https://liberty.ioCurated star highlights with fresh energy.en-usTue, 11 Aug 2026 18:47:47 -0400https://liberty.io/what-039-s-the-least-positive-integer-that-is-not-a-factor-of-25-and-i.htmlhttps://liberty.io/what-039-s-the-least-positive-integer-that-is-not-a-factor-of-25-and-i.htmlIt's more of a question of what's the definition of factor. To my understanding, for a number 25, 1,5,25 would be the factor because 1*25=25and 5*5=25. It is a question from GRE official guide book......Tue, 11 Aug 2026 22:40:53 -0400https://liberty.io/how-to-find-the-determinant-of-a-matrix-using-the-pa-lu-method.htmlhttps://liberty.io/how-to-find-the-determinant-of-a-matrix-using-the-pa-lu-method.htmlI am currently trying to write a code in matlab for a course I'm taking. My program needs to implement the PA = LU method to solve a matrix and compute it's determinant. I am not allowed to use the......Tue, 11 Aug 2026 22:11:57 -0400https://liberty.io/conjugates-of-complex-numbers.htmlhttps://liberty.io/conjugates-of-complex-numbers.htmlThis is from a textbook: $w=i\frac{1-z}{1+z}$ where $z$ is also a complex number Then $$\mathrm{Im}(w)=\frac{w-\bar w}{2i}$$ Then $$\mathrm{Im}(w)=\frac 1 {2i} \left(i\frac{1-z}{1+z}+i......Tue, 11 Aug 2026 22:11:08 -0400https://liberty.io/how-do-i-simplify-the-expression-a-1-b-1-1-closed.htmlhttps://liberty.io/how-do-i-simplify-the-expression-a-1-b-1-1-closed.htmlHow do I simplify the expression ... $$(a^{-1} + b^{-1}) ^{-1}$$ ?...Tue, 11 Aug 2026 22:07:01 -0400https://liberty.io/composite-trapezoidal-rule-and-simpson-rule-question.htmlhttps://liberty.io/composite-trapezoidal-rule-and-simpson-rule-question.htmlWe have the following integral $$\int_{1}^{3} \frac x{x^2+4}\; dx$$ and $n=6$. I have to approximate it using composite trapezoidal rule and then composite simpson rule. Using trapezoidal rule: $$\...Tue, 11 Aug 2026 22:06:12 -0400https://liberty.io/circle-puzzle-geometry.htmlhttps://liberty.io/circle-puzzle-geometry.htmlTwo friends are playing a game. One friend stands in the middle of a circle radius 100m. His objective is to leave the circle. He may take one step at a time, distance 1m, in any direction. However......Tue, 11 Aug 2026 22:05:41 -0400https://liberty.io/chain-rule-partial-derivatives-dw-dt.htmlhttps://liberty.io/chain-rule-partial-derivatives-dw-dt.htmlExpress dw/dt as function of t, if w = xy, x = cos t, y = sen t, z = t My first step is to sketch the tree. w - x - t w - y - t w - z - t dw/dx = y dw/dy = x dw/dz = 1 Then: dx/dt = -sen......Tue, 11 Aug 2026 22:00:38 -0400https://liberty.io/writing-a-general-formula-for-nth-derivative-of-cos-ax.htmlhttps://liberty.io/writing-a-general-formula-for-nth-derivative-of-cos-ax.htmlSo as stated in the title I have to create a general formula for nth derivative of $f(x) = \cos(ax)$ and proving it with induction. I get $f(x) = \cos(ax)$ $f'(x) = -a\sin(ax)$ $f''(x) = -a^2\......Tue, 11 Aug 2026 21:55:20 -0400https://liberty.io/multiplying-reciprocal-fractions-with-exponents.htmlhttps://liberty.io/multiplying-reciprocal-fractions-with-exponents.htmlThe question is as follows: If we multiply a fraction with its reciprocal then we get 1. So by this logic this should be equal to 1^(a/a+b). But, the solutions I have been provided with states the ...Tue, 11 Aug 2026 21:54:59 -0400https://liberty.io/what-039-s-the-difference-between-early-transcendentals-and-late-trans.htmlhttps://liberty.io/what-039-s-the-difference-between-early-transcendentals-and-late-trans.htmlAnton, Bivens, and Davis have written calculus books with late transcendentals and Stewart has a calculus book with early transcendentals. What's this all about? edit 1: (Both terms show up in ......Tue, 11 Aug 2026 21:45:10 -0400https://liberty.io/what-is-the-base-of-log-x-duplicate.htmlhttps://liberty.io/what-is-the-base-of-log-x-duplicate.htmlI've seen "$\log x$" being used in some papers (and by Wolfram|Alpha), and I was confused because so far I have only ever seen the $\log$ used with a base ( so e.g. $\log_y x$). Am I correct that ......Tue, 11 Aug 2026 21:43:37 -0400https://liberty.io/why-does-the-mandelbrot-shape-show-up-in-other-fractals.htmlhttps://liberty.io/why-does-the-mandelbrot-shape-show-up-in-other-fractals.htmlIn the pictures below, the Collatz map fractal includes parts resembling the Mandelbrot set. Why? Do other fractals do so? The Mandelbrot set From Wikimedia Commons Part of the Collatz map fract......Tue, 11 Aug 2026 21:41:21 -0400https://liberty.io/how-many-subsets-does-the-set-1-2-3-n-have-that-contain-no-three-conse.htmlhttps://liberty.io/how-many-subsets-does-the-set-1-2-3-n-have-that-contain-no-three-conse.htmlQ: How many subsets does the set $\{1,2,3,...n\}$ have that contain no three consecutive integers? Find a recurrence. I don’t think I understand the solution. According to the solution if I take......Tue, 11 Aug 2026 21:38:43 -0400https://liberty.io/cyclic-and-circular-permutation-are-they-different-concepts.htmlhttps://liberty.io/cyclic-and-circular-permutation-are-they-different-concepts.html"Cyclic" and "Circular" permutations - are these two different concepts? I have been reading about permutation and encountering them in many places. What are the definitions of them, in simple Engl......Tue, 11 Aug 2026 21:37:26 -0400https://liberty.io/good-books-on-complex-numbers.htmlhttps://liberty.io/good-books-on-complex-numbers.htmlI searched on this site but I found posts asking for books on complex analysis which I guess is not what I want. I want a book on problems concerning with roots of unity, argand plane, exponentia......Tue, 11 Aug 2026 21:36:52 -0400https://liberty.io/if-we-have-a-n-times-n-and-ax-xa-a-how-do-we-show-that-det-a-0.htmlhttps://liberty.io/if-we-have-a-n-times-n-and-ax-xa-a-how-do-we-show-that-det-a-0.htmlIf we have $A_{n \times n}$ and $AX-XA=A$, how do we show that $\det(A)=0$? I've tried taking the trace on both sides to get $$tr(AX)-tr(XA)=tr(A)$$ $$tr(AX)-tr(AX)=tr(A)$$ $$0=tr(A)$$ However, that ...Tue, 11 Aug 2026 21:07:46 -0400https://liberty.io/calculate-the-divergence-of-the-polar-coordinate-vector-field-partial-.htmlhttps://liberty.io/calculate-the-divergence-of-the-polar-coordinate-vector-field-partial-.htmlI have to solve this problem: $v=\partial_\phi$ on $M=\mathbb{R}^2\backslash{0}$ where the components of $v$ are in polar coordinates. Calculate the divergence of $v$. Even with the help of comme......Tue, 11 Aug 2026 21:05:21 -0400https://liberty.io/probability-of-guessing-m-m-colors-correctly.htmlhttps://liberty.io/probability-of-guessing-m-m-colors-correctly.htmlSuppose there are 5 M and M beans of different colors. You randomly choose 3. What is the probability that you guess 2 out of the 3 numbers correctly? Apparently the answer is $12/125$, but I don't......Tue, 11 Aug 2026 21:05:21 -0400https://liberty.io/infinite-countable-union-of-finite-sets.htmlhttps://liberty.io/infinite-countable-union-of-finite-sets.htmlIf $A_n$ is finite $\forall n\in \mathbb N,\;$ then countable infinite union of these sets is infinite, right?...Tue, 11 Aug 2026 21:00:46 -0400https://liberty.io/definition-of-bounded-set-in-mathbb-r-n.htmlhttps://liberty.io/definition-of-bounded-set-in-mathbb-r-n.htmlI have a question regarding the definition of a bounded set in $\mathbb{R}^n$ here: http://mathworld.wolfram.com/BoundedSet.html Specifically, the definition in the link says that "A set in $\math......Tue, 11 Aug 2026 20:58:16 -0400https://liberty.io/what-is-wrong-with-my-derivation-of-the-exponential-form-of-sinh-x.htmlhttps://liberty.io/what-is-wrong-with-my-derivation-of-the-exponential-form-of-sinh-x.htmlI know the exponential form of $\sin(x)=\dfrac{e^{ix}-e^{-ix}}{2i}$. I wish to derive the exponential form of $\sinh(x)=\dfrac{e^x-e^{-x}}{2}$, and I am stuck. I try to replace $\sin(x)$ with $\sin......Tue, 11 Aug 2026 20:56:38 -0400https://liberty.io/using-matlab-to-find-the-max-elements-and-their-positions-closed.htmlhttps://liberty.io/using-matlab-to-find-the-max-elements-and-their-positions-closed.htmlMATLAB documentation says "[C,I] = max(...) finds the indices of the maximum values of A, and returns them in output vector I." But that does not seem to work. See below. X = [2 8 4; 7 3 9]; ......Tue, 11 Aug 2026 20:52:26 -0400https://liberty.io/how-many-points-are-necessary-to-determine-a-cube-in-3-dimensions.htmlhttps://liberty.io/how-many-points-are-necessary-to-determine-a-cube-in-3-dimensions.htmlIf you have 2 points in 3D, there is only one line passing through them, but infinite planes can pass through them. If you have 3 points, only one plane can pass through all 3, but infinitely many......Tue, 11 Aug 2026 20:47:44 -0400https://liberty.io/if-s-is-a-multiple-of-both-a-and-b-then-s-is-a-multiple-of-operatornam.htmlhttps://liberty.io/if-s-is-a-multiple-of-both-a-and-b-then-s-is-a-multiple-of-operatornam.htmlI came across this as I was doing work for one of my classes. We just use this property, proved presumably in number theory, which we didn't need to take. Could someone help me? Prove that if $a$, ......Tue, 11 Aug 2026 20:47:32 -0400https://liberty.io/denomination-of-cards-probability.htmlhttps://liberty.io/denomination-of-cards-probability.htmlWe have a condensed deck of 40 cards containing only the denominations from Ace through 10. We draw four cards uniformly at random (without replacement). What is the probability that we draw four ...Tue, 11 Aug 2026 20:44:49 -0400https://liberty.io/finding-product-of-maxterms-form.htmlhttps://liberty.io/finding-product-of-maxterms-form.htmlI need help to resolve this problem, i have the following boolean function: [(A.!C)+!(A.!C)].!(A.!B) The Truth table is: (please see this LINK TO wolframalpha for more detail) Then the Sum-of-...Tue, 11 Aug 2026 20:36:08 -0400https://liberty.io/is-there-an-english-translation-of-diophantus-039-s-arithmetica-availa.htmlhttps://liberty.io/is-there-an-english-translation-of-diophantus-039-s-arithmetica-availa.htmlIt should be in the public domain (obviously), so I'd thought I could find the English text on the web somewhere. Apparently not?...Tue, 11 Aug 2026 20:35:58 -0400https://liberty.io/is-it-possible-to-have-a-spherical-object-with-only-hexagonal-faces.htmlhttps://liberty.io/is-it-possible-to-have-a-spherical-object-with-only-hexagonal-faces.htmlIf so, what would be the most efficient algorithm for generating spheres with different number of hexagonal faces at whatever interval required to make them fit uniformly or how might you calculate......Tue, 11 Aug 2026 20:31:28 -0400https://liberty.io/system-of-equations-has-no-solution.htmlhttps://liberty.io/system-of-equations-has-no-solution.htmlIf the system of linear equations, \begin{cases} x &+ ay &+ z &= 3\\ x &+ 2y &+ 2z &= 6\\ x &+ 5y &+ 3z &= b\\ \end{cases} has no solution, then: $a......Tue, 11 Aug 2026 20:29:53 -0400https://liberty.io/why-does-1-0-5-equal-2-duplicate.htmlhttps://liberty.io/why-does-1-0-5-equal-2-duplicate.htmlI would like to know, in $1/0.5 =2$ why does $1/0.5$ yield $2$. I want to know a conceptual explanation, not merely a method how to get the answer. If you are able to explain this in example u......Tue, 11 Aug 2026 20:18:18 -0400https://liberty.io/the-caret-symbol-means-exponentiation-informally-in-math-why-not-a-sym.htmlhttps://liberty.io/the-caret-symbol-means-exponentiation-informally-in-math-why-not-a-sym.htmlPlus, minus, multiply, divide, and exponentiation all have symbols in math (+, -, *, /, ^ ) . But why isn't there the missing log symbol too? Here's how it would work: 4 ^ 5 = 1024 (as is standard......Tue, 11 Aug 2026 20:17:02 -0400https://liberty.io/how-to-evaluate-the-gradient-of-a-function-at-a-point.htmlhttps://liberty.io/how-to-evaluate-the-gradient-of-a-function-at-a-point.htmlI have a problem where I am to create a function in terms of $x$ and $y$ and compute the gradient at the point $(1,1)$. I computed the gradient but in order to evaluate it at the given point do I j......Tue, 11 Aug 2026 20:14:47 -0400https://liberty.io/how-to-determine-the-periods-of-a-periodic-function.htmlhttps://liberty.io/how-to-determine-the-periods-of-a-periodic-function.htmlI am aware of the other similar questions but was not able to figure out what I want to know from those question thus posting it here. Given a periodic function $f(x)=sin(x)$, Why is the period of......Tue, 11 Aug 2026 20:04:19 -0400https://liberty.io/matrix-columns-vs-rows.htmlhttps://liberty.io/matrix-columns-vs-rows.htmlI am confused over the geometric meaning of columns and rows in a matrix. My understanding from 3Blue1Brown's Linear Algebra series was that the elements of a matrix column are the coordinates whi......Tue, 11 Aug 2026 20:03:52 -0400https://liberty.io/angle-measure-angles-in-standard-position-is-it-a-useless-term.htmlhttps://liberty.io/angle-measure-angles-in-standard-position-is-it-a-useless-term.htmlI have a pre calculus test next week, and I have been going over the chapters to gain a deeper understanding; however, I find it difficult, or at least I find the concept of "Angles in Standard Pos......Tue, 11 Aug 2026 20:00:20 -0400https://liberty.io/what-is-the-meaning-of-mean-field.htmlhttps://liberty.io/what-is-the-meaning-of-mean-field.htmlIn lots of Bayesian papers, people use variational approximation. In lots of them they call it "mean-field variational approximation". Does anyone know what is the meaning of mean-field in this co......Tue, 11 Aug 2026 19:54:52 -0400https://liberty.io/orthogonal-position-and-velocity-vectors.htmlhttps://liberty.io/orthogonal-position-and-velocity-vectors.htmlIs it true that if the position and velocity vectors of a moving particle are always perpendicular the path of the particle is on a sphere? If so how do I prove it? Geometrically I believe it makes ...Tue, 11 Aug 2026 19:44:25 -0400https://liberty.io/lower-bound-on-tail-probabilities.htmlhttps://liberty.io/lower-bound-on-tail-probabilities.htmlInequalities such as Markov's and Chebyshev’s provide upper bounds on tail probabilities. Are there similar inequalities that give lower bounds in the form $P(X \geq \alpha)>\theta$?...Tue, 11 Aug 2026 19:44:08 -0400https://liberty.io/intuitive-and-memorable-way-to-see-n1-n1-n2-as-distinct-permutations-o.htmlhttps://liberty.io/intuitive-and-memorable-way-to-see-n1-n1-n2-as-distinct-permutations-o.htmlExample Problem: Count the number of distinct ways in which N objects, of which n1 are indistinguishably of one type and n2 of a second type, can be accomodated in a total of N = n1 + n2 ways. ...Tue, 11 Aug 2026 19:36:17 -0400https://liberty.io/eigenspace-definition.htmlhttps://liberty.io/eigenspace-definition.htmlThis is the definition I have for eigenspace: Let $\lambda$ be an eigenvalue of $A \in Mn(\mathbb C)$. Then {$v \in \mathbb C^n|Av= \lambda v$}$=W_\lambda$ is a subspace of $\mathbb C^n$, call......Tue, 11 Aug 2026 19:34:51 -0400https://liberty.io/proving-cos-4-theta-4-cos-2-theta-equiv-8-sin-4-theta-3.htmlhttps://liberty.io/proving-cos-4-theta-4-cos-2-theta-equiv-8-sin-4-theta-3.htmlProve the following identity: $$\cos(4\theta) - 4\cos(2\theta) \equiv 8\sin^4\theta - 3$$ How can I express $\cos(4\theta) $ in other terms?...Tue, 11 Aug 2026 19:25:37 -0400https://liberty.io/the-significance-and-acceptance-of-helfgott-s-proof-of-the-weak-goldba.htmlhttps://liberty.io/the-significance-and-acceptance-of-helfgott-s-proof-of-the-weak-goldba.htmlRecently I was browsing math Wikipedia, and found that Harald Helfgott announced the complete proof of the weak Goldbach Conjecture in 2013, a proof which has been accepted widely by the math commu......Tue, 11 Aug 2026 19:25:09 -0400https://liberty.io/proof-of-pythagorean-theorem-without-using-geometry-for-a-high-school-.htmlhttps://liberty.io/proof-of-pythagorean-theorem-without-using-geometry-for-a-high-school-.htmlThere are some proofs of Pythagoras theorem which don't even require high school maths to understand, but they all are using shapes to prove of the theorem. However, I am trying to find some proofs......Tue, 11 Aug 2026 19:21:40 -0400https://liberty.io/how-is-a-general-case-equivalent-to-a-special-case-and-how-showing-a-s.htmlhttps://liberty.io/how-is-a-general-case-equivalent-to-a-special-case-and-how-showing-a-s.html"The general theorem expressed by $\lambda a^2 = \lambda b^2 + \lambda c^2 $ is equivalent not only to the special case $a^2 = b^2 + c^2 $ but to any other special case. Therefore, if any such spe......Tue, 11 Aug 2026 19:18:19 -0400https://liberty.io/is-heat-equation-is-also-called-as-thermal-diffusion-equation.htmlhttps://liberty.io/is-heat-equation-is-also-called-as-thermal-diffusion-equation.htmlIs the following equation known as heat equation, also referred as thermal diffusion equation? Yes/No. $${\displaystyle {\frac {\partial u}{\partial t}}-\alpha \left({\frac {\partial ^{2}u}{\parti......Tue, 11 Aug 2026 19:12:24 -0400https://liberty.io/proof-of-yet-another-weyl-s-inequality-using-courant.htmlhttps://liberty.io/proof-of-yet-another-weyl-s-inequality-using-courant.htmlThe Courant-Fischer theorem states that $$\lambda_j=\max_{\dim(\mathbb{V})=j}\min_{v\in \mathbb{V},v\neq 0}\rho(v,A)=\min_{\dim(\mathbb{W})=n-j+1}\max_{w\in \mathbb{W},w\neq0}\rho(v,A)$$ where $\la......Tue, 11 Aug 2026 19:12:23 -0400https://liberty.io/polar-equation-of-a-circle.htmlhttps://liberty.io/polar-equation-of-a-circle.htmlA very long time ago in algebra/trig class we did polar equation of a circle where $r = 2a\cos\theta + 2b\sin\theta$ Now I forgot how to derive this. So I tried using the standard form of a circl......Tue, 11 Aug 2026 19:08:27 -0400https://liberty.io/numerically-calculate-the-second-left-hand-derivative.htmlhttps://liberty.io/numerically-calculate-the-second-left-hand-derivative.htmlThe Problem I have a series of measurements for which I have to calculate the first and second derivative numerically in a "live" fashion, i.e. using only previous data. This is easy for the first ...Tue, 11 Aug 2026 19:07:01 -0400https://liberty.io/divisibility-criteria-of-24-why-is-this.htmlhttps://liberty.io/divisibility-criteria-of-24-why-is-this.htmlI am currently familiar with the method of checking if a number is divisible by $2, 3, 4, 5, 6, 8, 9, 10, 11$. While Checking for divisibility for $24$ (online). I found out that the number has to ...Tue, 11 Aug 2026 18:52:47 -0400https://liberty.io/differential-topology-versus-differential-geometry.htmlhttps://liberty.io/differential-topology-versus-differential-geometry.htmlI have just finished my undergraduate studies. During last two semesters I've taken two subjects dealing with manifolds: Analysis on manifolds, containing: definition of manifold, tangent space (as ...Tue, 11 Aug 2026 18:48:31 -0400https://liberty.io/derivative-of-integral.htmlhttps://liberty.io/derivative-of-integral.htmlI'm having a little trouble with the following problem: Calculate $F'(x)$: $F(x)=\int_{1}^{x^{2}}(t-\sin^{2}t) dt$ It says we have to use substitution but I don't see why the answer can't just b......Tue, 11 Aug 2026 18:41:08 -0400https://liberty.io/toy-examples-for-kan-extensions.htmlhttps://liberty.io/toy-examples-for-kan-extensions.htmlBackground: If $\mathcal{C}$ is a cocomplete category and $f : I \to J$ is a functor between small categories, then $f^* : \mathrm{Hom}(J,\mathcal{C}) \to \mathrm{Hom}(I,\mathcal{C})$ has a left ad......Tue, 11 Aug 2026 18:38:24 -0400https://liberty.io/parabolic-points-and-curvature.htmlhttps://liberty.io/parabolic-points-and-curvature.htmlI have problems to solve this exercise: Let $p$ be a point of an oriented surface $S$ and assume that there is a neighborhood $U$ of $p$ in $S$ all points of which are parabolic. Prove that the (un......Tue, 11 Aug 2026 18:22:40 -0400https://liberty.io/find-the-perimeter-of-triangle-abc.htmlhttps://liberty.io/find-the-perimeter-of-triangle-abc.htmlThis practice engineering board exam which I've tried seems very tricky (question no.2). The inscribed circle of $\triangle ABC$ is tangent to $AB$ at $P$ . If the radius is $21$, $AP=23$ and $PB=2......Tue, 11 Aug 2026 18:21:06 -0400https://liberty.io/number-of-words-that-can-be-made-using-all-the-letters-of-the-word-w-i.htmlhttps://liberty.io/number-of-words-that-can-be-made-using-all-the-letters-of-the-word-w-i.htmlI am facing difficulty in solving the 18th Question from the above passage. Attempt: I have attempted it using principal of inclusion and exclusion. Let n be the required number of ways. $n = \...Tue, 11 Aug 2026 18:19:20 -0400https://liberty.io/square-of-a-differential-duplicate.htmlhttps://liberty.io/square-of-a-differential-duplicate.htmlJust wondering, is this valid: $$ \left(\frac{df}{dx}\right)^2=\frac{d^{2}f}{dx^{2}} $$...Tue, 11 Aug 2026 18:15:33 -0400https://liberty.io/expected-value-of-a-the-estimate-of-bernoulli-distribution.htmlhttps://liberty.io/expected-value-of-a-the-estimate-of-bernoulli-distribution.htmlI know that $E[x] = p$ for some Bernoulli random variable X with parameter $p$ where $P(X = 1) = p$ and $P(X = 0) = 1 − p$. Similarly $E[X^2] = p$ and $Var[X] = p − p^2 = p(1 − p)$. However, I'm ha......Tue, 11 Aug 2026 18:15:10 -0400https://liberty.io/find-from-the-first-principle-the-derivative-of-sqrt-sin-2x.htmlhttps://liberty.io/find-from-the-first-principle-the-derivative-of-sqrt-sin-2x.htmlFind, from the first principle, the derivative of: $$\sqrt {\sin (2x)}$$ My Attempt: $$f(x)=\sqrt {\sin (2x)}$$ $$f(x+\Delta x)=\sqrt {\sin (2x+2\Delta x)}$$ Now, $$f'(x)=\lim_{\Delta x\to 0} \dfr......Tue, 11 Aug 2026 18:06:29 -0400https://liberty.io/how-to-find-the-percent-variation-in-y-is-explained-by-x.htmlhttps://liberty.io/how-to-find-the-percent-variation-in-y-is-explained-by-x.htmlI know that the r^2 value for the data is 0.9832. Is there a way to use that value to find the percent variation in Y is explained by X? Or do I need to use the data given to me?...Tue, 11 Aug 2026 18:05:17 -0400https://liberty.io/list-of-common-or-useful-limits-of-sequences-and-series.htmlhttps://liberty.io/list-of-common-or-useful-limits-of-sequences-and-series.htmlThere are many sequences or series which come up frequently, and it's good to have a directory of the most commonly used or useful ones. I'll start out with some. Proofs are not required. $$\begin......Tue, 11 Aug 2026 18:03:51 -0400https://liberty.io/how-can-you-show-that-the-center-of-mass-of-a-triangle-lies-on-the-med.htmlhttps://liberty.io/how-can-you-show-that-the-center-of-mass-of-a-triangle-lies-on-the-med.htmlIn geometry class, it is usually first shown that the medians of a triangle intersect at a single point. Then is is explained that this point is called the centroid and that it is the balance poin......Tue, 11 Aug 2026 17:45:07 -0400https://liberty.io/finding-maximum-height-with-initial-velocity.htmlhttps://liberty.io/finding-maximum-height-with-initial-velocity.htmlAn object is projected from the surface of Earth with a velocity of one mile per second. What is the maximum height it will reach? My solution for solving this was to use the following equat......Tue, 11 Aug 2026 17:44:32 -0400https://liberty.io/identity-element-generating-a-cyclic-group.htmlhttps://liberty.io/identity-element-generating-a-cyclic-group.htmlA group is defined as a set that is: associative, contains the identity element $(e)$, and every element has an inverse under given operation $(*)$. By this definition, $\{e\}$ is a group over the ......Tue, 11 Aug 2026 17:25:02 -0400https://liberty.io/how-to-define-a-relation.htmlhttps://liberty.io/how-to-define-a-relation.htmlI was tasked with this question: Let $X = \{0,1,2,3,4\}$. Define a relation $R$ on $X$ such that $xRy$, if $x + y = 4$. I don't understand/know what syntax I should use to define this as a relat......Tue, 11 Aug 2026 17:14:32 -0400https://liberty.io/given-the-function-minimize-the-function-subject-to-the-constraint-lag.htmlhttps://liberty.io/given-the-function-minimize-the-function-subject-to-the-constraint-lag.htmlGiven the function $f(x, y, z) = (x-1)^2 + y^2 + (z+1)^2$: Minimize $f(x, y, z)$ subject to the constraint $x + y +z = 1$ I know that you need to use Lagrange's Multipliers and thus would need to......Tue, 11 Aug 2026 17:14:01 -0400https://liberty.io/equation-of-a-line-that-passes-halfway-between-two-points-in-other-wor.htmlhttps://liberty.io/equation-of-a-line-that-passes-halfway-between-two-points-in-other-wor.htmlIs there a formal proper way of finding the line between two points? By that I don't mean the line connecting the two points, I mean a line that runs the same distance away from point 1 and point ......Tue, 11 Aug 2026 16:59:07 -0400https://liberty.io/is-there-an-integral-that-proves-pi-333-106.htmlhttps://liberty.io/is-there-an-integral-that-proves-pi-333-106.htmlThe following integral, $$ \int_0^1 \frac{x^4(1-x)^4}{x^2 + 1} \mathrm{d}x = \frac{22}{7} - \pi $$ is clearly positive, which proves that $\pi < 22/7$. Is there a similar integral which prove......Tue, 11 Aug 2026 16:54:43 -0400https://liberty.io/curves-with-a-common-tangent-line.htmlhttps://liberty.io/curves-with-a-common-tangent-line.htmlQuestion Find the point where the curves $$\tag 1y = x^3 -3x + 4$$ and $$\tag 2 y = 3x^2 - 3x$$ are tangent to each other, that is, have a common tangent line. My approach Let $x = a$ and $x = b......Tue, 11 Aug 2026 16:29:20 -0400https://liberty.io/solving-for-given-variable-how-to-solve.htmlhttps://liberty.io/solving-for-given-variable-how-to-solve.htmlSo the question states: 'To convert degrees Celsius to Kelvin, the formula $K = C + 273.15$ is used. Solve this formula for $C$.' The answer I came up with was: $K - 273.15 = C$. So is this correc......Tue, 11 Aug 2026 16:17:44 -0400https://liberty.io/what-is-cumulative-logistic-distribution.htmlhttps://liberty.io/what-is-cumulative-logistic-distribution.htmlI read an article on Logistic Regression on Wikipedia: https://en.wikipedia.org/wiki/Logistic_regression. It was written that logistic function is a cumulative logistic distribution. My question......Tue, 11 Aug 2026 16:11:06 -0400https://liberty.io/how-do-you-derive-the-formula-of-sin-a-b-from-cos-a-b-cos-a-cos-b-sin-.htmlhttps://liberty.io/how-do-you-derive-the-formula-of-sin-a-b-from-cos-a-b-cos-a-cos-b-sin-.htmlI know that cos is even while sin is odd, and I know $\cos(\pi)=\sin((\pi/2)-x)$, but I still can't figure the derivation of $\sin (a+b)$ from $\cos(a+b)=\cos(a)\cos(b)-\sin(a)\sin(b)$. Could you ......Tue, 11 Aug 2026 15:38:22 -0400https://liberty.io/laplace-transform-to-solve-differential-equation-with-ivp-given-at-a-p.htmlhttps://liberty.io/laplace-transform-to-solve-differential-equation-with-ivp-given-at-a-p.htmlhow could I use Laplace Transform to solve the following differential equation: $$y''+2y'+y=0; \;\;\;\;\;y'(0)=2\;\;\;\;\;\mbox{and}\;\;\;\;\;\boldsymbol{y(1)=2}.$$ The solution may involve the ...Tue, 11 Aug 2026 15:34:12 -0400https://liberty.io/matrix-by-matrix-derivative-formula.htmlhttps://liberty.io/matrix-by-matrix-derivative-formula.htmlI need to derive $\frac{\delta(X^{T}MX)}{\delta X}$, where $X$ and $M$ are $n \times n$ matrices. I know that $\frac{\delta(AXB)}{\delta X}=B^{T} \otimes A$ but am having a hard time deriving what I ...Tue, 11 Aug 2026 15:31:07 -0400https://liberty.io/the-meaning-of-differentiation-of-x-with-respect-to-y.htmlhttps://liberty.io/the-meaning-of-differentiation-of-x-with-respect-to-y.htmlThe physical meaning of the differentiation of $x$ with respect to $y$ is the rate of change of $x$ with respect to $y$. But, I am finding it difficult in understanding the geometrical interpretati......Tue, 11 Aug 2026 15:28:43 -0400https://liberty.io/round-table-seating-probability.htmlhttps://liberty.io/round-table-seating-probability.htmlThere are $6$ people, let's call them - (a,b,c,d,e,f), to sit at a round table. The number of ways they can arrange themselves is $(6-1)! = 5! = 120$ ways. What is the probability that person 'a' ......Tue, 11 Aug 2026 15:26:42 -0400https://liberty.io/determinant-of-an-elementary-matrix.htmlhttps://liberty.io/determinant-of-an-elementary-matrix.htmlI read a very slick proof of determinant properties, in this case of the fact $\det A = \det A^T$, which says in one place It suffices to notice that for any elementary matrix $M$ we have $\det......Tue, 11 Aug 2026 15:07:13 -0400https://liberty.io/third-order-linear-parabolic-pde.htmlhttps://liberty.io/third-order-linear-parabolic-pde.htmlWhat's the best method to solve analytically an equation of the form $$f_t=f_x+af_{xx}+bf_{xxx}$$ with $a,b\in\mathbb{R}$ ?...Tue, 11 Aug 2026 14:59:52 -0400https://liberty.io/solving-system-of-differential-equations-using-runge-kutta-method.htmlhttps://liberty.io/solving-system-of-differential-equations-using-runge-kutta-method.htmlI have two differential equations: \begin{align}\frac{dx}{dt} &= 1 -(1+b)x+ax^2y\\ \frac{dy}{dt} &= bx - ax^2y\end{align} I have been asked to solve them on Python using the Runge Kutta (......Tue, 11 Aug 2026 14:52:12 -0400https://liberty.io/proving-that-pi-2x-2-pi-x.htmlhttps://liberty.io/proving-that-pi-2x-2-pi-x.htmlIn our analytic number theory class we were given the following problem as homework: prove rigorously that for large $x$ the number of primes in $(1,x]$ exceeds that in $(x,2x]$. In class we prove......Tue, 11 Aug 2026 14:46:14 -0400https://liberty.io/flipping-a-fraction-within-an-inequality.htmlhttps://liberty.io/flipping-a-fraction-within-an-inequality.htmlIs it possible to flip a faction within an inequality? Such that: $$\frac13 < x < \frac23$$ becomes, $$3 > \frac1x > \frac32$$...Tue, 11 Aug 2026 14:28:59 -0400https://liberty.io/tetrahedron-packing-in-cube.htmlhttps://liberty.io/tetrahedron-packing-in-cube.htmlI'm thinking about following solid geometry problem. Q: Suppose you have a box of "cube" shape with edge length 1. Then, How many regular tetrahedrons(with edge length 1) can be in the box? So, t......Tue, 11 Aug 2026 14:11:37 -0400https://liberty.io/how-can-you-visually-distinguish-between-a-parabola-and-hyperbola.htmlhttps://liberty.io/how-can-you-visually-distinguish-between-a-parabola-and-hyperbola.htmlI am fully aware of the mathematical definition of either and how they relate to conic sections. Still, I wonder whether there is an easy way to see that these are fundamentally different curves.......Tue, 11 Aug 2026 14:05:09 -0400https://liberty.io/solution-to-equations-in-3-variable-without-forming-a-cubic-polynomial.htmlhttps://liberty.io/solution-to-equations-in-3-variable-without-forming-a-cubic-polynomial.htmlCan we solve 3 equations:- $$a+b+c = x$$ $$abc = y$$ $$ab+bc+ac = z$$ to get the value of $a,b$ and $c$ where $x,y,z$ are constants As you can probably see I'm trying to find a solution to a cubic ...Tue, 11 Aug 2026 14:04:11 -0400https://liberty.io/symbol-for-kernel-and-range-of-a-linear-transformation.htmlhttps://liberty.io/symbol-for-kernel-and-range-of-a-linear-transformation.htmlI'm wondering what symbol is used for the kernel and the range of a linear transformation. I've seen them being written as such: $\ker(T)$ $\mathcal{R}(T)$ Are these the "correct" symbols for the ...Tue, 11 Aug 2026 13:58:40 -0400https://liberty.io/what-does-x-2-1-mean.htmlhttps://liberty.io/what-does-x-2-1-mean.htmlI am studying some inequality properties of absolute values and I bumped into some expressions like $|x-2| < 1$ that I just can't get the meaning of them. Lets say I have this expression $$ |......Tue, 11 Aug 2026 13:40:49 -0400https://liberty.io/is-there-a-geometric-interpretation-for-a-function-039-s-alpha-subleve.htmlhttps://liberty.io/is-there-a-geometric-interpretation-for-a-function-039-s-alpha-subleve.htmlIn Boyd and Vandenberghe's "Convex Optimization": The $\alpha$-sublevel set set of a function $f:\mathbb{R}^n \rightarrow \mathbb{R}$ is defined as $$C_\alpha=\{x \in \mathbf{dom} f|f(x)\leq\......Tue, 11 Aug 2026 13:40:07 -0400https://liberty.io/study-according-to-the-values-of-x-the-relative-position-of-line-curve.htmlhttps://liberty.io/study-according-to-the-values-of-x-the-relative-position-of-line-curve.htmlWe have $F(x) = x + \ln \left(\frac{1+2e^{-2x}}{1+e^{-x}}\right)$ represents curve $C$ and a line $d: y=x$ In an exercise it is required to study the relative position of these two functions, ......Tue, 11 Aug 2026 13:37:43 -0400https://liberty.io/finding-all-real-zeros-of-the-polynomial.htmlhttps://liberty.io/finding-all-real-zeros-of-the-polynomial.htmlOkay, so I need to find all real zeros in this polynomial... $$f(x) = 2x^3 + x^2 - 13x + 6$$ I know that the first step is to find the factors of 6 and 2, then see which when multiplied by the ot......Tue, 11 Aug 2026 13:36:04 -0400https://liberty.io/strategies-for-factoring-expressions-with-four-terms.htmlhttps://liberty.io/strategies-for-factoring-expressions-with-four-terms.htmlI'm trying to come up with a general strategy for factoring expressions with four terms on the basis of the symmetries of the expressions. One thought I had was the following: count up the number of ...Tue, 11 Aug 2026 13:33:22 -0400https://liberty.io/novikoff-039-s-proof-for-perceptron-convergence.htmlhttps://liberty.io/novikoff-039-s-proof-for-perceptron-convergence.htmlIn Machine Learning, the Perceptron algorithm converges on linearly separable data in a finite number of steps. One can prove that $(R/\gamma)^2$ is an upper bound for how many errors the algorithm......Tue, 11 Aug 2026 13:25:05 -0400https://liberty.io/simple-formula-for-a-sieries-like-1-2-5-10-20-50-100.htmlhttps://liberty.io/simple-formula-for-a-sieries-like-1-2-5-10-20-50-100.htmlI'm looking for a simple formula that will give a series that looks like this: $1; 2; 5; 10; 20; 50; 100; ...$ That means a function that will give this output: $f(1) = 1$ $f(2) = 2$ $f(3) = 5$ $f......Tue, 11 Aug 2026 13:20:43 -0400https://liberty.io/geometric-interpretation-of-eigendecomposition.htmlhttps://liberty.io/geometric-interpretation-of-eigendecomposition.htmlIf we have an orthogonal matrix $U$, then $U^Tx$ is essentially a rotation of the vector $x$. If we have a diagonal matrix $\Lambda$, then $\Lambda x$ is scaling the vector $x$ in each direction by......Tue, 11 Aug 2026 13:18:59 -0400https://liberty.io/how-to-determine-the-right-side-of-the-dice.htmlhttps://liberty.io/how-to-determine-the-right-side-of-the-dice.htmlsay I have a dice, with following numbering arrangements: top front right bottom back left 1 2 3 6 5 4 the dice can be rolled freely of course I think it is enough to give the ...Tue, 11 Aug 2026 12:57:09 -0400https://liberty.io/finding-partial-fractions-involving-complex-numbers-express-h-z-frac-2.htmlhttps://liberty.io/finding-partial-fractions-involving-complex-numbers-express-h-z-frac-2.htmlI'm attempting the following problem in the context of complex analysis: Express $h(z) = \dfrac{2z + 4i}{z^2 + 2z + 2}$ as partial fractions. I got $\dfrac{2z + 4i}{z^2 + 2z + 2} = \dfrac{2z + ......Tue, 11 Aug 2026 12:56:57 -0400https://liberty.io/cardinality-of-union-of-3-sets.htmlhttps://liberty.io/cardinality-of-union-of-3-sets.htmlI know there are answers to this specific question, but I'm trying to solve it through a specific line of thought which was used in a textbook on modern algebra for the case with 2 sets. It goes like ...Tue, 11 Aug 2026 12:52:28 -0400https://liberty.io/continuity-is-required-for-differentiability.htmlhttps://liberty.io/continuity-is-required-for-differentiability.htmlMy professor emphasized that: Differentiability implies continuity and Continuity is required for differentiability. Since a function like $\frac 1 x$ is differentiable but not continuous, I thou......Tue, 11 Aug 2026 12:48:10 -0400https://liberty.io/how-to-round-0-45-is-it-0-or-1.htmlhttps://liberty.io/how-to-round-0-45-is-it-0-or-1.htmlThis question is inspired by How to round 0.4999... ? Is it 0 or 1? I didn't quite understand the logic of the answer. It seems you round every decimal place no matter how far back it goes? In the ......Tue, 11 Aug 2026 12:47:14 -0400https://liberty.io/how-was-this-approximation-of-pi-involving-sqrt-5-arrived-at.htmlhttps://liberty.io/how-was-this-approximation-of-pi-involving-sqrt-5-arrived-at.htmlThe Wikipedia article for Approximations of $\pi$ contains this little gem: $$ \pi \approx \frac{63}{25}\times\frac{17 + 15\sqrt{5}}{7 + 15\sqrt{5}} $$ which is clearly in $\mathbb{Q[\sqrt{5}]}$. ...Tue, 11 Aug 2026 12:46:58 -0400https://liberty.io/measurable-space-vs-measure-space.htmlhttps://liberty.io/measurable-space-vs-measure-space.htmlFrom what I understood a Measurable Space is $(X,S)$ where $X$ is a set and $S\subset P(X)$ is a $\sigma$ algebra a Measure Space is $(X,S,\mu)$ where $X$ is a set and $S\subset P(X)$ is a $\sigma$ ...Tue, 11 Aug 2026 12:46:20 -0400https://liberty.io/using-conjugates-to-find-a-limit-with-a-cubic-root-lim-h-to-0-frac-sqr.htmlhttps://liberty.io/using-conjugates-to-find-a-limit-with-a-cubic-root-lim-h-to-0-frac-sqr.htmlI currently have $$\lim_{h\to 0}\frac{\sqrt[3]{h+1}-1}{h}$$ Now, I know that when you have square root instead of a cubic root it's easy. You just multiply by the conjugate and simplify afterwards......Tue, 11 Aug 2026 12:45:30 -0400