https://liberty.ioCurated star highlights with fresh energy.en-usThu, 03 Sep 2026 15:27:31 -0400https://liberty.io/proving-the-cauchy-schwarz-inequality-by-induction.htmlhttps://liberty.io/proving-the-cauchy-schwarz-inequality-by-induction.htmlI ran across this problem in some old notes, and I frustratingly can't figure out how to do it Let $a_i$ and $b_i$ be sequences of natural numbers, use induction to show $\sum_{i=1}^n (a_ib_i)^{......Thu, 03 Sep 2026 19:25:40 -0400https://liberty.io/how-can-i-find-the-equation-of-a-parabola-only-given-it-039-s-x-interc.htmlhttps://liberty.io/how-can-i-find-the-equation-of-a-parabola-only-given-it-039-s-x-interc.htmlI received a problem in my math class the other day that left me stumped. The problem went something like this. Mr. Lots-O-Cash would like to order a parabola that passes through the points $(-4......Thu, 03 Sep 2026 19:17:47 -0400https://liberty.io/a-jacobi-elliptic-integral-with-cosine.htmlhttps://liberty.io/a-jacobi-elliptic-integral-with-cosine.htmlAfter reduction of a problem, I find myself in front of these integrals, $$ \int_0^{2\pi} cn \left( \frac{2K(1/2)}{\pi} \theta, \frac{1}{2} \right) \cos(m \theta) \mathrm{d}\theta, $$ with $m$ posi......Thu, 03 Sep 2026 19:11:00 -0400https://liberty.io/using-shell-method-find-the-volume-of-the-solid-generated-when-y-3-x-2.htmlhttps://liberty.io/using-shell-method-find-the-volume-of-the-solid-generated-when-y-3-x-2.htmlQuestion Via the Shell Method, what is the volume of the solid of revolution formed by revolving a portion of $y=3-x^2$ about the x-axis bounded by $y=0$ and $y=3$. Context The answer from Wolfram......Thu, 03 Sep 2026 19:10:37 -0400https://liberty.io/intuitive-explanation-of-duhamel-039-s-principle.htmlhttps://liberty.io/intuitive-explanation-of-duhamel-039-s-principle.htmlThis is with regards to the first section of Wikipedia article Duhamel's principle (revision from July 2012). I want to see if I am understanding this. Basically the inhomogeneous equation says tha......Thu, 03 Sep 2026 18:58:39 -0400https://liberty.io/translate-to-first-order-logic.htmlhttps://liberty.io/translate-to-first-order-logic.htmlI have this two sentences: All men love the woman to whom they are married. No woman loves a man who does not respect all women. and in the exercise they are translated to First Order Logic as fo......Thu, 03 Sep 2026 18:53:30 -0400https://liberty.io/variance-of-x-y.htmlhttps://liberty.io/variance-of-x-y.htmlIf X and Y are random variables with correlation coefficient 0.7, each of which has variance 6, what is the variance of X−Y? Enter your answer as a decimal. Using the information given, I was abl......Thu, 03 Sep 2026 18:50:06 -0400https://liberty.io/ordinary-differential-equations-v-dx-dt.htmlhttps://liberty.io/ordinary-differential-equations-v-dx-dt.htmlThe question is: What ODE does v=dx/dt solve? And what is the limit as t approaches infinity of v(t). It sounds almost too easy, but I am confused about what to do exactly. Intuitively, as the ques......Thu, 03 Sep 2026 18:49:05 -0400https://liberty.io/how-the-product-of-two-integrals-is-iterated-integral-int-cdot-int-iin.htmlhttps://liberty.io/how-the-product-of-two-integrals-is-iterated-integral-int-cdot-int-iin.htmlLet $\large{I} = \Large \int\limits_{-\infty}^{\infty} \normalsize e^{\small -\frac{y^2}{2}}\ dy$. Then, my textbook says, $$\tag{1} I^2 = \left( \int\limits_{-\infty}^{\infty} \normalsize e^{\......Thu, 03 Sep 2026 18:47:24 -0400https://liberty.io/simplify-sqrt-sqrt-3-5-sqrt-3-4.htmlhttps://liberty.io/simplify-sqrt-sqrt-3-5-sqrt-3-4.htmlDenest $\sqrt {\sqrt[3]{5}-\sqrt[3]{4}}$. I have tried completing square by several method but all failed. Can anyone help me please? Thank you. p.s. I'm a poor question-tagger....Thu, 03 Sep 2026 18:46:11 -0400https://liberty.io/cauchy-schwarz-inequality-for-nonsymmetric-matrices.htmlhttps://liberty.io/cauchy-schwarz-inequality-for-nonsymmetric-matrices.htmlI am interested in Cauchy-Schwarz inequalities for inner products of the form $\langle Ax,y\rangle$ where $A$ is some matrix. It is rather easy to see that the Cauchy-Schwarz inequality continues t......Thu, 03 Sep 2026 18:45:59 -0400https://liberty.io/consistency-of-a-system-of-linear-equations-using-unknowns-in-the-matr.htmlhttps://liberty.io/consistency-of-a-system-of-linear-equations-using-unknowns-in-the-matr.htmlCan anyone tell me how would I answer these type of questions? I already know how to answer the normal way which doesn't include any variables such as Alpha in the question....Thu, 03 Sep 2026 18:40:20 -0400https://liberty.io/how-find-this-sum-sin-2-x-sin-2-2x-sin-2-3x-cdots-sin-2-nx.htmlhttps://liberty.io/how-find-this-sum-sin-2-x-sin-2-2x-sin-2-3x-cdots-sin-2-nx.htmlQuestion: Find the value $$f^{(2)}_{n}(x)=\sin^2{x}+\sin^2{(2x)}+\sin^2{(3x)}+\cdots+\sin^2{(nx)}$$ My solution: since $$\sin^2{x}=\dfrac{1}{2}(1-\cos{(2x)})$$ so $$f_{n}(x)=\dfrac{n}{2}-\dfra......Thu, 03 Sep 2026 18:37:29 -0400https://liberty.io/change-of-polynomial-basis-exercise.htmlhttps://liberty.io/change-of-polynomial-basis-exercise.htmlCan someone please check if my solution is correct? Thanks. Considering the following basis: $$ B= \left\{ x^2,1+x^2,x-1 \right\}\\ C= \left\{ x,1-x,x^2 \right\} $$ a) Find the change of basis mat......Thu, 03 Sep 2026 18:26:53 -0400https://liberty.io/derivative-of-matrix-exponential-w-r-t-to-each-element-of-the-matrix.htmlhttps://liberty.io/derivative-of-matrix-exponential-w-r-t-to-each-element-of-the-matrix.htmlI have $x= \exp(At)$ where $A$ is a matrix. I would like to find derivative of $x$ with respect to each element of $A$. Could anyone help with this problem?...Thu, 03 Sep 2026 18:20:43 -0400https://liberty.io/i-wonder-if-the-set-of-all-characteristic-function-on-a-set-a-is-simil.htmlhttps://liberty.io/i-wonder-if-the-set-of-all-characteristic-function-on-a-set-a-is-simil.htmlRecently I studied something known as characteristic function (Indicator function), after going through it deeply I want know if the set of all characteristic functions on A will be similar to its ......Thu, 03 Sep 2026 18:14:27 -0400https://liberty.io/riemann-sheets-for-combined-roots.htmlhttps://liberty.io/riemann-sheets-for-combined-roots.htmlSo, I am aware of the fact that if I do have a function like $$f(z) = z^{1/2} + z^{1/3}$$ being $2, 3$ relatively primes, I have $$\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z}$$ hence ......Thu, 03 Sep 2026 18:14:11 -0400https://liberty.io/proof-for-rational-roots.htmlhttps://liberty.io/proof-for-rational-roots.htmlLet $f(x) = a_0+a_1x+\cdots+a_nx^n$ be a polynomial of degree $n$ over $\Bbb Z$. A: If a rational number $\frac{p}{q}$ is a root of $f(X)$, show that $p\mid a_0$ and $q\mid a_n$. Assume $\gcd(p, q......Thu, 03 Sep 2026 18:03:50 -0400https://liberty.io/derivative-of-operatorname-arcsec-3x.htmlhttps://liberty.io/derivative-of-operatorname-arcsec-3x.htmlThis seems straightforward to me, but it was marked wrong on my test with no clarification and no ability to ask questions. $y=\operatorname{arcsec}(3x)$. Calculate the derivative. I take the se......Thu, 03 Sep 2026 17:57:58 -0400https://liberty.io/prove-that-in-any-triangle-abc-cos-2a-cos-2b-cos-2c-geq-frac-3-4.htmlhttps://liberty.io/prove-that-in-any-triangle-abc-cos-2a-cos-2b-cos-2c-geq-frac-3-4.htmlI have two similar looking questions. (1) Prove that in triangle $ABC$, $$\cos^2A+\cos^2B+\cos^2C\geq\frac{3}{4}.$$ (2) If $\Delta ABC$ is an acute angled, then prove that $$\cos^2A+\cos^2B+\cos^2......Thu, 03 Sep 2026 17:56:16 -0400https://liberty.io/can-i-subtract-infinity-from-infinity.htmlhttps://liberty.io/can-i-subtract-infinity-from-infinity.htmlI was stuck when solving a problem on limits. It was like----> $\lim_{x\to\infty} (x-x)$. What should I do now?...Thu, 03 Sep 2026 17:55:32 -0400https://liberty.io/aaa-similarity-theorem.htmlhttps://liberty.io/aaa-similarity-theorem.htmlDef: Two triangles are called similar if and only if one can be scaled into the other. AAA Theorem: If two triangles have the same angles if and only if they are similar. How does one prove the T......Thu, 03 Sep 2026 17:36:53 -0400https://liberty.io/how-to-find-the-nth-term-in-the-following-sequence-1-1-2-2-4-4-8-8-16-.htmlhttps://liberty.io/how-to-find-the-nth-term-in-the-following-sequence-1-1-2-2-4-4-8-8-16-.htmlI'm having difficulty in finding the formula for the sequence above, when I put this in WolframAlpha it gave me a rather complex formula which I'm not convinced even works properly but I'm sure the......Thu, 03 Sep 2026 17:34:17 -0400https://liberty.io/is-x-1-a-linear-function.htmlhttps://liberty.io/is-x-1-a-linear-function.htmlI want to know if $\frac1x$ is a linear function or not. I read that a linear function can also be defined as a function of the first degree. Since $\frac1x=x^{-1}$, the function is of the first d......Thu, 03 Sep 2026 17:28:59 -0400https://liberty.io/orthogonal-set-vs-orthogonal-basis.htmlhttps://liberty.io/orthogonal-set-vs-orthogonal-basis.htmlFor some reason my book distinguishes the two names. If a set is an orthogonal set, doesn't that make it immediately a basis for some subspace $W$ since all the vectors in the orthogonal set are ...Thu, 03 Sep 2026 17:26:11 -0400https://liberty.io/why-are-all-non-prime-numbers-divisible-by-a-prime-number.htmlhttps://liberty.io/why-are-all-non-prime-numbers-divisible-by-a-prime-number.htmlIn Euclid's infinite prime numbers proof, the logic is as follows: Assume a set $S$ of all prime numbers in existence is finite (there are a finite amount of primes) Then there must be a greatest......Thu, 03 Sep 2026 17:14:21 -0400https://liberty.io/double-pendulum-lagrangian.htmlhttps://liberty.io/double-pendulum-lagrangian.htmlcould someone help me with the following problem, for part $b$ i know that $$\Bbb L=T-U$$ but i've forgotten how i can calculate the kinetic energy and potential energies...Thu, 03 Sep 2026 17:14:07 -0400https://liberty.io/gram-schmidt-process-and-polynomial-basis-inner-product-exercise.htmlhttps://liberty.io/gram-schmidt-process-and-polynomial-basis-inner-product-exercise.htmlLet the vector space $P_2$ have the inner product: $\langle p,q\rangle=\int\limits_{-1}^{1}p(x)q(x)dx$ Apply the Gram-Schmidt process to transform the standard $S=\{1,x,x^2\}$ into an orthonormal b......Thu, 03 Sep 2026 17:09:20 -0400https://liberty.io/secant-line-and-diameter-of-a-circle.htmlhttps://liberty.io/secant-line-and-diameter-of-a-circle.htmlA secant line incident to a circle at points $A$ and $C$ intersects the circle's diameter at point $B$ with a $45^\circ$ angle. If the length of $AB$ is $1$ and the length of $BC$ is $7$, then what......Thu, 03 Sep 2026 17:04:40 -0400https://liberty.io/meaning-of-vector-derivative-of-velocity-with-respect-to-position.htmlhttps://liberty.io/meaning-of-vector-derivative-of-velocity-with-respect-to-position.htmlI would like to compute an expression for $\frac{\partial v}{\partial x}$, where $x$ is a position vector and $v = \dot{x}$ is a velocity vector. Using the Chain Rule, we should have: $$\frac{\par......Thu, 03 Sep 2026 16:45:53 -0400https://liberty.io/for-what-values-does-the-geothmetic-meandian-converge.htmlhttps://liberty.io/for-what-values-does-the-geothmetic-meandian-converge.htmlThe geothmetic meandian, $G_{MDN}$ is defined in this XKCD as $$F(x_1, x_2, ..., x_n) = \left(\frac{x_1 +x_2+\cdots+x_n}{n}, \sqrt[n]{x_1 x_2 \cdots x_n}, x_{\frac{n+1}{2}} \right)$$ $$G_{MDN}(x_1,......Thu, 03 Sep 2026 16:34:21 -0400https://liberty.io/how-do-you-find-the-turning-points-of-a-polynomial-without-using-calcu.htmlhttps://liberty.io/how-do-you-find-the-turning-points-of-a-polynomial-without-using-calcu.htmlI have a polynomial $P(x) = -x^3+12x+3$, and I am asked to find the turning points of it, and hence state how many zeroes it has. Since this chapter is separate from calculus, we are expected to so......Thu, 03 Sep 2026 16:29:20 -0400https://liberty.io/harvard-math-25-and-55-prerequisites-closed.htmlhttps://liberty.io/harvard-math-25-and-55-prerequisites-closed.htmli'll be attending harvard next year (deferred entry) and am planning to use this year to study calculus and linear algebra in preparation for either math 25 or 55. (See here for the various courses......Thu, 03 Sep 2026 16:28:05 -0400https://liberty.io/how-do-i-find-approximate-the-jacobian-of-this-ode-system.htmlhttps://liberty.io/how-do-i-find-approximate-the-jacobian-of-this-ode-system.html$$x_1'=(x_2-x_4)x_5-x_1+8$$ $$x_2'=(x_3-x_5)x_1-x_2+8$$ $$x_3'=(x_4-x_1)x_2-x_3+8$$ $$x_4'=(x_5-x_2)x_3-x_4+8$$ $$x_5'=(x_1-x_3)x_4-x_5+8$$ I am asked for either an exact or approximate solution. I ...Thu, 03 Sep 2026 16:27:13 -0400https://liberty.io/arma-1-2-model-auto-covariance-function.htmlhttps://liberty.io/arma-1-2-model-auto-covariance-function.htmlI am struggling with finding the Autocovariance function $\gamma(k)$, of the following ARMA(1,2) model: $x_t-0.9x_{t-1}=e_t+2e_{i-1}+0.5e_{t-2}$. I have already found this model to be stationary,......Thu, 03 Sep 2026 16:25:10 -0400https://liberty.io/complex-conjugate-clarification.htmlhttps://liberty.io/complex-conjugate-clarification.htmlSo I understand that something like $4-i$ is a complex conjugate of $4+i$, however are things like; (a) $-4+i = \overline{-4-i}$, (b) $-4+i = \overline{4+i}$, and (c) $-4-i = \overline{4+i}$. Gene......Thu, 03 Sep 2026 16:22:48 -0400https://liberty.io/3-sided-coin-odds-you-roll-two-heads-two-tails-and-two-feet.htmlhttps://liberty.io/3-sided-coin-odds-you-roll-two-heads-two-tails-and-two-feet.htmlAssume it's a fair coin. What are the odds you roll two heads, two tails, and two feet on six rolls? I got the following: $ 3 \times \binom{6}{2} \over 3^6 $ $3$ for the different sides, $6 \choo......Thu, 03 Sep 2026 16:16:42 -0400https://liberty.io/is-pi-equal-to-180-circ.htmlhttps://liberty.io/is-pi-equal-to-180-circ.html$$ \begin{array}{ccc} \sin{(\theta+180^{\circ})}=-\sin{\theta} & \cos{(\theta+180^{\circ})}=-\cos{\theta} & \tan{(\theta+180^{\circ})}=\tan{\theta} \\ \sin{(\theta+\pi)}=-\sin{\theta} &......Thu, 03 Sep 2026 16:15:48 -0400https://liberty.io/find-the-length-of-ad-to-the-nearest-whole-number.htmlhttps://liberty.io/find-the-length-of-ad-to-the-nearest-whole-number.htmlA sketch of a rectangular ground ABCD is given here. (i) Based on the information marked on the sketch, find the length of AD to the nearest whole number. Any Ideas on how to begin?...Thu, 03 Sep 2026 16:07:23 -0400https://liberty.io/logic-of-step-to-convert-rectangular-equation-of-circle-to-parametric.htmlhttps://liberty.io/logic-of-step-to-convert-rectangular-equation-of-circle-to-parametric.htmlI recently learned in math class how to convert the rectangular equation of a circle into a parametric equation: $x^2+y^2=1$ $\cos^2(t)+\sin^2(t)=1$ $x^2+y^2=\cos^2(t)+\sin^2(t)$ $x^2=\cos^2(t)......Thu, 03 Sep 2026 16:00:42 -0400https://liberty.io/how-to-find-the-anti-derivative-of-sin-2-x-closed.htmlhttps://liberty.io/how-to-find-the-anti-derivative-of-sin-2-x-closed.htmlI already know that this problem can be solved through a trigonometric identity. However, since this is a relatively simple expression, why can’t the antiderivative just be traced back? In such cas......Thu, 03 Sep 2026 15:48:01 -0400https://liberty.io/defining-the-l-2-norm-of-a-vector-valued-function.htmlhttps://liberty.io/defining-the-l-2-norm-of-a-vector-valued-function.htmlI am considering a collection of function of the type, $ f:[0,2\pi]\rightarrow \mathbb{R^2}$. I want to define the $L^2$ norm of the function in that space. I am defining the a norm of $f(=(f_1,f_......Thu, 03 Sep 2026 15:45:52 -0400https://liberty.io/finding-the-fourth-roots-of-a-complex-number.htmlhttps://liberty.io/finding-the-fourth-roots-of-a-complex-number.htmlHow would I find the fourth roots of $-81i$ in the complex numbers? Here is what I currently have: $w = -81i$ $r = 9$ $\theta = \arctan (-81)$? Although I am not sure it's correct or if ......Thu, 03 Sep 2026 15:43:14 -0400https://liberty.io/an-open-interval-is-an-open-set.htmlhttps://liberty.io/an-open-interval-is-an-open-set.htmlFirst, the question I have is very similar to this question, but I hope it doesn't get closed as duplicate since I'm stuck nevertheless. I'm trying to algebraically prove that an open interval is an ...Thu, 03 Sep 2026 15:35:15 -0400https://liberty.io/user-andrew-elkouri-mathematics-stack-exchange.htmlhttps://liberty.io/user-andrew-elkouri-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Thu, 03 Sep 2026 15:29:38 -0400https://liberty.io/convert-from-dfa-to-nfa.htmlhttps://liberty.io/convert-from-dfa-to-nfa.htmlFor this language $\{ w \mid w \text{ contains at least three 1's} \}$, its DFA diagram is defined as follows: While trying to convert it to NFA, but I realized that its NFA would be identical to ......Thu, 03 Sep 2026 15:07:03 -0400https://liberty.io/what-do-you-call-a-number-that-represents-20-percent-written-as-0-2-an.htmlhttps://liberty.io/what-do-you-call-a-number-that-represents-20-percent-written-as-0-2-an.htmlLets say I want to represent 20 Percent What do you call/name the "0.2" and "20%" notations I am calling them Fractional and WholeValue notations for the time being, but wondering if there was ...Thu, 03 Sep 2026 15:00:28 -0400https://liberty.io/need-help-with-the-integral-int-frac-2-tan-x-3-5-sin-2-x-4-dx.htmlhttps://liberty.io/need-help-with-the-integral-int-frac-2-tan-x-3-5-sin-2-x-4-dx.htmlI'm having a problem resolving the following integral, spent almost all day trying. Any help would be appreciated. $$\int \frac{2\tan(x)+3}{5\sin^2(x)+4}\,dx$$...Thu, 03 Sep 2026 14:41:05 -0400https://liberty.io/why-cos-2-2x-frac-1-2-1-cos-4x.htmlhttps://liberty.io/why-cos-2-2x-frac-1-2-1-cos-4x.htmlWhy: $$\cos ^2(2x) = \frac{1}{2}(1+\cos (4x))$$ I don't understand this, how I must to multiply two trigonometric functions? Thanks a lot....Thu, 03 Sep 2026 14:16:12 -0400https://liberty.io/what-is-the-isometry-and-isometry-group.htmlhttps://liberty.io/what-is-the-isometry-and-isometry-group.htmlCan someone explain with some simple examples what is meant by the isometry and isometry group? I'm a student of physics and I often come across this term in texts of general relativity for various ...Thu, 03 Sep 2026 14:15:15 -0400https://liberty.io/volume-of-a-pencil-closed.htmlhttps://liberty.io/volume-of-a-pencil-closed.htmlA pencil sharpened at both ends is formed of a cylindrical barrel together with two conical points. The pencil measures 110mm from tip to tip, has radius $r$ mm, and the cylindrical barrel is $h$ m......Thu, 03 Sep 2026 14:05:20 -0400https://liberty.io/prove-that-the-limit-exists-and-then-find-its-limit.htmlhttps://liberty.io/prove-that-the-limit-exists-and-then-find-its-limit.htmlWe are given a function $h(x)$ which strictly positive. The function $h$ is defined on $\mathbb{R}$ and that satisfies the following:\ $$\lim_{x\to 0}(h(x)+\frac{1}{h(x)})=2$$ The question is to ......Thu, 03 Sep 2026 14:02:35 -0400https://liberty.io/cubic-equation-in-pocket-calculator.htmlhttps://liberty.io/cubic-equation-in-pocket-calculator.htmlIs it possible to input and automaticly get solutions for the cubic equation using Texas Instruments TI-30X IIS calculator?...Thu, 03 Sep 2026 13:56:10 -0400https://liberty.io/how-to-find-values-from-1-equation-with-3-variables.htmlhttps://liberty.io/how-to-find-values-from-1-equation-with-3-variables.htmlIs there any way to get something out of: $50x + 5y - 18z = 0$ ? I found this trying to solve this: \begin{equation} xyz+xyz+xyz=zzz \end{equation} in which xyz are not multiplying themselves. T......Thu, 03 Sep 2026 13:51:12 -0400https://liberty.io/how-can-i-find-the-hypotenuse-and-opposite-sides-of-triangle.htmlhttps://liberty.io/how-can-i-find-the-hypotenuse-and-opposite-sides-of-triangle.htmlBased on the given information how can I find the pixel values of the hypotenuse and opposite sides of the triangle?...Thu, 03 Sep 2026 13:49:53 -0400https://liberty.io/how-do-you-draw-a-direction-field-for-2x2-matrix.htmlhttps://liberty.io/how-do-you-draw-a-direction-field-for-2x2-matrix.htmlI understand $AX = X'$ and by doing so, you get both equation for derivative of $x_1 and $x_2$. When I make a $x_1$ and $x_2$ plot, I am confused regarding which derivative of $x_1$ or $x_2$ to ch......Thu, 03 Sep 2026 13:45:06 -0400https://liberty.io/how-do-we-decide-the-direction-of-vector-that-is-orthogonal-to-some-ot.htmlhttps://liberty.io/how-do-we-decide-the-direction-of-vector-that-is-orthogonal-to-some-ot.htmlassume that two vector given with relationship below: $$\vec{n}\cdot \vec{u} = 0 $$ Then $\vec{n}$ and $\vec{u}$ are orthogonal vectors. Assume that we now have the direction of $\vec{n}$ Then t......Thu, 03 Sep 2026 13:36:48 -0400https://liberty.io/what-is-a-formal-polynomial.htmlhttps://liberty.io/what-is-a-formal-polynomial.htmlI'm starting to study Field Theory by myself, the books don't say explicitly what a polynomial is, I mean, what the $x$ of $f(x)$ in $F[x]$ is? $x\in F$? When I take $f(\alpha)$ am I taking the ele......Thu, 03 Sep 2026 13:31:33 -0400https://liberty.io/is-dodecahedron-tesselation-somehow-possible.htmlhttps://liberty.io/is-dodecahedron-tesselation-somehow-possible.htmlIn this video (at 3:25) there is an animation of planets inside a dodecahedron matrix (or any data-structure that best fit this 3d mosaic). I tried reproducing it with 12 sided dices, or in Blender......Thu, 03 Sep 2026 13:28:44 -0400https://liberty.io/definite-integral-word-problem.htmlhttps://liberty.io/definite-integral-word-problem.htmlThe sales of a plastic widget were estimated to be: $P(t)  =  8000te ^{−0.5t}$ where $t$ is in weeks, and $P(t)$ is in units per week. How many widgets were sold in the first $9$ weeks? To sta......Thu, 03 Sep 2026 13:27:52 -0400https://liberty.io/what-is-the-difference-between-the-dot-product-and-the-scalar-projecti.htmlhttps://liberty.io/what-is-the-difference-between-the-dot-product-and-the-scalar-projecti.htmlI don't understand the difference between the dot product of two vectors and the scalar projection of a vector onto another one. To me it looks like they are both (geometrically) the length of the ...Thu, 03 Sep 2026 13:21:31 -0400https://liberty.io/multivariable-differential-equation.htmlhttps://liberty.io/multivariable-differential-equation.htmlGiven $u=f(2x-y)+g(x-2y)$, show that $$2 \frac{\mathrm{d}^2 u}{\mathrm{d}x^2} + 5 \frac{\mathrm{d}^2 u}{\mathrm{d}x\,\mathrm{d}y} + 2\frac{\mathrm{d}^2 u}{\mathrm{d}y^2} = 0.$$ I'm not even sure wh......Thu, 03 Sep 2026 13:18:42 -0400https://liberty.io/from-randn-to-bivariate-gaussian-distribution-image.htmlhttps://liberty.io/from-randn-to-bivariate-gaussian-distribution-image.htmlIn Matlab, I generated a bivariate Gaussian distribution with mean 50 and standard deviation 50: G=50*randn((100*100)/2,2)+50; histogram2(G(:,1),G(:,2)); The 3D histogram gives me the output I am ...Thu, 03 Sep 2026 13:17:00 -0400https://liberty.io/calculate-line-integral-with-respect-to-arc-length.htmlhttps://liberty.io/calculate-line-integral-with-respect-to-arc-length.htmlCalculate the line integral with respect to arc length of $\int_Czds$, where $C$ is parametrized by $γ(t) = (tcost,tsint,t), 0 ≤ t ≤ t_0.$ I calculate the norm of $γ'(t)$ and I get $\sqrt {2+t^2}$.......Thu, 03 Sep 2026 13:11:19 -0400https://liberty.io/how-do-i-simplify-frac-sin-2x-1-cos-2x.htmlhttps://liberty.io/how-do-i-simplify-frac-sin-2x-1-cos-2x.html$$\dfrac{\sin(2x)}{1-\cos (2x)}$$ How do I simplify given expression? My attempt: We know the double-angle identity for $\sin(2x)$ and $\cos(2x)$ as shown below $$\sin(2x) = 2\sin (x)\cos(x)$$......Thu, 03 Sep 2026 13:07:23 -0400https://liberty.io/is-infinity-the-reciprocal-of-zero-is-zero-the-reciprocal-of-infinity.htmlhttps://liberty.io/is-infinity-the-reciprocal-of-zero-is-zero-the-reciprocal-of-infinity.htmlIs infinity the reciprocal of zero? Is zero the reciprocal of infinity? It would make sense that they would be--they behave in a similar way (anything multiplied by zero or infinity results in zero......Thu, 03 Sep 2026 13:06:56 -0400https://liberty.io/when-is-there-arbitrary-variable-in-system.htmlhttps://liberty.io/when-is-there-arbitrary-variable-in-system.htmli am just stuck right now. what was the theorem again about the arbitrary variables in the system of equations? if i have 2 equations with 3 variables, then last variable will be arbitrary, right? ......Thu, 03 Sep 2026 13:05:34 -0400https://liberty.io/fraction-multiplication-with-square-root-as-the-numerator-closed.htmlhttps://liberty.io/fraction-multiplication-with-square-root-as-the-numerator-closed.htmlremoved - unclear question.......Thu, 03 Sep 2026 12:55:01 -0400https://liberty.io/grad-f-in-index-notation.htmlhttps://liberty.io/grad-f-in-index-notation.htmlIf $f(\mathbf{r})=\vert\mathbf{r}\vert^4$ How would you calculate $\operatorname{grad} f$ in index notation? I get that $\vert\mathbf{r}\vert^2=x_ix_i$ but how do I represent $f$ in index notation?...Thu, 03 Sep 2026 12:54:03 -0400https://liberty.io/mode-of-lognormal-distribution.htmlhttps://liberty.io/mode-of-lognormal-distribution.htmlSuppose $$y=e^{x}$$ where x is normal with mean mu and variance sigma. Then I see how to derive mode of f(y) (distribution of y), as we need to find the value y that makes $$f'(y)==0$$ However, why......Thu, 03 Sep 2026 12:44:14 -0400https://liberty.io/calculating-the-address-of-an-element-in-an-n-dimensional-array.htmlhttps://liberty.io/calculating-the-address-of-an-element-in-an-n-dimensional-array.htmlIn a single dimensional array the address of an element of an array say A[i] is calculated using the following formula Address of $A[i] =B+W * (i–L_B)$ where $B$ is the base address of the array, W......Thu, 03 Sep 2026 12:31:56 -0400https://liberty.io/decompose-a-real-symmetric-matrix.htmlhttps://liberty.io/decompose-a-real-symmetric-matrix.htmlProve that, without using induction, A real symmetric matrix $A$ can be decomposed as $A = Q^T \Lambda Q$, where $Q$ is an orthogonal matrix and $\Lambda$ is a diagonal matrix with eigenvalues of $......Thu, 03 Sep 2026 12:16:49 -0400https://liberty.io/total-probability-theorem-partition.htmlhttps://liberty.io/total-probability-theorem-partition.htmlIn the Total Probability Theorem we assume that the sample space is partitioned into subsets. If we consider $B$ to be the sample space and $A_1$, $A_2$ to be the partition then the theorem says: $$\...Thu, 03 Sep 2026 12:12:54 -0400https://liberty.io/confused-about-limits-when-denominator-is-0.htmlhttps://liberty.io/confused-about-limits-when-denominator-is-0.htmlSo I thought that whenever the denominator of a function was $0$ then the limit does not exist. Until I read the arithmetic rules for limits of functions that states if $f$ and $g$ are functions, and ...Thu, 03 Sep 2026 12:08:53 -0400https://liberty.io/discontinuity-vs-not-continuity.htmlhttps://liberty.io/discontinuity-vs-not-continuity.htmlWhat is difference between Discontinuous and not continuous of a function $f$ at some point $x=a?$ I think there is not difference between the two. Is there some difference between these two relate......Thu, 03 Sep 2026 12:03:51 -0400https://liberty.io/how-to-factor-this-cubic-polynomial-closed.htmlhttps://liberty.io/how-to-factor-this-cubic-polynomial-closed.htmlI am asking for detailed steps how to factor the cubic polynomial $${x^3-3x+2}$$ The solution is $${(x-1)^2(x+2)}$$...Thu, 03 Sep 2026 11:57:11 -0400https://liberty.io/what-are-some-interesting-counterexamples-given-by-finite-topological-.htmlhttps://liberty.io/what-are-some-interesting-counterexamples-given-by-finite-topological-.htmlAccording to Wikipedia, 'finite topological spaces are often used to provide examples of interesting phenomena or counterexamples to plausible sounding conjectures.' I have been studying the book '...Thu, 03 Sep 2026 11:35:25 -0400https://liberty.io/prime-divisors-of-60.htmlhttps://liberty.io/prime-divisors-of-60.htmlHow to find all prime divisors of $60!$?, where $60!$ denotes the factorial of $60$, the product all integers from $1$ to $60$. Is there any solution to solve this? Thank you....Thu, 03 Sep 2026 11:31:40 -0400https://liberty.io/find-the-probability-that-a-randomly-chosen-applicant-who-was-approved.htmlhttps://liberty.io/find-the-probability-that-a-randomly-chosen-applicant-who-was-approved.htmlOf all the people applying for a certain job, $89\%$ are qualified. The personnel manager claims that she approves qualified people $61\%$ of the time and she approves an unqualified person $12\%$ ......Thu, 03 Sep 2026 11:30:30 -0400https://liberty.io/how-to-figure-of-the-laplace-transform-for-log-x.htmlhttps://liberty.io/how-to-figure-of-the-laplace-transform-for-log-x.htmlI was looking at a table of common Laplace transforms of functions when I came across the transform for $\log x$. Apparently, the transform is as follows: $$\mathcal{L} \left\{ \log x\right\}=-\f......Thu, 03 Sep 2026 11:29:05 -0400https://liberty.io/how-would-a-triangle-for-sin-90-degree-look.htmlhttps://liberty.io/how-would-a-triangle-for-sin-90-degree-look.htmlI am studying trigonometry in my school and learned that in a triangle the side opposite to angle theta should be taken as perpendicular side - hypotenuse remains the same and the third remaining s......Thu, 03 Sep 2026 11:28:03 -0400https://liberty.io/series-solution-to-y-039-039-xy-039-y-0.htmlhttps://liberty.io/series-solution-to-y-039-039-xy-039-y-0.htmlSo I'm learning to solve ODE's with series on my own using Boyce and DiPrima and exercise #3 is irking me...just looking for power series solutions around the ordinary point... $$y''-xy'-y=0$$ So I......Thu, 03 Sep 2026 11:27:44 -0400https://liberty.io/what-is-the-difference-between-lim-h-to0-frac-0-h-and-lim-h-to-infty-f.htmlhttps://liberty.io/what-is-the-difference-between-lim-h-to0-frac-0-h-and-lim-h-to-infty-f.htmlWhat is the difference (if any) between- $$\lim_{h\to0}\frac{0}{h} \text{ and } \lim_{h\to\infty}\frac{0}{h}$$ I argue that both must be $=0$ since the numerator is exactly $0$. But one fellow ...Thu, 03 Sep 2026 11:20:34 -0400https://liberty.io/compound-interest-with-regular-monthly-contributions-formula.htmlhttps://liberty.io/compound-interest-with-regular-monthly-contributions-formula.htmlI have discover formulas to build Investment Calculator by using compound interest (TVM) concept formula. The base of the formula Find Future Value: FV= P(1 +r/n) ^ (nt) + PMT * [(((1 + r/n) ^ (nt......Thu, 03 Sep 2026 11:16:19 -0400https://liberty.io/what-is-reform-calculus.htmlhttps://liberty.io/what-is-reform-calculus.htmlIn an answer to another question I asked, Isaac suggested a book that is the standard "reform calculus" book. In a comment, I asked what the phrase "reform calculus" means, and Isaac provided a lin......Thu, 03 Sep 2026 11:13:24 -0400https://liberty.io/book-recomendation-for-elliptic-curve-cryptography.htmlhttps://liberty.io/book-recomendation-for-elliptic-curve-cryptography.htmlI have already taken a course on Cryptography, The course focused mainly on the public-key cryptography based on the algebraic structure of elliptic curves over finite fields. Now, I would like to ...Thu, 03 Sep 2026 11:13:22 -0400https://liberty.io/is-division-allowed-in-rings-and-fields.htmlhttps://liberty.io/is-division-allowed-in-rings-and-fields.htmlIs division allowed in ring and field? The definition of ring I am using here does not require the presence of multiplicative inverse. I think in general, division is not a well-defined operation......Thu, 03 Sep 2026 11:08:07 -0400https://liberty.io/why-does-the-volume-element-in-spherical-polar-coordinates-contain-a-s.htmlhttps://liberty.io/why-does-the-volume-element-in-spherical-polar-coordinates-contain-a-s.htmlThe volume element in Cartesian coordinates is $dx dy dz$, the volume of a rectangular prism with side lengths being the length elements along the three rectangular axes. In spherical polar coordi......Thu, 03 Sep 2026 11:00:26 -0400https://liberty.io/triple-integral-in-a-sphere.htmlhttps://liberty.io/triple-integral-in-a-sphere.htmlHow can I integrate $\sqrt{x^2+y^2+z^2}$ over the region $x^2+y^2+z^2 \le 1$? The problem is I do not know how to set up the triple integral. I guess that I can integrate with respect to $x$ first......Thu, 03 Sep 2026 10:48:00 -0400https://liberty.io/prove-the-projection-transformation-is-linear.htmlhttps://liberty.io/prove-the-projection-transformation-is-linear.htmlLet $P_z : \mathbb R^3 \to \mathbb R^3$ be the function which orthogonally projects onto the $x$-$y$ linearity plane (that is, the set of vectors $(x, y, z) \in \mathbb R^3$ with $z = 0$). I......Thu, 03 Sep 2026 10:46:07 -0400https://liberty.io/why-the-column-space-of-a-matrix-is-useful.htmlhttps://liberty.io/why-the-column-space-of-a-matrix-is-useful.htmlI know what is the column space of a matrix: it is basically the subspace formed by the linear combinations of the columns (vectors) of a matrix. From wikipedia, we have the following nice picture: ...Thu, 03 Sep 2026 10:40:43 -0400https://liberty.io/line-integral-equals-zero.htmlhttps://liberty.io/line-integral-equals-zero.htmlCurrently going through some line integral problems and in the worked example provided, the line integral evaluated to become zero. The curve C that they have provided was a simple closed curve (i......Thu, 03 Sep 2026 10:35:37 -0400https://liberty.io/finding-int-x-xdx.htmlhttps://liberty.io/finding-int-x-xdx.htmlI'm trying to find $\int x^x \, dx$, but the only thing I know how to do is this: Let $u=x^x$. $$\begin{align} \int x^x \, dx&=\int u \, du\\[6pt] &=\frac{u^2}{2}\\[6pt] &=\dfrac{\lef......Thu, 03 Sep 2026 10:32:53 -0400https://liberty.io/duality-theorem-optimal-solution-of-both-primal-and-dual.htmlhttps://liberty.io/duality-theorem-optimal-solution-of-both-primal-and-dual.htmlThe Duality Theorem, in short, states that the optimal value of the Primal (P) and Dual (D) Linear Programs are the same if the solution, of either, is a basic feasible solution. My question is t......Thu, 03 Sep 2026 10:30:43 -0400https://liberty.io/what-does-it-mean-to-converge-in-law.htmlhttps://liberty.io/what-does-it-mean-to-converge-in-law.htmlIf $X_1, X_2, X_3, \cdots$ is a sequence of independent identically distributed random variables with $E[X_i] < \infty$ and $Var(X_i)< \infty$ such that the sequence $Y_n = 3 \frac{X_1 + X_2 + \...Thu, 03 Sep 2026 10:23:49 -0400https://liberty.io/how-to-approach-vector-space-of-matrices.htmlhttps://liberty.io/how-to-approach-vector-space-of-matrices.htmlThe following is from this Wikipedia page: Let $\mathbf{F}^{m\times n}$ denote the set of $m\times n$ matrices with entries in $\mathbf{F}$. Then $\mathbf{F}^{m\times n}$ is a vector space over $\...Thu, 03 Sep 2026 10:12:51 -0400https://liberty.io/find-how-many-terms-there-are-in-this-geometric-sequence-1-2-4-8-16777.htmlhttps://liberty.io/find-how-many-terms-there-are-in-this-geometric-sequence-1-2-4-8-16777.htmlFind how many terms there are in this geometric sequence: $-1, 2, -4, 8, ..., -16777216$ My attempt: $a_k=a.r^{k-1}$ And in this sequence: $a=-1$, $r=-2$ So $a_k=(-1){(-2)}^{k-1}$ $-16777216=(-1){......Thu, 03 Sep 2026 10:10:27 -0400https://liberty.io/construct-a-square-without-a-ruler-duplicate.htmlhttps://liberty.io/construct-a-square-without-a-ruler-duplicate.htmlHow can I construct a square using only a pencil and a compass, i.e. no ruler. Given: a sheet of paper with $2$ points marked on it, a pencil and a compass. Aim: plot $2$ vertices such that the $......Thu, 03 Sep 2026 10:09:17 -0400https://liberty.io/a-total-of-28-percent-of-american-males-smoke-cigarettes-7-smoke-cigar.htmlhttps://liberty.io/a-total-of-28-percent-of-american-males-smoke-cigarettes-7-smoke-cigar.htmlI was able to find out how many smoke neither cigars nor cigarettes as: $P(E \cup F)$ is the event that someone smokes a cigar OR a cigarette $P(E \cup F)^c$ is the event someone smokes neither ......Thu, 03 Sep 2026 10:09:15 -0400https://liberty.io/cof-ab-cof-a-cdot-cof-b.htmlhttps://liberty.io/cof-ab-cof-a-cdot-cof-b.htmlLet $A$ and $B$ be two $n\times n$ matrices and let cof$(A)$ and cof$(B)$ denote the cofactor matrix of A and B respectively. Then show that cof($AB$) = cof($A$)$\cdot$cof($B$)? I am trying to sh......Thu, 03 Sep 2026 10:07:39 -0400