https://liberty.ioCurated star highlights with fresh energy.en-usThu, 13 Aug 2026 11:32:06 -0400https://liberty.io/domain-of-rational-functions.htmlhttps://liberty.io/domain-of-rational-functions.htmlI have the following definition for rational functions: "Let $p(x), q(x)$ be polynomials and define $D=\{x\in \mathbb{R} : q(x) \neq 0 \}$ Then, the function $$f:D \rightarrow \mathbb{R}$$ $$f......Thu, 13 Aug 2026 15:23:56 -0400https://liberty.io/what-are-the-components-in-the-ito-process.htmlhttps://liberty.io/what-are-the-components-in-the-ito-process.htmlWe know that if W(t) is a Brownian Motion, then the Ito Process is given by, X(t)= X(0) + $\int_0^t \Delta(u)dW(u)$ + $\int_0^t \Phi(u)du$ Could someone explain briefly what the two components a......Thu, 13 Aug 2026 15:23:33 -0400https://liberty.io/what-is-undecidability.htmlhttps://liberty.io/what-is-undecidability.htmlWhat does it mean that some problem is undecidable? For instance the halting problem. Does it mean that humans can never invent a new technique that always decides whether a turing machine will h......Thu, 13 Aug 2026 15:19:41 -0400https://liberty.io/finding-moment-of-inertia-of-a-cone-why-can-039-t-i-integrate-like-thi.htmlhttps://liberty.io/finding-moment-of-inertia-of-a-cone-why-can-039-t-i-integrate-like-thi.htmlI have a cone (vertex at $(x,y,z)=0$) with height $H$ and radius $a$ (radius of the base). Here is a picture: I want to find the moment of inertia around the $z$-axis using calculus. I know the in......Thu, 13 Aug 2026 15:15:04 -0400https://liberty.io/equivalent-systems-of-linear-equation.htmlhttps://liberty.io/equivalent-systems-of-linear-equation.htmlI've just begun to re-learn linear algebra because is so important, the book that I chose is naturally the Hoffman's for a lot of reason. Well, In the first chapter I'm stuck with the following, ...Thu, 13 Aug 2026 15:13:56 -0400https://liberty.io/real-valued-analytic-function-defined-on-a-connected-set-is-constant.htmlhttps://liberty.io/real-valued-analytic-function-defined-on-a-connected-set-is-constant.htmlLet $G$ be a connected set and $f : G \rightarrow \mathbb{C}$ a real valued analytic function. Prove that $f$ is constant. My idea to prove the result is to prove a subset $A \neq \varnothing$ of......Thu, 13 Aug 2026 15:13:38 -0400https://liberty.io/four-non-coplanar-points-in-mathbb-r-3-determine-an-unique-sphere.htmlhttps://liberty.io/four-non-coplanar-points-in-mathbb-r-3-determine-an-unique-sphere.htmlShow that four non-coplanar points in $\mathbb{R}^3$ determine an unique sphere. I have no idea how to solve this exercise. Thank you for your help....Thu, 13 Aug 2026 15:07:00 -0400https://liberty.io/finding-the-value-of-k-duplicate.htmlhttps://liberty.io/finding-the-value-of-k-duplicate.html$$π1:( 1+k , -3 , 6 )$$ $$π2: ( 1 , 5+k , 3k)$$ Does anyone know the values of k that would make these parallel and perpendicular, ive been trying for hours but nothing seems to work The original ...Thu, 13 Aug 2026 15:05:55 -0400https://liberty.io/no-mode-or-all-mode.htmlhttps://liberty.io/no-mode-or-all-mode.htmlLet's say I have a data set:- $${1,2,3,4,5}$$ I decide to find the mode though it would not be an appropriate measure of central tendency. But still........ Now this is an exceptional case and for ...Thu, 13 Aug 2026 15:05:45 -0400https://liberty.io/is-a-set-contained-in-its-cartesian-product-with-itself.htmlhttps://liberty.io/is-a-set-contained-in-its-cartesian-product-with-itself.htmlGiven the Cartesian product $X = A \times A$, is $A \subset X$?...Thu, 13 Aug 2026 14:38:22 -0400https://liberty.io/find-the-derivative-dy-dx-given-some-value-for-x.htmlhttps://liberty.io/find-the-derivative-dy-dx-given-some-value-for-x.htmlI know how to find the derivative of a function by itself given a function. I can use multiple rules to determine the derivative. However in a situation where I must find the derivative when the x......Thu, 13 Aug 2026 14:10:51 -0400https://liberty.io/how-to-prove-if-something-is-a-linear-transformation.htmlhttps://liberty.io/how-to-prove-if-something-is-a-linear-transformation.html(I think) My textbook says something is a linear transformation if $L(ax) = aL(x)$ $L(x+y) = L(x) + L(y)$ $L(x) = A(x)$ But a lot of sites I've been on haven't proved these 3 things, so I just w......Thu, 13 Aug 2026 13:57:13 -0400https://liberty.io/proof-by-lewis-carroll-that-an-urn-cannot-contain-two-balls-duplicate.htmlhttps://liberty.io/proof-by-lewis-carroll-that-an-urn-cannot-contain-two-balls-duplicate.htmlI am self-learning probability theory and solving some problems from the book Probability and Random processes by Grimmett and Stirzaker. Exercise 1.4.4 poses a nice problem. I would like someone to ...Thu, 13 Aug 2026 13:54:13 -0400https://liberty.io/time-complexity-of-a-modulo-operation.htmlhttps://liberty.io/time-complexity-of-a-modulo-operation.htmlI am trying to prove that if $p$ is a decimal number having $m$ digits, then $p \bmod q$ can be performed in time $O(m)$ (at least theoretically), if $q$ is a prime number. How do I go about this? ......Thu, 13 Aug 2026 13:28:46 -0400https://liberty.io/subcoalgebra-generated-by-an-element.htmlhttps://liberty.io/subcoalgebra-generated-by-an-element.htmlLet $(A, \Delta, \epsilon)$ be a coalgebra and $f\in A$. What is the subcoalgebra generated by $f$ like? For example, if $A$ is the dual of the quaternions $\mathbb{H}$ (which is $\mathbb{R}$-alge......Thu, 13 Aug 2026 13:27:27 -0400https://liberty.io/a-basic-property-of-slowly-varying-functions.htmlhttps://liberty.io/a-basic-property-of-slowly-varying-functions.htmlIt seems that a fundamental property of a slowly varying function is that for all $\delta > 0$ $$ \lim_{x\to \infty} L(x) \, x^{-\delta} = 0.$$ How to prove this? The only book I found (that was......Thu, 13 Aug 2026 13:26:12 -0400https://liberty.io/weierstrass-m-test-proof.htmlhttps://liberty.io/weierstrass-m-test-proof.htmlLet (X,d) be a metric space. For each n $\epsilon$ N let $g_n$:X$\rightarrow$R be a continuous function. Let ($a_n$) be a sequence of positive real numbers such that the series $\sum_{n=1}^\infty a......Thu, 13 Aug 2026 13:24:53 -0400https://liberty.io/what-exactly-is-a-0-form.htmlhttps://liberty.io/what-exactly-is-a-0-form.htmlFrom what I understand, a k-form in the real numbers is essentially a mapping $\mathbb{R^k} \rightarrow \mathbb{R}$, but I can't seem to find a corresponding definition for a "0-form". Wikipedia se......Thu, 13 Aug 2026 13:24:21 -0400https://liberty.io/how-to-find-an-eulerian-circuit-in-a-complicated-graph.htmlhttps://liberty.io/how-to-find-an-eulerian-circuit-in-a-complicated-graph.htmlConsider the following graph G: (a) Give a decomposition of G into cycles. (b) Find an Eulerian circuit in G. This is a very complicated graph and each time I am trying to find the solution I am ...Thu, 13 Aug 2026 13:24:08 -0400https://liberty.io/how-do-i-convert-the-distance-between-two-lat-long-points-into-feet-me.htmlhttps://liberty.io/how-do-i-convert-the-distance-between-two-lat-long-points-into-feet-me.htmlI've been reading around the net and everything I find is really confusing. I just need a formula that will get me 95% there. I have a tool that outputs the distance between two lat/long points. P......Thu, 13 Aug 2026 13:18:52 -0400https://liberty.io/interpretation-of-log-differences.htmlhttps://liberty.io/interpretation-of-log-differences.htmlI have a very simple question. I am confused about the interpretation of log differences. Here a simple example: $$\log(2)-\log(1)=.3010$$ With my present understanding, I would interpret the resul......Thu, 13 Aug 2026 13:17:30 -0400https://liberty.io/how-to-rigorously-find-range-of-a-function.htmlhttps://liberty.io/how-to-rigorously-find-range-of-a-function.htmlWhat is a rigorous method/mechanism to find the range of a function $f(x)$? Is it acceptable to find $f^{-1}(x)$ and then its domain? I understand that $f^{-1}(x)$ does not always exist (at least not ...Thu, 13 Aug 2026 13:14:24 -0400https://liberty.io/determine-the-derivative-of-the-reciprocal-function-only-with-the-prod.htmlhttps://liberty.io/determine-the-derivative-of-the-reciprocal-function-only-with-the-prod.htmlFollowing is given: $( \alpha )$ the derivative of a inverse function: $$\left(\frac{1}{f}\right)' = -\frac{f'}{f^2}$$ $( \beta )$ The the product rule: $$({f}*g) = ({f'}*{g})+({f}*{g'})$$ Follo......Thu, 13 Aug 2026 13:04:56 -0400https://liberty.io/how-to-calculate-csc-2-85-in-calculator.htmlhttps://liberty.io/how-to-calculate-csc-2-85-in-calculator.htmlIn my calculator (TI-84), there are only $sin, cos,$ and $tan$ commands (and inverse sin, inverse cos, inverse tan). I had a question that was as follows: Calculate $csc(2.85)$ in which I was perm......Thu, 13 Aug 2026 12:55:56 -0400https://liberty.io/find-the-amplitude-of-the-steady-state-solution.htmlhttps://liberty.io/find-the-amplitude-of-the-steady-state-solution.htmlHeave motion of a point energy absorber in the form of a buoy due t incident wave of frequency $w$ and amplitude $a$ is described by $(m+m_a)y'' + (c_h + c_e)y' +ky = F_i(t)$ $F_i(t)=aF(w)\cos(wt......Thu, 13 Aug 2026 12:46:38 -0400https://liberty.io/how-many-proper-nontrivial-subgroups-do-d5-have.htmlhttps://liberty.io/how-many-proper-nontrivial-subgroups-do-d5-have.htmlDo I have to find out every element in D5 and draw a table to find out subgroups? I know how to find out every single element in D5, but can't think of how to find proper nontrivial subgroups...Thu, 13 Aug 2026 12:46:19 -0400https://liberty.io/conic-section-equation-of-section-plane-in-3d.htmlhttps://liberty.io/conic-section-equation-of-section-plane-in-3d.htmlAll of us know general equation for any conic section: $$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$$ If we'll change of parameters of this equation we get different type of conic section. If we imagine ...Thu, 13 Aug 2026 12:41:04 -0400https://liberty.io/is-a-temperature-change-in-celsius-larger-than-a-temperature-change-in.htmlhttps://liberty.io/is-a-temperature-change-in-celsius-larger-than-a-temperature-change-in.htmlIs a temperature change in Celsius larger than a temperature change in Fahrenheit? The teacher offers this second way of thinking about the question. If the temperature increases by 1 degree Ce......Thu, 13 Aug 2026 12:40:47 -0400https://liberty.io/simple-trig-equations-why-is-it-wrong-to-cancel-trig-terms.htmlhttps://liberty.io/simple-trig-equations-why-is-it-wrong-to-cancel-trig-terms.htmlIn the following problem, I first did it using a cancellation of $sin^2\theta$, working shown below, which gave the wrong answer. Having looked at the question again, I saw it could be solved by ...Thu, 13 Aug 2026 12:40:30 -0400https://liberty.io/specific-strings-that-cannot-be-found-in-pi.htmlhttps://liberty.io/specific-strings-that-cannot-be-found-in-pi.htmlConsidering that the decimals of PI aren't like prime numbers (random), they are pseudo-random (can be calculated through a formula or an infinite series, yet it is calculable). Keeping this in min......Thu, 13 Aug 2026 12:38:24 -0400https://liberty.io/finding-the-average-velocity-and-instantaneous-velocity.htmlhttps://liberty.io/finding-the-average-velocity-and-instantaneous-velocity.htmlA water balloon is launched upward with an initial velocity of $40 ft/s$. the height h at time $t$ is given $h(t)= 16t^2+40t$ a. find the avg. velocity of the balloon between $t=1$ and $t=2$ secon......Thu, 13 Aug 2026 12:37:32 -0400https://liberty.io/how-to-find-the-common-base-of-terms-in-an-expression.htmlhttps://liberty.io/how-to-find-the-common-base-of-terms-in-an-expression.htmlI'm teaching myself basic algebra from a book and am stuck on a question. In the current section it is about expressing numbers as powers of the same base. So $9$ maybe expressed as $3^2$. Another......Thu, 13 Aug 2026 12:32:40 -0400https://liberty.io/does-the-existence-of-a-free-variable-mean-there-are-infinitely-many-s.htmlhttps://liberty.io/does-the-existence-of-a-free-variable-mean-there-are-infinitely-many-s.htmllets say I have an augmented matrix $$\begin{bmatrix} 2&0&3\\ 0&1&0\\ 2&1&3 \end{bmatrix}$$ The question is: Is the matrix consistent with a unique solution, inconsistent,......Thu, 13 Aug 2026 12:32:11 -0400https://liberty.io/are-all-symmetric-matrices-invertible.htmlhttps://liberty.io/are-all-symmetric-matrices-invertible.htmlIs any symmetric matrix ​​invertible? I'm trying to prove this theoretical question, but I don't know what I need to do. I apologize for the simple question, but I'm in doubt and need clarification....Thu, 13 Aug 2026 12:01:44 -0400https://liberty.io/continuous-uniformly.htmlhttps://liberty.io/continuous-uniformly.htmlI need help to demonstrate the following or if you find a function I would appreciate it, I have already looked for many functions to see if any one fulfills it but none works for me and I can't th......Thu, 13 Aug 2026 11:47:05 -0400https://liberty.io/finding-mean-and-standard-deviation-of-normal-distribution-given-2-poi.htmlhttps://liberty.io/finding-mean-and-standard-deviation-of-normal-distribution-given-2-poi.htmlIn general, how do do you calculate the mean and standard deviation of a normal distribution given 2 values on the distribution with their respective probabilities? For Example: Suppose that the a......Thu, 13 Aug 2026 11:31:35 -0400https://liberty.io/comparing-the-growth-rates.htmlhttps://liberty.io/comparing-the-growth-rates.htmlHow can I go about comparing the growth rate of the following functions? $$\sqrt n,\quad 10^n,\quad n^{1.5},\quad 2^{\sqrt{\log n}},\quad n^{5/3}.$$ I am looking for a more generic answer on how ......Thu, 13 Aug 2026 11:26:49 -0400https://liberty.io/what-is-the-math-behind-the-game-spot-it.htmlhttps://liberty.io/what-is-the-math-behind-the-game-spot-it.htmlI just purchased the game Spot It. As per this site, the structure of the game is as follows: Game has 55 round playing cards. Each card has eight randomly placed symbols. There are a total of 50 ...Thu, 13 Aug 2026 11:26:06 -0400https://liberty.io/set-notation-inclusion-and-proper-inclusion.htmlhttps://liberty.io/set-notation-inclusion-and-proper-inclusion.htmlBy definition: If$ \ A$ and $B\ $are sets and every element of$ \ A\ $ is also an element of $ \ B \ $then we can say $ \ A\ $is a subset of $ \ B, \ $and denote this by $ \ A\subset B\ $or $ \ A\...Thu, 13 Aug 2026 11:18:26 -0400https://liberty.io/first-order-logic-vs-second-order-logic.htmlhttps://liberty.io/first-order-logic-vs-second-order-logic.htmlWikipedia describes the first-order vs. second-order logic as follows: First-order logic uses only variables that range over individuals (elements of the domain of discourse); second-order logic......Thu, 13 Aug 2026 11:18:17 -0400https://liberty.io/the-coefficient-of-x-100-in-the-expansion-of-1-x-3.htmlhttps://liberty.io/the-coefficient-of-x-100-in-the-expansion-of-1-x-3.htmlPlease solve it and tell me the technique so that I can solve it in examination in multiple choice questions....Thu, 13 Aug 2026 11:17:31 -0400https://liberty.io/unnecessary-condition-for-alternating-series-test.htmlhttps://liberty.io/unnecessary-condition-for-alternating-series-test.htmlGiven a series $$ \sum^\infty_{n=0} (-1)^n b_n$$ the Alternating Series test states that if $b_n$ is convergent if 1) $b_n$ is positive 2) $b_n$ is monotonically decreasing 3) $\lim_{n\to\infty}......Thu, 13 Aug 2026 11:16:21 -0400https://liberty.io/calculate-an-integral-with-riemann-sum.htmlhttps://liberty.io/calculate-an-integral-with-riemann-sum.htmlWe know that Riemann sum gives us the following formula for a function $f\in C^1$: $$\lim_{n\to \infty}\frac 1n\sum_{k=0}^n f\left(\frac kn\right)=\int_0^1f(x) dx.$$ I am looking for an example......Thu, 13 Aug 2026 11:09:59 -0400https://liberty.io/deriving-sin-ix-i-sinh-x.htmlhttps://liberty.io/deriving-sin-ix-i-sinh-x.htmlDerive $\sin{ix}=i\sinh{x}$ from $(5)$. What is $\sin{i}$? $$\cos{x}=\frac{1}{2}\left(e^{ix}+e^{-ix}\right)\quad\text{and}\quad\sin{x}=\frac{1}{2i}\left(e^{ix}-e^{-ix}\right)\tag 5$$ We have $$\......Thu, 13 Aug 2026 11:02:33 -0400https://liberty.io/what-are-the-uses-of-euler-039-s-number-e.htmlhttps://liberty.io/what-are-the-uses-of-euler-039-s-number-e.htmlPeople make such a big deal of the number $e$. I do not get why it is so important, other than the fact that $\ln(x)=\log_e(x)$. People say it is used all over mathematics and such, but they never ......Thu, 13 Aug 2026 11:01:38 -0400https://liberty.io/user-jialin-zhang-mathematics-stack-exchange.htmlhttps://liberty.io/user-jialin-zhang-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Thu, 13 Aug 2026 10:56:01 -0400https://liberty.io/find-the-domain-of-the-given-function.htmlhttps://liberty.io/find-the-domain-of-the-given-function.htmlI have the function $f(x)=\cos^{-1}(\frac{1}{2\cos(x)})$ and I have to find its domain. What I know is that the domain of $\cos^{-1}(x)$ is $[-1,1]$ so I think that $\frac{1}{2\cos(x)}$ should be at ...Thu, 13 Aug 2026 10:55:27 -0400https://liberty.io/what-is-the-time-complexity-of-euclid-039-s-algorithm-upper-bound-lowe.htmlhttps://liberty.io/what-is-the-time-complexity-of-euclid-039-s-algorithm-upper-bound-lowe.htmlI looked it up online in many sites but none give a clear answer. They all give a lot of complicated mathematical stuff which is not only hard for me to grasp but also irrelevant as I simply want t......Thu, 13 Aug 2026 10:53:21 -0400https://liberty.io/how-to-prove-that-n-frac-1-3-is-not-a-polynomial.htmlhttps://liberty.io/how-to-prove-that-n-frac-1-3-is-not-a-polynomial.htmlI'm reading Barbeau's Polynomials, there's an exercise: How to prove that $n^{\frac{1}{3}}$ is not a polynomial? I've made this question and with the first answer as an example, I guess I should ...Thu, 13 Aug 2026 10:48:38 -0400https://liberty.io/solve-for-a-and-b-of-this-piecewise-function-such-that-the-function-f-.htmlhttps://liberty.io/solve-for-a-and-b-of-this-piecewise-function-such-that-the-function-f-.htmlConsider the function defined by \begin{align*} f(x) = \begin{cases} ax^{2} + x - b, & x < 2\\ ax+b, & 2\leq x \leq 5\\ 2ax - 7, & x > 5. \end{cases} \end{align*} I was able to ......Thu, 13 Aug 2026 10:48:13 -0400https://liberty.io/expectation-of-absolute-difference-of-two-uniform-random-variables.htmlhttps://liberty.io/expectation-of-absolute-difference-of-two-uniform-random-variables.htmlLet $X \sim U(0, 1)$ and $Y \sim U(0, 2)$ be independent random variables. Find $\mathbb{E}[|X - Y|]$. I found $f_{X}(x) = 1$ and $f_{Y}(y) = 1/2$, and I took an integral to find $P(Y > X) = 3......Thu, 13 Aug 2026 10:45:45 -0400https://liberty.io/determine-the-fourth-roots-of-16-duplicate.htmlhttps://liberty.io/determine-the-fourth-roots-of-16-duplicate.htmlDetermine the fourth roots of -16 in the form $x +iy$ where $x$ and $y$ are not trigonometric functions. I do not even know what they really want from me in this question. My initial thought wass......Thu, 13 Aug 2026 10:45:44 -0400https://liberty.io/a-lot-of-confusion-in-the-polynomial-remainder-theorem.htmlhttps://liberty.io/a-lot-of-confusion-in-the-polynomial-remainder-theorem.htmlLately I've been reading about Polynomial Remainder Theorem from various sources, mainly from the wikipedea article, this post and some high school books. Wikipedea says that if we divide a polynom......Thu, 13 Aug 2026 10:43:38 -0400https://liberty.io/lie-group-and-lie-algebra-homomorphisms.htmlhttps://liberty.io/lie-group-and-lie-algebra-homomorphisms.htmlI am reading Brian C Hall's Lie Groups, Lie Algebras, and Representations and trying to do one of the exercises about showing that isomorphic matrix Lie groups give rise to isomorphic Lie Algebras.......Thu, 13 Aug 2026 10:33:00 -0400https://liberty.io/find-closest-power-of-2-to-a-specific-number.htmlhttps://liberty.io/find-closest-power-of-2-to-a-specific-number.htmlThis question is weird and it's not a homework question. I couldn't come up with anything substantial, so sorry if you think that I should have posted my tried methods (I have tried but most of the......Thu, 13 Aug 2026 10:31:51 -0400https://liberty.io/you-can-039-t-find-an-equation-for-the-cross-sectional-shape-when-give.htmlhttps://liberty.io/you-can-039-t-find-an-equation-for-the-cross-sectional-shape-when-give.htmlWhy not? I'm trying to design a calculus I problem. I want them to eventually prove that the volume of a cone (oblique or even with irregular base) is $\frac{\pi r^2 h}{3}$, I also want them to know ...Thu, 13 Aug 2026 10:18:42 -0400https://liberty.io/is-a-omega-spectrum-a-connective-one.htmlhttps://liberty.io/is-a-omega-spectrum-a-connective-one.htmlI can't find this result anywhere, but it seems pretty straightforward. I want to avoid silly mistakes, but I can't see any fault. I'd love to receive some feedback Let $X$ be a $\Omega$-spectrum ......Thu, 13 Aug 2026 10:18:21 -0400https://liberty.io/what-are-we-really-doing-when-we-integrate-a-complex-exponential.htmlhttps://liberty.io/what-are-we-really-doing-when-we-integrate-a-complex-exponential.htmlFirstly, I have read this related question from this site, but it does not answer what I am asking here. What I like to know is why $$\int_{-n}^n e^{ix}dx \ne 0$$ for any finite $n \in \mathbb N$ ......Thu, 13 Aug 2026 10:04:47 -0400https://liberty.io/proper-suffix-suffix-and-empty-string.htmlhttps://liberty.io/proper-suffix-suffix-and-empty-string.htmlI found this definition on Wikipedia. Suffix. A suffix of a string is any substring of the string which includes its last letter, including itself. A proper suffix of a string is not equal to the ......Thu, 13 Aug 2026 10:03:14 -0400https://liberty.io/deriving-the-hamilton-product-of-two-quaternions-for-spatial-rotation.htmlhttps://liberty.io/deriving-the-hamilton-product-of-two-quaternions-for-spatial-rotation.htmlI work in computational chemistry, and I want make a simple Python program that rotates a molecular structure by some angle $\alpha$ around some axis defined by the unit vector $\vec{v}$. I define......Thu, 13 Aug 2026 09:57:41 -0400https://liberty.io/is-this-derivation-of-the-path-length-formula-actually-correct.htmlhttps://liberty.io/is-this-derivation-of-the-path-length-formula-actually-correct.htmlSaw this in a physics lecture. This all assumes we have some function, $y=f(x)$. First he defined $$ds = \sqrt{dx^2+dy^2},$$ where the professor drew a picture and seemed to be using dx and dy t......Thu, 13 Aug 2026 09:41:15 -0400https://liberty.io/probability-that-exactly-k-bins-are-empty-given-m-balls-and-n-bins.htmlhttps://liberty.io/probability-that-exactly-k-bins-are-empty-given-m-balls-and-n-bins.htmlI've searched for an understandable answer to this exact question and have failed to find it. How do you find the probability that exactly $k$ bins are empty, given $m$ balls and $n$ bins? (Each b......Thu, 13 Aug 2026 09:33:22 -0400https://liberty.io/definition-of-adjoint-operator-for-quantum-mechanics.htmlhttps://liberty.io/definition-of-adjoint-operator-for-quantum-mechanics.htmlWhile learning about adjoint operators for quantum mechanics, I encountered two definitions. The first definition is given by Shankar in The Principle of Quantum Mechanics: Given a ket $$ A\lve......Thu, 13 Aug 2026 09:28:58 -0400https://liberty.io/expected-value-of-e-tx-moment-generating-function.htmlhttps://liberty.io/expected-value-of-e-tx-moment-generating-function.htmlThis is a question from a Statistics class exam that I still am not sure on how to complete. The number of accidents in a week follows a discrete random variable X with a probability function x|......Thu, 13 Aug 2026 09:11:53 -0400https://liberty.io/what-does-50-choose-4-mean-in-statistics.htmlhttps://liberty.io/what-does-50-choose-4-mean-in-statistics.htmlI have a test tomorrow in statistics and was wondering what the following means? $$\binom{50}{4}$$ My professor along with most of my classmates have a calculator they can just plug that into. The ...Thu, 13 Aug 2026 09:06:57 -0400https://liberty.io/summing-digits-of-powers-of-2-to-get-1-2-4-8-7-5-pattern.htmlhttps://liberty.io/summing-digits-of-powers-of-2-to-get-1-2-4-8-7-5-pattern.htmlIf you add the decimal digits of multiples of 9,possibly repeatedly, you get 9. For instance, with 18 we have $$18\mapsto 1+8=9$$ and with 909 we have $$ 909\mapsto 9+0+9=18\mapsto 1+8=9.$$ This is ...Thu, 13 Aug 2026 08:59:23 -0400https://liberty.io/why-does-operatorname-var-x-e-x-2-e-x-2.htmlhttps://liberty.io/why-does-operatorname-var-x-e-x-2-e-x-2.html$ \operatorname{Var}(X) = E[X^2] - (E[X])^2 $ I have seen and understand (mathematically) the proof for this. What I want to understand is: intuitively, why is this true? What does this formula te......Thu, 13 Aug 2026 08:55:21 -0400https://liberty.io/covariance-of-normal-distribution.htmlhttps://liberty.io/covariance-of-normal-distribution.htmlSuppose we have two normal distribution $\mathcal{N}(0,s)$, $\mathcal{N}(0,t)$. how can I find the covariance if I don't have the distribution function $f(x,y)$ or anything, not given the correla......Thu, 13 Aug 2026 08:55:15 -0400https://liberty.io/how-to-transform-fraction-powers-between-numerator-and-denominator.htmlhttps://liberty.io/how-to-transform-fraction-powers-between-numerator-and-denominator.htmlI apologize in advance if this question is trivial to most of you but I'd like to verify if my understanding is correct. I want to verify the following rules: If we have $\dfrac{1}{\sqrt{x}}$, the......Thu, 13 Aug 2026 08:32:30 -0400https://liberty.io/how-to-integrate-a-4th-power-of-sine-and-cosine.htmlhttps://liberty.io/how-to-integrate-a-4th-power-of-sine-and-cosine.htmlI'm having some trouble figuring out the right substitutions to make to integrate $$\int \sin^4(\theta)d\theta$$ and $$\int \cos^4(\theta)d\theta$$ Any hints or suggestions are welcome. Thanks,...Thu, 13 Aug 2026 08:31:49 -0400https://liberty.io/discriminant-of-multivariable-function.htmlhttps://liberty.io/discriminant-of-multivariable-function.htmlHey I'm doing an online course and I just can't figure out what I'm doing wrong on one of the questions. The question is to find the discriminant of the function $$f(x,y) = 5x^2y^2 + 8x^2 + 9y^2$$ ......Thu, 13 Aug 2026 08:28:22 -0400https://liberty.io/finding-a-logarithmic-function-from-a-graph.htmlhttps://liberty.io/finding-a-logarithmic-function-from-a-graph.htmlHere's the graph. When I use the points $(-1,1)$ or $(-3,2)$ to use in the equation $a\log(-x-1)+k$, I can't find a finite value for k. Any ideas?...Thu, 13 Aug 2026 08:25:19 -0400https://liberty.io/writing-an-objective-function-such-that-the-problem-has-an-optimal-sol.htmlhttps://liberty.io/writing-an-objective-function-such-that-the-problem-has-an-optimal-sol.htmli know it's basic but I'm struggling with how to find the objective function of a linear program. How do i write an objective function given only two constraints of a linear program in three variab......Thu, 13 Aug 2026 08:24:18 -0400https://liberty.io/what-are-the-formulas-for-topological-transformations-how-to-obtain-th.htmlhttps://liberty.io/what-are-the-formulas-for-topological-transformations-how-to-obtain-th.htmlI'm reading Flegg's From Geometry to Topology, the author says that in Euclidean geometry, translation and rotation are: $$T:(x,y)\to(x+a,y+b)$$ $$R:(x,y)\to(x \cos \phi - y \sin \phi, x \sin \phi......Thu, 13 Aug 2026 08:23:24 -0400https://liberty.io/what-is-the-probability-of-selecting-a-red-marble-then-a-blue-marble-i.htmlhttps://liberty.io/what-is-the-probability-of-selecting-a-red-marble-then-a-blue-marble-i.htmlA bag contains 4 green marbles, 9 red marbles, 10 orange marbles, 9 brown marbles, and 11 blue marbles. You choose a marble, replace it, and choose again. What is P (red, then blue) This is my que......Thu, 13 Aug 2026 08:20:23 -0400https://liberty.io/when-to-use-normal-approximation.htmlhttps://liberty.io/when-to-use-normal-approximation.htmlDo we use normal approximation when discrete distributions are hard to solve? For example, $P(X\ge 7000)$ where is $X\sim\operatorname{Binomial}(13000, 0.7)$. Obviously, summing each case manually ...Thu, 13 Aug 2026 08:14:27 -0400https://liberty.io/finding-c-in-a-definite-integral-with-no-reference-point.htmlhttps://liberty.io/finding-c-in-a-definite-integral-with-no-reference-point.htmlHow to find C in this equation? $$\int_0^x f(t)~dt=\int_x^1t^2f(t)~dt + x^2/4 + x^4/8 +C$$ I attempted to move over $\int_0^x f(t)~dt$ to the right side, so that I could solve a definite integral......Thu, 13 Aug 2026 07:51:39 -0400https://liberty.io/problem-comprehending-the-representation-of-complex-numbers-by-a-unit-.htmlhttps://liberty.io/problem-comprehending-the-representation-of-complex-numbers-by-a-unit-.htmlI'm having a very basic problem with understanding the representation of complex numbers using the unit circle. An example of what I'm struggling with is shown in the initial diagram on this page ......Thu, 13 Aug 2026 07:45:34 -0400https://liberty.io/how-do-i-evaluate-the-following-derivative.htmlhttps://liberty.io/how-do-i-evaluate-the-following-derivative.htmlHow can I evaluate the derivative $$\left. \frac{d}{dx} \left( x \, \ln(x^5) \right)\right|_{x=1} \, ?$$ I have this math problem, and the biggest thing I do not understand, is which derivative ru......Thu, 13 Aug 2026 07:23:48 -0400https://liberty.io/describe-the-galois-group-of-a-field-extension.htmlhttps://liberty.io/describe-the-galois-group-of-a-field-extension.htmlI'm struggling to understand the basics of Galois theory. One of the things I don't understand is how to actually derive automorphisms of a field extension. Let's say you had a simple problem: $x^......Thu, 13 Aug 2026 07:17:45 -0400https://liberty.io/linear-transformations-and-polynomials.htmlhttps://liberty.io/linear-transformations-and-polynomials.htmlI've been having a really hard time recently understanding linear transformations. I would really like to know conceptually what exactly is going on. Let $T: P_3 \rightarrow P_2$ be defined by ......Thu, 13 Aug 2026 07:16:02 -0400https://liberty.io/user-christopher-d-long-mathematics-stack-exchange.htmlhttps://liberty.io/user-christopher-d-long-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Thu, 13 Aug 2026 07:14:02 -0400https://liberty.io/definition-of-finite-elements-ciarlet.htmlhttps://liberty.io/definition-of-finite-elements-ciarlet.htmlI dont understand this definition of finite elements: The triple $(K,P,\Sigma)$ with $K \subset \mathbb{R}^n$ connected lipschitz domain with non-empty interior $P$ finite dimensional vector spa......Thu, 13 Aug 2026 07:03:01 -0400https://liberty.io/what-is-the-approximate-area-of-the-shaded-region-of-the-given-figure.htmlhttps://liberty.io/what-is-the-approximate-area-of-the-shaded-region-of-the-given-figure.htmlHow do i find the area of the black shaded portion of the circle? I noticed the 4 so i think that's the radius. The formula to find the area $$A=πr^2$$ so I thought of using that to find the area ......Thu, 13 Aug 2026 07:00:12 -0400https://liberty.io/calculus-i-optimization-problem-maximization-of-profit.htmlhttps://liberty.io/calculus-i-optimization-problem-maximization-of-profit.htmlCalculus problem that I've been trying to get my head around. Problem: A company can sell 20 products if it charges $40 per product. For each dollar decrease or increase in the price, the compan......Thu, 13 Aug 2026 06:54:32 -0400https://liberty.io/integral-test-proof.htmlhttps://liberty.io/integral-test-proof.htmlCould someone explain what does it mean by "(and hence $\sum a_n$)"? Isn't $a_{n+1} \leq a_n$ for all $n$? How does the convergence of $\sum a_{n+1}$ imply $\sum a_n$ converges when we have $\sum ......Thu, 13 Aug 2026 06:48:48 -0400https://liberty.io/why-is-the-gradient-always-perpendicular-to-level-curves.htmlhttps://liberty.io/why-is-the-gradient-always-perpendicular-to-level-curves.htmlLet $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be a function. A level set is a set of points: $$L(c) = \{x \in \mathbb{R}^n | f(x) = c\}$$ Two vectors $a, b \in \mathbb{R}^n$ are perpendicular, when ...Thu, 13 Aug 2026 06:47:46 -0400https://liberty.io/why-is-the-area-of-an-equilateral-triangle-frac-sqrt-3-4-b-2.htmlhttps://liberty.io/why-is-the-area-of-an-equilateral-triangle-frac-sqrt-3-4-b-2.htmlWhy is the area of an equilateral triangle $$\frac{\sqrt{3}}{4}b^2$$ and not $$\frac12b^2$$ or $$bh$$...Thu, 13 Aug 2026 06:40:40 -0400https://liberty.io/question-on-applying-jensen-inequality-on-logarithm-of-a-sum.htmlhttps://liberty.io/question-on-applying-jensen-inequality-on-logarithm-of-a-sum.htmlI am confused by the meaning of the $t \in (0,1)$ parameter in Jensen's inequality $$ f( tx_1+(1-t)x_2) \le tf(x_1)+(1-t)f(x_2) $$ When I apply this to the logarithm $$ \log( tx_1+(1-t)x_2) \ge t \......Thu, 13 Aug 2026 06:39:30 -0400https://liberty.io/the-range-of-arc-cotangent-function-arccot-1.htmlhttps://liberty.io/the-range-of-arc-cotangent-function-arccot-1.htmlWe know that the range of arc-cotangent function is $(0,π)$ and we I calculate the value of $cot^{-1}(-1)$ by a calculator, I get ($-π/4$) Which is clearly not included in the range !! Why isn't......Thu, 13 Aug 2026 06:37:21 -0400https://liberty.io/explanation-of-proof-of-logarithm-power-rule.htmlhttps://liberty.io/explanation-of-proof-of-logarithm-power-rule.htmlCan someone explain how to get from steps $(1)$ to $(2)$ in this proof of the logarithm power rule? How is the "equivalent exponential expression found? The power rule: For any positive $x$, ......Thu, 13 Aug 2026 06:35:08 -0400https://liberty.io/imaginary-and-complex-dimensions.htmlhttps://liberty.io/imaginary-and-complex-dimensions.htmlSo I saw a really intriguing question here, and I thought, "Why stop at integers? Why stop at rationals? Why even stop at real numbers?" So I'm asking this: Is there such a thing as imaginary and c......Thu, 13 Aug 2026 06:33:04 -0400https://liberty.io/to-evaluate-the-limit-as-h-approaches-0-how-do-i-rewrite-the-limit-in-.htmlhttps://liberty.io/to-evaluate-the-limit-as-h-approaches-0-how-do-i-rewrite-the-limit-in-.htmlI found a Question from the textbook which is not well explained...or not at all explained. And it would be interesting to know how to solve it. Or perhaps I am simply that bad at understanding squ......Thu, 13 Aug 2026 06:32:40 -0400https://liberty.io/is-v-tav-diagonal-in-general.htmlhttps://liberty.io/is-v-tav-diagonal-in-general.htmlI want to know if the following is a true statement: If $A$ is real a symmetric and $V$ is orthogonal then $V^TAV=D$ where $D$ is a diagonal matrix. To prove it I can say $$(V^TAV)^T=V^TA^T......Thu, 13 Aug 2026 06:32:25 -0400https://liberty.io/how-do-i-plot-the-curve-of-a-conical-spiral-on-an-unwrapped-2d-conic-s.htmlhttps://liberty.io/how-do-i-plot-the-curve-of-a-conical-spiral-on-an-unwrapped-2d-conic-s.htmlExample images I need a real world solution to a reconstructive surgery problem! A fairly commonly encountered clinical scenario results in the need to create a tube (from skin) in the shape of an ...Thu, 13 Aug 2026 06:30:00 -0400https://liberty.io/is-a-single-line-line-segment-or-ray-a-valid-angle.htmlhttps://liberty.io/is-a-single-line-line-segment-or-ray-a-valid-angle.htmlAfter online research and several definitions of the term angle it seems they all fit a short set of restrictions for what defines an angle, "a shape formed by two rays that intersect at a point ca......Thu, 13 Aug 2026 06:16:05 -0400https://liberty.io/any-shortcut-to-calculate-factorial-of-a-number-without-calculator-or-.htmlhttps://liberty.io/any-shortcut-to-calculate-factorial-of-a-number-without-calculator-or-.htmlI've been searching the internet for quite a while now to find anything useful that could help me to figure out how to calculate factorial of a certain number without using calculator but no luck ...Thu, 13 Aug 2026 05:59:32 -0400https://liberty.io/work-done-pulling-a-leaky-bucket.htmlhttps://liberty.io/work-done-pulling-a-leaky-bucket.htmlA bucket that weighs $4.6$ pounds and a rope of negligible weight are used to draw water from a well that is $80$ feet deep. The bucket is filled with $35$ pounds of water and is pulled up at a rat......Thu, 13 Aug 2026 05:59:22 -0400https://liberty.io/why-does-10-35-2-7.htmlhttps://liberty.io/why-does-10-35-2-7.htmlMy question is not meant to be dumb. I'm working on a much more complicated problem, but I forgot some of the basics. Often we perform steps in math without really knowing why we can do a certain......Thu, 13 Aug 2026 05:53:53 -0400https://liberty.io/let-x-and-y-be-two-vectors-in-mathbb-r-d-with-mid-x-mid-mid-y-mid-find.htmlhttps://liberty.io/let-x-and-y-be-two-vectors-in-mathbb-r-d-with-mid-x-mid-mid-y-mid-find.htmlLet $u \in \mathbb{R}^d = V$ be a unit vector and set $W = \text{span}(u) ^{\bot}$ (with respect to the dot product). The reflector across W is $P_u = I_d - 2uu^T$. Let $x$ and $y$ be two vectors ......Thu, 13 Aug 2026 05:52:55 -0400