https://liberty.ioCurated star highlights with fresh energy.en-usFri, 04 Sep 2026 18:20:48 -0400https://liberty.io/quadratic-formula-and-factoring-are-leading-to-different-answers.htmlhttps://liberty.io/quadratic-formula-and-factoring-are-leading-to-different-answers.html$$x^{ 2 }-2x-15=0$$ By factoring, I get: $$(x-5)(x+3)$$ Which has the solutions: $$x=5, x=-3$$ However when I use the quadratic formula (which is what the book saids to use), I get $$\frac ......Fri, 04 Sep 2026 22:08:20 -0400https://liberty.io/why-can-a-venn-diagram-for-4-sets-not-be-constructed-using-circles.htmlhttps://liberty.io/why-can-a-venn-diagram-for-4-sets-not-be-constructed-using-circles.htmlThis page gives a few examples of Venn diagrams for $4$ sets. Some examples: Thinking about it for a little, it is impossible to partition the plane into the $16$ segments required for a complet......Fri, 04 Sep 2026 22:07:07 -0400https://liberty.io/why-does-soh-cah-toa-hold.htmlhttps://liberty.io/why-does-soh-cah-toa-hold.htmlEven looking on the Wiki pages I have a hard time figuring this one out. But why does SOH CAH TOA hold? As in, why is $\sin(x)$ the same as the opposite over hypotenuse of a right triangle? Why is......Fri, 04 Sep 2026 21:54:41 -0400https://liberty.io/invertibility-and-rank.htmlhttps://liberty.io/invertibility-and-rank.htmlHow do you formally prove that a matrix A is invertible if and only if it has full rank, without using determinants?...Fri, 04 Sep 2026 21:52:32 -0400https://liberty.io/baker-hausdorff-lemma-from-sakurai-039-s-book.htmlhttps://liberty.io/baker-hausdorff-lemma-from-sakurai-039-s-book.htmlI'd like to show that, given to hermitian operators $A,G$ on a Hilbert space $\mathscr{H}$, the following identity holds: $$ e^{iG\lambda}A e^{-iG\lambda} = A + i\lambda [G,A] + \frac{\left(i\lambda\...Fri, 04 Sep 2026 21:50:49 -0400https://liberty.io/is-a-proof-required-for-the-division-algorithm-for-polynomials.htmlhttps://liberty.io/is-a-proof-required-for-the-division-algorithm-for-polynomials.htmlI understand that the Division Algorithm can be applied to polynomials. Namely, for polynomials, for any polynomials $f,g$, there exist polynomials $q,r$ such that $f = gq + r$, with $\deg(f) > ......Fri, 04 Sep 2026 21:49:37 -0400https://liberty.io/deriving-probability-from-logit-or-log-odds.htmlhttps://liberty.io/deriving-probability-from-logit-or-log-odds.htmlHow can I motivate the derivation of the $p$ below? From: https://en.wikipedia.org/wiki/Logistic_regression#Logistic_model $l = log_b\frac{p}{1-p}=\beta_0 + \beta_1x_1+\beta_2x_2 $ $p = \frac{b^{\......Fri, 04 Sep 2026 21:31:11 -0400https://liberty.io/user-cortizol-mathematics-stack-exchange.htmlhttps://liberty.io/user-cortizol-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Fri, 04 Sep 2026 21:14:57 -0400https://liberty.io/how-to-use-the-aops-books.htmlhttps://liberty.io/how-to-use-the-aops-books.htmlI just acquired the art of problem solving prealgebra book. I would like to ask, how does one use the AOPS books, are they meant to be supplementary to a full textbook? Or are they used for introd......Fri, 04 Sep 2026 21:09:10 -0400https://liberty.io/parametrization-of-the-hyperboloid-of-two-sheets.htmlhttps://liberty.io/parametrization-of-the-hyperboloid-of-two-sheets.htmlFind the parametrization for the hyperboloid of two sheets${(x,y,z) \in \mathbb{R}^3}; -x^2-y^2+z^2=1$. Ok so I saw two answers for this question: $x(u,v)=(\sinh u \cos v, \sinh u \sin v, \cosh u)......Fri, 04 Sep 2026 21:08:56 -0400https://liberty.io/integrate-sin-x-cos-x-using-trig-identity.htmlhttps://liberty.io/integrate-sin-x-cos-x-using-trig-identity.htmlBook tells me the answer is: $$ \int \sin(x)\cos(x) dx = \frac{1}{2} \sin^{2}(x) + C $$ however, I get the result: $$ \sin(A)\cos(B) = \frac{1}{2} \sin(A-B)+\frac{1}{2}\sin(A+B) $$ $$ \begin{s......Fri, 04 Sep 2026 21:07:15 -0400https://liberty.io/what-is-the-uv-pair-or-uv-plane-exactly.htmlhttps://liberty.io/what-is-the-uv-pair-or-uv-plane-exactly.htmlMaybe the answer to this question is easier than computing $1+1$, but I often find this $uv$ pair on pretty much all the parametric equations that have something to do with 3D geometry and all the ...Fri, 04 Sep 2026 21:04:20 -0400https://liberty.io/grothendieck-topology-and-relation-with-usual-topologies.htmlhttps://liberty.io/grothendieck-topology-and-relation-with-usual-topologies.htmlRecently I stumbled upon the definition of $\textbf{Grothendieck}$ $\textbf{topologies}$ of a category $\mathcal{C}$. I do know that is one of the most interesting parts of the contemporary algebraic ...Fri, 04 Sep 2026 20:53:47 -0400https://liberty.io/where-does-the-ln-c-came-from.htmlhttps://liberty.io/where-does-the-ln-c-came-from.htmlI'm trying to solve problems which is about the application of indefinite integral. I do quite understand how to solve it but at some point I'm curious where does the $ln(C)$ came from. On the first ...Fri, 04 Sep 2026 20:49:25 -0400https://liberty.io/books-on-logic-proof-theory-and-set-theory.htmlhttps://liberty.io/books-on-logic-proof-theory-and-set-theory.htmlI graduated in Computer Science at University of Bologna in Italy some years ago. For various reasons now I am discovering a back interest in mathematic logic higher than I was a student. I have o......Fri, 04 Sep 2026 20:44:15 -0400https://liberty.io/integral-cohomology-ring-of-real-projective-space.htmlhttps://liberty.io/integral-cohomology-ring-of-real-projective-space.htmlWhat is the cohomology ring $$ H^*(\mathbb{R}P^\infty;\mathbb{Z})?$$ $$ H^*(\mathbb{R}P^n;\mathbb{Z})?$$ for mod 2 coefficient, the answer is on Hatcher's book and Proving that the cohomology rin......Fri, 04 Sep 2026 20:44:00 -0400https://liberty.io/in-the-principle-of-mathematical-induction-why-do-we-take-the-base-cas.htmlhttps://liberty.io/in-the-principle-of-mathematical-induction-why-do-we-take-the-base-cas.htmlI kinda understand the logic and motivation behind the proof, but what bothers me is the fact, why is the base case (the first statement that we write) is always $P(1)$ when we are proving a propos......Fri, 04 Sep 2026 20:30:12 -0400https://liberty.io/solve-the-system-x1-x2-x3.htmlhttps://liberty.io/solve-the-system-x1-x2-x3.html$$\left\{\begin{align}x_{1} + x_{2} +4x_{3} &= 1 \\ 3x_{1} +2x_{2} -5x_{3} &= -8 \end{align}\right.$$ $$\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} \quad \\ \quad \\ \...Fri, 04 Sep 2026 20:28:55 -0400https://liberty.io/fooling-set-method.htmlhttps://liberty.io/fooling-set-method.htmlIn the book Computational Complexity (By Sanjeev Arora and Boaz Barak) a method called "The fooling set" is mentioned, when trying to bound the communication complexity of a function from below. A ......Fri, 04 Sep 2026 20:27:17 -0400https://liberty.io/solving-an-ivp-using-undetermined-coefficients.htmlhttps://liberty.io/solving-an-ivp-using-undetermined-coefficients.htmlI'm trying to solve the following Initial value problem using the method of undetermined coefficients, but I keep getting the wrong answer. Any pointers? $y''-9y=20e^{2t} - 81\quad\quad y(0)=10\qu......Fri, 04 Sep 2026 19:59:35 -0400https://liberty.io/probability-of-exactly-three-of-a-kind-in-a-roll-of-5-dice.htmlhttps://liberty.io/probability-of-exactly-three-of-a-kind-in-a-roll-of-5-dice.htmlThe answer according to edx course HarvardX: FC1x Fat Chance: Probability from the Ground Up is $$\frac{6*5*5* \left(^5_2\right)}{6^5} =\frac{1500}{6^5}$$ The remaining two dice can be same say......Fri, 04 Sep 2026 19:50:23 -0400https://liberty.io/on-the-proof-that-every-positive-continuous-random-variable-with-the-m.htmlhttps://liberty.io/on-the-proof-that-every-positive-continuous-random-variable-with-the-m.htmlThe theorem to prove is: $X$ is a positive continuous random variable with the memoryless property, then $X \sim Expo(\lambda)$ for some $\lambda$. The proof is explained in this video, but I will ......Fri, 04 Sep 2026 19:46:15 -0400https://liberty.io/vector-line-parallel-to-x-axis.htmlhttps://liberty.io/vector-line-parallel-to-x-axis.htmlThe points $P$ and $Q$ have position vectors, relative to the origin $O$, given by $$ \overrightarrow{OP} = 7\mathbf{i} + 7\mathbf{j} - 5\mathbf{k} \quad\text{and}\quad \overrightarrow{OQ} = -5\...Fri, 04 Sep 2026 19:46:12 -0400https://liberty.io/leading-coefficient-of-polynomial-with-more-than-one-variable.htmlhttps://liberty.io/leading-coefficient-of-polynomial-with-more-than-one-variable.htmlWhat is the leading coefficient of a polynomial with more than one variable, when two or more terms have the same degree but different coefficients? For example: $3x^2y^2 + 5xy^3$. The degree is ......Fri, 04 Sep 2026 19:39:33 -0400https://liberty.io/taylor-series-expansion-of-f-x-2-1-x-centered-at-x-3-give-the-interval.htmlhttps://liberty.io/taylor-series-expansion-of-f-x-2-1-x-centered-at-x-3-give-the-interval.htmlFind the Taylor Expansion for the function f(x) = 2/(1-x) centered at x = 3. Give the interval of convergence for this series. So if I remember correctly, we first take the first four or so ...Fri, 04 Sep 2026 19:24:18 -0400https://liberty.io/which-method-works-to-determine-the-degrees-of-operatorname-arccot-tan.htmlhttps://liberty.io/which-method-works-to-determine-the-degrees-of-operatorname-arccot-tan.htmlHere, I want to convert $\tan(x)$ (let's say $\tan(-37^\circ)=\tan(x)$) to $\cot(x)$. Now, there is one identity$$\tan(\frac{\pi}{2}\pm x)=\mp\cot(x).$$If I choose $\tan(\frac{\pi}{2}+x)$ I will ge......Fri, 04 Sep 2026 19:21:22 -0400https://liberty.io/how-do-i-solve-such-logarithm.htmlhttps://liberty.io/how-do-i-solve-such-logarithm.htmlI understand that $\log_b n = x \iff b^x = n$ But all examples I see is with values that I naturally know how to calculate (like $2^x = 8, x=3$) What if I don't? For example, how do I solve for ......Fri, 04 Sep 2026 19:19:35 -0400https://liberty.io/iterative-trapezoidal-method-for-differential-equations.htmlhttps://liberty.io/iterative-trapezoidal-method-for-differential-equations.htmlI am studying numerical methods for differential equations. I came accros the trapezoidal method in two forms, an explicit and an iterative one. I would like to know the advantages and disadvantage......Fri, 04 Sep 2026 19:18:44 -0400https://liberty.io/solve-the-complex-equation.htmlhttps://liberty.io/solve-the-complex-equation.htmlThe equation is $$z^2 -4z +4+ 2i = 0$$ I know that i am supposed to use $$(a+bi)^2 = a^2 + 2abi + bi^2$$ to solve the equation but i am stuck on how to expand the equation. Can you help out with......Fri, 04 Sep 2026 19:17:29 -0400https://liberty.io/solve-for-x-in-cos-2-sin-1-x-0.htmlhttps://liberty.io/solve-for-x-in-cos-2-sin-1-x-0.htmlSolve $$\cos(2 \sin ^{-1}(- x)) = 0$$ I get the answer $\frac{-1}{ \sqrt2}$ By solving like this \begin{align}2\sin ^{-1} (-x )&= \cos ^{-1} 0\\ 2\sin^{-1}(- x) &= \frac\pi2\\ \sin^{-1}......Fri, 04 Sep 2026 19:13:24 -0400https://liberty.io/how-to-write-x-is-the-set-of-all-positive-integers-greater-than-or-equ.htmlhttps://liberty.io/how-to-write-x-is-the-set-of-all-positive-integers-greater-than-or-equ.htmlHow to write "$X$ is the set of all positive integers greater than or equal to $10$" with mathematical notations? I was thinking $X=\{n:n\ge10\in\mathbb Z^+\}$, but I'm not too sure if t......Fri, 04 Sep 2026 19:03:25 -0400https://liberty.io/proof-for-any-integer-n-with-n-ge-1-the-number-of-permutations-of-a-se.htmlhttps://liberty.io/proof-for-any-integer-n-with-n-ge-1-the-number-of-permutations-of-a-se.htmlI am trying to use mathematical induction to prove the following theorem: For any integer $n$ with $n \ge 1$, the number of permutations of a set with $n$ elements is $n!$. Proof Let $P(n)$ be......Fri, 04 Sep 2026 19:02:18 -0400https://liberty.io/what-is-an-envelope-of-a-function.htmlhttps://liberty.io/what-is-an-envelope-of-a-function.htmlWhat is the definition of an envelope of a function ? For example if we multiply a sinusoid function of certain frequency ($1/f$ < support of bump) with a bump function we get a function whose ...Fri, 04 Sep 2026 18:59:52 -0400https://liberty.io/what-does-2x-p-x-mean.htmlhttps://liberty.io/what-does-2x-p-x-mean.htmlI am learning sets. I have seen preposition like {2x|P(x)}. I wanted to ask what does 2x mean here? I would be thankful if someone could make it simple to understand. Thanks....Fri, 04 Sep 2026 18:51:32 -0400https://liberty.io/reasoning-behind-holes-in-rational-functions.htmlhttps://liberty.io/reasoning-behind-holes-in-rational-functions.htmlI am having some confusion about holes in rational functions. As I'm aware, a hole is where both the numerator and denominator become zero due to some discontinuity. For example, f(x) = (x+1)(x-1......Fri, 04 Sep 2026 18:50:58 -0400https://liberty.io/binomial-probability-of-airline-overbooking.htmlhttps://liberty.io/binomial-probability-of-airline-overbooking.htmlBecause not all airline passengers show up for their reserved seat, an airline sells 125 tickets for a flight that holds 120 passengers. The probability that a passenger does not show up is 0.10, a......Fri, 04 Sep 2026 18:48:39 -0400https://liberty.io/understanding-this-proof-for-sin-a-b-sin-a-cos-b-sin-b-cos-a-from-gelf.htmlhttps://liberty.io/understanding-this-proof-for-sin-a-b-sin-a-cos-b-sin-b-cos-a-from-gelf.htmlWe would like to have a small question concerning this proof for $\sin(a+b)=\sin(a)\cos(b)+\sin(b)\cos(a)$ from Gelfand's Trigonometry Why is it true that $\frac{q}{d}=\frac{a}{d}$. Is it necessar......Fri, 04 Sep 2026 18:39:20 -0400https://liberty.io/sum-of-the-first-n-prime-numbers.htmlhttps://liberty.io/sum-of-the-first-n-prime-numbers.htmlLet $P_i$ denote the i-th prime number. Is there any formula for expressing $$S= \sum_{i=1}^m P_i.$$ We know that there are around $\frac{P_m}{\ln(P_m)}$ prime numbers less than or equal to $P_m......Fri, 04 Sep 2026 18:32:42 -0400https://liberty.io/properties-of-the-euler-totient-function.htmlhttps://liberty.io/properties-of-the-euler-totient-function.htmlWhy is it that the euler totient function has the following condition true based on its definition? $$ \phi(p^k)=p^{k-1}(p-1) = p^k(1-\frac{1}{p}) = p^k-p^{k-1} $$ A proof would be awesome and an ...Fri, 04 Sep 2026 18:29:28 -0400https://liberty.io/the-boundary-is-a-closed-set.htmlhttps://liberty.io/the-boundary-is-a-closed-set.htmlA point $p$ in a metric space $X$ is a boundary point of the set $A$, if any neighbourhood of $p$ has points of both $A$ and $X-A$.Prove that the set of all boundary points of $A$ is closed. My at......Fri, 04 Sep 2026 18:28:56 -0400https://liberty.io/statistical-distance-between-two-points.htmlhttps://liberty.io/statistical-distance-between-two-points.htmlI read the concept of statistical distance and understand somewhat. Given the point $P(x_{1},x_{2})$ in 2-D space, the euclidean distance of point P from origin is given by: $d(O,P) = \sqrt{\frac{......Fri, 04 Sep 2026 18:14:40 -0400https://liberty.io/p-a-given-that-p-a-b-and-p-b-are-known.htmlhttps://liberty.io/p-a-given-that-p-a-b-and-p-b-are-known.htmlGiven that $P(A \mid B)$ is known, and $P(B)$ is known, how can $P(A)$ be calculated? I tried using the definition of conditional probability - $P(A \mid B) = P(A \cap B)/ P(B)$ - to solve for $P(A......Fri, 04 Sep 2026 18:10:31 -0400https://liberty.io/show-that-a-vector-that-is-orthogonal-to-every-other-vector-is-the-zer.htmlhttps://liberty.io/show-that-a-vector-that-is-orthogonal-to-every-other-vector-is-the-zer.htmlI have the following question, and I'd like to get some tips on how to write the proof. I know why it is, but I'm still not so great at writing it mathematically. If $u$ is a vector in $\mathbb{......Fri, 04 Sep 2026 17:52:28 -0400https://liberty.io/what-does-basic-solution-mean.htmlhttps://liberty.io/what-does-basic-solution-mean.htmlLinear programming: basic solution? If the matrix consists of $$\begin{bmatrix}1&-2&0&0&0\\-3&6&1&3&0\\0&0&2&6&-1\end{bmatrix},$$ how is it that th......Fri, 04 Sep 2026 17:44:08 -0400https://liberty.io/2019-amc10a-25-combinatorics-question.htmlhttps://liberty.io/2019-amc10a-25-combinatorics-question.htmlI'm going over some AMC problem. It's 2019 AMC 10A #25. I was going over some solutions and I got stuck in one part. They say: $\frac{(n^2)!}{(n!)^{n+1}}\cdot\frac{n!}{n^2}$ is an integer, if $\fra......Fri, 04 Sep 2026 17:40:44 -0400https://liberty.io/is-a-constant-function-periodic-closed.htmlhttps://liberty.io/is-a-constant-function-periodic-closed.htmlCan we regard a constant function "$f(x)=\text{constant}$" to be a periodic function? If yes, what is its period?...Fri, 04 Sep 2026 17:00:39 -0400https://liberty.io/approximating-the-cosine-by-taylor-polynomial.htmlhttps://liberty.io/approximating-the-cosine-by-taylor-polynomial.htmlLet $f:=\cos(x)$ I'm asked to find for which values of $x$ we can be sure the 4th degree Taylor polynomial will give an error lesser than $\frac{1}{1000}$. Now, $\cos(x)=1-\frac{x^2}{2}+\frac{x^4}......Fri, 04 Sep 2026 16:39:37 -0400https://liberty.io/complete-probability-spaces.htmlhttps://liberty.io/complete-probability-spaces.htmlI am taking a course on Stochastic Analysis, and the introduction of the course covers Measure Theory\ Probability Theory. I am very new to this area of abstract maths and am trying to get to grips......Fri, 04 Sep 2026 16:36:07 -0400https://liberty.io/finite-sum-of-cosecant-squared.htmlhttps://liberty.io/finite-sum-of-cosecant-squared.htmlI need help with that sum $$\sum_{k=1}^N \csc^2\left(\frac{k\pi}{2N}\right)$$ - it is given in many tables as a standard result, but I can't figure it out. I know how to do $\sin^2$ and $\cos^2$, ......Fri, 04 Sep 2026 16:31:15 -0400https://liberty.io/positive-and-negative-integer-that-is-congruent-to-0-mod-5-and-incongr.htmlhttps://liberty.io/positive-and-negative-integer-that-is-congruent-to-0-mod-5-and-incongr.htmlI'm kind of confused by this because I thought 0 mod 5 = 0, and 0 mod 6 = 0 as well. So what's an integer that is congruent to one but not the other?...Fri, 04 Sep 2026 16:02:43 -0400https://liberty.io/proofs-of-the-properties-of-jacobi-symbol.htmlhttps://liberty.io/proofs-of-the-properties-of-jacobi-symbol.htmlThe definition and properties of Jacobi symbol are stated in this article. I don't have a textbook handy containing the proofs of the following properties of Jacobi symbol. It seems to me that not ......Fri, 04 Sep 2026 15:50:29 -0400https://liberty.io/integral-int-0-1-dx-frac-ln-x-ln-2-1-x-ln-1-x-x.htmlhttps://liberty.io/integral-int-0-1-dx-frac-ln-x-ln-2-1-x-ln-1-x-x.htmlI am trying to calculate $$ I:=\int_0^1 dx \frac{\ln x \ln^2(1-x)\ln(1+x)}{x}$$ Note, the closed form is beautiful (yes beautiful) and is given by $$ I=−\frac{3}{8}\zeta_2\zeta_3 -\frac{2}{3}\zeta_......Fri, 04 Sep 2026 15:47:45 -0400https://liberty.io/differentiate-y-e-ax-3.htmlhttps://liberty.io/differentiate-y-e-ax-3.html$y=e^{ax^3}$ What in a problem indicates I should use the laws of logarithm? Here is how I differentiated the function: $\ln(y)=\ln{e^{ax^3}}$ $\ln(y)=ax^3\ln(e)$ $\frac{dy}{dx}\cdot\frac{1}{......Fri, 04 Sep 2026 15:40:47 -0400https://liberty.io/last-digit-on-3-100-duplicate.htmlhttps://liberty.io/last-digit-on-3-100-duplicate.htmlHow to find the last digit on $3^{100}$? Is there any proper method to solve such questions without calculator of course?...Fri, 04 Sep 2026 15:33:45 -0400https://liberty.io/does-zero-curl-imply-a-conservative-field.htmlhttps://liberty.io/does-zero-curl-imply-a-conservative-field.htmlA field that is conservative must have a curl of zero everywhere. However, I was wondering whether the opposite holds for functions continuous everywhere: if the curl is zero, is the field conserva......Fri, 04 Sep 2026 15:33:27 -0400https://liberty.io/evaluate-6-log-8-4-without-using-a-calculator.htmlhttps://liberty.io/evaluate-6-log-8-4-without-using-a-calculator.htmlI am to evaluate $6\log_8(4)$ without a calculator. The answer is provided as 4 but I cannot see how to arrive at 4. Ignoring the 6 at the beginning of the expression, $log_8(4)$ can be written as ......Fri, 04 Sep 2026 15:32:05 -0400https://liberty.io/solve-simple-xor-equation.htmlhttps://liberty.io/solve-simple-xor-equation.htmlI have to solve few simple equation like this: $$ 117 = 86 \oplus x $$ I don't know how to move x to one side and calculate its value. How to do it?...Fri, 04 Sep 2026 15:30:59 -0400https://liberty.io/what-is-the-039-implicit-function-theorem-039.htmlhttps://liberty.io/what-is-the-039-implicit-function-theorem-039.htmlPlease give me an intuitive explanation of 'implicit function theorem'. I read some bits and pieces of information from some textbook, but they look too confusing, especially I do not understand wh......Fri, 04 Sep 2026 15:18:22 -0400https://liberty.io/what-does-the-derivative-of-area-with-respect-to-length-signify.htmlhttps://liberty.io/what-does-the-derivative-of-area-with-respect-to-length-signify.htmlSuppose that we have a square sheet of edge length $L$. Its area $A=L^2$. Differentiating $A$ w.r.t. L, we get $$\dfrac{dA}{dL}=2L$$ I do understand what it means to differentiate, graphicall......Fri, 04 Sep 2026 15:17:45 -0400https://liberty.io/why-does-grapher-a-mac-application-plot-the-3d-inequality-x-1-using-sp.htmlhttps://liberty.io/why-does-grapher-a-mac-application-plot-the-3d-inequality-x-1-using-sp.htmlGrapher (sometimes called Grapher.app) is a mathematics graphing application that comes bundled with Mac computers (see https://en.wikipedia.org/wiki/Grapher). I was recently playing around with G......Fri, 04 Sep 2026 15:16:21 -0400https://liberty.io/what-is-a-combined-function-and-what-factors-affect-its-domain.htmlhttps://liberty.io/what-is-a-combined-function-and-what-factors-affect-its-domain.htmlCan someone please explain to me what a combined function is and how it is different from a composite function. From my understanding a combined function would be $(f+g)(x) = f(x)+g(x)$ and a com......Fri, 04 Sep 2026 15:03:41 -0400https://liberty.io/example-of-a-subgroup-that-is-not-normal.htmlhttps://liberty.io/example-of-a-subgroup-that-is-not-normal.htmlCan you kindly provide an example of a subgroup that is not normal? I have been told many times that, for coset multiplication to be defined, the subgroup must be normal. I have seen the proof and ...Fri, 04 Sep 2026 15:02:29 -0400https://liberty.io/how-to-find-the-dual-cone.htmlhttps://liberty.io/how-to-find-the-dual-cone.htmlThe similar question is here , but I do not see the desired answer. Assume a cone $K=\{(x,y) |\ x+y=0\}$, find the dual cone of $K$. The definition of dual cone is here: $K^*=\{y|x^{T}y\geq0, \fora......Fri, 04 Sep 2026 14:56:32 -0400https://liberty.io/square-root-of-a-binary-number-and-beyond.htmlhttps://liberty.io/square-root-of-a-binary-number-and-beyond.htmlhow to find (efficiently) binary expansion of a square root of a number which is given as a binary number. Is there any general method to find the n-nary expansion of m-th root of a number which is ...Fri, 04 Sep 2026 14:56:32 -0400https://liberty.io/what-is-the-best-book-to-learn-probability.htmlhttps://liberty.io/what-is-the-best-book-to-learn-probability.htmlQuestion is quite straight... I'm not very good in this subject but need to understand at a good level....Fri, 04 Sep 2026 14:55:33 -0400https://liberty.io/multivariable-chain-rule-formula-doesn-039-t-make-sense-to-me.htmlhttps://liberty.io/multivariable-chain-rule-formula-doesn-039-t-make-sense-to-me.htmlConsider $z=f(x(t),y(t))$, then its chain rule formula is: $\frac{dz}{dt} = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt}$ What I cannot make sense of, a......Fri, 04 Sep 2026 14:46:48 -0400https://liberty.io/do-matrices-ab-and-ba-have-the-same-minimal-and-characteristic-polynom.htmlhttps://liberty.io/do-matrices-ab-and-ba-have-the-same-minimal-and-characteristic-polynom.htmlLet $ A, B $ be two square matrices of order $n$. Do $ AB $ and $ BA $ have same minimal and characteristic polynomials? I have a proof only if $ A$ or $ B $ is invertible. Is it true for all cases?...Fri, 04 Sep 2026 14:29:30 -0400https://liberty.io/definition-of-measure-preserving-why-inverse-image.htmlhttps://liberty.io/definition-of-measure-preserving-why-inverse-image.htmlIn the definition of measure-preserving dynamical system, the crucial equation is $$ \mu \left(T^{-1} \left(A\right)\right) = \mu\left(A\right) . $$ Why is it not the seemingly more natural $......Fri, 04 Sep 2026 14:13:40 -0400https://liberty.io/realification-and-complexification-of-vector-spaces.htmlhttps://liberty.io/realification-and-complexification-of-vector-spaces.htmlI am interested in a good comprehensive resource on realification and complexification of vector spaces over the reals or complexes (and the interplay of these structures on the 'same' space in gen......Fri, 04 Sep 2026 14:06:30 -0400https://liberty.io/finding-derivative-of-frac-1-sqrt-x-2-using-only-the-definition-of-the.htmlhttps://liberty.io/finding-derivative-of-frac-1-sqrt-x-2-using-only-the-definition-of-the.htmlI need to find $\frac{d}{dx}(\frac{1}{\sqrt{x+2}})$ only using the basic definition of the derivative $f'(x)=\lim_{h \to 0}\frac{f(x+h)-f(x)}{h}$, but I'm not having any luck with the algebra. Any ......Fri, 04 Sep 2026 14:00:56 -0400https://liberty.io/probability-of-getting-3-cards-in-the-same-suit-from-a-deck.htmlhttps://liberty.io/probability-of-getting-3-cards-in-the-same-suit-from-a-deck.htmlWhen three cards are randomly selected at a time from a standard deck of 52 playing cards, what is the probability that all of these three cards are in the same suit (heart, diamond, spade, o......Fri, 04 Sep 2026 13:56:34 -0400https://liberty.io/how-do-you-find-an-angle-between-two-points-on-the-edge-of-a-circle.htmlhttps://liberty.io/how-do-you-find-an-angle-between-two-points-on-the-edge-of-a-circle.htmlI have two points on the circumference of circle, and I also know the center of the circle. I want to calculate the angle between those two points which are on the circumference of circle. Is this ...Fri, 04 Sep 2026 13:55:51 -0400https://liberty.io/definition-of-convex-function.htmlhttps://liberty.io/definition-of-convex-function.htmlSpivak's book states first this definition of Convex Function: Definition 1: A function $f$ is convex on an interval, if for all $a$ and $b$ in the interval, the line segment joining $(a, f(a))......Fri, 04 Sep 2026 13:52:40 -0400https://liberty.io/user-andrew-stacey-mathematics-stack-exchange.htmlhttps://liberty.io/user-andrew-stacey-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Fri, 04 Sep 2026 13:52:28 -0400https://liberty.io/finding-n-value-basic-statistics.htmlhttps://liberty.io/finding-n-value-basic-statistics.htmlI have some practice homework questions. I did the first one I will go over the steps please tell me If I am doing it right. a) As mentioned earlier, it is claimed that 70% of households in Ontar......Fri, 04 Sep 2026 13:49:06 -0400https://liberty.io/given-q-and-cos-q-pi-to-be-rational-find-all-possible-values-of-cos-q-.htmlhttps://liberty.io/given-q-and-cos-q-pi-to-be-rational-find-all-possible-values-of-cos-q-.htmlQuestion: Given $q$ and $\cos(q\pi)$ to be rational, find all possible values of $\cos(q\pi)$ After some trial and error I think the possible values are $\{0,\pm 1,\pm 1/2 \}$. I could show that $......Fri, 04 Sep 2026 13:49:04 -0400https://liberty.io/proving-that-a-point-is-a-midpoint.htmlhttps://liberty.io/proving-that-a-point-is-a-midpoint.htmlI'm trying to solve this using elementary geometry. Let $\triangle ABC$ be a triangle and consider the squares $CBFG$ and $ACDE$. Draw $\overline{EF}$ and let $M$ be its midpoint. Draw the perpendi......Fri, 04 Sep 2026 13:39:51 -0400https://liberty.io/conditions-for-cyclic-quotient-group.htmlhttps://liberty.io/conditions-for-cyclic-quotient-group.htmlLet $G$ be an arbitrary finite group and $H$ a normal subgroup. What are some good conditions on $H$ that make the quotient $G/H$ cyclic? I want to avoid any further restriction on $G$....Fri, 04 Sep 2026 13:39:08 -0400https://liberty.io/modulo-question.htmlhttps://liberty.io/modulo-question.htmlI thought you could only mod positive numbers but then I saw this and became confused...How does this even work ? how can you have negatives? $$\begin{align*} -8 &\equiv 7 \pmod{5}\\ 2 &\...Fri, 04 Sep 2026 13:37:50 -0400https://liberty.io/inscribed-rhombus.htmlhttps://liberty.io/inscribed-rhombus.htmlCan a rhombus that is NOT a square be inscribed in a circle? A quadrilaterals opposite angles must add up to 180 in order to be inscribed in a circle, but a rhombuses opposite angles are equal and ......Fri, 04 Sep 2026 13:37:43 -0400https://liberty.io/what-is-the-symbol-for-undefined.htmlhttps://liberty.io/what-is-the-symbol-for-undefined.htmlJust how there is a symbol for no real solutions (the empty set) and there is DNE for does not exist, what symbol or abbreviation would you use for undefined?...Fri, 04 Sep 2026 13:36:38 -0400https://liberty.io/puzzle-where-did-the-extra-dollar-come-from-duplicate.htmlhttps://liberty.io/puzzle-where-did-the-extra-dollar-come-from-duplicate.htmlPossible Duplicate: Riddle (simple arithmetic problem/illusion) I have this small story (sorry about my bad English) : One man (M) took 25 dollar from man (A), and another 25 dollar from man (......Fri, 04 Sep 2026 13:24:15 -0400https://liberty.io/unclear-about-what-the-diagonal-relation-means.htmlhttps://liberty.io/unclear-about-what-the-diagonal-relation-means.htmlI am reading through Skornjakov's Elements of Lattice Theory (1977), and am having trouble understanding what is meant by a diagonal relation. I am also unclear about how the diagonal differs from ......Fri, 04 Sep 2026 13:15:30 -0400https://liberty.io/solve-the-ode-y-039-039-y-039-y-2.htmlhttps://liberty.io/solve-the-ode-y-039-039-y-039-y-2.htmlSolve the ODE $$ y''+ y' = y^2$$ I tried to solve, but I don't know any algorithm except for numerical methods....Fri, 04 Sep 2026 13:07:39 -0400https://liberty.io/how-to-solve-x-5-2x-6-1.htmlhttps://liberty.io/how-to-solve-x-5-2x-6-1.html$|x-5|=|2x+6|-1$. The answer is $0$ or $-12$, but how would I solve it by algebraically solving it as opposed to sketching a graph? $|x-5|=|2x+6|-1\\ (|x-5|)^2=(|2x+6|-1)^2\\ ...\\ 9x^4+204x^3+1......Fri, 04 Sep 2026 13:03:27 -0400https://liberty.io/derivative-of-an-exponentially-weighted-moving-average.htmlhttps://liberty.io/derivative-of-an-exponentially-weighted-moving-average.htmlIt has been a while since my university math courses, so let me apologize right off the bat... I'm using GSL to perform non-linear regression analysis and am mostly happy with the outcome, however......Fri, 04 Sep 2026 13:02:34 -0400https://liberty.io/what-do-trivial-and-non-trivial-solution-of-homogeneous-equations-mean.htmlhttps://liberty.io/what-do-trivial-and-non-trivial-solution-of-homogeneous-equations-mean.htmlSuppose I have system of 3 equations $$a_1x+b_1y+c_1z=0$$ $$a_2x+b_2y+c_2z=0$$ $$a_3x+b_3y+c_3z=0$$ and cofficient matrix $A=\begin{equation} \begin{pmatrix} a_1 & b_1 & c_1 \\ a_2 & b_......Fri, 04 Sep 2026 13:02:04 -0400https://liberty.io/find-the-maximum-displacement-given-trajectory.htmlhttps://liberty.io/find-the-maximum-displacement-given-trajectory.htmlI'm trying to find the maximum displacement of a object given the trajectory. The trajectory is given by the equation: $$p(t)=-11.79t^2+25.9t+4.35$$ Looking around online, the closest article I've ......Fri, 04 Sep 2026 12:57:12 -0400https://liberty.io/why-negating-universal-quantifier-gives-existential-quantifier.htmlhttps://liberty.io/why-negating-universal-quantifier-gives-existential-quantifier.htmlNegating a universal quantifier gives the existential quantifier, and vice versa: $\neg \forall x = \exists x \neg \\ \neg \exists x = \forall x \neg $ Why is this, and is there a proof for it (i......Fri, 04 Sep 2026 12:51:02 -0400https://liberty.io/range-two-039-s-complement.htmlhttps://liberty.io/range-two-039-s-complement.htmlThe most used formula's to calculate the range of numbers in a two's complement system are + $2^{n-1}-1$ for the highest number and $-2^{n-1}$ for the lowest number. The problem is that this only ......Fri, 04 Sep 2026 12:45:17 -0400https://liberty.io/dual-of-a-positive-semidefinite-cone.htmlhttps://liberty.io/dual-of-a-positive-semidefinite-cone.htmlThe PSD cone is the set of all positive semidefinite matrices. The dual is the set of all matrices $A$ such that tr($A^T X$) $\geq 0$ for all positive semidefinite matrices $X$. How to prove that......Fri, 04 Sep 2026 12:44:36 -0400https://liberty.io/finding-velocity-from-a-table.htmlhttps://liberty.io/finding-velocity-from-a-table.htmlI have a homework question I am seeking an alternative solution to. Basically, the question is... "The table provided below shows the position of a particle S, at several times, t. as the particle......Fri, 04 Sep 2026 12:42:03 -0400https://liberty.io/find-a-particular-solution-for-the-differential-equation-by-the-method.htmlhttps://liberty.io/find-a-particular-solution-for-the-differential-equation-by-the-method.htmlFind a particular solution for the differential equation by the method of undetermined coefficients. $$2y'' - 16y' + 32y = -e^{4x}$$ Also, find the general solution of this equation. The step......Fri, 04 Sep 2026 12:41:25 -0400https://liberty.io/how-well-does-probability-work-in-the-real-world.htmlhttps://liberty.io/how-well-does-probability-work-in-the-real-world.htmlWhen I was first introduced to probability at the age of about 13, me and my classmates always used to ridicule the subject in a most immature manner. We used to take a deck of well shuffled cards ......Fri, 04 Sep 2026 12:38:41 -0400https://liberty.io/what-is-the-centroid-of-a-hollow-spherical-cap.htmlhttps://liberty.io/what-is-the-centroid-of-a-hollow-spherical-cap.htmlI have a unit hollow sphere which I cut along a diameter to generate two equivalent hollow hemispheres. I place one of these hemispheres on an (x,y) plane, letting it rest on the circular planar f......Fri, 04 Sep 2026 12:32:21 -0400https://liberty.io/symbol-for-the-shape-of-a-matrix.htmlhttps://liberty.io/symbol-for-the-shape-of-a-matrix.htmlI'm writing a technical document and need to describe some mathematics involving the shape of a matrix. I realize that I can write $X_{m\times n}$ and someone can simply see the dimensions. I'm ...Fri, 04 Sep 2026 12:31:07 -0400https://liberty.io/example-of-strictly-concave-functions.htmlhttps://liberty.io/example-of-strictly-concave-functions.htmlI'm learning concave/convex functions. Wikipedia article provides us with example functions of concave or convex. However, it does not provide strictly concave function. Do you have some examples of ...Fri, 04 Sep 2026 12:15:14 -0400https://liberty.io/what-does-the-d-represent-in-the-equation-of-a-plane-ax-by-cz-d-0.htmlhttps://liberty.io/what-does-the-d-represent-in-the-equation-of-a-plane-ax-by-cz-d-0.htmlLook at the following problem: Consider the plane $\alpha$ defined by $x-2y+z+3=0$ and $A(0;0;2)$. Write an equation of the plane that is parallel to $\alpha$ and goes through A. My book ...Fri, 04 Sep 2026 12:06:06 -0400https://liberty.io/questions-about-relations-on-directed-and-possibly-cyclical-graphs.htmlhttps://liberty.io/questions-about-relations-on-directed-and-possibly-cyclical-graphs.htmlLet $(P, E)$ be a directed graph with $P$ being the set of nodes and $E \subset P^2$ the set of directed edges. It is well-known that the transitive closure of $E$ is a preorder $\leq$, which is more ...Fri, 04 Sep 2026 11:58:21 -0400https://liberty.io/finding-a-b-c-d-in-a-quartic-expression.htmlhttps://liberty.io/finding-a-b-c-d-in-a-quartic-expression.htmlLet $p(x)=x^4+ax^3+bx^2+cx+d$ where a,b,c,d are constants. If $p(1)=10$, $p(2)=20$, $p(3)=30$, compute $\frac {p(12)+p(-8)}{10}$. I have tried so far. \begin{align} a+b+c+d=&9\\8a+4b+2c+d=&amp......Fri, 04 Sep 2026 11:51:15 -0400