https://liberty.ioCurated star highlights with fresh energy.en-usSat, 22 Aug 2026 16:23:19 -0400https://liberty.io/fourier-transform-of-sinc-function.htmlhttps://liberty.io/fourier-transform-of-sinc-function.htmllet us consider fourier transform of sinc function,as i know it is equal to rectangular function in frequency domain and i want to get it myself,i know there is a lot of material about this,but i w......Sat, 22 Aug 2026 20:22:30 -0400https://liberty.io/evaluating-the-limit-of-lim-x-to-9-frac-x-9-sqrt-x-3.htmlhttps://liberty.io/evaluating-the-limit-of-lim-x-to-9-frac-x-9-sqrt-x-3.htmlHow do you evaluate the limit of $$\lim_{x \to 9} \frac {x-9} {\sqrt{x} - 3}$$ by first principle? I was thinking of this way to evaluate but it seems to evaluate to $\frac {0} {0}$. $$\lim_{x \......Sat, 22 Aug 2026 20:20:17 -0400https://liberty.io/showing-a-binary-operation-is-well-defined.htmlhttps://liberty.io/showing-a-binary-operation-is-well-defined.htmlLet $G=\{x\in\mathbb{R}~|~0\le x<1\}$ and define $\star$ as follows $$x\star y=x+y-[x+y]$$ I want to show this is a well defined operation. I think it is this easy; If $x=x', y=y'$, then $$......Sat, 22 Aug 2026 20:16:48 -0400https://liberty.io/what-is-the-number-of-dimensions-of-a-scalar.htmlhttps://liberty.io/what-is-the-number-of-dimensions-of-a-scalar.htmlAm I correct that a matrix has two dimensions, and a vector has one dimension? What is the number of dimensions of a scalar? zero? Thanks....Sat, 22 Aug 2026 20:09:18 -0400https://liberty.io/definition-of-the-approx-symbol.htmlhttps://liberty.io/definition-of-the-approx-symbol.htmlTake an unending number, say e.g $π$. If we want to show $π$'s value, should we use the approximately notation or equal sign when writing: $π = 3.14...$ or $π ≈ 3.14...$ This might be a really s......Sat, 22 Aug 2026 20:06:29 -0400https://liberty.io/find-the-dual-of-a-linear-programming-problem.htmlhttps://liberty.io/find-the-dual-of-a-linear-programming-problem.htmlI have a very simple linear programming problem with the following constraints: Minimize $3x_1 + 2x_2 - 33x_3$ subject to $x_1 - 4x_2 + x_3 \leq 15$ $9x_1 + 6x_3 \leq 12$ $5x_1 + 9x_2 \geq 3$ ......Sat, 22 Aug 2026 19:58:27 -0400https://liberty.io/what-is-the-average-roll-on-dice-while-re-rolling-a-result-of-1.htmlhttps://liberty.io/what-is-the-average-roll-on-dice-while-re-rolling-a-result-of-1.htmlThe average roll on a six sided die (d6) is 3.5. If the result is a 1 you are allowed a single re-roll accepting the second result. What does the average roll increase to? And how to calculate this? ...Sat, 22 Aug 2026 19:55:25 -0400https://liberty.io/derivative-of-x-x-2.htmlhttps://liberty.io/derivative-of-x-x-2.htmlStudying past exam problems for my exam in ~$4$ weeks, and I came across this derivative as one of the questions. I actually have no idea how to solve it. $$\frac{d}{dx} (x^{x^2})$$ Using the chain ...Sat, 22 Aug 2026 19:54:35 -0400https://liberty.io/finding-maclaurin-series-of-integral.htmlhttps://liberty.io/finding-maclaurin-series-of-integral.htmlI wish to find the Maclaurin-series for the function $$L(x) = \int_0^x \cos(t^2)\space dt$$ Alternative 1 Set $f(t) = \cos(t)$. We have that $f'(t) = -\sin(t)$, $f''(t) = -\cos(t)$ and $f^{(3)}(t)......Sat, 22 Aug 2026 19:47:02 -0400https://liberty.io/two-definitions-of-bounded-variation-function.htmlhttps://liberty.io/two-definitions-of-bounded-variation-function.htmlAs far as I know, a function $f$ defined on an interval $[a, b]$ is said to be of bounded variation if $$\tag{1}V_a^b(f)=\sup\left\{\sum_{P} \lvert f(x_{j+1})-f(x_j)\rvert \ :\ P\ \text{partition ......Sat, 22 Aug 2026 19:32:16 -0400https://liberty.io/find-the-equivalent-value-of-radians.htmlhttps://liberty.io/find-the-equivalent-value-of-radians.htmlA function $f$ is called periodic with period $T$ if $f (x + T ) = f (x)$ (for all x) and $T$ is the smallest positive number with this property. The sine and cosine functions are periodic with per......Sat, 22 Aug 2026 19:28:49 -0400https://liberty.io/linear-algebra-how-to-interpret-vectors-and-equations-of-a-system-of-e.htmlhttps://liberty.io/linear-algebra-how-to-interpret-vectors-and-equations-of-a-system-of-e.htmlFrom Strang's 18.06 Linear Algebra OCW Course I realized that there is a row view and column view to interpret a matrix and the solution to Ax = b. Well the view made sense but it was hard understa......Sat, 22 Aug 2026 19:19:31 -0400https://liberty.io/what-does-ln-ln-x-simplify-to.htmlhttps://liberty.io/what-does-ln-ln-x-simplify-to.htmlWhat does $\ln(\ln(x))$ simplify to? First instinct was to write $\ln^2x$, but that looked odd, because $\ln(\ln(x))$ is not the same as $\ln(x)\ln(x)$ or $(\ln(x))^2$. Or is it ?...Sat, 22 Aug 2026 19:12:35 -0400https://liberty.io/ax-by-c-c-is-always-0.htmlhttps://liberty.io/ax-by-c-c-is-always-0.htmlFind an equation for the given line in the form $ax +by =​c$, where​ a, b, and c are integers with no factor common to all three and a ≥0. Through $(-40, 35)$; parallel to $7x + 8y = 13$ I put th......Sat, 22 Aug 2026 19:10:08 -0400https://liberty.io/suppose-you-draw-two-cards-from-a-deck-of-52-cards-without-replacement.htmlhttps://liberty.io/suppose-you-draw-two-cards-from-a-deck-of-52-cards-without-replacement.htmlSuppose you draw two cards from a deck of 52 cards without replacement. 1) What is the probability that both cards are hearts? 2) What is the probability that exactly one of the cards is hearts......Sat, 22 Aug 2026 19:07:39 -0400https://liberty.io/if-x-4-x-4-then-x-3-4-x-3-is.htmlhttps://liberty.io/if-x-4-x-4-then-x-3-4-x-3-is.htmlIf $x+(4/x)=4$ then $x^3+(4/x)^3 $is. Note: $(x+(4/x))^3=4^3$ assuming $4/x=y$ then $(x+y)^3=64$ $x^3+y^3+3xy^2+3yx^2=64$ $x^3+y^3+3xy(x+y)=64$ $x^3+y^3+3*4(4)=64$ so $x^3+y^3=16$...Sat, 22 Aug 2026 19:04:04 -0400https://liberty.io/how-to-prove-cos-4x-sin-4x-cos-2x-sin-2x-is-always-0.htmlhttps://liberty.io/how-to-prove-cos-4x-sin-4x-cos-2x-sin-2x-is-always-0.htmlSo I have a small problem here where I have to prove the following : $$\cos^4x - \sin^4x - \cos^2x + \sin^2x = 0 $$ I know that the 2nd part is always $1$, so I need to prove that the first par......Sat, 22 Aug 2026 19:01:16 -0400https://liberty.io/norm-of-finite-flat-morphism.htmlhttps://liberty.io/norm-of-finite-flat-morphism.htmlLet $X,Y$ be two schemes and $f : X \to Y$ be a finite flat map. In the following document http://www.ms.u-tokyo.ac.jp/~t-saito/talk/eepr.pdf page 20, the author talks about the norm map $$f_* : f_*\...Sat, 22 Aug 2026 18:57:45 -0400https://liberty.io/if-x-is-the-remainder-when-a-multiple-of-4-is-divided-by-6-and-y-is-th.htmlhttps://liberty.io/if-x-is-the-remainder-when-a-multiple-of-4-is-divided-by-6-and-y-is-th.htmlThe question is: if $x$ is the remainder when a multiple of $4$ is divided by $6$, and $y$ is the remainder when a multiple of $2$ is divided by $3$, what is the greatest possible value of $x+y$? ......Sat, 22 Aug 2026 18:43:45 -0400https://liberty.io/predator-prey-dynamical-system-find-the-vector-x-k-for-a-certain-k.htmlhttps://liberty.io/predator-prey-dynamical-system-find-the-vector-x-k-for-a-certain-k.htmlConsider a predator-pray model with the matrix $A= \begin{bmatrix} 0.5 & 0.6\\ -0.2 & 1.4 \end{bmatrix}$ ($A$ is the population distribution) and the initial state is $\vec{x}_0 = \begin{...Sat, 22 Aug 2026 18:29:06 -0400https://liberty.io/linear-regression-ssr-equation.htmlhttps://liberty.io/linear-regression-ssr-equation.htmlI've studied SST and the others but I saw the equation that $SSR=(n-1)(β_1)^2(S_x)^2$. I want to know how the equation is done. Thnks!!...Sat, 22 Aug 2026 18:28:40 -0400https://liberty.io/definition-of-an-axiom.htmlhttps://liberty.io/definition-of-an-axiom.htmlWhat would be some convenient and general enough definition of an axiom? If we go through mathematical theories we can observe that axioms either demand that something exists, or tell us how some ...Sat, 22 Aug 2026 18:17:22 -0400https://liberty.io/what-about-simon-donaldson-039-s-riemann-surfaces.htmlhttps://liberty.io/what-about-simon-donaldson-039-s-riemann-surfaces.htmlI am studying Riemann surfaces now. How about Donaldson's book "Riemann surfaces"? Could you recommend some references, and point out the required mathematical knowledge? Many thanks in advance!...Sat, 22 Aug 2026 18:16:10 -0400https://liberty.io/normal-operators-self-adjoint.htmlhttps://liberty.io/normal-operators-self-adjoint.htmlThis is one of my homework questions and I was hoping someone could check my proof: Prove that a normal operator with real eigenvalues is self-adjoint. So what I thought to do was, for some normal ...Sat, 22 Aug 2026 18:06:25 -0400https://liberty.io/how-do-i-find-the-base-and-height-of-a-triangle-when-given-only-the-hy.htmlhttps://liberty.io/how-do-i-find-the-base-and-height-of-a-triangle-when-given-only-the-hy.htmlMy hypotenuse is 1 and the area is .125... can someone find out how to get the base + height when just given the hypotenuse and area of a triangle? I know finding the area of a triangle is 1/2base(......Sat, 22 Aug 2026 17:54:58 -0400https://liberty.io/difference-between-each-every-any-etc.htmlhttps://liberty.io/difference-between-each-every-any-etc.htmlI'm having a bit of trouble understanding the use of these different phrases. Are there any certain rules for when which of the word is to be used? I feel like they overlap a lot, but can bear slig......Sat, 22 Aug 2026 17:47:06 -0400https://liberty.io/multiplicative-inverse-of-modulo-n-is-unique.htmlhttps://liberty.io/multiplicative-inverse-of-modulo-n-is-unique.htmlI am trying to prove this: $x,b,n\in \mathbb{Z}, \space xb\;(\mod{n}\;) = 1 \implies $ b is unique. I have tried by using proof by contradiction. That is, assume there are integers $b,c$ where the ...Sat, 22 Aug 2026 17:35:30 -0400https://liberty.io/calculating-the-average-speed-using-only-velocities.htmlhttps://liberty.io/calculating-the-average-speed-using-only-velocities.htmlFor the following question: Pedro travels by bus to school at an average speed of $40$ km/hr. He is driven home by the same route by a friend's car at an average speed of $50$ km/hr. Which ......Sat, 22 Aug 2026 17:30:42 -0400https://liberty.io/definition-of-a-compact-operator.htmlhttps://liberty.io/definition-of-a-compact-operator.htmlOperator compactness is characterized by maps the send the unit ball to relatively compact sets. Does anyone have a good justification for why we call this property compactness? The best justifica......Sat, 22 Aug 2026 17:28:15 -0400https://liberty.io/how-to-solve-x-times-0-n-where-n-does-not-equal-0.htmlhttps://liberty.io/how-to-solve-x-times-0-n-where-n-does-not-equal-0.htmlSo recently I was watching some videos on complex numbers. I was surprised to learn the uses of $i$ in different problems. For example, I never thought it would make sense that $x^2 + 1 = 0$ has any ...Sat, 22 Aug 2026 17:27:21 -0400https://liberty.io/proper-notation-for-distinct-sets.htmlhttps://liberty.io/proper-notation-for-distinct-sets.htmlConsider I have two sets that neither one is a subset of the other (for example, the set of prime numbers and the set of odd integers). Is there a specific notation that combines the meanings of bo......Sat, 22 Aug 2026 17:24:47 -0400https://liberty.io/when-applying-the-master-theorem-how-to-decide-on-case-1-2-or-3.htmlhttps://liberty.io/when-applying-the-master-theorem-how-to-decide-on-case-1-2-or-3.htmlWhen you are given a relation, such as $$T(n)=8T\Big(\frac{n}{2}\Big)+160n,$$ what are the logical steps to consider when deciding whether to apply case 1, 2, or 3? Or do you have to apply all 3 and ...Sat, 22 Aug 2026 17:17:14 -0400https://liberty.io/proper-definition-of-a-proper-map.htmlhttps://liberty.io/proper-definition-of-a-proper-map.htmlWhat is the proper definition of a proper map? Here are a few definitions I've come across: A continuous map $f: X \to Y$ is proper if preimages of compact subsets are compact. A continuous map......Sat, 22 Aug 2026 17:13:16 -0400https://liberty.io/is-aa-t-a-positive-definite-symmetric-matrix.htmlhttps://liberty.io/is-aa-t-a-positive-definite-symmetric-matrix.htmlLet $A\in M_{n\times n}(\mathbb{R})$ be a matrix. Is it true that $AA^{T}$ is positive-definite? Clearly $AA^{T}$ is symmetric. I have shown that a symmetric matrix $S\in M_{n\times n}(\mathbb{R})......Sat, 22 Aug 2026 17:06:25 -0400https://liberty.io/dot-product-is-an-euclidean-norm-in-euclidean-space.htmlhttps://liberty.io/dot-product-is-an-euclidean-norm-in-euclidean-space.htmlI might be missing something, but from what I have learned in linear algebra, for a matrix or vector multiplication to be valid you need to have number of columns in pre-multiplying vector to be th......Sat, 22 Aug 2026 16:59:08 -0400https://liberty.io/definition-simple-stochastic-process.htmlhttps://liberty.io/definition-simple-stochastic-process.htmlCan anybody explain what concretely is a simple (stochastic) process? (...and yes I tried to google it but without success)...Sat, 22 Aug 2026 16:57:55 -0400https://liberty.io/what-is-graphical-addition-closed.htmlhttps://liberty.io/what-is-graphical-addition-closed.html"Use the graphs of f and g and the method of graphical addition to sketch the graph of $f + g$". $f(x) = x, g(x) = \frac{1}{x}$...Sat, 22 Aug 2026 16:56:59 -0400https://liberty.io/finding-formula-for-a-cubic-polynomial.htmlhttps://liberty.io/finding-formula-for-a-cubic-polynomial.htmlSo I need to find a formula for a cubic polynomial with the given conditions: Local max at $x=1$ Local min at $x=3$ $Y$ intercept of $5$ $x^3$ term whose coefficient is $1$ So all I really have i......Sat, 22 Aug 2026 16:55:07 -0400https://liberty.io/how-can-i-calculate-the-running-time-of-my-compressor-knowing-the-cfm-.htmlhttps://liberty.io/how-can-i-calculate-the-running-time-of-my-compressor-knowing-the-cfm-.htmlSo, I actually have a real world application for the things my calc teacher tried to shove into my head, unfortunately it's been ten years since I took the class, and it all leaked out. That being......Sat, 22 Aug 2026 16:44:23 -0400https://liberty.io/is-borel-field-different-from-sigma-field.htmlhttps://liberty.io/is-borel-field-different-from-sigma-field.htmlMy mathematical statistics book denotes $\sigma$-field as following: Let $\Bbb B$ be the collection of subsets of $\Bbb C$ where $\Bbb C$ denotes sample space which is the collection of all possible ...Sat, 22 Aug 2026 16:39:35 -0400https://liberty.io/consider-the-following-system-of-linear-equations.htmlhttps://liberty.io/consider-the-following-system-of-linear-equations.htmlStory cut short, I have an exam in a weeks time and this is a question off a previous exam paper - I'm unsure as to how I should go about it as there are 4 variables with only three linear equation......Sat, 22 Aug 2026 16:36:15 -0400https://liberty.io/application-of-double-expectation-theorem.htmlhttps://liberty.io/application-of-double-expectation-theorem.htmlA component in system and a backup unit both have mean lifetime $\mu$. if component fails then it is in ediately replaced by backup, and there is probability $p$ that backup works. Find the expected ...Sat, 22 Aug 2026 16:34:11 -0400https://liberty.io/what-shape-is-a-bean.htmlhttps://liberty.io/what-shape-is-a-bean.htmlHow can I mathematically describe the shape of an idealised bean? (In two dimensions and in threes dimensions) At the moment I'm calling the shape I refer to an ellipse/ellipsoid on a curved major......Sat, 22 Aug 2026 16:33:49 -0400https://liberty.io/chain-rule-and-vector-valued-functions.htmlhttps://liberty.io/chain-rule-and-vector-valued-functions.htmlLet $f: R^n \to R$ be given by $f(x) = \frac{||x||^4} {1 + ||x||^2}$ . Use the chain rule to show that $f$ is differentiable at each $x \in R^n$ and compute $Df(x)$. This vector valued stuff just......Sat, 22 Aug 2026 16:31:49 -0400https://liberty.io/what-is-the-derivative-of-digamma-function.htmlhttps://liberty.io/what-is-the-derivative-of-digamma-function.htmlI want to know $(\frac{\Gamma'(\alpha)}{\Gamma(\alpha)})'$. My goal is to show $\alpha $ times this derivative of digamma is greater than 1. The background of question is to show $\bar{x}$ is not ...Sat, 22 Aug 2026 16:28:54 -0400https://liberty.io/is-a-spinor-an-element-of-the-spin-group-or-an-object-that-transforms-.htmlhttps://liberty.io/is-a-spinor-an-element-of-the-spin-group-or-an-object-that-transforms-.htmlI've been researching spinors, and I'm a bit confused by some of the terminology. In some cases, spinors seem to be presented as elements of the Spin group, whereas in others they seem to be presen......Sat, 22 Aug 2026 16:18:09 -0400https://liberty.io/determine-if-a-region-is-horizontally-or-vertically-simple.htmlhttps://liberty.io/determine-if-a-region-is-horizontally-or-vertically-simple.htmlI have an assignment problem which asks me to decide if the regions provided are horizontally or vertically simple. I am stuck on one of these which is the following: $$ A = \{(x,y)\in \Bbb R^2 | ......Sat, 22 Aug 2026 16:14:00 -0400https://liberty.io/little-oh-notation.htmlhttps://liberty.io/little-oh-notation.htmlAssume I have a function $ f:\mathbb{R}^{m}\to\mathbb{R}^n $, and let $ h\in \mathbb{R}^m $. Does the notation $ f\left(x_0+h\right)=o\left(h\right) $ (for fixed $x_0 $) mean that $$ \lim_{h\to0}\f......Sat, 22 Aug 2026 16:06:26 -0400https://liberty.io/what-are-the-advantages-of-studying-the-dual-problem-in-linear-program.htmlhttps://liberty.io/what-are-the-advantages-of-studying-the-dual-problem-in-linear-program.htmlI am studying linear programming and I came across primal-dual algorithm in Linear Programming. I understood it but I am unable to understand why there is a need to calculate a dual, if problem can......Sat, 22 Aug 2026 15:59:37 -0400https://liberty.io/1-2-5-12-27-what-will-be-the-formula-to-calculate-nth-term-for-this-se.htmlhttps://liberty.io/1-2-5-12-27-what-will-be-the-formula-to-calculate-nth-term-for-this-se.htmlHere, the difference of the digits at level 2 are in G.P., so is there any generalised formulae for calculating nth term if difference of digits at any level are in G.P....Sat, 22 Aug 2026 15:34:53 -0400https://liberty.io/convergence-of-student-039-s-t-distribution-to-a-standard-normal.htmlhttps://liberty.io/convergence-of-student-039-s-t-distribution-to-a-standard-normal.htmlI was looking at this question where it is shown that a Student's t-distribution converges to a standard normal distribution as the degrees of freedom tend to infinity.We start with the Student's t-...Sat, 22 Aug 2026 15:32:03 -0400https://liberty.io/number-of-4-digit-numbers-with-no-repeated-digit.htmlhttps://liberty.io/number-of-4-digit-numbers-with-no-repeated-digit.htmlNumber of $4$ digit numbers with no repeated digit is $4536$ $3024$ $5040$ $4823$ Well, I am very much weak in combinatorics. Please help....Sat, 22 Aug 2026 15:30:35 -0400https://liberty.io/calculate-order-of-magnitude-difference.htmlhttps://liberty.io/calculate-order-of-magnitude-difference.htmlSay if I have two Areas of size a = 6 km$^2$ and b = 0.1 km$^2$ how would I find how many orders of magnitude one is greater or less than the other. So, the way I would work this is by saying th......Sat, 22 Aug 2026 15:19:37 -0400https://liberty.io/user-eric-auld-mathematics-stack-exchange.htmlhttps://liberty.io/user-eric-auld-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Sat, 22 Aug 2026 15:13:08 -0400https://liberty.io/what-is-the-difference-between-a-sequence-of-functions-f-n-and-a-seque.htmlhttps://liberty.io/what-is-the-difference-between-a-sequence-of-functions-f-n-and-a-seque.htmlWhat is the difference between a sequence of functions $(f_n)$ and a sequence of functions $f_n(x)$? I am reading my textbook on analysis, and it seems to use 'sequence of functions' to describe bo......Sat, 22 Aug 2026 15:12:00 -0400https://liberty.io/number-with-finite-binary-representation-and-infinite-decimal-represen.htmlhttps://liberty.io/number-with-finite-binary-representation-and-infinite-decimal-represen.htmlOne can easily find numbers with finite decimal representation with infinite binary representation. (Like $0.3$ and $0.01010101..$) I assume there is an opposite case, meaning a number with finite ...Sat, 22 Aug 2026 15:11:27 -0400https://liberty.io/correct-interpretation-of-exponential-decay-and-decay-factor.htmlhttps://liberty.io/correct-interpretation-of-exponential-decay-and-decay-factor.htmlmy question is about exponential decay and its factor. English isn't my native language and therefore I'm not sure about the precise definition in my particular case. I'm reading a specific paper......Sat, 22 Aug 2026 15:10:29 -0400https://liberty.io/what-is-the-correct-way-to-self-learn-from-a-textbook.htmlhttps://liberty.io/what-is-the-correct-way-to-self-learn-from-a-textbook.htmlThey taught me just to calculate, and not to think or ask. So now I just regurgitate, and bullsh*t just to pass. They taught me how to integrate, but not the reasons why. As well, I learned to der......Sat, 22 Aug 2026 14:53:09 -0400https://liberty.io/is-there-a-common-symbol-for-concatenating-two-finite-sequences.htmlhttps://liberty.io/is-there-a-common-symbol-for-concatenating-two-finite-sequences.htmlSay we have two finite sequences $X = (x_0,...,x_n)$ and $Y = (y_0,...,y_n)$. Is there a more or less common notation for the concatenation of these sequences, like $\sum (X,Y) = (x_0,...,x_n,y_0,.......Sat, 22 Aug 2026 14:52:54 -0400https://liberty.io/how-would-one-evaluate-int-frac-x-sin-x-x-2-1-over-the-real-line.htmlhttps://liberty.io/how-would-one-evaluate-int-frac-x-sin-x-x-2-1-over-the-real-line.htmlAnother exam problem I'm looking at is to evaluate the following integral. $$ \int_{-\infty}^{\infty} \frac{x\hspace{-0.04 in}\cdot\hspace{-0.04 in}\sin(x)}{x^{\hspace{.02 in}2}+1} dx $$ This is a ...Sat, 22 Aug 2026 14:52:06 -0400https://liberty.io/good-book-for-an-introduction-to-differential-equations-for-engineers.htmlhttps://liberty.io/good-book-for-an-introduction-to-differential-equations-for-engineers.htmlI will be leading a discussion class on differential equations for engineers this coming semester and I am wondering if anyone has a book that they could recommend. The book that will be used in the ...Sat, 22 Aug 2026 14:50:01 -0400https://liberty.io/inverse-transformation-of-arctan2.htmlhttps://liberty.io/inverse-transformation-of-arctan2.htmlSimilar to this and this I'm trying to recover the original 2D field after the arctan2 operation. Here's a MWE in python: import numpy as np theta = np.random.uniform(low=-3*np.pi, high=3*np.pi, si......Sat, 22 Aug 2026 14:45:29 -0400https://liberty.io/when-does-a-system-of-equations-have-no-solution.htmlhttps://liberty.io/when-does-a-system-of-equations-have-no-solution.htmlI have performed Gaussian elimination on this matrix to reduce it to $$ \left[ \begin{array}{@{}ccc|c@{}} -3&-1&2 & 1 \\ 0& \frac{-5}{3}& \frac{10}{3} & \frac{8}{3} \\ 0&a......Sat, 22 Aug 2026 14:44:25 -0400https://liberty.io/maximizing-cross-sectional-area-of-trapezoid.htmlhttps://liberty.io/maximizing-cross-sectional-area-of-trapezoid.htmlThe task is to fold a piece of sheet metal that measures 60 cm across in such a way as to form a trapezoidal "gutter" (a trough for carrying rainwater) with the maximum possible cross-sectional are......Sat, 22 Aug 2026 14:35:15 -0400https://liberty.io/using-proof-by-contradiction-vs-proof-of-the-contrapositive.htmlhttps://liberty.io/using-proof-by-contradiction-vs-proof-of-the-contrapositive.htmlWhat is the difference between a "proof by contradiction" and "proving the contrapositive"? Intuitive, it feels like doing the exact same thing. And when I compare an exercise, one person proves by ...Sat, 22 Aug 2026 14:34:20 -0400https://liberty.io/ratios-of-median-mean-and-standard-deviation-iqr-in-a-normal-distribut.htmlhttps://liberty.io/ratios-of-median-mean-and-standard-deviation-iqr-in-a-normal-distribut.htmlI have some queries on the following question For a normal distribution, find the ratios of: (a) $\frac {\mbox{median}}{\mbox{mean}}$ (b) $\frac {\mbox{standard deviation}}{\mbox{interquartile r......Sat, 22 Aug 2026 14:33:09 -0400https://liberty.io/what-is-observation.htmlhttps://liberty.io/what-is-observation.htmlOften in mathematical writing one encounters texts like ''..we observe that this-and-that..''. Also one may find a review report basically saying ''..the paper is just a chain of observations...''.......Sat, 22 Aug 2026 14:29:53 -0400https://liberty.io/how-to-efficiently-compute-17-23-mod-31-by-hand.htmlhttps://liberty.io/how-to-efficiently-compute-17-23-mod-31-by-hand.htmlI could use that $17^{2} \equiv 10 (\mod 31)$ and express $17^{23}$ as $17^{16}.17^{4}.17^{3} = (((17^2)^2)^2)^2.(17^2)^2.17^2.17$ and take advantage of the fact that I can more easily work with po......Sat, 22 Aug 2026 14:24:11 -0400https://liberty.io/show-that-2222-5555-5555-2222-is-divisible-by-7-duplicate.htmlhttps://liberty.io/show-that-2222-5555-5555-2222-is-divisible-by-7-duplicate.htmlShow that $2222^ {5555} + 5555 ^ {2222}$ is divisible by $7$. I tried factorizing but it didnt lead to anything. Can divisibility rules be used? Any ideas please tell me....Sat, 22 Aug 2026 14:22:56 -0400https://liberty.io/how-to-make-a-geodesic-dome-2v.htmlhttps://liberty.io/how-to-make-a-geodesic-dome-2v.htmlI am trying to make a geodesic dome 2v in Solidwork just for fun. I use this dome calculator : http://www.domerama.com/calculators/2v-geodesic-dome-calculator/ The problem is once I start the ass......Sat, 22 Aug 2026 14:17:10 -0400https://liberty.io/example-of-degenerate-scalar-product.htmlhttps://liberty.io/example-of-degenerate-scalar-product.htmlSuppose I have a vector space $V$ over $\mathbb{R}$. A scalar product on $V$ is a symmetric bilinear form, so such that $\langle v,w\rangle = \langle w,v\rangle~ \forall v,w \in V$. To my ...Sat, 22 Aug 2026 14:11:00 -0400https://liberty.io/finite-measure-space-sigma-finite-measure-space.htmlhttps://liberty.io/finite-measure-space-sigma-finite-measure-space.htmlA measure space $(X, \Sigma, \mu)$ is finite if $\mu(X)<\infty$. It is equivalent to saying that $(X, \Sigma, \mu)$ is finite if $\mu(E)<\infty$ for all $E \in \Sigma$ A measure space $(X, \S......Sat, 22 Aug 2026 13:54:43 -0400https://liberty.io/general-gaussian-integrals-over-the-positive-real-axis.htmlhttps://liberty.io/general-gaussian-integrals-over-the-positive-real-axis.htmlEveryone has a special memory from their multivariable calc class deriving the famous Gaussian integral: $$ \int_0^\infty e^{-x^2} \,dx = \frac{\sqrt\pi}{2}$$ A more general case is easy to find on......Sat, 22 Aug 2026 13:37:35 -0400https://liberty.io/example-of-functions-that-are-onto-but-not-one-to-one.htmlhttps://liberty.io/example-of-functions-that-are-onto-but-not-one-to-one.htmlI have been preparing for my exam tomorrow and I just can't think of a function that is onto but not one-to-one. I know an absolute function isn't one-to-one or onto. And an example of a one-to-one ...Sat, 22 Aug 2026 13:21:10 -0400https://liberty.io/matrix-derivative-ax-b-t-ax-b.htmlhttps://liberty.io/matrix-derivative-ax-b-t-ax-b.htmlI am trying to find the minimum of $(Ax-b)^T(Ax-b)$ but I am not sure whether I am taking the derivative of this expression properly. What I did is the following: \begin{align*} \frac{\delta}{\del......Sat, 22 Aug 2026 13:18:51 -0400https://liberty.io/prove-that-1-2-1-4-1-8-1-duplicate.htmlhttps://liberty.io/prove-that-1-2-1-4-1-8-1-duplicate.htmlI've often heard that instead of adding up to a little less than one, 1/2 + 1/4 + 1/8... = 1. Is there any way to prove this using equations without using Sigma, or is it just an accepted fact? I n......Sat, 22 Aug 2026 13:12:44 -0400https://liberty.io/what-are-some-intuitive-ways-to-find-a-3-times-3-permutation-matrix-wi.htmlhttps://liberty.io/what-are-some-intuitive-ways-to-find-a-3-times-3-permutation-matrix-wi.htmlFind a $3\times3$ permutation matrix with $P^3 = I$, $P \ne I$? I reduced the above problem to $P^T = P^2$ and tried solving for all $6$ $3 \times 3$ permutation matrices which yielded $$P = \begin{...Sat, 22 Aug 2026 13:10:52 -0400https://liberty.io/4-gon-with-two-opposite-sides-congruent-and-two-opposite-angles-congru.htmlhttps://liberty.io/4-gon-with-two-opposite-sides-congruent-and-two-opposite-angles-congru.htmlMy kid's geometry classmate tried to use the following "theorem" in a proof: If a quadrilateral has a pair of opposite sides congruent and a pair of opposite angles congruent, it's a parallelogr......Sat, 22 Aug 2026 12:57:45 -0400https://liberty.io/compactness-of-function-space.htmlhttps://liberty.io/compactness-of-function-space.htmlLet $X$ be the set of all continuous functions from the unit interval $[0, 1]$ to itself. (1) Prove that $X$ is not compact under the pointwise convergence topology. (2) Prove that $X$ is compact ...Sat, 22 Aug 2026 12:48:15 -0400https://liberty.io/proof-of-intermediate-value-theorem-using-bisection.htmlhttps://liberty.io/proof-of-intermediate-value-theorem-using-bisection.htmlProof of the special case of the Intermediate Value Theorem: Let $f$ be a continuous function on $[a, b]$ and suppose that: $$ f(a) < 0 < f(b) $$ Then there exists a number $c$ in $(a, b)$ such ...Sat, 22 Aug 2026 12:46:47 -0400https://liberty.io/proof-of-c-a-b-for-a-right-triangle.htmlhttps://liberty.io/proof-of-c-a-b-for-a-right-triangle.htmlI am asked to "prove or disprove the following statement using contradiction" --> If a, b, c are the sides of a right traingle and c is the hypotenuse then $c<a+b$. Here is my proposed proof: P......Sat, 22 Aug 2026 12:35:20 -0400https://liberty.io/surface-area-of-cone.htmlhttps://liberty.io/surface-area-of-cone.htmlthanks for any help. I'm trying to find the surface area of a cone via integration. I know that the parametric equation of a cone is $$x=u\cos(p) \\ y=u\sin(p) \\ z=u$$ So as a vector, $\vec{R} = \...Sat, 22 Aug 2026 12:20:52 -0400https://liberty.io/solution-for-exponential-function-039-s-functional-equation-by-using-a.htmlhttps://liberty.io/solution-for-exponential-function-039-s-functional-equation-by-using-a.htmllet $f(0)=1$ and $f'(0)=1$. and $f(x+y)=f(x)f(y)$ for $x,y\in R$. How can I found $f(x)$ by using a definition of derivative?...Sat, 22 Aug 2026 12:15:48 -0400https://liberty.io/finding-unbiased-point-estimate-of-population-variance.htmlhttps://liberty.io/finding-unbiased-point-estimate-of-population-variance.htmlQ.The contents of each of a random sample of 100 cans of a soft drink are measured. The results have a mean of 331.28 ml and a standard deviation of 2.97 ml. Show that an unbiased estimate of the ...Sat, 22 Aug 2026 12:13:32 -0400https://liberty.io/definition-of-power-set-uses-strict-subset-duplicate.htmlhttps://liberty.io/definition-of-power-set-uses-strict-subset-duplicate.htmlIn the following lecture notes one finds the following: Examples. The following are all examples of σ-algebras. (HW: check this in each case) • Let P(X) denote the collection of all subsets of X, ......Sat, 22 Aug 2026 11:53:42 -0400https://liberty.io/how-to-express-sqrt-x-1.htmlhttps://liberty.io/how-to-express-sqrt-x-1.htmlHow would one express a solution to $\sqrt{x} =-1$? I just read that a solution to the above equation cannot be expressed in the form of complex numbers, really interested in any additional inform......Sat, 22 Aug 2026 11:52:26 -0400https://liberty.io/what-039-s-a-proof-that-the-angles-of-a-triangle-add-up-to-180.htmlhttps://liberty.io/what-039-s-a-proof-that-the-angles-of-a-triangle-add-up-to-180.htmlBack in grade school, I had a solution involving "folding the triangle" into a rectangle half the area, and seeing that all the angles met at a point: However, now that I'm in university, I'm not ...Sat, 22 Aug 2026 11:47:23 -0400https://liberty.io/how-to-find-the-inverse-cosine-without-a-calculator.htmlhttps://liberty.io/how-to-find-the-inverse-cosine-without-a-calculator.htmlHow to find the inverse of: $$\cos(c)=\frac{1}{3}$$ In other words, i'm trying to solve for c and without a calculator. If it's hard or not possible, then how would you go about solving inverses in ...Sat, 22 Aug 2026 11:39:18 -0400https://liberty.io/how-does-e-i-x-produce-rotation-around-the-imaginary-unit-circle.htmlhttps://liberty.io/how-does-e-i-x-produce-rotation-around-the-imaginary-unit-circle.htmlEuler’s formula states that $e^{i x} = \cos(x) + i \sin(x)$. I can see from the MacLaurin Expansion that this is indeed true; however, I don’t intuitively understand how raising $e$ to the power ......Sat, 22 Aug 2026 11:29:17 -0400https://liberty.io/what-is-additive-homomorphic-encryption-and-multiplicative-homomorphic.htmlhttps://liberty.io/what-is-additive-homomorphic-encryption-and-multiplicative-homomorphic.htmlExplain in brief about additive homomorphic encryption and multiplicative homomorphic encryption. I have read about them but I could not properly understand. So would you please elaborate additive ...Sat, 22 Aug 2026 11:28:20 -0400https://liberty.io/number-of-surjective-functions-from-a-to-b.htmlhttps://liberty.io/number-of-surjective-functions-from-a-to-b.htmlAm I on the right track? I am not sure about my reasoning... Number of surjective functions from $A$ to $B$ $$A = \{1,2,3,4\} ; B = \{a,b,c\}$$ We must count the surjective functions, meaning the ...Sat, 22 Aug 2026 11:27:59 -0400https://liberty.io/convolution-integral-limits.htmlhttps://liberty.io/convolution-integral-limits.htmlThere are 2 types of convolution: The limit of the integral is from minus infinity to plus infinity The limit is from zero to t. When we use the first and when we use the second? $$\int f(\tau)g......Sat, 22 Aug 2026 11:27:48 -0400https://liberty.io/definition-of-curl-in-mathbb-r-n.htmlhttps://liberty.io/definition-of-curl-in-mathbb-r-n.htmlIn calculus we saw that how to define curl in $\mathbb{R}^3$. My question is about the generalization; I mean is there any way that one can define curl in $\mathbb{R}^n$? If yes, how can I do it?...Sat, 22 Aug 2026 11:24:06 -0400https://liberty.io/is-my-interpretation-of-direct-comparison-test-correct.htmlhttps://liberty.io/is-my-interpretation-of-direct-comparison-test-correct.htmlI had been studying convergence and divergence of series in Calculus. This is the problem that I've to prove using direct comparison test. Prove that the series $\sum_{n=1}^{\infty} \frac{1}{2^......Sat, 22 Aug 2026 11:18:28 -0400https://liberty.io/why-does-0-0-mean-there-it-a-hole-and-not-an-asymptote.htmlhttps://liberty.io/why-does-0-0-mean-there-it-a-hole-and-not-an-asymptote.htmlhttps://www.khanacademy.org/math/algebra2/rational-expressions/rational-function-graphing/v/finding-asymptotes-example At around 8 minutes he says that we want a number which makes only the denomi......Sat, 22 Aug 2026 11:07:57 -0400https://liberty.io/log-of-a-negative-number.htmlhttps://liberty.io/log-of-a-negative-number.htmlWhy is the log of $x$ equal to the log of the absolute value of $x$ plus $i$ times $\pi$? $$\log(x)=\log(|x|)+i\pi\text{, for }x < 0$$ Where does the $\pi$ come from? Is it from a logarithmic ...Sat, 22 Aug 2026 11:03:54 -0400https://liberty.io/what-is-the-relationship-between-graphs-graph-theory-and-logic.htmlhttps://liberty.io/what-is-the-relationship-between-graphs-graph-theory-and-logic.htmlI have read that Graph theory is useful to mathematical logic, but I don’t understand how. How exactly does a logic of graphs differ from standard predicate logic? Here is an example from an arti......Sat, 22 Aug 2026 10:57:22 -0400https://liberty.io/perfect-solutions-to-the-kissing-number-problem-besides-in-dimensions-.htmlhttps://liberty.io/perfect-solutions-to-the-kissing-number-problem-besides-in-dimensions-.htmlThe kissing number problem asks how many n dimensional unit spheres can fit around a central one with no overlapping; a natural question is in what dimensions can this be done so that there is no e......Sat, 22 Aug 2026 10:53:34 -0400https://liberty.io/opposite-of-mod-in-a-equation.htmlhttps://liberty.io/opposite-of-mod-in-a-equation.htmlI am trying to solve this module equation (due to a cryptography program that I am coding) but cannot figure it out how: X + 7 % 10 = 6 How Do i solve this? Thanks in advance....Sat, 22 Aug 2026 10:52:54 -0400https://liberty.io/definition-clarification-of-graph-embeddings.htmlhttps://liberty.io/definition-clarification-of-graph-embeddings.htmlRecently I started reading about graph embeddings, but I am unable to grasp its definition from Wikipedia. Can anyone explain this term with an example....Sat, 22 Aug 2026 10:48:37 -0400