group, show that $ba = ca \implies b=c $ and $ab = ac \implies b = c$ For the first p......Wed, 19 Aug 2026 04:22:34 -0400https://liberty.io/what-039-s-laplace-transform-of-u-t-1-delta-t.htmlhttps://liberty.io/what-039-s-laplace-transform-of-u-t-1-delta-t.htmlIf I took LT : $L(u(t-1)\delta(t)) = u((0)-1)$ as Laplace of $\delta(t)=1$, and we substitute $t$ with zero don't we get a zero answer as $u(t-1) = 0$ when $t<1$? but what if I did it the othe......Wed, 19 Aug 2026 04:13:27 -0400https://liberty.io/minimal-polynomial-of-a-6x6-matrix.htmlhttps://liberty.io/minimal-polynomial-of-a-6x6-matrix.htmlfor the matrix $$ \begin{pmatrix} 4 & 1 & 0 & 0 & 0 & 0 \\ -1 & 2 & 0 & 0 & 0 & 0\\ 0 & 0 & -1 & -2 & -2 &amp......Wed, 19 Aug 2026 04:11:02 -0400https://liberty.io/why-does-pert-work.htmlhttps://liberty.io/why-does-pert-work.htmle is the limit of (1+1/n)^n. So how come we do Pe^rt to calculate continuously compounded interest? The regular formula for compound interest is (1+r/n)^tn, with rate being part of the base, not the ...Wed, 19 Aug 2026 04:06:48 -0400https://liberty.io/probability-notation.htmlhttps://liberty.io/probability-notation.htmlHey guys, I was just wondering why in my textbook(A First Course in Probability, 8th edition) and basically everywhere I've looked at when we have some random variable(assume for the sake of the ...Wed, 19 Aug 2026 04:05:04 -0400https://liberty.io/how-to-convert-kj-kmole-to-kj-kg-closed.htmlhttps://liberty.io/how-to-convert-kj-kmole-to-kj-kg-closed.htmlSo say I have CO2 and a value of 50 000 kJ/kmole. This is how ive converted just want to check if im right. so.. kJ/kmole = J/mole 1 mole of CO2 = 12 +16x2 = 44g J/.044 Kg then what would I d......Wed, 19 Aug 2026 04:04:00 -0400https://liberty.io/how-many-fair-dice-exist.htmlhttps://liberty.io/how-many-fair-dice-exist.htmlWe know a coin is a fair die with a 50-50 probability for two alternatives. Similarly, all five Platonic solids are fair dice. That makes six solids that can be fair dice, but can there be more? One ...Wed, 19 Aug 2026 04:00:42 -0400https://liberty.io/is-every-real-root-also-a-complex-root-complex-zeros-and-fundamental-t.htmlhttps://liberty.io/is-every-real-root-also-a-complex-root-complex-zeros-and-fundamental-t.htmlStupid question about this topic: I don't understand the rule that says "A poly with degree N will have N complex roots". eg: $P(X)=x^5 + x^3 - 1$ is a 5th degree polynomial function, so P(x) has ...Wed, 19 Aug 2026 03:57:13 -0400https://liberty.io/is-there-a-technique-to-find-a-hamilton-circuit-in-a-graph.htmlhttps://liberty.io/is-there-a-technique-to-find-a-hamilton-circuit-in-a-graph.htmlMy textbook goes in depth about how to show that a graph does NOT have a hamilton circuit, but one of its practice problems asks you to find a hamilton circuit in a graph that does have a hamilton ...Wed, 19 Aug 2026 03:47:25 -0400https://liberty.io/proving-a-removable-discontinuity-exists.htmlhttps://liberty.io/proving-a-removable-discontinuity-exists.htmlSo I made up the following function as a thought exercise (for myself). $$f(x) = 1\hspace{0.3cm} \forall\hspace{0.1cm} x = 1,2,3 \hspace{0.4cm}\text{and} \hspace{0.4cm} -1 \hspace{0.3cm}\forall \......Wed, 19 Aug 2026 03:46:36 -0400https://liberty.io/how-to-solve-0-cos-2x-cos-x.htmlhttps://liberty.io/how-to-solve-0-cos-2x-cos-x.htmlI'm desperately trying to solve $$0=\cos{2x}+\cos x$$ Am I on the right track when I'm this far? $$\cos x(2\cos x+1)=1$$ I don't know where to go from here. What method can I use to further solve......Wed, 19 Aug 2026 03:23:46 -0400https://liberty.io/how-to-tell-if-a-multivariable-limit-really-exists.htmlhttps://liberty.io/how-to-tell-if-a-multivariable-limit-really-exists.htmlHi i'm having this doubt about this multivariable limit $$ \lim\limits_{(x,y)\to(0,0)} \frac{2x^2y}{x^4+y^2} $$ If I use these two paths like let $x = y^3$ and $y=x$ I get that the limit is going......Wed, 19 Aug 2026 03:17:34 -0400https://liberty.io/last-digits-of-a-card-are-the-same.htmlhttps://liberty.io/last-digits-of-a-card-are-the-same.htmlI have a question If I have a certain card, with for example 10 digits. Do you think that is it possible that someone else have the same last 4 digits of my card? I think that is possible. Beca......Wed, 19 Aug 2026 03:01:54 -0400https://liberty.io/the-sides-of-a-pentagon-are-represented-in-centimeters-by-x-10-2x-1-an.htmlhttps://liberty.io/the-sides-of-a-pentagon-are-represented-in-centimeters-by-x-10-2x-1-an.htmlThe sides of a pentagon are represented in centimeters by $x$, $10$, $2x$, $1$ and $3$. Determine how many even values of $x$ ​​are there that satisfy this pentagon. The answer is 5. How can I solv......Wed, 19 Aug 2026 02:52:00 -0400https://liberty.io/what-is-the-mutual-information-i-x-x.htmlhttps://liberty.io/what-is-the-mutual-information-i-x-x.html$X$ is a random variable with normal distribution, assume $Y=X$, what is the mutual information $I(X;Y)$? I guess that $h(Y|X)=0$ since when $X$ is known, $Y$ is completely known, so $$I(X;Y)=h(Y)......Wed, 19 Aug 2026 02:44:11 -0400https://liberty.io/l0-norm-l1-norm-and-l2-norm.htmlhttps://liberty.io/l0-norm-l1-norm-and-l2-norm.htmlFor vector $\boldsymbol{x},\boldsymbol{y} \in \mathcal{R}^{n}$, if \begin{equation} \| \boldsymbol{x} \|_0 = \| \boldsymbol{y} \|_0 \end{equation} What relationship will $\| \boldsymbol{x} \|_1$ a......Wed, 19 Aug 2026 02:43:20 -0400https://liberty.io/what-does-a-compact-set-look-like.htmlhttps://liberty.io/what-does-a-compact-set-look-like.htmlSo for a writing assignment in one of my classes we are asked to discuss and prove some basic results about compact sets in general topological spaces. I like proving these things, but they dont he......Wed, 19 Aug 2026 02:40:13 -0400https://liberty.io/help-me-to-understand-the-potato-paradox.htmlhttps://liberty.io/help-me-to-understand-the-potato-paradox.htmlFrom wiki - Fred brings home $100$ kg of potatoes, which (being purely mathematical potatoes) consist of $99\%$ water. He then leaves them outside overnight so that they consist of $98\%$ water. Wh......Wed, 19 Aug 2026 02:37:16 -0400https://liberty.io/proving-de-morgan-039-s-generalized-laws.htmlhttps://liberty.io/proving-de-morgan-039-s-generalized-laws.htmlHow can I prove the generalized De Morgan laws using mathematical writing, and not induction (I am not allowed). This is what I concluded: $\bigcap_{i\in I}A_i' = (\bigcup_{i\in I}(A_i)).$ and its ...Wed, 19 Aug 2026 02:25:16 -0400https://liberty.io/what-is-meant-by-passing-to-a-subsequence.htmlhttps://liberty.io/what-is-meant-by-passing-to-a-subsequence.htmlLet $\{f_k(x)\}_{k=1}^\infty$ be a sequence of differentiable functions defined on $[0,1]$. Suppose $f_k(0)=0$ and $|f_k'(x)|\le M$ for all $x\in[0,1]$ and for all $k\ge1$. Prove that after passing......Wed, 19 Aug 2026 02:22:33 -0400https://liberty.io/explanation-of-the-formula-for-combinations-without-repetition-probabi.htmlhttps://liberty.io/explanation-of-the-formula-for-combinations-without-repetition-probabi.htmlThe formula for combinations without the repetitions is as follows: $$ \frac{n!}{r!(n-r)!}$$ This is achieved by doing $$\frac{n!}{(n-r)!}*\frac{1}{r!}$$ What I don't understand where $\frac{1}{r......Wed, 19 Aug 2026 02:21:36 -0400https://liberty.io/why-do-you-need-three-perpendicular-bisectors-to-circumscribe-a-triang.htmlhttps://liberty.io/why-do-you-need-three-perpendicular-bisectors-to-circumscribe-a-triang.htmlAll the perpendicular bisectors of the sides of a triangle intersect at one point -- the circumcenter. So, why do we typically find all three perpendicular bisectors when trying to circumscribe a ...Wed, 19 Aug 2026 02:11:21 -0400https://liberty.io/homeomorphic-graphs.htmlhttps://liberty.io/homeomorphic-graphs.htmlAre these graphs homeomorphic? a) and the Peterson graph? b) I think both are not homeomorphic.Is it correct?Is there a way I can show that they are not homemorphic Also two graphs being isomo......Wed, 19 Aug 2026 02:09:55 -0400https://liberty.io/muddy-children-puzzle-when-more-than-2-kids-are-muddy.htmlhttps://liberty.io/muddy-children-puzzle-when-more-than-2-kids-are-muddy.htmlThis is about famous Muddy Children puzzle mentioned in section 1.1. I am trying to understand the significance of rounds when number of muddy children are 3 or more than 3. http://www.cs.cornell.......Wed, 19 Aug 2026 01:53:43 -0400https://liberty.io/how-to-find-an-angle-in-degrees-in-a-right-triangle-given-its-sides.htmlhttps://liberty.io/how-to-find-an-angle-in-degrees-in-a-right-triangle-given-its-sides.htmlI need to find out a degree of an angle. Pretty simple, or so I thought. I remember doing a crap-ton of these in high-school, sadly the details did not remain. Anyway, let's take a look at this tr......Wed, 19 Aug 2026 01:50:17 -0400https://liberty.io/finding-minimum-value-of-sqrt-x-2-y-2-duplicate.htmlhttps://liberty.io/finding-minimum-value-of-sqrt-x-2-y-2-duplicate.htmlFind the minimum value of $\sqrt{x^2+y^2}$, given $15x+8y=120$. My attempt: From $15x+8y=120$, I get $y=\frac{120-15x}{8}$. I substitute this value into $\sqrt{x^2+y^2}$, getting $\sqrt{\frac{28......Wed, 19 Aug 2026 01:49:26 -0400https://liberty.io/questions-tagged-definite-integrals.htmlhttps://liberty.io/questions-tagged-definite-integrals.htmlQ&A for people studying math at any level and professionals in related fields...Wed, 19 Aug 2026 01:36:31 -0400https://liberty.io/is-there-a-name-for-a-3d-shape-that-looks-like-a-circle-when-viewed-fr.htmlhttps://liberty.io/is-there-a-name-for-a-3d-shape-that-looks-like-a-circle-when-viewed-fr.htmlThe shape can be constructed by taking a cylinder with its height equal to its diameter and cutting a triangle out of it when viewing it from the side with the base of the triangle matching one end......Wed, 19 Aug 2026 01:28:36 -0400https://liberty.io/is-there-a-three-dimensional-version-of-curl-in-which-the-result-is-a-.htmlhttps://liberty.io/is-there-a-three-dimensional-version-of-curl-in-which-the-result-is-a-.htmlCurl is an operation from a vector field to a vector field in 3D. Is there an interpretation of Divergence that doesn't produce a scalar field but rather a vector field, such that at any point $|\t......Wed, 19 Aug 2026 01:23:40 -0400https://liberty.io/domain-of-exponential-function.htmlhttps://liberty.io/domain-of-exponential-function.htmlI have $$e^{\frac{x²+2}{x-1}}.$$ As the domain is the values for which the function is defined, I thought of making the denominator of the power equal to zero. So $x-1=0$ and the function is undefi......Wed, 19 Aug 2026 01:16:39 -0400https://liberty.io/bounded-vs-unbounded-linear-programs.htmlhttps://liberty.io/bounded-vs-unbounded-linear-programs.htmlConsider a (linear) optimization problem of the form "maximize $c^{\top}x$ subject to $\varphi(x)$". Consider the following definitions: The program is called unbounded iff it is feasible but its ...Wed, 19 Aug 2026 01:14:51 -0400https://liberty.io/real-analysis-continuity-of-a-composition-function.htmlhttps://liberty.io/real-analysis-continuity-of-a-composition-function.htmlSuppose $f$ and $g$ are functions such that $g$ is continuous at $a$, and $f$ is continuous at $g(a)$. Show the composition $f(g(x))$ is continuous at $a$. My idea: Can I go straight from definiti......Wed, 19 Aug 2026 01:12:20 -0400https://liberty.io/how-to-show-cubic-function-is-one-to-one-using-assumption.htmlhttps://liberty.io/how-to-show-cubic-function-is-one-to-one-using-assumption.htmlOkay I know the function $f(x)= x^3 +2x+3$ is one to one but how do I show that using the assumption that $f(a)=f(b)$ where $a=b$? So if I do the process I have $ a^3+2a+3 = b^3+2b+3$ my 3's ca......Wed, 19 Aug 2026 01:07:46 -0400https://liberty.io/integrate-over-an-area-of-a-parabola.htmlhttps://liberty.io/integrate-over-an-area-of-a-parabola.htmlHow do I figure out the integral bonds for r when integrating over the area M under a parabola y=2-x^2? Do I need to solve a quadratic equation? I think there must be an easier solution. Clarific......Wed, 19 Aug 2026 01:06:24 -0400https://liberty.io/centroid-root-locus.htmlhttps://liberty.io/centroid-root-locus.htmlI can't figure out how to find the root locus centroid for the poles of this simple equation in a positive feedback system. $$ H(s)=\frac{s}{s^2+3s+1} $$ I have read in many places that the centr......Wed, 19 Aug 2026 01:02:44 -0400https://liberty.io/why-is-establishing-absolute-consistency-of-zfc-impossible.htmlhttps://liberty.io/why-is-establishing-absolute-consistency-of-zfc-impossible.htmlWhy is establishing the absolute consistency of ZFC impossible? What are the fundamental limitations that prohibit us with coming up with a proof? EDIT: This post seems to make the most sense. In ......Wed, 19 Aug 2026 00:57:45 -0400https://liberty.io/what-is-the-conjugate-of-i-iota.htmlhttps://liberty.io/what-is-the-conjugate-of-i-iota.htmlAfter solving a bunch of stuff I got to $\ arg(1/-i)$. What does this mean geometrically, and moreover, will the conjugate of $i$ be $-i$ or just $i$ itself?...Wed, 19 Aug 2026 00:35:46 -0400https://liberty.io/prove-the-derivative-of-sin-1-x-exists.htmlhttps://liberty.io/prove-the-derivative-of-sin-1-x-exists.htmlHow do I prove the derivative of $$\sin(1/x)=-\frac{1}{x^2}\cos(1/x)$$? I understand that you use $$f'(x_0) = \lim_{x \to x_0} \frac{\sin(1/x) - \sin(1/x_0)}{x-x_0} = -\frac{1}{x_0^2}\cos(1/x_0)$$ ...Wed, 19 Aug 2026 00:26:12 -0400https://liberty.io/how-many-possible-3-character-combinations-using-letters-closed.htmlhttps://liberty.io/how-many-possible-3-character-combinations-using-letters-closed.htmlHow many combinations can be using 3 alphabet letters (from A to Z)? Note that string doesn't have to be unique, it can be eg. RPP or AZZ...Wed, 19 Aug 2026 00:08:29 -0400