https://liberty.ioCurated star highlights with fresh energy.en-usTue, 18 Aug 2026 04:25:10 -0400https://liberty.io/help-with-proof-of-injection-and-surjection.htmlhttps://liberty.io/help-with-proof-of-injection-and-surjection.htmlFor the record, I am sorry, I haven't yet learnt how to use LaTeX I have a function $f(x) = 2x^3 - 1$ My proof of injection is as follows: $f$ is one to one for all $x_1,x_2$ element $X$, if $f(x......Tue, 18 Aug 2026 08:14:51 -0400https://liberty.io/relationship-between-primes-and-fibonacci-sequence.htmlhttps://liberty.io/relationship-between-primes-and-fibonacci-sequence.htmlI recently stumbled across an unexpected relationship between the prime numbers and the Fibonacci sequence. We know a lot about Fibonacci numbers but relatively little about primes, so this connect......Tue, 18 Aug 2026 07:58:37 -0400https://liberty.io/integral-definition-of-e.htmlhttps://liberty.io/integral-definition-of-e.htmlI know that $e$ can be defined via a convergent series: $$ e = \sum_{n=0}^\infty {1\over n!}$$ Or as a limit: $$ e = \lim_{n \to \infty} { \left(1 + {1 \over n}\right)^n }$$ Or as the value which ...Tue, 18 Aug 2026 07:54:49 -0400https://liberty.io/how-many-zeros-are-in-all-natural-numbers-from-1-to-2019.htmlhttps://liberty.io/how-many-zeros-are-in-all-natural-numbers-from-1-to-2019.htmlProblem : How many zeros are in all natural numbers from $1$ to $2019$ ? My attempt : From $1$ to $100$ we have $11$ zeros Now from $101$ to $110$ we have $10$ zeros So from $20$ zeros ...Tue, 18 Aug 2026 07:43:23 -0400https://liberty.io/optimizing-svd-factors.htmlhttps://liberty.io/optimizing-svd-factors.htmlProblem Description My aim is to find the SVD decomposition of the minimizer of $F:\mathbb{R}^{m\times n}\to\mathbb{R}$ $$ U_*\Sigma_* V^\top_* = A_*\qquad \text{where} \qquad A_* = \arg\min_A F(A)......Tue, 18 Aug 2026 07:37:53 -0400https://liberty.io/ratio-of-triangle-a-and-b-if-the-length-of-the-sides-are-a-25-25-30-an.htmlhttps://liberty.io/ratio-of-triangle-a-and-b-if-the-length-of-the-sides-are-a-25-25-30-an.htmlIf $A$ triangle's side length are $25,25$ and $30$ and $B$ triangle's length are $25,25$ and $40$ what is the ratio between the two areas of the triangles? #math...Tue, 18 Aug 2026 07:36:12 -0400https://liberty.io/i-have-a-doubt-in-roman-numericals.htmlhttps://liberty.io/i-have-a-doubt-in-roman-numericals.htmlThe below question, I am looking at the 80, They translated it as LXXX, but I can also write it as XXC right?...Tue, 18 Aug 2026 07:20:21 -0400https://liberty.io/how-do-inverse-functions-exist-for-exponential-functions.htmlhttps://liberty.io/how-do-inverse-functions-exist-for-exponential-functions.htmlI know that they exist for exponential functions (we currently have them in class), but to me it doesn't seem "reasonable" when I look at the definition of what an inverse function is. The inverse is ...Tue, 18 Aug 2026 07:14:30 -0400https://liberty.io/deriving-the-variance-of-the-bernoulli-distribution.htmlhttps://liberty.io/deriving-the-variance-of-the-bernoulli-distribution.htmlFor a Bernoulli distribution, $\mu_X = p$. I can easily derive this from the general equation for mean of a discrete random variable: $$ \mu_X=\sum_{i=1}^kx_iPr(X=x) $$ $$ \mu_X=1(p)+0(1-p)=p $$ I ......Tue, 18 Aug 2026 07:09:06 -0400https://liberty.io/why-do-hyperbolic-trig-functions-seem-to-be-encountered-rarely.htmlhttps://liberty.io/why-do-hyperbolic-trig-functions-seem-to-be-encountered-rarely.htmlHyperbolic "trig" functions such as $\sinh$, $\cosh$, have close analogies with regular trig functions such as $\sin$ and $\cos$. Yet the hyperbolic versions seem to be encountered relatively rarel......Tue, 18 Aug 2026 07:07:26 -0400https://liberty.io/mathematical-symbol-for-and.htmlhttps://liberty.io/mathematical-symbol-for-and.htmlI have found some pretty complete lists (I think) of mathematical symbols here and here, but I don't see a symbol for the word "and" on either list. A person could easily just write the word "and"......Tue, 18 Aug 2026 06:58:17 -0400https://liberty.io/geometrical-interpretation-of-trigonometric-antiderivative.htmlhttps://liberty.io/geometrical-interpretation-of-trigonometric-antiderivative.htmlI know about geometrical explanation of [definite] integral as an area under the curve, and I wonder if there are some ideas, which may give similar insight in taking antiderivatives [indefinite ...Tue, 18 Aug 2026 06:56:00 -0400https://liberty.io/are-the-graphs-of-inverse-functions-always-reflections-of-the-function.htmlhttps://liberty.io/are-the-graphs-of-inverse-functions-always-reflections-of-the-function.htmlFor an invertible function, is the graph of its inverse always the mirror image of the function's graph in the line $y=x$? And, if yes, then why is it so?...Tue, 18 Aug 2026 06:53:27 -0400https://liberty.io/what-books-are-prerequisites-for-spivak-039-s-calculus.htmlhttps://liberty.io/what-books-are-prerequisites-for-spivak-039-s-calculus.htmlFor financial reasons, I dropped out my senior year of college as a piano performance major. I will be returning to college to dual major in mathematics and computer science. I've taught myself to ...Tue, 18 Aug 2026 06:48:21 -0400https://liberty.io/what-calculation-shortcuts-exist-to-help-or-speed-up-mental-or-paper-c.htmlhttps://liberty.io/what-calculation-shortcuts-exist-to-help-or-speed-up-mental-or-paper-c.htmlAnything to speed up or simplify calculations. A simple example might be to get a multiple of $19$, for instance, $38 \cdot 19 = 38 \cdot 20 - 38$. (This is hard to tag with so few tags in play!) ...Tue, 18 Aug 2026 06:44:32 -0400https://liberty.io/using-operators-and-4-4-4-4-digits-find-all-formulas-that-would-resolv.htmlhttps://liberty.io/using-operators-and-4-4-4-4-digits-find-all-formulas-that-would-resolv.htmlI had a conversation with a colleague of mine and he brought up an interesting problem. Using the + - * / operators and four 4 4 4 4 digits, create an algorithm that will output all the formulas that ...Tue, 18 Aug 2026 06:44:31 -0400https://liberty.io/help-to-clarify-proof-of-euler-039-s-theorem-on-homogenous-equations.htmlhttps://liberty.io/help-to-clarify-proof-of-euler-039-s-theorem-on-homogenous-equations.htmlWhy is the last step (setting $\lambda = 1$ allowed? I have trouble accepting this because if I set $\lambda =1 $ at the very start, then: $f(\lambda x , \lambda y)=\lambda^r f(x,y)$ becomes $f......Tue, 18 Aug 2026 06:34:24 -0400https://liberty.io/how-do-i-find-the-composition-of-two-functions.htmlhttps://liberty.io/how-do-i-find-the-composition-of-two-functions.htmlIn a question sheet for my university course I am given the sets: $A = \{1, 2, 3, 4\}, B = \{x, y, z, w\}$ and $C = \{a, b, c\}$. I am then told to work out the functions: $f: A → B$ whi......Tue, 18 Aug 2026 06:33:22 -0400https://liberty.io/how-to-approximate-partial-derivatives-from-contour-map.htmlhttps://liberty.io/how-to-approximate-partial-derivatives-from-contour-map.htmlA problem of approximating partial derivatives from a given contour map Hi, I would like to know the connection between a contour map and partial derivatives. I understand that partial derivatives ......Tue, 18 Aug 2026 06:26:58 -0400https://liberty.io/creating-link-matrix.htmlhttps://liberty.io/creating-link-matrix.htmlI have a matrix called $ A $ with elements equal to $ 0 $ and $ 1 $. if element $ A _ { i j } $ is equal to $ 1 $, it means that member $ i $ is connected to member $ j $ directly. if elements $ A ......Tue, 18 Aug 2026 06:21:01 -0400https://liberty.io/i-can-039-t-find-a-way-to-simplify-this-boolean-algebra.htmlhttps://liberty.io/i-can-039-t-find-a-way-to-simplify-this-boolean-algebra.htmlxyz' + x'y + x'z the answer from boolean expression calculator on the internet is x'z + yz' I know the end x'z same like the answer. but I can't find a way to simplify xyz' + x'y = yz'...Tue, 18 Aug 2026 06:18:10 -0400https://liberty.io/how-many-integers-between-1-and-1000-use-exactly-three-digits.htmlhttps://liberty.io/how-many-integers-between-1-and-1000-use-exactly-three-digits.htmlThere is a problem in which I am obtaining a different answer than my professor. The problem is as follows: How many integers between $1$ and $1000$ use exactly three digits? The professor shows......Tue, 18 Aug 2026 06:15:36 -0400https://liberty.io/what-should-i-do-if-no-one-answers-my-question.htmlhttps://liberty.io/what-should-i-do-if-no-one-answers-my-question.htmlQ&A for people studying math at any level and professionals in related fields...Tue, 18 Aug 2026 06:14:34 -0400https://liberty.io/sum-of-total-scores-odds-will-end-even-or-odd.htmlhttps://liberty.io/sum-of-total-scores-odds-will-end-even-or-odd.htmlFriend and i are betting on the ending total scores of basketball game, he says odds are greater it will end with an even sum, i say the odds are equal. Who is right?...Tue, 18 Aug 2026 06:09:46 -0400https://liberty.io/this-projection-is-a-covering-map.htmlhttps://liberty.io/this-projection-is-a-covering-map.htmlIf $\tilde X$ is the product of $X$ with a discrete space, the projection $\tilde X \to X$ is a covering map. This question seems really easy, but as I'm a beginner there are some things a little ......Tue, 18 Aug 2026 06:05:03 -0400https://liberty.io/what-is-so-special-about-klein-4-group.htmlhttps://liberty.io/what-is-so-special-about-klein-4-group.htmlThis is my first course in abstract algebra and so far I am only learning about groups. So is there anyone who can explain to me why Klein 4-group is so special that it warrants a category of its o......Tue, 18 Aug 2026 05:59:44 -0400https://liberty.io/wrong-wolfram-alpha-result.htmlhttps://liberty.io/wrong-wolfram-alpha-result.htmlI have this function $$ f(x,y) = \left\{ \begin{array}{ll} \frac{x^3}{x^2 + y^2} & \mbox{if } (x,y) \neq (0,0) \\ 0 & \mbox{if } (x,y) = (0,0) \end{array} \right. $$ And I want to ......Tue, 18 Aug 2026 05:44:07 -0400https://liberty.io/find-a-vector-function-that-represents-the-curve-of-intersection-of-th.htmlhttps://liberty.io/find-a-vector-function-that-represents-the-curve-of-intersection-of-th.htmlThe cone $z=\sqrt{x^2+y^2}$ and the plane $z=1+y$ I parameterized the plane and put it into vector form: $t=1+y \to y=t-1$ $z= 1+t-1 \to z = t$ $y= t-1$, $z=t$ Since I'm finding the intersecti......Tue, 18 Aug 2026 05:32:38 -0400https://liberty.io/what-is-the-length-of-ef.htmlhttps://liberty.io/what-is-the-length-of-ef.htmlIn triangle $ABC$ , $AD$ is perpendicular to $BC$. It is given that $BC$=8 cm and $AD$=6 cm. If $E$ and $F$ are the mid points of $BD$ and $AC$ respectively, find $EF$. I tried using the mid point ...Tue, 18 Aug 2026 05:21:55 -0400https://liberty.io/mexican-hat-function-that-converges-to-1.htmlhttps://liberty.io/mexican-hat-function-that-converges-to-1.htmlThe Ricker wavelet, also known as the Mexican Hat function, converges to 0 for negative and positive infinity. I'm looking for a function with the same shape in the center, but that converges back......Tue, 18 Aug 2026 05:10:18 -0400https://liberty.io/reduction-of-order-method-for-3rd-order-de.htmlhttps://liberty.io/reduction-of-order-method-for-3rd-order-de.htmlUse the Reduction of Order method to solve $$(2-x)y'''+(2x-3)y''-xy'+y=0$$ (such that $x<2$) using $u(x)=e^x$. How do you use the method for $3$rd order... I have seen it been used on $2$nd but......Tue, 18 Aug 2026 04:58:54 -0400https://liberty.io/a-problem-with-progressive-percentage-incrementation.htmlhttps://liberty.io/a-problem-with-progressive-percentage-incrementation.htmlI have absolutely no clue if any of the terms I used in the title actually exist or make sense. I'm usually good at math (relatively) but I am facing this problem today that I just cannot solve. J......Tue, 18 Aug 2026 04:56:01 -0400https://liberty.io/is-zero-an-element-of-the-empty-set.htmlhttps://liberty.io/is-zero-an-element-of-the-empty-set.htmlI have a question about the empty set. Is $0$ (zero) an element of $\varnothing$? And what is the cardinality of {0}?...Tue, 18 Aug 2026 04:48:07 -0400https://liberty.io/rotate-a-point-in-circle-about-an-angle.htmlhttps://liberty.io/rotate-a-point-in-circle-about-an-angle.htmlHow should I rotate a point $(x,y)$ to location $(a,b)$ on the co-ordinate by any angle?...Tue, 18 Aug 2026 04:44:56 -0400https://liberty.io/equivalent-of-solutions-of-ivp.htmlhttps://liberty.io/equivalent-of-solutions-of-ivp.htmlConsider the IVP $y''-2y'+26y=0$, $y(0)=1$, $y'(0)=2$. From the characteristic equation $m^2-2m+26=0$, i found the roots as $m_1=1-5i$ and $m_2=1+5i$. Then when i use the basis solutions $y_1(x......Tue, 18 Aug 2026 04:37:38 -0400https://liberty.io/finding-new-standard-deviation-and-mean-after-adding-an-element.htmlhttps://liberty.io/finding-new-standard-deviation-and-mean-after-adding-an-element.htmlSay I have a mean and standard deviation for a dataset of 5 elements. I now add a sixth element. Is there a way to calculate the new mean and standard deviation using the information we had prior ......Tue, 18 Aug 2026 04:26:41 -0400https://liberty.io/two-holes-in-the-graph-when-dividing-polynomials.htmlhttps://liberty.io/two-holes-in-the-graph-when-dividing-polynomials.htmlSorry if this seems like a dumb question, or if it has been answered already. I did a quick search and didn't turn up anything, so here goes. I'm teaching a high school algebra class, and the book......Tue, 18 Aug 2026 04:21:36 -0400https://liberty.io/archimedes-trisection-of-equilateral-triangle.htmlhttps://liberty.io/archimedes-trisection-of-equilateral-triangle.htmlEquilateral triangle paradox Consider triangle ABC: Each side is trisected - AE, FC, GB each = 1/3 of their sides. First we established that triangle HIJ was equilateral. This was accomplished ......Tue, 18 Aug 2026 04:15:37 -0400https://liberty.io/tanget-to-the-curve-but-point-not-on-curve.htmlhttps://liberty.io/tanget-to-the-curve-but-point-not-on-curve.htmlQuestion is Find the $x$-coordinate of all points on the curve $$y=22x\sin(5x)+55\sqrt{3}x^2+68,\quad \frac\pi{10}<x<\frac{3\pi}{10}$$ where the tangent line passes through the point $P(0,68)......Tue, 18 Aug 2026 04:07:48 -0400https://liberty.io/minimum-number-of-steps-needed-to-solve-a-rubik-cube.htmlhttps://liberty.io/minimum-number-of-steps-needed-to-solve-a-rubik-cube.htmlLong time ago I've seen a book on group theory and there was an appendix about rubik cube. I remember there were only three steps that enabled me to solve my cube (three strings with letters encoding ...Tue, 18 Aug 2026 03:56:39 -0400https://liberty.io/how-to-graph-x-sin-x.htmlhttps://liberty.io/how-to-graph-x-sin-x.htmlGraph $x\sin(x)$ I am having problem with the period section of the graph. I can see that the function is even, and that as $x \rightarrow \pm\infty$, the amplitude $\rightarrow \infty$. I am use......Tue, 18 Aug 2026 03:56:24 -0400https://liberty.io/non-circular-ellipse-nomenclature.htmlhttps://liberty.io/non-circular-ellipse-nomenclature.htmlA circle a special case of an ellipse. Is there a general word to describe an ellipse that is specifically not a circle?...Tue, 18 Aug 2026 03:43:15 -0400https://liberty.io/why-does-the-integral-of-cot-x-have-absolute-value.htmlhttps://liberty.io/why-does-the-integral-of-cot-x-have-absolute-value.htmlThere's a proof of that integral on this website: http://math2.org/math/integrals/more/cot.htm I just don't understand why we need the absolute value....Tue, 18 Aug 2026 03:34:28 -0400https://liberty.io/formula-for-the-nth-partial-sum-of-a-telescoping-series.htmlhttps://liberty.io/formula-for-the-nth-partial-sum-of-a-telescoping-series.htmlFind the $n$th term for the sequence of partial sums for the series $$\sum_{n=1}^{\infty} \frac{5}{n(n+3)} =\sum_{n=1}^{\infty} \left(\frac{5}{3n}-\frac{5}{3(n+3)} \right)$$ and find $\lim\limits_{n\...Tue, 18 Aug 2026 03:32:02 -0400https://liberty.io/what-is-a-slower-decay-rate-than-hyperbolic.htmlhttps://liberty.io/what-is-a-slower-decay-rate-than-hyperbolic.htmlHyperbolic decay $$f_\alpha(x)=\frac{1}{\alpha x + 1} $$ is slower than exponential decay $$f(x) = e^{-x}$$ where $\alpha > 0$ is a scaling factor. The larger that $\alpha$ is, the steeper the d......Tue, 18 Aug 2026 03:29:57 -0400https://liberty.io/get-the-numbers-from-0-30-by-using-the-number-2-four-times.htmlhttps://liberty.io/get-the-numbers-from-0-30-by-using-the-number-2-four-times.htmlHow can I get the numbers from (0-30) by using the number $2$ four times.Use any common mathematical function and the (+,-,*,/,^) I tried to solve this puzzle, but I couldn't solve it completely. S......Tue, 18 Aug 2026 03:28:44 -0400https://liberty.io/expectation-on-1-x.htmlhttps://liberty.io/expectation-on-1-x.htmlIn general can one say that for a random variable X: $E[\frac{1}{X}] = \frac{1}{E[X]}$ ? I've worked out a few examples where this works but I'm not sure how widely this is useful......Tue, 18 Aug 2026 03:23:51 -0400https://liberty.io/too-simple-to-be-true.htmlhttps://liberty.io/too-simple-to-be-true.htmlAs indicates the title, this question is about "proofs" of true statements which are short and/or look elegant but are wrong. I mean example like Cayley-Hamilton's theorem, which states that for ......Tue, 18 Aug 2026 03:21:51 -0400https://liberty.io/why-are-box-topology-and-product-topology-different-on-infinite-produc.htmlhttps://liberty.io/why-are-box-topology-and-product-topology-different-on-infinite-produc.htmlWhy are box topology and product topology different on infinite products of topological spaces ? I'm reading Munkres's topology. He mentioned that fact but I can't see why it's true that they are ...Tue, 18 Aug 2026 03:13:33 -0400https://liberty.io/is-hubbard-and-hubbard-039-s-vector-calculus-linear-algebra-and-differ.htmlhttps://liberty.io/is-hubbard-and-hubbard-039-s-vector-calculus-linear-algebra-and-differ.htmlSince I have taken single variable calculus, proof based linear algebra and proof courses before multivariable calculus, I am used to some proof writing. Since I have taken about a year off due to ...Tue, 18 Aug 2026 03:09:41 -0400https://liberty.io/how-can-i-solve-this-hard-system-of-equations.htmlhttps://liberty.io/how-can-i-solve-this-hard-system-of-equations.htmlSolve the system below \begin{align} &\sqrt {3x} \left( 1+\frac {1}{x+y} \right) =2\\ &\sqrt {7y} \left( 1-\frac{1}{x+y} \right) =4\sqrt{2} \end{align} Frankly I am disappointed, beca......Tue, 18 Aug 2026 03:02:44 -0400https://liberty.io/express-this-curve-in-the-rectangular-form.htmlhttps://liberty.io/express-this-curve-in-the-rectangular-form.htmlExpress the curve $r = \dfrac{9}{4+\sin \theta}$ in rectangular form. And what is the rectangular form? If I get the expression in rectangular form, how am I able to convert it back to polar ...Tue, 18 Aug 2026 02:58:56 -0400https://liberty.io/how-to-prove-that-lim-limits-x-to0-frac-sin-x-x-1.htmlhttps://liberty.io/how-to-prove-that-lim-limits-x-to0-frac-sin-x-x-1.htmlHow can one prove the statement $$\lim_{x\to 0}\frac{\sin x}x=1$$ without using the Taylor series of $\sin$, $\cos$ and $\tan$? Best would be a geometrical solution. This is homework. In my math c......Tue, 18 Aug 2026 02:47:50 -0400https://liberty.io/prove-1-tan-2-theta-sec-2-theta.htmlhttps://liberty.io/prove-1-tan-2-theta-sec-2-theta.htmlProve the following trigonometric identity: $$1 + \tan^2\theta = \sec^2\theta$$ I'm curious to know of the different ways of proving this depending on different characterizations of tangent and se......Tue, 18 Aug 2026 02:39:17 -0400https://liberty.io/find-the-values-of-sin-69-circ-sin-18-circ-tan-23-circ.htmlhttps://liberty.io/find-the-values-of-sin-69-circ-sin-18-circ-tan-23-circ.htmlCalculate $\sin 69^{\circ},\sin 18^{\circ} , \tan 23^{\circ}$. accurate upto two decimal places or in surds . $\begin{align}\sin 69^{\circ}&=\sin (60+9)^{\circ}\\~\\ &=\sin (60^{\circ})\co......Tue, 18 Aug 2026 02:31:05 -0400https://liberty.io/how-to-find-the-boundary-conditions-of-the-differential-equation.htmlhttps://liberty.io/how-to-find-the-boundary-conditions-of-the-differential-equation.htmlI have a very complicated differential equation that can not be solved analytically and I show a on the simple example: Example 1: $$y''(x)-y(x)=0\tag{1}$$ with boundary conditions: $$y(1)=1,y'(......Tue, 18 Aug 2026 02:13:51 -0400https://liberty.io/what-are-the-units-of-cyclotomic-integers.htmlhttps://liberty.io/what-are-the-units-of-cyclotomic-integers.htmlThis question made me realize I had a misconception about the cyclotomic integers: I thought the units were exactly the roots of unity. There are only finitely many units but infinitely many intege......Tue, 18 Aug 2026 02:12:01 -0400https://liberty.io/why-is-039-catastrophic-cancellation-039-called-so.htmlhttps://liberty.io/why-is-039-catastrophic-cancellation-039-called-so.htmlI was studying Numerical Analysis by K. Mukherjee; there he discussed Loss of Significant Figures by Subtraction, as followed: In the subtraction of two approximate numbers, a serious type of er......Tue, 18 Aug 2026 02:04:55 -0400https://liberty.io/negative-mixed-fractions.htmlhttps://liberty.io/negative-mixed-fractions.htmlI'm comfortable with fractions like $\frac{-3}{8}$ being the same as $\frac{3}{-8}$ (though I'd think the latter anachronistic and would in any case probably prefer to write either of those two as $-\...Tue, 18 Aug 2026 01:59:17 -0400https://liberty.io/how-many-ways-can-you-make-change-for-a-dollar.htmlhttps://liberty.io/how-many-ways-can-you-make-change-for-a-dollar.htmlThis is a generic generating functions problem, and you are left with 1/(x-1)(x^5-1)....(x^50-1) however now I want to find the coefficient of the x^100 term of this expression... how can I do this......Tue, 18 Aug 2026 01:58:55 -0400https://liberty.io/chance-of-overlapping-events-in-milliseconds-per-day.htmlhttps://liberty.io/chance-of-overlapping-events-in-milliseconds-per-day.htmlI've got a server in which about 100 times per day a specific event happens. We now wanted to number those events using the unix time in milliseconds (the number of milliseconds that have passed si......Tue, 18 Aug 2026 01:52:06 -0400https://liberty.io/lax-wendroff-and-crank-nicolson-matlab-simulation-of-linear-advection.htmlhttps://liberty.io/lax-wendroff-and-crank-nicolson-matlab-simulation-of-linear-advection.htmlI am implementing two schemes: Lax-Wendroff and Crank Nicolson to solve the PDE linear advection in Matlab. My simulation seems to run; however, between the transition in time space the numerical ...Tue, 18 Aug 2026 01:47:21 -0400https://liberty.io/find-the-extreme-values-of-the-given-function-subject-to-constraints-f.htmlhttps://liberty.io/find-the-extreme-values-of-the-given-function-subject-to-constraints-f.htmlFind the extreme values of the function, subject to constraints $$f(x,y) = y^2 - x^2 ; x^2 + y^2 = 36$$ Here is what I have done: Using the substitution method we can say: $$y = \sqrt{-x^2 + 36}$$......Tue, 18 Aug 2026 01:42:40 -0400https://liberty.io/solving-ode-problem-with-calculator.htmlhttps://liberty.io/solving-ode-problem-with-calculator.htmlI'm trying to solve a below ODE problem with TI-nspire cas: $$ y''-4y'+3y=cos^2x $$ If I solve the above problem by hand (or using the wolfram alpha), I get the $$ y=c_1e^x+c_2e^{3x}+\frac16-\f......Tue, 18 Aug 2026 01:41:56 -0400https://liberty.io/multivariable-limit-of-a-piecewise-function.htmlhttps://liberty.io/multivariable-limit-of-a-piecewise-function.htmlConsider the following expression: $$\lim_{(x,y) \to (0,0)} g(x,y)= \begin{cases} \frac{\sin x}{x} y \text{ $\hspace{0.5in}$if $x \neq 0$}\\ y \text{ $\hspace{0.81in}$if $x=0$}\end{cases}$$ I am se......Tue, 18 Aug 2026 01:36:57 -0400https://liberty.io/why-is-the-graph-of-2logx-different-from-log-x-2-duplicate.htmlhttps://liberty.io/why-is-the-graph-of-2logx-different-from-log-x-2-duplicate.htmlAccording to Logarithmic exponent rules: log(x^y) = y ∙ log(x) For example: 2 log x = log x^2 However take the following graphs: y=2logx y=logx^2 The graphs seems to display something that sho......Tue, 18 Aug 2026 01:30:07 -0400https://liberty.io/general-solution-of-the-differential-equation-in-explicit-form.htmlhttps://liberty.io/general-solution-of-the-differential-equation-in-explicit-form.htmlFind the general solution of the differential equation in explicit form, $\dfrac{dr}{dt} =\dfrac{ k}{r^2}, (t>0, 0 < r < 10)$ I've split it into two so I have $r^2 dr = k dx$ so when ...Tue, 18 Aug 2026 01:11:07 -0400https://liberty.io/the-equation-1-x-2-frac-dy-dx-xy-x-2y-2.htmlhttps://liberty.io/the-equation-1-x-2-frac-dy-dx-xy-x-2y-2.htmlI am very grateful for all your comments and answers to my previous question concerning ODEs ( The equation $(x-2xy-y^2)\frac{dy}{dx}+y^2=0$ ). Now I am struggling with this one $$(1+x)^2\frac{dy}......Tue, 18 Aug 2026 01:08:30 -0400https://liberty.io/limits-as-x-approaches-one-from-the-left.htmlhttps://liberty.io/limits-as-x-approaches-one-from-the-left.htmlHow do I find $$\lim_{x \to 1^-} -\frac{1}{x-1}$$ Can someone please show me how to work it out algebraically?...Tue, 18 Aug 2026 00:57:00 -0400https://liberty.io/prove-that-sqrt-45-is-irrational-using-euclid-s-lemma.htmlhttps://liberty.io/prove-that-sqrt-45-is-irrational-using-euclid-s-lemma.htmlProve that $\sqrt{45}$ is irrational (using Euclid’s Lemma) Assume $\sqrt{45}$ is rational. By definition of rational : $\sqrt{p}=\frac{a}{b}$= $\sqrt{45}$=$\frac{a}{b}$ for some $a,b$ integers a......Tue, 18 Aug 2026 00:56:05 -0400https://liberty.io/how-do-i-use-interpolation-with-the-z-table.htmlhttps://liberty.io/how-do-i-use-interpolation-with-the-z-table.htmlMy textbook has an example of interpolation, but I am not sure how the book did it since it doesn't explain it. It says if we want $P(Z < 1.246)$ we must use interpolation and the steps given a......Tue, 18 Aug 2026 00:55:21 -0400https://liberty.io/find-the-value-of-sin-left-cos-1-frac-1-2-sin-1-frac-3-5-right.htmlhttps://liberty.io/find-the-value-of-sin-left-cos-1-frac-1-2-sin-1-frac-3-5-right.htmlFind the value of $\sin \left(\cos^{-1} \left(\dfrac {1}{2}\right) + \sin^{-1} \left(\dfrac {3}{5}\right)\right)$ My Attempt: \begin{align*} \sin \left(\cos^{-1} \left(\dfrac {1}{2}\right) + \sin......Tue, 18 Aug 2026 00:53:43 -0400https://liberty.io/partial-derivative-of-integral-leibniz-rule.htmlhttps://liberty.io/partial-derivative-of-integral-leibniz-rule.htmlThe Leibniz rule is as follows: $$\frac{d}{d\alpha} \int_{a(\alpha)}^{b(\alpha)} f(x, \alpha) dx = \frac{db(\alpha)}{d\alpha} f(b(\alpha), \alpha) - \frac{da(\alpha)}{d\alpha} f(a(\alpha), \alpha)......Tue, 18 Aug 2026 00:40:42 -0400https://liberty.io/intuition-behind-an-integral-identity.htmlhttps://liberty.io/intuition-behind-an-integral-identity.htmlA proof for the identity $$\int_{-\infty}^{\infty} f(x)\, dx=\int_{-\infty}^{\infty} f\left(x-\frac{1}{x}\right)\, dx,$$ has been asked before (for example, here), and one answer to that question ...Tue, 18 Aug 2026 00:40:35 -0400https://liberty.io/volume-of-tetrahedron-using-cross-and-dot-product.htmlhttps://liberty.io/volume-of-tetrahedron-using-cross-and-dot-product.htmlConsider the tetrahedron in the image: Prove that the volume of the tetrahedron is given by $\frac16 |a \times b \cdot c|$. I know volume of the tetrahedron is equal to the base area times height......Tue, 18 Aug 2026 00:34:01 -0400https://liberty.io/how-is-a-decibel-a-tenth-of-a-bel.htmlhttps://liberty.io/how-is-a-decibel-a-tenth-of-a-bel.htmlBy definition, one Bel is log10(P2/P1) (in the power example) and one dB is 10log10(P2/P1), so it appears to me that one dB is ten Bels, not one-tenth of a Bel. Yet, many online publications, univ......Tue, 18 Aug 2026 00:28:12 -0400https://liberty.io/why-does-the-higher-order-derivative-test-work.htmlhttps://liberty.io/why-does-the-higher-order-derivative-test-work.htmlI'm an AP Calculus BC student, so all I know about derivatives is the increasing/decreasing/ relative max/min function relation with first derivative (first derivative test), concavity (second deri......Tue, 18 Aug 2026 00:27:02 -0400https://liberty.io/work-to-pump-water-from-a-cylindrical-tank.htmlhttps://liberty.io/work-to-pump-water-from-a-cylindrical-tank.htmlWe have a cylindrical tank with radius 2m and length 10m filled with water. How much work does it tank to pump the water out of the tank from the top? My attempt at the problem goes as follow. $g......Tue, 18 Aug 2026 00:23:19 -0400https://liberty.io/subtracting-multiple-logarithms.htmlhttps://liberty.io/subtracting-multiple-logarithms.htmlI am asked to simplify $$J=\ln (x^2-16)-\ln x-\ln(x-4), \quad x>4$$ Since each logarithm's argument is non-negative, I can use $$\ln x -\ln y=\ln\frac{x}{y}$$ and obtain the correct answer $$\ln......Tue, 18 Aug 2026 00:20:02 -0400https://liberty.io/is-the-equator-a-line.htmlhttps://liberty.io/is-the-equator-a-line.htmlThe equator is a circle that goes around the earth. Does that mean that it is a line? I'm guessing that it is because it is infinite. At the same time, it's overlapping. The real question is, is a ...Tue, 18 Aug 2026 00:07:20 -0400https://liberty.io/what-is-an-empty-set.htmlhttps://liberty.io/what-is-an-empty-set.htmlWe define the term "Set" as, A set is a collection of objects. And an "Empty set" as, An empty set is a set which contains nothing. First problem I encountered: How the definition of "Emp......Tue, 18 Aug 2026 00:02:47 -0400https://liberty.io/dragon-curve-formula.htmlhttps://liberty.io/dragon-curve-formula.htmlCan someone please explain to me this formula?...Mon, 17 Aug 2026 23:51:13 -0400https://liberty.io/is-determinant-of-matrix-multiplied-its-transpose-always-positive.htmlhttps://liberty.io/is-determinant-of-matrix-multiplied-its-transpose-always-positive.htmlAssume $A$ is an arbitrary $m\times n$ real matrix. Is $\det(AA^T)$ always positive? Is it non-negative or it can have any value? Edit: It seems I have to emphasis that $m \ne n$ i.e. matrix is non-...Mon, 17 Aug 2026 23:44:23 -0400https://liberty.io/the-equation-of-the-line-pass-through-2-points-in-3d-space-closed.htmlhttps://liberty.io/the-equation-of-the-line-pass-through-2-points-in-3d-space-closed.htmlI'm looking for a line equation passes through $2$ points in a $3$-dimensional space, and use it to determine the intersection between sphere and line. Any help would be appreciated...Mon, 17 Aug 2026 23:35:35 -0400https://liberty.io/what-do-french-mathematicians-call-a-ray.htmlhttps://liberty.io/what-do-french-mathematicians-call-a-ray.htmlIn the USA we are taught at a young age that a ray is a line with one end, a compromise between a line and a line segment, something like $[0,\infty)$. Is there a word for this in french? I have ta......Mon, 17 Aug 2026 23:33:02 -0400https://liberty.io/the-graphs-of-f-x-y-x-2-y-2-and-g-x-y-x-2-y-2-xy-3-are-tangent-to-each.htmlhttps://liberty.io/the-graphs-of-f-x-y-x-2-y-2-and-g-x-y-x-2-y-2-xy-3-are-tangent-to-each.htmlWhy can we say that the graphs of $f(x,y)=x^2+y^2$ and $g(x,y)=-x^2-y^2+xy^3$ "are tangent to each other" at $(0,0)$ ? I have done the following: The tangent plane at the graph of $f(x,y)=x^2+y^2......Mon, 17 Aug 2026 23:22:31 -0400https://liberty.io/whats-is-the-meaning-of-the-derivative-of-a-vector-function.htmlhttps://liberty.io/whats-is-the-meaning-of-the-derivative-of-a-vector-function.htmlAssuming we have a continuous and well behaved vector function $$ R(t) = \langle f(t), g(t), h(t) \rangle $$ then its derivative at an arbitrary point a is $R'(a)$, which can be computed (I'm not ......Mon, 17 Aug 2026 23:21:48 -0400https://liberty.io/relation-between-a-convex-set-and-convex-function.htmlhttps://liberty.io/relation-between-a-convex-set-and-convex-function.htmlI was reading about convex sets and convex functions. I would like to know if there is a relation between convex functions and convex sets. Like, a function whose domain and range is a convex set is ...Mon, 17 Aug 2026 23:05:28 -0400https://liberty.io/find-the-average-velocity-and-the-instantaneous-velocity.htmlhttps://liberty.io/find-the-average-velocity-and-the-instantaneous-velocity.htmlFind the average velocity of an object whose position is given by: $$s(t)=\frac{13}{t+2}$$ Using the following intervals: i.[11,12] ii. [11, 11.1] iii. [11, 11.001] iv. [11, 1.00001] Then gu......Mon, 17 Aug 2026 22:58:43 -0400https://liberty.io/constructing-an-isomorphism.htmlhttps://liberty.io/constructing-an-isomorphism.htmlI need to construct an isomorphism: $T: W \to C^3$ where W is a subspace of $P_3(C)$ I got bases (for W and $C^3$ respectively) $\alpha = \{(ix + 1),\ \ (x^3 + 2x), \ (2ix^2 + ix -1)\}$ $\beta ......Mon, 17 Aug 2026 22:58:34 -0400https://liberty.io/trick-with-3-digit-numbers-always-get-1089.htmlhttps://liberty.io/trick-with-3-digit-numbers-always-get-1089.htmlWhen I was in primary school a teacher showed us the following exercise in arithmetic. Take any 3 digit number between 201 and 998 provided that the hundreds digit is bigger than the ones digit an......Mon, 17 Aug 2026 22:54:33 -0400https://liberty.io/how-to-calculate-moment-of-inertia-of-parabolic-spandrel.htmlhttps://liberty.io/how-to-calculate-moment-of-inertia-of-parabolic-spandrel.htmlI want to find the moment of inertia of this spandrel along $DD'$ $$I=\int r^2 \,\mathrm dm$$ The area of the spandrel is $\frac{3m}{ab}$ so $I=\sigma\int r^2\mathrm dA $ where $\sigma$ is aerial ...Mon, 17 Aug 2026 22:53:04 -0400https://liberty.io/simple-proof-show-that-4-n-1-is-divisible-by-3-guided-proof-task.htmlhttps://liberty.io/simple-proof-show-that-4-n-1-is-divisible-by-3-guided-proof-task.htmlFirst part of the task is just to show that $(4^n-1)$ actually is divisible by 3 for n=1,2,3,4. No problem. Second step: is to show that $(4^n -1) = (2^n-1)(2^n+1)$ No problem, just algebra. Third ...Mon, 17 Aug 2026 22:45:55 -0400https://liberty.io/context-sensitive-grammar-for-the-copy-language.htmlhttps://liberty.io/context-sensitive-grammar-for-the-copy-language.htmlI need to find a context-sensitive grammar for the copy language $$L = \{ww \; | w \in \{0,1\}^* \}$$ This is what I got so far: $\begin{eqnarray} S & \Rightarrow & \lambda \; | \; X \\ X......Mon, 17 Aug 2026 22:41:06 -0400https://liberty.io/continuous-projections-on-ell-1-with-norm-1.htmlhttps://liberty.io/continuous-projections-on-ell-1-with-norm-1.htmlI was trying to find papers and articles about non-contractive continuous projections on $\ell_1(S)$ where $S$ is an arbitrary set. If it is not studied yet, I would like to know results for the ca......Mon, 17 Aug 2026 22:37:08 -0400https://liberty.io/understanding-the-unit-circle.htmlhttps://liberty.io/understanding-the-unit-circle.htmlI'm going to need to know this (the unit circle, English) for a test, but I'm so horrible with both charts and memorization, the whole thing is failing to stick into my brain. Using non-complicated ...Mon, 17 Aug 2026 22:27:30 -0400https://liberty.io/a-different-type-of-infinite-power-tower-function.htmlhttps://liberty.io/a-different-type-of-infinite-power-tower-function.htmlThe question is about the following: Let a function be defined such that $$f_n(x) = x \uparrow \uparrow n$$ Where $n$ is a natural number Now, it is reported at many places that the function $$F_......Mon, 17 Aug 2026 22:24:32 -0400https://liberty.io/how-to-compute-the-similarity-transformation-matrix.htmlhttps://liberty.io/how-to-compute-the-similarity-transformation-matrix.htmlStuck on this question: Let $$A=\begin{pmatrix} 2&1\\ -1&-1 \end{pmatrix}$$$$B=\begin{pmatrix} -2&5\\ -1&3 \end{pmatrix}$$$$C=\begin{pmatrix} 5&2\\ 4&1 \end{pmatrix}$$ ......Mon, 17 Aug 2026 22:23:12 -0400https://liberty.io/a-die-is-rolled-and-a-coin-is-flipped-what-is-wrong-with-the-following.htmlhttps://liberty.io/a-die-is-rolled-and-a-coin-is-flipped-what-is-wrong-with-the-following.htmlA die is rolled and a coin is flipped. What is wrong with the following reasoning?: Let $A=\{1,3,6\}$ (event from die roll), let $B=\{H\}$ (event of heads) then, $A\cap B=\emptyset$ (since they ha......Mon, 17 Aug 2026 22:15:21 -0400https://liberty.io/how-to-calculate-carmichael-function-of-a-large-number.htmlhttps://liberty.io/how-to-calculate-carmichael-function-of-a-large-number.htmlHow would I go about calculating $\lambda(49392)$ by hand? I have $\lambda$ defined in a book as $$ \lambda(n) = max{\{\bar{a} | \bar{a} \in Z_n^*\}} \ \cdot \ ord_{Z_n^{*}} (\bar{a}) $$ where $......Mon, 17 Aug 2026 22:14:12 -0400