https://liberty.ioCurated star highlights with fresh energy.en-usSat, 22 Aug 2026 12:19:43 -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 -0400https://liberty.io/figuring-out-the-amount-of-039-straight-edge-039-pieces-in-a-puzzle.htmlhttps://liberty.io/figuring-out-the-amount-of-039-straight-edge-039-pieces-in-a-puzzle.htmlI was wondering if there was any set way to determine the number of 'straight edge' pieces in a puzzle, assuming the pieces are all in neat rows and columns? Does the ratio of edge pieces to middle ...Sat, 22 Aug 2026 10:48:20 -0400https://liberty.io/what-is-the-largest-prime-number-in-the-denominator-of-a-fraction-that.htmlhttps://liberty.io/what-is-the-largest-prime-number-in-the-denominator-of-a-fraction-that.htmlI was studying a Target question for Math League competitions, and after a few hours of pondering, I finally came up with the following method of solving the mentioned problem: For any decimal, it......Sat, 22 Aug 2026 10:46:55 -0400https://liberty.io/positive-definiteness-of-the-schur-complement.htmlhttps://liberty.io/positive-definiteness-of-the-schur-complement.htmlLet $M$ to be a real-valued symmetric and positive-definite (PD) matrix (also sparse and banded if it helps) $$ M= \begin{bmatrix} A & B\\ B^T & D \end{bmatrix} $$ Under what conditions the ...Sat, 22 Aug 2026 10:24:17 -0400https://liberty.io/calculating-the-derivative-of-csc-2-4x.htmlhttps://liberty.io/calculating-the-derivative-of-csc-2-4x.htmlI am having problems calculating derivatives of squared trigonometric functions. $$f(x) = \csc^2(4x)$$ Let's see... This is equivalent to $$f(x) = \csc(4x)\cdot \csc(4x)$$ Now, to get the deriv......Sat, 22 Aug 2026 10:17:07 -0400https://liberty.io/how-to-explain-what-is-wrong-in-this.htmlhttps://liberty.io/how-to-explain-what-is-wrong-in-this.htmlMy friend showed this to me and I instantly know that this is wrong. However, I cannot explain why this is wrong to my friend. Question. Prove $\displaystyle \frac{100-100}{100-100} = 2.$ A......Sat, 22 Aug 2026 09:59:43 -0400https://liberty.io/what-is-the-meaning-of-c-0-0-1-c-0-1-and-c-1-0-1.htmlhttps://liberty.io/what-is-the-meaning-of-c-0-0-1-c-0-1-and-c-1-0-1.htmlSome difficulty to understand the symbol what is the meaning of $C^0[0,1]$ , $C[0,1]$ and $C^1[0,1]$ ?...Sat, 22 Aug 2026 09:56:45 -0400https://liberty.io/online-visualization-tool-for-planes-spans-in-linear-algebra.htmlhttps://liberty.io/online-visualization-tool-for-planes-spans-in-linear-algebra.htmlI would like to visualize planes in 3D as I start learning linear algebra, to build a solid foundation. Surprisingly, I have been unable to find an online tool (website/web app) to visualize planes......Sat, 22 Aug 2026 09:54:08 -0400https://liberty.io/how-to-find-the-area-under-a-semicircle-using-integration-closed.htmlhttps://liberty.io/how-to-find-the-area-under-a-semicircle-using-integration-closed.htmlHow would I go about finding the area under a semicircle? I know that to use integration the formula is $\int_a^b f(x) \mathrm{d}x.$However, when I put this into my graphing calculator it doesn't w......Sat, 22 Aug 2026 09:32:37 -0400https://liberty.io/proving-monge-039-s-theorem-using-menelaus-039.htmlhttps://liberty.io/proving-monge-039-s-theorem-using-menelaus-039.htmlMonge's theorem states that for any three circles in a plane, none of which is completely inside one of the others, the intersection points of each of the three pairs of external tangent lines are ...Sat, 22 Aug 2026 09:28:35 -0400https://liberty.io/finding-a-joint-moment-generating-function.htmlhttps://liberty.io/finding-a-joint-moment-generating-function.htmlPlease consider the following problem and my solution to it: Problem: Let $(X,Y)$ be a continues bivariate r.v. with joint pdf \begin{eqnarray*} f_{XY}(x,y) &=& \begin{cases} e^{-(x+y)} &a......Sat, 22 Aug 2026 09:10:25 -0400https://liberty.io/norm-of-gradient-in-gradient-descent.htmlhttps://liberty.io/norm-of-gradient-in-gradient-descent.htmlThis question discusses the size of gradient in gradient descent. Some examples were pointed to show it is not necessarily the case that gradient will decrease, for example, $f(x) = \sqrt{|x|}$ or ......Sat, 22 Aug 2026 09:07:56 -0400https://liberty.io/how-does-the-reduced-cone-look-like.htmlhttps://liberty.io/how-does-the-reduced-cone-look-like.htmlI have to find in terms of standard spaces (that is interval, circle, disk, sphere, cone, etc.) the reduced cone (i.e. cone in category of pointed spaces) $\operatorname{Cone}(*)$ where $\{*\}$ is ......Sat, 22 Aug 2026 09:03:59 -0400https://liberty.io/binomial-distribution-word-problem.htmlhttps://liberty.io/binomial-distribution-word-problem.htmlA student gets at least 8 hours of sleep 45% of the nights; the sleeping schedule is independent from night to night. Let Xi indicate whether the student gets at least 8 hours of sleep during the n......Sat, 22 Aug 2026 09:00:50 -0400https://liberty.io/what-is-the-symbol-for-coincident-in-geometry.htmlhttps://liberty.io/what-is-the-symbol-for-coincident-in-geometry.htmlI am looking for a symbol to say that one geometrical figure coincides with another without writing the phrase "is coincident with." For example, the altitude $a$ of an equilateral triangle coinci......Sat, 22 Aug 2026 08:55:19 -0400https://liberty.io/get-polynomial-function-from-3-points.htmlhttps://liberty.io/get-polynomial-function-from-3-points.htmlI need to understand how to define a polynomial function from 3 given points. Everything I found on the web so far is either too complicated or the reversed way around. (how to get points with a gi......Sat, 22 Aug 2026 08:54:41 -0400https://liberty.io/how-does-squaring-both-sides-of-an-equation-lead-to-extraneous-solutio.htmlhttps://liberty.io/how-does-squaring-both-sides-of-an-equation-lead-to-extraneous-solutio.htmlLet’s say I have $x = x + 1$, which is a false statment for real $x$; why can I solve for real $x$ when I square both sides of the equation, giving $x^2=(x+1)^2$?...Sat, 22 Aug 2026 08:50:43 -0400https://liberty.io/convert-y-2-4-x-1-to-a-polar-equation.htmlhttps://liberty.io/convert-y-2-4-x-1-to-a-polar-equation.htmlI'm trying to convert the rectangular cartesian equation $$ y^2 = 4(x + 1) $$ to a polar equation. After replacing $y = r \sin \theta$ and $x = r \cos \theta$, I get $$ r^2 \sin^2 \theta = 4(r \......Sat, 22 Aug 2026 08:40:36 -0400https://liberty.io/what-is-the-perimeter-of-a-sector.htmlhttps://liberty.io/what-is-the-perimeter-of-a-sector.htmlI don't understand this. So we have: \begin{align} r &= 12 \color{gray}{\text{ (radius of circle)}} \\ d &= 24 \text{ (r}\times2) \color{gray}{\text{ (diameter of circle)}} \\ c &= 24\......Sat, 22 Aug 2026 08:37:00 -0400https://liberty.io/symbols-for-odd-and-even.htmlhttps://liberty.io/symbols-for-odd-and-even.htmlLet $A$ be a sequence of letters $\langle a,b,c,d,e,f \rangle$. I want to create two subsequences, one with the values with odd index and other with the values with even index: $A_\mathrm{odd} = \l......Sat, 22 Aug 2026 08:33:11 -0400https://liberty.io/context-free-grammar-for-palindrome-over-all-terminals.htmlhttps://liberty.io/context-free-grammar-for-palindrome-over-all-terminals.htmlI've found various examples of context-free grammars for palindromes but they all seem to hardcode the rules to be of the form "terminal Statement (same) terminal" For example, if $T \...Sat, 22 Aug 2026 08:32:49 -0400https://liberty.io/what-is-the-square-root-of-infinity-and-what-is-infinity-2.htmlhttps://liberty.io/what-is-the-square-root-of-infinity-and-what-is-infinity-2.htmlIn rigorous mathematics, how can one formalize the notion of a "square root of infinity", as well as that of "infinity squared"?...Sat, 22 Aug 2026 08:27:23 -0400https://liberty.io/what-is-the-best-calculus-book-for-a-strong-introdutory-course.htmlhttps://liberty.io/what-is-the-best-calculus-book-for-a-strong-introdutory-course.htmlI'm teaching for a small group of future undergraduate science students. I'm from Brazil, so here, we don't see any calculus in school, Just definition of function and some properties of most known ...Sat, 22 Aug 2026 08:26:03 -0400https://liberty.io/coin-toss-probability-using-bayes-theorem.htmlhttps://liberty.io/coin-toss-probability-using-bayes-theorem.htmlIn a box, there are the same number of two kinds of coins: the fair coins (50% chance for head) and the biased coins (70% chance for head). One person randomly selected a coin and tosses it twice. ......Sat, 22 Aug 2026 08:23:09 -0400https://liberty.io/2-easy-gre-questions.htmlhttps://liberty.io/2-easy-gre-questions.htmlI've been having trouble with these two questions. The first is simple interest, the second is rate. I'm sure they're easy but I can't focus on getting the solution because I'm terrible at focusing......Sat, 22 Aug 2026 08:19:15 -0400https://liberty.io/does-sum-limits-x-1-infty-sin-x-converge.htmlhttps://liberty.io/does-sum-limits-x-1-infty-sin-x-converge.htmlI received a task to find out whether the following series converges: $$\sum_{x=1}^\infty\sin(x)$$ On first look it seems simple, but as I keep thinking about it, there's not a single lemma or ...Sat, 22 Aug 2026 08:19:04 -0400https://liberty.io/what-is-the-minimum-possible-exterior-surface-area-of-an-open-glassed-.htmlhttps://liberty.io/what-is-the-minimum-possible-exterior-surface-area-of-an-open-glassed-.htmlThis is a question from Khan Academy I do not understand. An open-topped glass aquarium with a square base is designed to hold $32$ space cubic feet of water. What is the minimum possible exterior ...Sat, 22 Aug 2026 08:13:46 -0400https://liberty.io/finding-integer-solutions-to-y-2-x-3-2.htmlhttps://liberty.io/finding-integer-solutions-to-y-2-x-3-2.htmlI have the equation: $$y^2=x^3-2$$ It seems to be deceivingly simple, yet I simply cannot crack it. It is obviously equivalent to finding a perfect cube that is two more than a perfect square, and a ...Sat, 22 Aug 2026 08:11:45 -0400https://liberty.io/terminal-value-formula-in-the-discounted-cash-flow-dcf-valuation.htmlhttps://liberty.io/terminal-value-formula-in-the-discounted-cash-flow-dcf-valuation.htmlI have just started with Finance and have a physics background. However, I am struggling to understand the formula for the terminal value in a discounted cash flow valuation (DCF valuation) accordi......Sat, 22 Aug 2026 08:06:34 -0400https://liberty.io/disconnected-connected-graphs.htmlhttps://liberty.io/disconnected-connected-graphs.htmlDetermine whether the statements below are true or false. If the statement is true, then prove it; and if it is false, give a counterexample. (a) Every disconnected graph has a vertex of degree 0.......Sat, 22 Aug 2026 08:06:19 -0400https://liberty.io/finding-the-number-of-surfaces-containing-a-curve-given-its-parametriz.htmlhttps://liberty.io/finding-the-number-of-surfaces-containing-a-curve-given-its-parametriz.htmlIf there is a parametrized curve: $$r(t) = t^2\,i + ln(t)\,j + \frac{1}{t}\,k$$ then as I understand it, to find surfaces that contain the curve, you can solve for $t$ and find three equations, each ...Sat, 22 Aug 2026 08:00:20 -0400https://liberty.io/is-there-something-like-a-normal-distribution-model-for-discrete-proba.htmlhttps://liberty.io/is-there-something-like-a-normal-distribution-model-for-discrete-proba.htmlIf one wants to find the probability that a continuous random variable will fall within a range of $a \leq X \leq b$, based on a mean value $\mu$, and a deviation of $\sigma$, he would integrate the ...Sat, 22 Aug 2026 07:59:51 -0400https://liberty.io/finding-moment-of-inertia-rotational-inertia-i-using-integration.htmlhttps://liberty.io/finding-moment-of-inertia-rotational-inertia-i-using-integration.htmlI just came back from my Introduction to Rotational Kinematics class, and one of the important concepts they described was Rotational Inertia, or Moment of Inertia. It's basically the equivalent of......Sat, 22 Aug 2026 07:50:01 -0400https://liberty.io/conditional-introduction-rule.htmlhttps://liberty.io/conditional-introduction-rule.htmlIn the derivation (the image below) the author shows that given the premise $\neg S \land \neg J$, the conclusion is $S \implies J$. All these deductive maneuver for concluding implications I find ...Sat, 22 Aug 2026 07:48:29 -0400https://liberty.io/how-can-we-make-11-non-isomorphic-graphs-on-4-vertices.htmlhttps://liberty.io/how-can-we-make-11-non-isomorphic-graphs-on-4-vertices.htmlHow can we draw all the non-isomorphic graphs on $4$ vertices ? But it is mentioned that $ 11 $ graphs are possible....Sat, 22 Aug 2026 07:47:34 -0400https://liberty.io/how-are-asymptotes-actually-defined-in-rigorous-mathematics.htmlhttps://liberty.io/how-are-asymptotes-actually-defined-in-rigorous-mathematics.htmlThis question is coming from the analytic geometry viewpoint. Please ignore the viewpoint of algebraic geometry here, unless that viewpoint is somehow able to handle non-algebraic curves like $x \m......Sat, 22 Aug 2026 07:29:24 -0400https://liberty.io/find-the-value-of-the-expression.htmlhttps://liberty.io/find-the-value-of-the-expression.htmlThe expression $ax^2 + bx + 1$ takes the values $1$ and $4$ when $x$ takes the values $2$ and $3$ respectively. How can we find the value of the expression when $x$ takes the value of $4$?...Sat, 22 Aug 2026 07:16:04 -0400https://liberty.io/determining-whether-cartesian-first-order-tensor.htmlhttps://liberty.io/determining-whether-cartesian-first-order-tensor.htmlI'm struggeling to understand when we call a certain tuple a vector or first order Cartesian tensor $\mathbf v$. Riley, Hobson and Riley (3rd) states that when you apply a rotation, the new compone......Sat, 22 Aug 2026 07:13:08 -0400https://liberty.io/why-is-the-determinant-of-a-symplectic-matrix-1.htmlhttps://liberty.io/why-is-the-determinant-of-a-symplectic-matrix-1.htmlSuppose $A \in M_{2n}(\mathbb{R})$. and$$J=\begin{pmatrix} 0 & E_n\\ -E_n&0 \end{pmatrix}$$ where $E_n$ represents identity matrix. if $A$ satisfies $$AJA^T=J.$$ How to figure out $$\det......Sat, 22 Aug 2026 07:09:18 -0400https://liberty.io/the-interior-of-a-connected-set-in-mathbb-r-k.htmlhttps://liberty.io/the-interior-of-a-connected-set-in-mathbb-r-k.htmlIs the interior of a connected set in $\mathbb R^k$ connected?...Sat, 22 Aug 2026 07:02:22 -0400https://liberty.io/intuitively-what-is-the-difference-between-eigendecomposition-and-sing.htmlhttps://liberty.io/intuitively-what-is-the-difference-between-eigendecomposition-and-sing.htmlI'm trying to intuitively understand the difference between SVD and eigendecomposition. From my understanding, eigendecomposition seeks to describe a linear transformation as a sequence of three b......Sat, 22 Aug 2026 07:02:16 -0400https://liberty.io/i-have-a-d3-three-sided-die-and-i-m-trying-to-get-the-highest-possible.htmlhttps://liberty.io/i-have-a-d3-three-sided-die-and-i-m-trying-to-get-the-highest-possible.htmlAs the title says, if I roll a 2 on a d3 and am allowed to reroll it, is there an advantage to re-rolling? It seems like the choice doesn’t matter, or is purely preferential but what is wrong with ...Sat, 22 Aug 2026 07:01:38 -0400https://liberty.io/empty-and-trivial-graph-diestel-039-s-book.htmlhttps://liberty.io/empty-and-trivial-graph-diestel-039-s-book.htmlThe number of vertices of a graph $G$ is its order, written as $|G|$; For the empty graph $(\varnothing, \varnothing)$ we simply write $\varnothing$. A graph of order $0$ or $1$ is called trivial. ......Sat, 22 Aug 2026 06:54:17 -0400https://liberty.io/closure-of-integers.htmlhttps://liberty.io/closure-of-integers.htmlFellow Mathers, is the fact that the Integers are closed under addition and multiplication... ∀ a,b ∈ Z , a + b = c where c ∈ Z ∀ a,b ∈ Z , ab = c where c ∈ Z ...universal axioms, theorems, depen......Sat, 22 Aug 2026 06:50:13 -0400https://liberty.io/dense-sequence-definition.htmlhttps://liberty.io/dense-sequence-definition.htmlA subset $A$ of a space $X$ is called dense if every point $x$ in $X$ either belongs to $A$ or is a limit point of $A$ ; that is, the closure of $A$ constitutes the whole set $X$. That the definiti......Sat, 22 Aug 2026 06:45:56 -0400https://liberty.io/2-norm-vs-operator-norm.htmlhttps://liberty.io/2-norm-vs-operator-norm.htmlI have read that we define the "2-norm" of a matrix as $$\max_i \,{|\sigma_i|},$$ which I have also heard called the "operator norm" (here $\sigma_i$ are the singular values). Also we have the......Sat, 22 Aug 2026 06:33:32 -0400