https://liberty.ioCurated star highlights with fresh energy.en-usMon, 10 Aug 2026 14:02:12 -0400https://liberty.io/what-is-the-meaning-of-a-discriminant-graphically.htmlhttps://liberty.io/what-is-the-meaning-of-a-discriminant-graphically.htmlI know that if $b^2-4ac>0$, then there are real solutions and so on. But for any quadratic function, where is the discriminant present in the plot? For example, $x^2-8x+7=f(x)$. Now how is $......Mon, 10 Aug 2026 17:57:28 -0400https://liberty.io/proving-that-the-second-derivative-of-a-convex-function-is-nonnegative.htmlhttps://liberty.io/proving-that-the-second-derivative-of-a-convex-function-is-nonnegative.htmlMy task is as follows: Let $f:\mathbb{R}\to\mathbb{R}$ be a twice-differentiable function, and let $f$'s second derivative be continuous. Let $f$ be convex with the following definition of ...Mon, 10 Aug 2026 17:57:23 -0400https://liberty.io/roll-a-4-sided-die-whatever-value-appears-flip-exactly-that-many-fair-.htmlhttps://liberty.io/roll-a-4-sided-die-whatever-value-appears-flip-exactly-that-many-fair-.htmlRoll a $4$-sided die. Whatever value appears, flip exactly that many fair coins. [For instance, if the die shows $3$, then flip $3 $fair coins.] Let $X$ denote the number of heads that appear on the ...Mon, 10 Aug 2026 17:46:12 -0400https://liberty.io/what-is-the-probability-of-multiple-independent-events-occurring-in-a-.htmlhttps://liberty.io/what-is-the-probability-of-multiple-independent-events-occurring-in-a-.htmlI thought about this problem the other day and my limited knowledge in basic probability theory was not enough to help me figure this one out. Say you have 3 probabilities: You have a 50% chance of ...Mon, 10 Aug 2026 17:43:31 -0400https://liberty.io/the-prolate-cycloid.htmlhttps://liberty.io/the-prolate-cycloid.htmlA cycloid is given by the parametric equations: $ x = 2 - \pi \cos(t)$ and $ y = 2t - \pi \sin(t)$. The problem asks for the slope of the tangents on the cycloid at a point where the cycloid inte......Mon, 10 Aug 2026 17:43:12 -0400https://liberty.io/proving-that-the-inverse-function-exists-and-finding-it.htmlhttps://liberty.io/proving-that-the-inverse-function-exists-and-finding-it.htmlLet $f$ be a function defined as $$f:=\sqrt[3]{x^{3}+bx^{2}}, b\in \mathbb{R}$$ for all$$x\leq 0$$ In order for a function ta have an inverse, it must be both injective and surjective(hope I spe......Mon, 10 Aug 2026 17:42:33 -0400https://liberty.io/how-to-find-the-stabilizer-of-a-subgroup-of-s-4.htmlhttps://liberty.io/how-to-find-the-stabilizer-of-a-subgroup-of-s-4.htmlGiven $H = \{ id , (12) , (34) , (12)(34)\}$ find $Stab(H)$ and the bijection from the set of left cosets of $Stab(H) $ in $ S_4 $ to $ \mathcal{O} (H) $ ; $S_4$ acts on $H$ by conjugation. ie: $$\...Mon, 10 Aug 2026 17:25:13 -0400https://liberty.io/for-a-matrix-what-is-a-solution-vector.htmlhttps://liberty.io/for-a-matrix-what-is-a-solution-vector.htmlIf given a value $N$ and told to generate the following: A random matrix $A[N][N]$ A solution vector $S[N]$ What would be a "solution vector" of a randomly generated matrix? I am not sure I have ......Mon, 10 Aug 2026 17:20:31 -0400https://liberty.io/given-a-latitude-how-many-miles-is-the-corresponding-longitude.htmlhttps://liberty.io/given-a-latitude-how-many-miles-is-the-corresponding-longitude.htmlOK so lines of longitude (the distance/circumference around the earth horizontally) differ based on what latitude you are at (0 at north and south poles up to ~25k at the equator.) So given a lati......Mon, 10 Aug 2026 17:16:20 -0400https://liberty.io/why-does-the-geometric-series-formula-intuitively-work.htmlhttps://liberty.io/why-does-the-geometric-series-formula-intuitively-work.htmlI understand the proof for the geometric series formula, but I don't understand how the formula, $S_n = a_1\frac{(r^n -1)}{(r-1)}$ actually relates to the sum of all the terms. What operations are ...Mon, 10 Aug 2026 17:14:58 -0400https://liberty.io/find-the-derivative-of-f-gmm-over-r-2.htmlhttps://liberty.io/find-the-derivative-of-f-gmm-over-r-2.htmlNewton's Law of Gravitation says that the magnitude F of the force exerted by a body of mass M on a body of mass m is $F =$ $(GmM)\over r^2$ Where G is the gravitational constant and r is the dis......Mon, 10 Aug 2026 16:57:45 -0400https://liberty.io/is-it-possible-to-calculate-the-standard-deviation-with-a-given-mean-a.htmlhttps://liberty.io/is-it-possible-to-calculate-the-standard-deviation-with-a-given-mean-a.htmlI have been going in rounds with this problem... I may be thinking "complicated", any advice? I have the mean and total sample size (=number of data points) and I need to know what is the standard ...Mon, 10 Aug 2026 16:47:13 -0400https://liberty.io/how-to-convert-from-cartesian-to-polar-equation.htmlhttps://liberty.io/how-to-convert-from-cartesian-to-polar-equation.htmlI am trying to convert the equation $y=4/x$ into a polar equation. I have done this work but I am not sure if it is right. I just subsituted $r\sin(\theta)$ for $y$ and $r\cos(\theta)$ for $x$ ......Mon, 10 Aug 2026 16:45:14 -0400https://liberty.io/scaling-rigid-or-non-rigid-transformation.htmlhttps://liberty.io/scaling-rigid-or-non-rigid-transformation.htmlI am trying to look for a precise definition of what rigid and non-rigid transformation is, and to which categories does 'scaling' belong. This is connected to a Point-Set registration problem that......Mon, 10 Aug 2026 16:43:34 -0400https://liberty.io/how-to-find-the-vector-equation-of-a-plane-given-the-scalar-equation-c.htmlhttps://liberty.io/how-to-find-the-vector-equation-of-a-plane-given-the-scalar-equation-c.htmlHow would I find the vector equation of the plane: $x + 2y + 7z - 3 = 0$ So far, I found the normal vector: it's $(1, 2, 7)$....Mon, 10 Aug 2026 16:43:06 -0400https://liberty.io/the-coercivity-of-uniformly-positive-definite-matrix-of-sobolev-functi.htmlhttps://liberty.io/the-coercivity-of-uniformly-positive-definite-matrix-of-sobolev-functi.htmlFor $u=(u^1,\ldots, u^N)\in W^{1,2}(\Omega,R^N)$ where $\Omega$ is bounded. We define $$ E[u]=\int_\Omega g_{ij}(u)\nabla u^i\nabla u^jdx$$ where $G=(g_{ij})_{1\leq i,j\leq N}$ is an given uniformly ...Mon, 10 Aug 2026 16:39:28 -0400https://liberty.io/in-english-is-there-a-difference-between-square-and-squared.htmlhttps://liberty.io/in-english-is-there-a-difference-between-square-and-squared.htmlFor example, if I say "one over $x$ square/squared" Do I mean: $1/\sqrt{x}$ or $1/x^2$? Is there some other way to distinguish between the English pronounciation between $x^2$ and $\sqrt{......Mon, 10 Aug 2026 16:37:06 -0400https://liberty.io/real-analysis-folland-problem-1-5-29-lebesgue-measurable-set.htmlhttps://liberty.io/real-analysis-folland-problem-1-5-29-lebesgue-measurable-set.html1.5.29 - Let $E$ be a Lebesgue measurable set. a.) If $E\subset N$ where $N$ is the nonmeasurable set described in section 1.1, then $m(E) = 0$. b.) If $m(E) > 0$, then $E$ contains a nonmeasura......Mon, 10 Aug 2026 16:32:06 -0400https://liberty.io/evaluating-surface-integral-using-stokes-039-theorem.htmlhttps://liberty.io/evaluating-surface-integral-using-stokes-039-theorem.htmlLet $\vec{F}$ be the vector field $\vec{F}\left(x,y,z\right)=\left(z,x,y\right)$. Let $\rm S$ be portion of the surface $x^{2}+y^{2}+z=1$ lying above the $\rm XY$-plane, oriented upward. Evaluate the ...Mon, 10 Aug 2026 16:30:34 -0400https://liberty.io/prove-that-the-planar-curve-obtained-by-projecting-alpha-into-its-oscu.htmlhttps://liberty.io/prove-that-the-planar-curve-obtained-by-projecting-alpha-into-its-oscu.htmlLet $\alpha(s)$ be a regular curve with $\kappa\neq0$ at P, where $\kappa$ is the curvature. Prove that the planar curve obtained by projecting $\alpha(s)$ into its osculating plane at $P$ has the ......Mon, 10 Aug 2026 16:29:25 -0400https://liberty.io/write-2-log-3-x-log-9-x-4-log-27-x-5-into-one-single-logarithm.htmlhttps://liberty.io/write-2-log-3-x-log-9-x-4-log-27-x-5-into-one-single-logarithm.htmlI'm trying to answer this problem: Write $$2+\log_3 x+\log_9 x^4-\log_{27} x^5$$ into one single logarithm. This is what I'm stuck with: $$\frac{\log x}{\log 3}+ \frac{\log x^4}{\log 3^2}+\frac......Mon, 10 Aug 2026 16:25:04 -0400https://liberty.io/how-to-find-the-potential-function-of-a-vector-field.htmlhttps://liberty.io/how-to-find-the-potential-function-of-a-vector-field.htmlI calculated that $\frac{dP}{dy} = \cos(y) = \frac{dQ}{dx}$ This tells me that the potential function exists, however I can't figure out what it is. So far I have found that $\frac{df}{dx} = \si......Mon, 10 Aug 2026 16:23:31 -0400https://liberty.io/is-the-term-monotone-used-fairly-consistently-to-mean-non-decreasing-o.htmlhttps://liberty.io/is-the-term-monotone-used-fairly-consistently-to-mean-non-decreasing-o.htmlIn BBFSK, (~1960, Germany) at least in the section I am currently reading, the authors use the term monotone increasing (decreasing) to mean what I often see called strictly increasing (decreasing)......Mon, 10 Aug 2026 16:18:54 -0400https://liberty.io/orthogonal-projection-on-an-eigenspace.htmlhttps://liberty.io/orthogonal-projection-on-an-eigenspace.htmlFor different symmetric matrix $A$ ( eigenspaces are orthogonals ), I've noticed that when I build the matrix of the orthogonal projection of an eigenspace I get identity. Are my calculation wrong ......Mon, 10 Aug 2026 16:18:31 -0400https://liberty.io/how-to-solve-sqrt-3-x-cos-2x-0-for-sketching-the-graph-of-y-sqrt-3-x-c.htmlhttps://liberty.io/how-to-solve-sqrt-3-x-cos-2x-0-for-sketching-the-graph-of-y-sqrt-3-x-c.htmlSolve the following question. $$\begin{align} \\\sqrt{3}x-\cos 2x=0\\ \end{align}$$ I have no idea how to solve the above question with x inside and outside the cos. Since I have to sketch the......Mon, 10 Aug 2026 16:07:36 -0400https://liberty.io/software-for-testing-relational-algebra.htmlhttps://liberty.io/software-for-testing-relational-algebra.htmlDoes anyone know of any software to let you test relational algebra queries? By this I don't mean a database such as MySQL, something where the query can be input in some for of mathematical notation ...Mon, 10 Aug 2026 16:07:17 -0400https://liberty.io/using-power-series-to-evaluate-a-limit.htmlhttps://liberty.io/using-power-series-to-evaluate-a-limit.htmlIf we want to evaluate the limit of $$\lim_{x\to 0} \frac{\ln(1-8x)+8x+32x^2}{3x^2}$$ I would imagine we need to use the power series expansion of the function $${\ln(1-8x)}$$ and then determine ......Mon, 10 Aug 2026 15:57:48 -0400https://liberty.io/finding-a-derivative-with-imaginary-numbers.htmlhttps://liberty.io/finding-a-derivative-with-imaginary-numbers.htmlIf I have some function $f = (1 + 7i - x)(7 + 5i - x)(3 + 1i - x)$ How do I find its derivative? I know it is $(13/3 + 11i/3 - x)(3 + 5i - x)$ but I don't know how to find it....Mon, 10 Aug 2026 15:45:19 -0400https://liberty.io/what-is-the-meaning-of-a-s.htmlhttps://liberty.io/what-is-the-meaning-of-a-s.htmlWhat is the meaning of a.s. behind a limit formula (I found this in a paper about stochastic processes) , or sometimes P-a.s.?...Mon, 10 Aug 2026 15:42:34 -0400https://liberty.io/why-are-there-12-pentagons-and-20-hexagons-on-a-soccer-ball.htmlhttps://liberty.io/why-are-there-12-pentagons-and-20-hexagons-on-a-soccer-ball.htmlEdge-attaching many hexagons results in a plane. Edge-attaching pentagons yields a dodecahedron. Is there some insight into why the alternation of pentagons and hexagons yields an approximated sph......Mon, 10 Aug 2026 15:40:55 -0400https://liberty.io/explanation-of-method-for-finding-the-intersection-of-two-parabolas.htmlhttps://liberty.io/explanation-of-method-for-finding-the-intersection-of-two-parabolas.htmlI am trying to understand the steps in a modified version of the Fortune's algorithm for voronoi generation (http://blog.ivank.net/fortunes-algorithm-and-implementation.html). I've stumbled on the ...Mon, 10 Aug 2026 15:40:01 -0400https://liberty.io/how-to-find-the-inverse-of-y-x-3-5x-2-3x-c.htmlhttps://liberty.io/how-to-find-the-inverse-of-y-x-3-5x-2-3x-c.htmlI know how to find the inverse of $y = x^3$, but once you add/subtract another term, that is where I become lost. I have used wolframalpha to get an answer. I am lost upon how this was obtained....Mon, 10 Aug 2026 15:37:25 -0400https://liberty.io/easy-optimization-problem-checking-to-see-my-reasoning-maximum-volume-.htmlhttps://liberty.io/easy-optimization-problem-checking-to-see-my-reasoning-maximum-volume-.htmlA cylindrical package to be sent by a postal service can have a maximum combined length and girth (perimeter of a cross section) of 108 inches. Find the dimensions of the package of maximum volume ......Mon, 10 Aug 2026 15:30:27 -0400https://liberty.io/probability-of-getting-at-least-2-heads.htmlhttps://liberty.io/probability-of-getting-at-least-2-heads.htmlLet assume, I through a fair coin three times, and I want to get at least two heads. I want to find the the probability. To find that, At first I evaluated the probability of getting 1 head whic......Mon, 10 Aug 2026 15:19:55 -0400https://liberty.io/gauss-sum-in-regular-7-gon.htmlhttps://liberty.io/gauss-sum-in-regular-7-gon.htmlThe heptagon on the picture is a regular heptagon with side 1. What is the length of the dashed interval? This is a (kind of) 'geometric version of a quadratic Gauss sum for p=7' (this observation......Mon, 10 Aug 2026 15:12:52 -0400https://liberty.io/symbol-for-such-that-not-in-set.htmlhttps://liberty.io/symbol-for-such-that-not-in-set.htmlIf $A$ is a set, we can use the set notation $$A= \{ b \mid\text{property $p_1$ of $b$}\}$$ But say $A$ is an element like $b$, $$A = b \mid \text{property $p_1$ of $b$}$$ is this a usual nota......Mon, 10 Aug 2026 15:01:32 -0400https://liberty.io/what-is-the-difference-between-sigma-sigma-bar-x-s-s-and-s-bar-x.htmlhttps://liberty.io/what-is-the-difference-between-sigma-sigma-bar-x-s-s-and-s-bar-x.htmlWhat is the difference between $\sigma, \sigma_{\bar{x}}, S, s,$ and $s_{\bar{x}}$? My textbook uses lots of different symbols, and it's not clear to me what the difference between all of them are......Mon, 10 Aug 2026 14:57:31 -0400https://liberty.io/what-is-cos-pi-12-i-sin-pi-12-16-cos-pi-12-i-sin-pi-12-16.htmlhttps://liberty.io/what-is-cos-pi-12-i-sin-pi-12-16-cos-pi-12-i-sin-pi-12-16.htmlWhat is $[\cos(\pi/12)+i\sin(\pi/12)]^{16}+[\cos(\pi/12)-i\sin(\pi/12)]^{16}$? I can use De Moivre's formula for the left part: $[\cos(\pi/12)+i\sin(\pi/12)]^{16} = \cos(4\pi/3) + i\sin(4\pi/3) = -\...Mon, 10 Aug 2026 14:54:58 -0400https://liberty.io/absolute-value-theorem.htmlhttps://liberty.io/absolute-value-theorem.htmlWhen trying to prove the inequality $$ |a +b| \leq |a| + |b| \text{, for any real numbers a and b} $$ I manage to use the absolute value definition to get to following inequality: $$ -\big(|a|+|......Mon, 10 Aug 2026 14:54:16 -0400https://liberty.io/decimal-form-of-irrational-numbers.htmlhttps://liberty.io/decimal-form-of-irrational-numbers.htmlIn the decimal form of an irrational number like: $$\pi=3.141592653589\ldots$$ Do we have all the numbers from $0$ to $9$. I verified $\pi$ and all the numbers are there. Is this true in general for ...Mon, 10 Aug 2026 14:51:46 -0400https://liberty.io/tessellate-the-plane-with-arbitrary-surfaces.htmlhttps://liberty.io/tessellate-the-plane-with-arbitrary-surfaces.htmlIt is known that every surface can be represented by a polygon and glueing the sides appropriately (A source). It is also known that the only tessellations of the plane are by squares, triangles and ...Mon, 10 Aug 2026 14:30:45 -0400https://liberty.io/markov-039-s-inequality-tight-in-general.htmlhttps://liberty.io/markov-039-s-inequality-tight-in-general.htmlFrom here: Even though Markov’s and Chebyshev’s Inequality only use information about the expectation and thevariance of the random variable under consideration, they are essentially tight for a ...Mon, 10 Aug 2026 14:29:18 -0400https://liberty.io/what-did-whitehead-and-russell-039-s-principia-mathematica-achieve.htmlhttps://liberty.io/what-did-whitehead-and-russell-039-s-principia-mathematica-achieve.htmlIn philosophical contexts, the Principia Mathematica is sometimes held in high regard as a demonstration of a logical system. But what did Whitehead and Russell's Principia Mathematica achieve for ...Mon, 10 Aug 2026 14:19:01 -0400https://liberty.io/how-does-the-difference-quotient-with-a-square-root-in-the-numerator-e.htmlhttps://liberty.io/how-does-the-difference-quotient-with-a-square-root-in-the-numerator-e.htmlI don't understand when I apply the difference quotient to: $f(x) = \sqrt{x} $ , to get: $$\frac{\sqrt{x+h} - \sqrt{x}}{h}$$ To simplify it.. How does it end up like this?: $$\frac{x + h - x}......Mon, 10 Aug 2026 14:16:08 -0400https://liberty.io/square-brackets-in-indices.htmlhttps://liberty.io/square-brackets-in-indices.htmlWhat do these brackets within the indices mean in an equation like $$ \delta ^\mu _{[\alpha} \eta _{\beta ]\nu} ?$$ I can't find a text, which uses this notation, that explains it....Mon, 10 Aug 2026 14:11:04 -0400https://liberty.io/one-sided-limits-of-f-039-x-at-a-point-vs-one-sided-limits-of-the-diff.htmlhttps://liberty.io/one-sided-limits-of-f-039-x-at-a-point-vs-one-sided-limits-of-the-diff.htmlI know that if a function is piecewise, then when differentiating it one needs to deal separately with the points of intersection of the pieces, because the derivative may not be defined there. Al......Mon, 10 Aug 2026 14:04:29 -0400https://liberty.io/can-the-delta-function-be-even-or-odd.htmlhttps://liberty.io/can-the-delta-function-be-even-or-odd.htmlSay I have the function, $$ \delta\left(\tau-\frac{T}{2}\right)\mathrm d\tau $$ where T = $$\frac{2π}{ω}$$ Is this even or odd? Or neither? Reason I ask is to find out whether I can cancel some ...Mon, 10 Aug 2026 14:03:15 -0400https://liberty.io/confusion-between-the-definition-of-increasing-function-and-a-theorem-.htmlhttps://liberty.io/confusion-between-the-definition-of-increasing-function-and-a-theorem-.htmlThe definition of increasing function given in my school maths text book is Let $I$ be an open interval contained in the domain of a real valued function $f$. Then $f$ is said to be increasing on......Mon, 10 Aug 2026 13:59:46 -0400https://liberty.io/forward-euler-stability.htmlhttps://liberty.io/forward-euler-stability.htmlI am trying to understand the stability of the forward Euler method. I read there's a model problem to see the stability. $$y'(t) = \lambda y(t) \qquad t \in (0, \infty)$$ $$y(0) = 1$$ then the b......Mon, 10 Aug 2026 13:55:43 -0400https://liberty.io/definition-of-closed-subscheme.htmlhttps://liberty.io/definition-of-closed-subscheme.htmlLet $(X,\mathcal O_X)$ be a scheme. I'm using the following definition from Görtz-Wedhorn. A closed subscheme of $(X,\mathcal O_X)$ is a scheme $(Z,\mathcal O_Z)$, where $i\colon Z\hookrightarrow X......Mon, 10 Aug 2026 13:52:22 -0400https://liberty.io/how-to-take-the-derivative-of-a-matrix-with-respect-to-itself.htmlhttps://liberty.io/how-to-take-the-derivative-of-a-matrix-with-respect-to-itself.htmlCould someone please explain how to take the derivative of matrix with respect to itself? $$\frac{\partial \textbf{X}}{\partial \textbf{X}}$$ where $\textbf{X}$ is an M x N matrix...Mon, 10 Aug 2026 13:52:13 -0400https://liberty.io/finding-the-fourth-vertex-of-a-square.htmlhttps://liberty.io/finding-the-fourth-vertex-of-a-square.htmlI was given this problem: Without drawing a graph and given the following 3 vertices, find the coordinates of the last vertex of the square: $(3,2)$$(0,5)$$(-3,2)$ So first, I found the leng......Mon, 10 Aug 2026 13:34:16 -0400https://liberty.io/tangent-of-a-straight-line.htmlhttps://liberty.io/tangent-of-a-straight-line.htmlI just got back a math test in my EF Calc class, and I disagree with my teacher on this one problem. We are using derivatives to determine equations of lines tangent to a given equation. The equation ...Mon, 10 Aug 2026 13:33:13 -0400https://liberty.io/is-square-root-a-function.htmlhttps://liberty.io/is-square-root-a-function.htmlI just began the topic of functions in my Mathematics textbook and in the first paragraph itself, two conditions about a relation being a function were mentioned. They were : 1) For every a∈A, t......Mon, 10 Aug 2026 13:28:30 -0400https://liberty.io/the-speed-of-light-in-a-vacuum-is-2-998-times-10-8-m-sec-what-is-this-.htmlhttps://liberty.io/the-speed-of-light-in-a-vacuum-is-2-998-times-10-8-m-sec-what-is-this-.htmlPeace to all. Studying and trying to get a handle on how to calculate like-minded questions, I came across this question from the web. I don't understand how exactly it was solved. How were the u......Mon, 10 Aug 2026 13:19:10 -0400https://liberty.io/does-the-relation-in-follow-the-axiom-of-substitution-for-equality.htmlhttps://liberty.io/does-the-relation-in-follow-the-axiom-of-substitution-for-equality.htmlI am trying to self-learn set theory from Analysis 1 by Terence Tao. I am facing a problem understanding how he shows the relation $\in$ for sets follows the axiom of substitution. He begins by ...Mon, 10 Aug 2026 13:18:11 -0400https://liberty.io/show-that-pi-3-arccos-frac-5-sqrt-28-3-arctan-frac-sqrt-3-2.htmlhttps://liberty.io/show-that-pi-3-arccos-frac-5-sqrt-28-3-arctan-frac-sqrt-3-2.htmlQuestion: Show that, $$\pi =3\arccos(\frac{5}{\sqrt{28}}) + 3\arctan(\frac{\sqrt{3}}{2}) ~~~~~~ (*)$$ My proof method for this question has received mixed responses. Some people say it's fine, ot......Mon, 10 Aug 2026 13:17:35 -0400https://liberty.io/rationalize-the-denominator-with-4-terms-frac-1-1-sqrt-12-sqrt-3-3-sqr.htmlhttps://liberty.io/rationalize-the-denominator-with-4-terms-frac-1-1-sqrt-12-sqrt-3-3-sqr.htmlRationalize the denominator of this fraction $$\frac{1}{1+\sqrt{12}-\sqrt[3]{3}-\sqrt[3]{9}}$$...Mon, 10 Aug 2026 13:17:03 -0400https://liberty.io/find-all-the-relative-extrema-of-the-function-f-x-2x-24x-1-3.htmlhttps://liberty.io/find-all-the-relative-extrema-of-the-function-f-x-2x-24x-1-3.htmlFind all the relative extrema of the function $f(x)=2x-24x^{1/3}$ Solution: Step 1: Find the values of $x$ where $f'(x)=0$ and $f'(x)$ DNE. $f'(x)=2-8x^{\frac{-2}{3}}=2-\frac{8}{x^{\frac{2}{3}}}......Mon, 10 Aug 2026 13:12:40 -0400https://liberty.io/trapezoids-similarity.htmlhttps://liberty.io/trapezoids-similarity.htmlI found this geometry problem in an IGCSE text. Can't seem to find the missing lengths that they are asking for. I got an estimate for f (f=2.6666... cm) by drawing a line parallel to the left side......Mon, 10 Aug 2026 13:03:37 -0400https://liberty.io/how-to-implement-subscript-and-superscript-in-legend-matlab.htmlhttps://liberty.io/how-to-implement-subscript-and-superscript-in-legend-matlab.htmlI am wondering how to implement text in a legend with both a superscript and a subscript, and with an horizontal bar, something like this: Image...Mon, 10 Aug 2026 12:58:39 -0400https://liberty.io/proof-about-cardinality-of-sets-r-z-r.htmlhttps://liberty.io/proof-about-cardinality-of-sets-r-z-r.htmlProve that $|R-Z|=|R|$ where $R$ is reals and $Z$ is integers. New approach Maybe the problem is reduced to finding a bijection between (0,1) and [0,1] if we find a bijection between these interv......Mon, 10 Aug 2026 12:55:22 -0400https://liberty.io/why-are-self-adjoint-operators-important.htmlhttps://liberty.io/why-are-self-adjoint-operators-important.htmlI am learning about self-adjoint and normal operators. So far, they have come up in the Spectral theorem, which says self-adjoint operators have an eigenvalue basis and a corresponding diagonal ma......Mon, 10 Aug 2026 12:49:52 -0400https://liberty.io/cartesian-form-of-a-line.htmlhttps://liberty.io/cartesian-form-of-a-line.htmlI have a general question: when given the parametric form of a line such that: $$ x = a_1 + \lambda t_1 $$ $$ y = a_2 + \lambda t_2 $$ $$ z = a_3 + \lambda t_3 $$ The Cartesian form is: $$ \fr......Mon, 10 Aug 2026 12:49:31 -0400https://liberty.io/chinese-remainder-theorem-for-three-equations.htmlhttps://liberty.io/chinese-remainder-theorem-for-three-equations.htmlIs there a straightforward approach for solving the Chinese Remainder Theorem with three congruences? $$x \equiv a \bmod A$$ $$x \equiv b \bmod B$$ $$x \equiv c \bmod C$$ Assuming all values are ...Mon, 10 Aug 2026 12:45:46 -0400https://liberty.io/can-someone-explain-why-x-log-a-a-log-x-duplicate.htmlhttps://liberty.io/can-someone-explain-why-x-log-a-a-log-x-duplicate.htmlI'm trying to see why the below is true. $$ x^{\log(a)} = a^{\log(x)} $$ Anyone here know why this is? Thank you....Mon, 10 Aug 2026 12:40:14 -0400https://liberty.io/find-angle-between-two-curves-at-point-of-intersection-closed.htmlhttps://liberty.io/find-angle-between-two-curves-at-point-of-intersection-closed.htmlFind angle between these two curves at point of intersection : $$K_1: x^2y^2 + y^4 = 1$$ and $$K_2 : x^2 + y^2 = 4 $$ Thanks!...Mon, 10 Aug 2026 12:29:57 -0400https://liberty.io/fractal-fundamentals.htmlhttps://liberty.io/fractal-fundamentals.htmlI am a programmer by trade, and am very interested in fractals. To be very basic about the concept, one might say a 'circle of circles' is a fractal. Where each circle is made up of circles, and t......Mon, 10 Aug 2026 12:25:20 -0400https://liberty.io/equal-spaced-points-in-a-logarithmic-graph.htmlhttps://liberty.io/equal-spaced-points-in-a-logarithmic-graph.htmlI am plotting a graph with the x-axis as logarithmic. I want to select 10 point that are equally spaced in a logarithm scale. How can I determine the values if we have the range from 100 to 10000?...Mon, 10 Aug 2026 12:18:30 -0400https://liberty.io/when-a-fraction-is-raised-to-a-negative-exponent-do-you-normally-trans.htmlhttps://liberty.io/when-a-fraction-is-raised-to-a-negative-exponent-do-you-normally-trans.htmlMy text shows that $$\left(\frac{3a^2}{4b}\right)^{-3}=\frac{1}{\left(\frac{3a^2}{4b}\right)^{3}}.$$ It also shows that $$\frac{1}{\frac{144}{b}}=\frac{b}{144}.$$ In the first equation, it se......Mon, 10 Aug 2026 12:12:18 -0400https://liberty.io/questions-tagged-linear-programming.htmlhttps://liberty.io/questions-tagged-linear-programming.htmlQ&A for people studying math at any level and professionals in related fields...Mon, 10 Aug 2026 11:59:08 -0400https://liberty.io/finding-a-basis-for-the-columnspace-of-a-matrix.htmlhttps://liberty.io/finding-a-basis-for-the-columnspace-of-a-matrix.htmlFind a linearly independent set of vectors that spans the same subspace of $R^4$ as that spanned by the vectors - $$ \begin{bmatrix} 2 \\ -4 \\ -1 \\ -2 \\ \end{bmatrix} , \begin{bmatrix}......Mon, 10 Aug 2026 11:46:04 -0400https://liberty.io/questions-tagged-prime-numbers.htmlhttps://liberty.io/questions-tagged-prime-numbers.htmlQ&A for people studying math at any level and professionals in related fields...Mon, 10 Aug 2026 11:08:47 -0400https://liberty.io/ryll-nardzewski-theorem-and-types-with-parameters.htmlhttps://liberty.io/ryll-nardzewski-theorem-and-types-with-parameters.htmlBy Ryll-Nardzewski theorem we know that a theory $T$ is $\omega$-categorical iff for each $n$, every type in $S_n(T)$ is isolated. Question. What can we say about types with parameters. Does $\omega$-...Mon, 10 Aug 2026 11:07:03 -0400https://liberty.io/how-can-i-find-a-polynomial-that-fits-a-given-table-closed.htmlhttps://liberty.io/how-can-i-find-a-polynomial-that-fits-a-given-table-closed.htmlThe first difference is 1, 4, 7, 10, 13, 16 The second difference is 3, 3, 3, 3, 3, 3 Since the second difference is constant this would be quadratic and I would have $\frac{3}{2}n^{2}$ So now I ...Mon, 10 Aug 2026 10:58:54 -0400https://liberty.io/why-do-exponents-not-distribute-over-addition-closed.htmlhttps://liberty.io/why-do-exponents-not-distribute-over-addition-closed.htmlI understand that exponents don't distribute over addition and have seen plenty of examples i.e. $$ (x + y)^2\neq x^2 + y^2 $$ but I'm wondering why that is. Multiplication distributes over additio......Mon, 10 Aug 2026 10:54:12 -0400https://liberty.io/solving-trigonometric-equations-of-the-form-a-sin-x-b-cos-x-c.htmlhttps://liberty.io/solving-trigonometric-equations-of-the-form-a-sin-x-b-cos-x-c.htmlSuppose that there is a trigonometric equation of the form $a\sin x + b\cos x = c$, where $a,b,c$ are real and $0 < x < 2\pi$. An example equation would go the following: $\sqrt{3}\sin x + \c......Mon, 10 Aug 2026 10:43:10 -0400https://liberty.io/second-order-ode-in-matlab.htmlhttps://liberty.io/second-order-ode-in-matlab.htmlI am given an example file for solving a second order ODE using ODE 45 function ex_with_2eqs t0 = 0; tf = 20; y0 = [10;60]; a = .8; b = .01; c = .6; d = .1; [t,y] = ode45(@f,[t0,tf],y0,[],a,......Mon, 10 Aug 2026 10:38:22 -0400https://liberty.io/expected-value-finding-the-price-of-a-raffle-ticket.htmlhttps://liberty.io/expected-value-finding-the-price-of-a-raffle-ticket.htmlThis question has been bugging me for a while now and I want to know where I'm going wrong. There are $20$ tickets in a raffle with one prize. What should each ticket cost if the prize is \$80 ......Mon, 10 Aug 2026 10:37:47 -0400https://liberty.io/jacobian-matrix-in-039-polar-coordinates-039.htmlhttps://liberty.io/jacobian-matrix-in-039-polar-coordinates-039.htmlI'm pretty familiar with the del operator and I've just become familiar with the Jacobian Matrix. From my basic understanding of what the Jacobian Matrix is, the gradient is a subset of the Jacobian ...Mon, 10 Aug 2026 10:37:04 -0400https://liberty.io/why-can-039-t-a-rational-function-have-both-a-horizontal-and-an-obliqu.htmlhttps://liberty.io/why-can-039-t-a-rational-function-have-both-a-horizontal-and-an-obliqu.htmlWhy if a rational function has a horizontal asymptote, it excludes the possibility of this function to have an oblique azymptote, or viceversa?...Mon, 10 Aug 2026 10:26:49 -0400https://liberty.io/shannon-entropy-of-a-fair-dice.htmlhttps://liberty.io/shannon-entropy-of-a-fair-dice.htmlThe formula for Shannon entropy is as follows, $$\text{Entropy}(S) = - \sum_i p_i \log_2 p_i $$ Thus, a fair six sided dice should have the entropy, $$- \sum_{i=1}^6 \dfrac{1}{6} \log_2 \dfrac{1......Mon, 10 Aug 2026 10:17:16 -0400https://liberty.io/what-does-echelon-mean.htmlhttps://liberty.io/what-does-echelon-mean.htmlWhen you solve a system of linear equations, you write down the augmented matrix and reduce this to reduced row echelon form. What is the meaning of the word echelon?...Mon, 10 Aug 2026 10:06:31 -0400https://liberty.io/does-every-nonzero-turing-degree-adimit-a-nonzero-incompatible-degree.htmlhttps://liberty.io/does-every-nonzero-turing-degree-adimit-a-nonzero-incompatible-degree.htmlIt is well known that every nonzero Turing degree $\bf x$ adimits an incomparable degree $\bf y$, that is a degree such that $\bf x\not\le_T\bf y$ and $\bf y\not\le_T\bf x$. I was wondering whethe......Mon, 10 Aug 2026 10:01:14 -0400https://liberty.io/problems-with-simplifying-using-factoring-of-binomial-expressions.htmlhttps://liberty.io/problems-with-simplifying-using-factoring-of-binomial-expressions.htmlI am running into problems simplifying using factoring of binomial expressions. The problem at hand is this: $(x-1)^3*(2x-3)-(2x+12)*(x-1)^2$ I first expanded the left side of the minus sign, lik......Mon, 10 Aug 2026 09:57:49 -0400https://liberty.io/can-we-ever-get-an-irrational-number-by-dividing-two-rational-numbers.htmlhttps://liberty.io/can-we-ever-get-an-irrational-number-by-dividing-two-rational-numbers.htmlIf we try to divide any two random arbitrarily long rational numbers like 103850.2387209375029375092730958297836958623986868349693868398659825528365... and 127.123123123... Is it guaranteed th......Mon, 10 Aug 2026 09:53:18 -0400https://liberty.io/finding-measuring-points-in-three-point-perspective-drawing.htmlhttps://liberty.io/finding-measuring-points-in-three-point-perspective-drawing.htmlIn his Complete Guide to Perspective Drawing (page 39), Craig Attebery places Measuring Points of two-point perspective in a line perperdicular to that crossing the Station Point and the Center of ......Mon, 10 Aug 2026 09:19:03 -0400https://liberty.io/difference-in-the-definitions-of-glb-and-lub-in-real-analysis-and-abst.htmlhttps://liberty.io/difference-in-the-definitions-of-glb-and-lub-in-real-analysis-and-abst.htmlThe following text is from the book Abstract Algebra by T. W. Judson : Let $X = {\{1,2,3,4,6,8,12,24}\}$ be the set of divisors of $24$ with the partial order defined by $a\preceq b$ if $a | b$.......Mon, 10 Aug 2026 09:09:34 -0400https://liberty.io/in-a-parallelogram-does-the-diagonal-bisect-the-angles-that-they-meet.htmlhttps://liberty.io/in-a-parallelogram-does-the-diagonal-bisect-the-angles-that-they-meet.htmlI understand the following properties of the parallelogram: Opposite sides are parallel and equal in length. Opposite angles are equal. Adjacent angles add up to 180 degrees therefore adjacent ang......Mon, 10 Aug 2026 09:04:21 -0400https://liberty.io/two-circles-with-a-common-external-tangent-problem-closed.htmlhttps://liberty.io/two-circles-with-a-common-external-tangent-problem-closed.htmlTwo circles are tangent externally at A, and a common external tangent touches them at B and C. The line segment BA is extended, meeting the second circle at D. Prove that CD is a diameter....Mon, 10 Aug 2026 08:47:43 -0400https://liberty.io/find-normal-vector-of-a-3d-vector.htmlhttps://liberty.io/find-normal-vector-of-a-3d-vector.htmlI need to find the normal vector for the following 3d vector presented in the vectorial equation because I need to find a plane that is orthogonal to the following line: $(x,y,z)=(1,0,0)+k(1,2,3......Mon, 10 Aug 2026 08:37:46 -0400https://liberty.io/rank-theorem-on-manifolds.htmlhttps://liberty.io/rank-theorem-on-manifolds.htmlRank theorem on manifolds says that : Suppose $M$ and $N$ are two smooth manifolds of dimensions $m$ and $n$, respectively, and $F:M\rightarrow N$ be a smooth map with constant rank $r$. For eac......Mon, 10 Aug 2026 08:17:08 -0400https://liberty.io/prove-operatorname-rank-a-leq-1-iff-a-xy-t-for-some-x-y-in-mathbb-r-n-.htmlhttps://liberty.io/prove-operatorname-rank-a-leq-1-iff-a-xy-t-for-some-x-y-in-mathbb-r-n-.htmlLet $A \in \mathbb{R}^{n \times n}$. Then $\operatorname{rank} A \leq 1$ if and only if $A = xy^T$ for some $x, y \in \mathbb{R}^{n \times 1}$. Need help. I can't find sufficient facts nor theorem......Mon, 10 Aug 2026 08:15:48 -0400https://liberty.io/is-a-function-that-is-one-to-one-necessarily-onto.htmlhttps://liberty.io/is-a-function-that-is-one-to-one-necessarily-onto.htmlI'd would like to know that, because I don't want to prove a function is onto if I don't have to. If the answer is no, is there any particular case where it is true?...Mon, 10 Aug 2026 08:13:47 -0400https://liberty.io/orientation-geometry-closed.htmlhttps://liberty.io/orientation-geometry-closed.htmlA xy-plane is rotated about x and y-axis. But when it is rotated along z-axis, it doesn't change the orientation of xy-plane, why? I've tried to find the answer but those answer that i found didn't......Mon, 10 Aug 2026 08:10:21 -0400https://liberty.io/complex-variable-limit-at-infinity.htmlhttps://liberty.io/complex-variable-limit-at-infinity.htmlIs $\lim\limits_{z\to\infty} \frac{4z^2}{(z-1)^2}$, $z\in\mathbb{C}$, evaluated the same way as a real variable function limit? Or does one need to show separate cases for $x\to\infty$ and $y\to\in......Mon, 10 Aug 2026 08:09:03 -0400https://liberty.io/how-to-determine-what-these-vectors-are.htmlhttps://liberty.io/how-to-determine-what-these-vectors-are.htmlSo I'm not sure how to solves these problems. The Question asks to Determine whether the given vectors are orthogonal, parallel or neither. 1) a = <-5, 3, 7> and b = <6, -8, 2> 2) a = &lt......Mon, 10 Aug 2026 08:04:10 -0400https://liberty.io/solving-for-upper-limit-of-integration-given-a-lower-limit-and-integra.htmlhttps://liberty.io/solving-for-upper-limit-of-integration-given-a-lower-limit-and-integra.htmlConsider $f$ a real integrable function, usually we want to evaluate the integral $\int_a^bf(x)dx$ for some $a<b$ given. Now, suppose we know $a$ but we don't know $b$, further, we know the valu......Mon, 10 Aug 2026 08:00:53 -0400https://liberty.io/general-formula-for-sin-left-k-arcsin-x-right.htmlhttps://liberty.io/general-formula-for-sin-left-k-arcsin-x-right.htmlI'm wondering if there's a simple way to rewrite this in terms of $k$ and $x$, especially as a polynomial. It seems to me to crop up every so often, especially for $k=2$, when I integrate with trig ...Mon, 10 Aug 2026 07:57:29 -0400https://liberty.io/prove-that-a-parallelogram-is-1-rectangle-2-rhombus-3-square.htmlhttps://liberty.io/prove-that-a-parallelogram-is-1-rectangle-2-rhombus-3-square.htmlThe question is a follows. Midpoints of the sides of a quadrilateral are the vertices of a paralleogram. Determine under what conditions this parallelogram will be (1) a rectangle, (2) a rhombus,......Mon, 10 Aug 2026 07:39:21 -0400