https://liberty.ioCurated star highlights with fresh energy.en-usWed, 19 Aug 2026 16:27:26 -0400https://liberty.io/finding-the-mean-and-median-of-a-probability-density-function.htmlhttps://liberty.io/finding-the-mean-and-median-of-a-probability-density-function.htmlI suspect this is super-easy, but I haven't done any math in about ten years and I'm working with concepts that have been woefully explained... I need to find the mean and median of a continuous r......Wed, 19 Aug 2026 20:27:12 -0400https://liberty.io/express-if-true-then-1-else-0-in-a-formula-suitable-for-desmos-calcula.htmlhttps://liberty.io/express-if-true-then-1-else-0-in-a-formula-suitable-for-desmos-calcula.htmlIn programming, often the value of True is also 1, and False is 0. This means that: (x>5)*4 will return 4 if x is greater than 5 (because (x>5)==1), else 0. I need to accomplish a similar......Wed, 19 Aug 2026 20:26:04 -0400https://liberty.io/l-039-hospital-rule-in-multivariable-calculus.htmlhttps://liberty.io/l-039-hospital-rule-in-multivariable-calculus.htmlI was just wondering. Given functions $f(x,y), g(x,y) $, and the corresponding limit $\lim_{(x,y) \to (0,0) } \frac{f(x,y)}{g(x,y)} $ , where $\lim_{(x,y) \to (0,0) } f = \lim_{(x,y) \to (0,0) }g =......Wed, 19 Aug 2026 20:19:18 -0400https://liberty.io/what-039-s-the-intuition-behind-a-split-exact-sequence.htmlhttps://liberty.io/what-039-s-the-intuition-behind-a-split-exact-sequence.htmlI've learned the following definition for a split exact sequence: A short exact sequence of $R$-module homomorphisms $0\to A \stackrel{f}{\to} B \stackrel{g}{\to} C \to 0$ is split if there exists a ...Wed, 19 Aug 2026 20:01:58 -0400https://liberty.io/torque-and-angular-frequency-of-a-merry-go-round.htmlhttps://liberty.io/torque-and-angular-frequency-of-a-merry-go-round.htmlI have been trying to approach this by finding the gravitational torque of the person on the merry-go-round, however I can't find a way to relate that to angular velocity in the given circumstances......Wed, 19 Aug 2026 19:57:35 -0400https://liberty.io/finding-the-vertex-of-a-quadratic-function.htmlhttps://liberty.io/finding-the-vertex-of-a-quadratic-function.htmlI'm in the process of relearning math and came across a problem when calculating the vertex for $5x^2-20x+15$ function. I took the following steps to find the vertex: I divided every term by 5 to ......Wed, 19 Aug 2026 19:46:43 -0400https://liberty.io/is-there-such-a-thing-as-implicit-integration.htmlhttps://liberty.io/is-there-such-a-thing-as-implicit-integration.htmlA problem I'm working on asks me at a certain point to integrate the following: $$x=2\left(\sin\left(y\right)\right)^2.$$ I've never integrated anything in this form. I can't isolate for $y$ eith......Wed, 19 Aug 2026 19:44:56 -0400https://liberty.io/what-are-equations-with-a-degree-more-than-3-called.htmlhttps://liberty.io/what-are-equations-with-a-degree-more-than-3-called.htmlSorry if I am using the incorrect terminology. I am beginning to learn Mathematics. I had it in school and college but have forgotten most of it, so please be gentle with me if my questions are stu......Wed, 19 Aug 2026 19:43:19 -0400https://liberty.io/line-integral-around-a-triangle.htmlhttps://liberty.io/line-integral-around-a-triangle.htmlLet $R$ be the interior of the triangle with vertices $(0,0), (4,2),$ and $(0,2)$. Let $C$ be the boundary of $R$, oriented counterclockwise. Now evaluate the integral below. $$\int_C(y+e^\sqrt{x......Wed, 19 Aug 2026 19:39:05 -0400https://liberty.io/039-normally-distributed-random-numbers-039-vs-039-uniformly-distribut.htmlhttps://liberty.io/039-normally-distributed-random-numbers-039-vs-039-uniformly-distribut.htmlwhat is the difference between 'normally distributed random numbers' and 'uniformly distributed random number'? A answer in a layman language is appreciated :)...Wed, 19 Aug 2026 19:38:41 -0400https://liberty.io/biased-coin-probability-uneven-probability.htmlhttps://liberty.io/biased-coin-probability-uneven-probability.htmlGetting at least $2$ heads when flipping a coin $3$ times but the coin is biased so that heads are $3$ times more likely than tails. Can anyone explain how uneven probability works? Thank you...Wed, 19 Aug 2026 19:38:12 -0400https://liberty.io/graph-of-y-log-1-x.htmlhttps://liberty.io/graph-of-y-log-1-x.htmlTo draw the graph of $|y|=|\log(1-|x|)|$, I see that $x=0, y=0$ satisfies it. Also, the asymptotes would be $x=1, x=-1$. I have made a graph which has sort of eight curves in it but I don't know ho......Wed, 19 Aug 2026 19:31:23 -0400https://liberty.io/why-can-039-t-there-be-a-quintic-formula-duplicate.htmlhttps://liberty.io/why-can-039-t-there-be-a-quintic-formula-duplicate.htmlPossible Duplicate: Why is it so hard to find the roots of polynomial equations? For polynomials (with real coefficients), in degrees 2, 3, 4, there are the quadratic, cubic, and quartic formu......Wed, 19 Aug 2026 19:30:35 -0400https://liberty.io/solve-tan2x-1.htmlhttps://liberty.io/solve-tan2x-1.htmlAre there any errors in my work? Thanks in advance! (Sorry for the bad format. I'm still new to this) $\tan2x=1$ $\frac{2\tan x}{1-\tan^2x}=1$ $2\tan x=1-\tan^2x$ $0=1-\tan^2x-2\tan x$ $0 =-\t......Wed, 19 Aug 2026 19:17:09 -0400https://liberty.io/what-local-system-really-is.htmlhttps://liberty.io/what-local-system-really-is.htmlI know a local system is a locally constant sheaf. But why does a local system on the topological space $X$ correspond to $\tilde{X}\times_G V$, where $G$ is the fundamental group of $X$, $\tilde{X......Wed, 19 Aug 2026 19:10:34 -0400https://liberty.io/why-h-p-coprod-i-in-i-x-i-g-cong-prod-i-in-i-h-p-x-i-g.htmlhttps://liberty.io/why-h-p-coprod-i-in-i-x-i-g-cong-prod-i-in-i-h-p-x-i-g.htmlMy question is the following; why $\displaystyle H^p\big(\coprod_{i\in I}X_i;G\big)$ is isomorphic to $\displaystyle\prod_{i\in I}H^p(X_i;G)$? I have tried the following: one can prove the followin......Wed, 19 Aug 2026 19:08:01 -0400https://liberty.io/what-does-exterior-algebra-actually-mean.htmlhttps://liberty.io/what-does-exterior-algebra-actually-mean.htmlThis question may be too basic and even silly, but I am new to exterior algebra and reading Wikipedia. Given $e_1, e_2,\cdots, e_n$ is a standard basis for a vector space $V$, what does $e_1\wedge......Wed, 19 Aug 2026 19:01:38 -0400https://liberty.io/definite-integral-in-matlab.htmlhttps://liberty.io/definite-integral-in-matlab.htmlI have just started learning how to solve definite integrals in matlab, and from the documentation, I picked up this example: $\int^{\infty}_{0}(e^{-x^2} )dx$ but when I run it in my matlab, I ge......Wed, 19 Aug 2026 18:58:06 -0400https://liberty.io/fast-algorithm-for-computing-integer-square-roots-on-machines-that-doe.htmlhttps://liberty.io/fast-algorithm-for-computing-integer-square-roots-on-machines-that-doe.htmlFast algorithm for computing integer square roots on machines that doesn't support floating-point arithmetic? I'm looking for a fast algorithm for computing the integer square root of an integer $......Wed, 19 Aug 2026 18:54:45 -0400https://liberty.io/mathematical-structures-and-signature.htmlhttps://liberty.io/mathematical-structures-and-signature.htmlFrom Wikipedia: In mathematics, a structure on a set, or more generally a type, consists of additional mathematical objects that in some manner attach (or relate) to the set, making it easier to ...Wed, 19 Aug 2026 18:46:24 -0400https://liberty.io/square-roots-modulo-powers-of-2.htmlhttps://liberty.io/square-roots-modulo-powers-of-2.htmlExperimentally, it seems like every $a\equiv1 \pmod 8$ has $4$ square roots mod $2^n$ for all $n \ge 3$ (i.e. solutions to $x^2\equiv a \pmod {2^n}$) Is this true? If so, how can I prove it? If n......Wed, 19 Aug 2026 18:44:04 -0400https://liberty.io/constructing-a-confidence-interval-for-population-variance.htmlhttps://liberty.io/constructing-a-confidence-interval-for-population-variance.htmlGiven the following data set, how does one construct a 99% confidence interval for the population variance? I should mention now I don't have a very good understanding of when to divide by n and wh......Wed, 19 Aug 2026 18:43:39 -0400https://liberty.io/what-does-the-symbol-nabla-indicate.htmlhttps://liberty.io/what-does-the-symbol-nabla-indicate.htmlFirst up, this question differs from the other ones on this site as I would like to know the isolated meaning of nabla if that makes sense. Meanwhile, other questions might ask what it means in rel......Wed, 19 Aug 2026 18:40:18 -0400https://liberty.io/calculate-the-confidence-interval-of-parameter-of-exponential-distribu.htmlhttps://liberty.io/calculate-the-confidence-interval-of-parameter-of-exponential-distribu.htmlHow can I calculate the confidence interval for parameter $\alpha$ of exponential distribution ? I think I can use test-t. Knowing that: $$mean = {1\over\alpha}$$ I found that : $${1\over {\bar......Wed, 19 Aug 2026 18:34:59 -0400https://liberty.io/list-of-functions-not-integrable-in-elementary-terms.htmlhttps://liberty.io/list-of-functions-not-integrable-in-elementary-terms.htmlWhen teaching integration to beginning calculus students I always tell them that some integrals are "impossible" (with a bit of expansion on what that actually means). However I must admit that the ...Wed, 19 Aug 2026 18:32:52 -0400https://liberty.io/find-5th-degree-polynomial-equation-from-points.htmlhttps://liberty.io/find-5th-degree-polynomial-equation-from-points.htmlI have a set of estimated points, listed below, and I would like to know how to calculate the polynomial. I've looked at Wikipedia, and I don't quite understand how at works. Example 1 on http://en....Wed, 19 Aug 2026 18:28:54 -0400https://liberty.io/rational-parametrization-of-circle-in-wikipedia.htmlhttps://liberty.io/rational-parametrization-of-circle-in-wikipedia.htmlIn http://en.wikipedia.org/wiki/Circle but also in the corresponding article in the German Wikipedia I find this formulation ( sorry, I exchange x and y as I am accustomed to it in this way ) : "An ...Wed, 19 Aug 2026 18:19:57 -0400https://liberty.io/find-the-domain-of-root-of-a-logarithmic-function.htmlhttps://liberty.io/find-the-domain-of-root-of-a-logarithmic-function.htmlI'm a little confused about this question since output of a logarithmic function varies from $ -\infty $ to $\infty$ .I should find the domain of this function: $ y=\sqrt{\log_x(10-x^2)} $ . How c......Wed, 19 Aug 2026 17:57:03 -0400https://liberty.io/how-many-32-digit-binary-number-combinations-are-possible.htmlhttps://liberty.io/how-many-32-digit-binary-number-combinations-are-possible.htmlHow many 32 digit binary number combinations are possible? For example: $$00000000-00000000-00000000-00000000$$ $$00000000-00000000-00000000-00000001$$ $$00000000-00000000-00000000-00000010$$ $$.$......Wed, 19 Aug 2026 17:54:25 -0400https://liberty.io/propositional-equivalence.htmlhttps://liberty.io/propositional-equivalence.htmlAre the following two propositions equivalent? p IMPLIES (q IMPLIES r) p IMPLIES (q AND r) From what I can tell, using the logical equivalences, this should be false, correct? p IMPLIES (q IMPLI......Wed, 19 Aug 2026 17:51:14 -0400https://liberty.io/how-lim-h-0-sin-h-h-is-equal-to-1-duplicate.htmlhttps://liberty.io/how-lim-h-0-sin-h-h-is-equal-to-1-duplicate.htmlHow $\lim_{h\to 0} \sin h$ when divided by $h$ gives the value $1$ ? Does it also follows with other ratios that for example. $\lim_{h\to 0} \cos x/x = 1$...Wed, 19 Aug 2026 17:39:27 -0400https://liberty.io/intuitive-way-to-calculate-volume-of-the-solid-bounded-by-a-plane.htmlhttps://liberty.io/intuitive-way-to-calculate-volume-of-the-solid-bounded-by-a-plane.htmlFor a National Board Exam Review: What is the volume of the solid bounded by the plane $3x+4y+6z=12$ and the coordinate axes? Answer is $4$. I am looking for a quick and intuitive way to so......Wed, 19 Aug 2026 17:35:42 -0400https://liberty.io/elliptic-coordinates-inverting-the-transformation.htmlhttps://liberty.io/elliptic-coordinates-inverting-the-transformation.htmlThe standard way to transform elliptic coordinates $(\mu, \nu)$ $\ to$ Cartesian coordinates $(x,y)$: $x = a \cosh(\mu) \cos(\nu)$ $y = a \sinh(\mu) \sin(\nu)$ Is there any way to get the ...Wed, 19 Aug 2026 17:31:07 -0400https://liberty.io/system-of-linear-equations-how-many-solutions.htmlhttps://liberty.io/system-of-linear-equations-how-many-solutions.htmlFor which real values of t does the following system of linear equations: $$ \left\{ \begin{array}{c} tx_1 + x_2 + x_3 = 1 \\ x_1 + tx_2 + x_3 = 1 \\ x_1 + x_2 + tx_3 = 1 \end{array} \right. $$ ...Wed, 19 Aug 2026 17:22:49 -0400https://liberty.io/geometric-interpretation-of-element-wise-vector-multiplication.htmlhttps://liberty.io/geometric-interpretation-of-element-wise-vector-multiplication.htmlGeometrically speaking, vector dot product is interpreted as a projection of one vector on the other, while vector cross product gives another vector that is orthogonal to the multiplied vectors(ri......Wed, 19 Aug 2026 17:02:41 -0400https://liberty.io/how-do-we-use-derivatives-in-our-daily-lives-closed.htmlhttps://liberty.io/how-do-we-use-derivatives-in-our-daily-lives-closed.htmlHow can we use derivatives in our real lives, I know that we have a lot of formulas to find the derivatives of a function,but why do we need them ?...Wed, 19 Aug 2026 16:43:27 -0400https://liberty.io/how-to-graph-sec-x-2-by-manipulating-the-cosine-function.htmlhttps://liberty.io/how-to-graph-sec-x-2-by-manipulating-the-cosine-function.html$\sec(x) = 1/\cos(x)$. Usually, I graph $\sec(x)$ by graphing $\cos(x)$ first. $\sec x/2= 1/2\cdot (1/\cos x) = (1/2)\cos x$. Following the same logic above, I graphed $2\cos(x)$. However, what I ...Wed, 19 Aug 2026 16:42:00 -0400https://liberty.io/is-it-true-that-a-3rd-order-polynomial-must-have-at-least-one-real-roo.htmlhttps://liberty.io/is-it-true-that-a-3rd-order-polynomial-must-have-at-least-one-real-roo.htmlWhile solving a problem a friend said - this polynomial is $3^{rd}$ order ($ax^3+bx^2+cx+d$), with $\{a,b,c,d\}$ real coefficients, so it must have a real root. I didn't want to sound stupid and I ......Wed, 19 Aug 2026 16:39:23 -0400https://liberty.io/is-the-product-of-uniquely-decipherable-codes-uniquely-decipherable.htmlhttps://liberty.io/is-the-product-of-uniquely-decipherable-codes-uniquely-decipherable.htmlA code $C$ is prefix-free if no codeword in $C$ is a prefix of another codeword in $C$. For example, $C=\{0,10,110\}$ is prefix-free but $D=\{0,10,100\}$ is not. A code $C$ is uniquely decipherabl......Wed, 19 Aug 2026 16:38:10 -0400https://liberty.io/is-the-given-ode-system-a-gradient-system.htmlhttps://liberty.io/is-the-given-ode-system-a-gradient-system.htmlConsider the planar system $$\dot{x} = 2x-y-\frac{2x^3+2xy^2-xy}{\sqrt{x^2+y^2}} \\ \dot{y} = 2y+x-\frac{2x^2y+2y^3+x^2}{\sqrt{x^2+y^2}}$$ which in polar coordinates is given by $$\ \dot{r}=2r(1......Wed, 19 Aug 2026 16:13:33 -0400https://liberty.io/coherent-sheaves-vs-sheaves-locally-of-finite-presentation.htmlhttps://liberty.io/coherent-sheaves-vs-sheaves-locally-of-finite-presentation.htmlI think it is clear for most people that finitely generated and finitely presented modules are important and, in a certain sense, the "right" notions. However, the corresponding notions for ...Wed, 19 Aug 2026 16:11:38 -0400https://liberty.io/how-do-you-obtain-the-2-3-7-triangle-group-as-a-quotient-of-the-modula.htmlhttps://liberty.io/how-do-you-obtain-the-2-3-7-triangle-group-as-a-quotient-of-the-modula.htmlSo, at various places on Wikipedia (like here), its claimed that the $(2,3,7)$ triangle group can be obtained as quotient of the modular group. How is this done? Treating the integers as mod $7$ ......Wed, 19 Aug 2026 16:01:42 -0400https://liberty.io/is-this-the-notation-you-use.htmlhttps://liberty.io/is-this-the-notation-you-use.htmlI've noticed that my terminology is a bit haggard. I do math on my own so I'm not entirely sure how everyone else refers to things and so I need a check. so is this correct: $\lim\limits_{\delta ......Wed, 19 Aug 2026 15:54:52 -0400https://liberty.io/is-1-3-and-or-17-in-the-set-of-integers-modulo-7.htmlhttps://liberty.io/is-1-3-and-or-17-in-the-set-of-integers-modulo-7.htmlIs $1/3$ and/or $-17$ in the set of integers modulo $7$? So I say that $1/3$ is in the set of integers modulo $7$ because $3 \cdot 1/3 = 1$, so $3x \bmod 7 = 1$. For $x = 5$, this works, so $1/3$ is ...Wed, 19 Aug 2026 15:49:35 -0400https://liberty.io/in-triangle-abc-find-the-value-of-cos-a-cos-b.htmlhttps://liberty.io/in-triangle-abc-find-the-value-of-cos-a-cos-b.htmlThe sides of $\triangle$ABC are in Arithmetic Progression (order being $a$, $b$, $c$) and satisfy $\dfrac{2!}{1!9!}+\dfrac{2!}{3!7!}+\dfrac{1}{5!5!}=\dfrac{8^a}{(2b)!}$, Then prove that the value ......Wed, 19 Aug 2026 15:40:41 -0400https://liberty.io/example-of-a-real-life-graph-with-a-hole.htmlhttps://liberty.io/example-of-a-real-life-graph-with-a-hole.htmlAnyone ever come across a real non-textbook example of a graph with a hole in it? In Precalc, you get into graphing rational expressions, some of which reduce to a non-rational. The cancelled f......Wed, 19 Aug 2026 15:33:45 -0400https://liberty.io/why-is-the-inductive-hypothesis-assumed-to-be-true.htmlhttps://liberty.io/why-is-the-inductive-hypothesis-assumed-to-be-true.htmlThe logic of the Inductive proof seems circular, the whole proof seems to hinge on whether or not the Inductive Hypothesis is true. Sure you can show that p(k+1) is true if p(k) is true, you have ......Wed, 19 Aug 2026 15:16:24 -0400https://liberty.io/why-does-limit-not-exist.htmlhttps://liberty.io/why-does-limit-not-exist.htmlI don't understand how the limit does not exist for the composite function. The limit as x approaches -2 for g(x) is zero. So, the last step is to evaluate h(0), which is -1. Yes, there is a hole a......Wed, 19 Aug 2026 15:14:18 -0400https://liberty.io/difference-between-weak-or-martingale-and-strong-solutions-to-sdes.htmlhttps://liberty.io/difference-between-weak-or-martingale-and-strong-solutions-to-sdes.htmlHi Im fairly new to SDE theory and am struggling with the difference between a weak ( or martingale ) solution and a strong solution to an SDE : $$ d(X_{t})=b(t,X_{t})dt + \sigma(t,X_{t})dW_{t} ......Wed, 19 Aug 2026 15:08:17 -0400https://liberty.io/graph-x-2-y-2-z-2-4-1.htmlhttps://liberty.io/graph-x-2-y-2-z-2-4-1.htmlGraph $x^2+y^2+z^2/4=1$ I want to know what would be the graph of the equation...Wed, 19 Aug 2026 15:06:10 -0400https://liberty.io/proving-that-sin-z-2-sin-x-2-sinh-y-2.htmlhttps://liberty.io/proving-that-sin-z-2-sin-x-2-sinh-y-2.htmlGoal: Prove $|\sin(z)|^2 = \sin(x)^2+\sinh(y)^2$ I have \begin{align*} \sin(x+iy) &=\sin(x)\cos(iy)+\cos(x)\sin(iy)\\ &=\sin(x)\cosh(y)+i\cos(x)\sinh(y)\\ |\sin(z)|^2 &=\sin(x)^2 \cosh......Wed, 19 Aug 2026 14:56:21 -0400https://liberty.io/mean-and-variance-of-squared-gaussian-y-x-2-where-x-sim-mathcal-n-0-si.htmlhttps://liberty.io/mean-and-variance-of-squared-gaussian-y-x-2-where-x-sim-mathcal-n-0-si.htmlWhat is the mean and variance of Squared Gaussian: $Y=X^2$ where: $X\sim\mathcal{N}(0,\sigma^2)$? It is interesting to note that Gaussian R.V here is zero-mean and non-central Chi-square Distribut......Wed, 19 Aug 2026 14:51:44 -0400https://liberty.io/what-is-the-best-strategy-for-cookie-clicker-esque-games.htmlhttps://liberty.io/what-is-the-best-strategy-for-cookie-clicker-esque-games.htmlToday, I stumbled across the game Cookie Clicker, which I recommend you avoid until you have at least a few hours of time to waste. The basic idea behind the game is this: You have a large stash......Wed, 19 Aug 2026 14:48:12 -0400https://liberty.io/find-the-area-enclosed-by-one-loop-of-this-lemniscate.htmlhttps://liberty.io/find-the-area-enclosed-by-one-loop-of-this-lemniscate.htmlFind the area enclosed by one loop of the lemniscate with equation $r^2 = 81cos(2\theta)$. Choose your limits carefully. When I was solving this problem, my intuition was to take the integral from......Wed, 19 Aug 2026 14:47:33 -0400https://liberty.io/how-to-prove-that-the-following-function-has-exactly-two-zeroes-in-a-p.htmlhttps://liberty.io/how-to-prove-that-the-following-function-has-exactly-two-zeroes-in-a-p.htmlI am practicing exam questions for my own exam in complex analysis. This was one I couldn't figure out. Let $U = \mathbb{C} \setminus \{ x \in \mathbb{R} : x \leq 0 \} $ en let $\log : U \to \math......Wed, 19 Aug 2026 14:47:31 -0400https://liberty.io/how-to-test-whether-a-set-of-four-points-can-form-a-parallelogram.htmlhttps://liberty.io/how-to-test-whether-a-set-of-four-points-can-form-a-parallelogram.htmlGiven four points $(x_1,y_1)$,$(x_2,y_2)$,$(x_3,y_3)$,$(x_4,y_4)$. How can we efficiently test them to make sure whether they are vertices of a parallelogram? I think evaluating and comparing the ...Wed, 19 Aug 2026 14:46:05 -0400https://liberty.io/finding-area-between-two-curves-below-and-above-x-axis.htmlhttps://liberty.io/finding-area-between-two-curves-below-and-above-x-axis.htmlI'm trying to find the area between: $y = 2x^2 - 1$ and $y = x^2$ I have found that the intersection points are at $(-1,1)$ and $(1,1)$. But the part that confuses me is that $y = 2x^2 - 1$ goes......Wed, 19 Aug 2026 14:45:56 -0400https://liberty.io/direct-integration-method.htmlhttps://liberty.io/direct-integration-method.htmlI want to directly integrate the following equation: $\frac{d^2f}{dt^2} - af = 0$ . I am wondering whether I have used a correct method. I denote $\frac{df}{dt} = f'$. Here are my steps: $f'' - ......Wed, 19 Aug 2026 14:42:10 -0400https://liberty.io/using-truth-tables-verify-that-the-contrapositive-and-original-stateme.htmlhttps://liberty.io/using-truth-tables-verify-that-the-contrapositive-and-original-stateme.htmlHow do I approach this? I understand how to construct a truth table, such as this one: However, is this enough to prove that they are equivalent statements?...Wed, 19 Aug 2026 14:34:49 -0400https://liberty.io/find-probability-density-function-of-y-sigma-x-mu.htmlhttps://liberty.io/find-probability-density-function-of-y-sigma-x-mu.htmlX has the pdf $f_x(x)$ Consider $Y=\sigma X+\mu$ where $\sigma$ and $\mu$ are constant. Find the PDF of $Y$? I find that $\mu_Y= \mu+\sigma \mu_x$ And $\sigma_Y^2=\sigma^2\sigma_x^2$ But I cannot f......Wed, 19 Aug 2026 14:28:44 -0400https://liberty.io/reference-request-for-gibbs-measure.htmlhttps://liberty.io/reference-request-for-gibbs-measure.htmlUnderstanding the Gibbs measure seems to be essential to fulling understanding statistical mechanics. However, I don't know of any mathematically rigorous textbooks/lecture notes that talk about such ...Wed, 19 Aug 2026 14:25:14 -0400https://liberty.io/int-infty-infty-e-ikx-dx-equals-what.htmlhttps://liberty.io/int-infty-infty-e-ikx-dx-equals-what.htmlWhat would $\int\limits_{-\infty}^\infty e^{ikx}dx$ be equal to where $i$ refers to imaginary unit? What steps should I go over to solve this integral? I saw this in the Fourier transform, and am ...Wed, 19 Aug 2026 14:22:49 -0400https://liberty.io/topological-basis-vs-local-basis.htmlhttps://liberty.io/topological-basis-vs-local-basis.htmlGiven a topological space $(X, \tau)$ and a basis $B$ for $X$, every open set $A \in \tau$ can be written as $A = \cup B_\alpha$, where $B_\alpha \in B$. The definition of a local basis is supposed......Wed, 19 Aug 2026 14:19:28 -0400https://liberty.io/i-need-help-writing-the-expression-5-sin-2x-cos-2x-in-terms-of-sine-on.htmlhttps://liberty.io/i-need-help-writing-the-expression-5-sin-2x-cos-2x-in-terms-of-sine-on.htmlI need help writing the expression $5(\sin 2x- \cos 2x)$ in terms of sine only. Using the double angle identity formula, I was able to get it to $5(\sin 2x - 1 + 2\sin^2x)$, but not sure how to move ...Wed, 19 Aug 2026 14:13:47 -0400https://liberty.io/spivak-ch-11-theorem-7.htmlhttps://liberty.io/spivak-ch-11-theorem-7.htmlCan someone please explain how to supply a rigorous $\epsilon,\delta$ argument for this theorem as Spivak says ? My argument is: $f'(a)=\lim_{h\to 0} f'(\alpha_h)$ equivalent to $|f'(\alpha_h)-f'......Wed, 19 Aug 2026 14:04:30 -0400https://liberty.io/what-is-the-significance-of-integrating-a-function.htmlhttps://liberty.io/what-is-the-significance-of-integrating-a-function.htmlNow i understand how important these things can be in terms of very small changes or finding area under curves and otherwise. However, when we integrate a function such as y = x we get (x^2)/2, and......Wed, 19 Aug 2026 14:04:20 -0400https://liberty.io/defective-matrix-and-eigenvalues.htmlhttps://liberty.io/defective-matrix-and-eigenvalues.htmlA matrix $A$ is called defective if $A$ has an eigenvalue $\lambda$ of multiplicity $m > 1$ for which the associated eigenspace has a basis of fewer than $m$ vectors; that is, the dimension of the ...Wed, 19 Aug 2026 13:46:36 -0400https://liberty.io/how-to-find-the-center-of-a-parallelogram.htmlhttps://liberty.io/how-to-find-the-center-of-a-parallelogram.htmlI want to find the center of a parallelogram in order to use it in my java program. I have four coordinates of the parallelogram and I want to find the center coordinate of the parallelogram. It se......Wed, 19 Aug 2026 13:43:25 -0400https://liberty.io/integral-notation-confusion-x-vs-t.htmlhttps://liberty.io/integral-notation-confusion-x-vs-t.htmlOne of our worksheets contains the following integrals (areas) to find: $$\int_0^x 1 \, dx$$ $$\int_0^x x \, dx$$ I am skeptical. Although this is our first day learning about integrals in class......Wed, 19 Aug 2026 13:35:16 -0400https://liberty.io/how-do-we-find-out-angle-from-x-y-coordinates.htmlhttps://liberty.io/how-do-we-find-out-angle-from-x-y-coordinates.htmlI found the following sentence. To find the angle you use the arctangent function like this, angle $=\tan^{-1}\left(\frac{y}{x}\right)$. But I am curious, is this the only way to know the angle......Wed, 19 Aug 2026 13:31:40 -0400https://liberty.io/how-do-percentiles-work.htmlhttps://liberty.io/how-do-percentiles-work.htmlIf I have 5 students (A-E) that score 80%, 70%, 70%, 60% and 50% on a test what percentiles do they fall in? A - 20th percentile (80%) B - 40th percentile (70%) C - 40th percentile (70%) D - 80th ...Wed, 19 Aug 2026 13:27:46 -0400https://liberty.io/problem-similar-to-folland-chapter-2-problem-51.htmlhttps://liberty.io/problem-similar-to-folland-chapter-2-problem-51.htmlLet $(X\mathcal M,\mu)$ and $(Y,\mathcal N,v)$ be $\sigma$-finite measure spaces. If $f:X\to\Bbb R$ is $\mathcal M$-measurable, $g:Y\to\Bbb R$ is $\mathcal N$-measurable, and $h(x,y)=f(x)g(......Wed, 19 Aug 2026 13:21:39 -0400https://liberty.io/maclaurin-polynomial-of-tan-x.htmlhttps://liberty.io/maclaurin-polynomial-of-tan-x.htmlThe method used to find the Maclaurin polynomial of sin(x), cos(x), and $e^x$ requires finding several derivatives of the function. However, you can only take a couple derivatives of tan(x) before it ...Wed, 19 Aug 2026 13:17:21 -0400https://liberty.io/how-to-find-the-point-on-the-sphere-that-is-closest-to-a-plane.htmlhttps://liberty.io/how-to-find-the-point-on-the-sphere-that-is-closest-to-a-plane.htmlConsider the plane $x+2y+2z=4$, how to find the point on the sphere $x^2+y^2+z^2=1$ that is closest to the plane? I could find the distance from the plane to the origin using the formula $D=\frac{|1\...Wed, 19 Aug 2026 13:05:23 -0400https://liberty.io/what-is-the-reflection-across-a-parabola.htmlhttps://liberty.io/what-is-the-reflection-across-a-parabola.htmlReflection across a line is well familiar, reflection across a circle is the inversion, the point at a distance $d$ from the center is reflected into a point on the same ray through the center, but......Wed, 19 Aug 2026 13:02:21 -0400https://liberty.io/user-david-yang-mathematics-stack-exchange.htmlhttps://liberty.io/user-david-yang-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Wed, 19 Aug 2026 12:59:09 -0400https://liberty.io/what-is-so-linear-about-linear-combination.htmlhttps://liberty.io/what-is-so-linear-about-linear-combination.htmlWhat is linear about a linear combination of things?. In linear algebra, the "things" we are dealing with are usually vectors and the linear combination gives the span of the vectors. Or it could b......Wed, 19 Aug 2026 12:55:27 -0400https://liberty.io/physical-significance-of-knot-vector-in-b-spline.htmlhttps://liberty.io/physical-significance-of-knot-vector-in-b-spline.htmlA B-spline blending curve formulation is: $P(u)=\sum_{k=0}^np_k B_{k,d}(u)$ Given $n+1$ control points, B-spline blending functions are polynomials of degree $d-1$, $(1<d<=n+1)$. ......Wed, 19 Aug 2026 12:51:45 -0400https://liberty.io/product-of-cosines-prod-r-1-7-cos-left-frac-r-pi-15-right.htmlhttps://liberty.io/product-of-cosines-prod-r-1-7-cos-left-frac-r-pi-15-right.htmlEvaluate $$ \prod_{r=1}^{7} \cos \left({\dfrac{r\pi}{15}}\right) $$ I tried trigonometric identities of product of cosines, i.e, $$\cos\text{A}\cdot\cos\text{B} = \dfrac{1}{2}[ \cos(A+B)+\cos(A-B)......Wed, 19 Aug 2026 12:49:43 -0400https://liberty.io/calculate-expectation-of-a-geometric-random-variable.htmlhttps://liberty.io/calculate-expectation-of-a-geometric-random-variable.htmlWhen you download a file from a website, the file gets corrupted with probability 0.8. What is the expected number of downloads to get an uncorrupted file? I have no idea how to do this. I only kn......Wed, 19 Aug 2026 12:48:26 -0400https://liberty.io/what-is-an-intuitive-explanation-of-the-combinations-formula-duplicate.htmlhttps://liberty.io/what-is-an-intuitive-explanation-of-the-combinations-formula-duplicate.htmlI perfectly understand the permutations formula i.e. if you have $n$ things how many ways can you rearrange it if taken $k$ at a time (or if you have $k$ slots)? So you draw the following tree. And......Wed, 19 Aug 2026 12:43:31 -0400https://liberty.io/final-value-theorem-for-closed-system.htmlhttps://liberty.io/final-value-theorem-for-closed-system.htmlWe have a system with output given by $\frac{Y(s)}{R(s)} = \frac{F(s)G(s)}{1+F(s)G(s)}$ where $F(s)G(s) = K\frac{s+1}{s^2+s+1}$. Let $K=4$ and $R(s) = 10/s$. Using the final value theorem, calcu......Wed, 19 Aug 2026 12:36:57 -0400https://liberty.io/prove-that-9-n-4-n-1-is-a-multiple-of-5-for-all-n-in-mathbb-n.htmlhttps://liberty.io/prove-that-9-n-4-n-1-is-a-multiple-of-5-for-all-n-in-mathbb-n.htmlProve that $9^n+4^{n+1}$ is a multiple of $5$ for all $n \in \mathbb{N}$ Proof. So i'm going to prove this by induction. The first case when $n=1$ is trivial since: $$9+16=25,$$ implying $ 5 \mid......Wed, 19 Aug 2026 12:27:15 -0400https://liberty.io/limit-of-logarithm.htmlhttps://liberty.io/limit-of-logarithm.htmlI came across this relation a lot,when evaluating lim of function like natural logarithm I could find that evaluating limit for logarithm equal to evaluating logarithm for limit. Why this is true? ......Wed, 19 Aug 2026 12:17:33 -0400https://liberty.io/area-of-a-trapezoid-without-the-height.htmlhttps://liberty.io/area-of-a-trapezoid-without-the-height.htmlHow would I find the area of a non-iscoceles trapezoid and without the height? The trapezoid's bases are $30$ and $40$, and the legs $14$ and $16$. Thanks...Wed, 19 Aug 2026 12:13:08 -0400https://liberty.io/why-do-we-use-decimal-system-duplicate.htmlhttps://liberty.io/why-do-we-use-decimal-system-duplicate.htmlWhy do we use decimal system? Some people give the answer that we have 10 fingers and it is therefore convenient for us to use decimal system, but I am not satisfy with this as I think this is too......Wed, 19 Aug 2026 12:06:51 -0400https://liberty.io/what-is-a-binary-expansion-of-a-real-number.htmlhttps://liberty.io/what-is-a-binary-expansion-of-a-real-number.htmlRelated to this question I asked. I want to know what is exactly meant by a binary expansion of a number, real or natural. Can someone show me via example, how would you binary expand a real number ...Wed, 19 Aug 2026 12:04:53 -0400https://liberty.io/using-the-comparison-test-prove-that-the-infinite-series-of-n-2-1-n-3-.htmlhttps://liberty.io/using-the-comparison-test-prove-that-the-infinite-series-of-n-2-1-n-3-.htmlThe conditions of the comparison sum state that if $0\le a_n\le b_n$ and if $b_n$ converges, then $a_n$ also converges and if $a_n$ diverges, then $b_n$ also diverges. I'm not sure how to go abou......Wed, 19 Aug 2026 12:02:39 -0400https://liberty.io/a-spectral-theorem-for-a-skew-hermitian-complex-matrix.htmlhttps://liberty.io/a-spectral-theorem-for-a-skew-hermitian-complex-matrix.htmlSorry if this is a duplicate, but I tried searching on this but only found results dealing with skew symmetric real matrices. So $A\in\mathbb{C}^{n\times n}$ is called skew-Hermitian if $A^*=-A$. I'm ...Wed, 19 Aug 2026 11:58:07 -0400https://liberty.io/how-to-compute-two-039-s-complement-of-a-negative-number.htmlhttps://liberty.io/how-to-compute-two-039-s-complement-of-a-negative-number.htmlI made the search on here and on google and couldn't find anything that answered the topic title. From my bit of understanding, two's complement can be used to make a decimal number, negative. Wh......Wed, 19 Aug 2026 11:39:09 -0400https://liberty.io/pseudo-arclength-continuation-scheme.htmlhttps://liberty.io/pseudo-arclength-continuation-scheme.htmlI have implemented a simple parameter continuation scheme to find the stationary solutions of a nonlinear problem at different parameter values. However, my scheme cannot handle bifurcations - it f......Wed, 19 Aug 2026 11:36:51 -0400https://liberty.io/similarity-between-trapezoidal-sum-and-riemann-sums.htmlhttps://liberty.io/similarity-between-trapezoidal-sum-and-riemann-sums.htmlIs it accurate to say that a Trapezoidal Riemann Sum includes one more term than the corresponding Left, Right, or Midpoint Riemann Sum? Using the following notation for the latter three: Rieman......Wed, 19 Aug 2026 11:36:01 -0400https://liberty.io/calculate-maximum-velocity-given-accel-decel-initial-v-final-position.htmlhttps://liberty.io/calculate-maximum-velocity-given-accel-decel-initial-v-final-position.htmlI'm trying to calculate the maximum velocity of an object given: acceleration deceleration starting position (0) ending position initial velocity I've been trying to use formulas from: https://...Wed, 19 Aug 2026 11:34:34 -0400https://liberty.io/show-the-series-an-1-an-converges-given-that-series-an-converges.htmlhttps://liberty.io/show-the-series-an-1-an-converges-given-that-series-an-converges.htmlGiven that $0\le a_n\lt 1$ the series $\sum_{n=0}^{\infty} (a_n)$ converges. Show that the series $\sum_{n=0}^{\infty} \frac{a_n}{1-a_n}$ converges. This question is supposed to be solved from first ...Wed, 19 Aug 2026 11:12:26 -0400https://liberty.io/are-category-theory-and-set-theory-on-the-equal-foundational-footing.htmlhttps://liberty.io/are-category-theory-and-set-theory-on-the-equal-foundational-footing.htmlSet-theory is widely taken to be foundational to the rest of mathematics. So is category-theory. My question is: Are they two alternative, rival candidates for the role of a foundational theory of ...Wed, 19 Aug 2026 10:56:09 -0400https://liberty.io/what-do-finite-infinite-countable-not-countable-countably-infinite-mea.htmlhttps://liberty.io/what-do-finite-infinite-countable-not-countable-countably-infinite-mea.htmlPossible Duplicate: Different kinds of infinities? Those terms keep confusing me. Could somebody give a clear and unambiguous definition? How do we prove a set is countable, finite, ...? I kno......Wed, 19 Aug 2026 10:54:37 -0400https://liberty.io/is-there-a-symbol-for-difference-like-sum-for-summation.htmlhttps://liberty.io/is-there-a-symbol-for-difference-like-sum-for-summation.htmlIm curious, if there is a symbol for summation: $$\sum$$ then is there a symbol for difference? If not, if not how could I represent the difference of a sequence of numbers $(1-2-3-4-...)$?...Wed, 19 Aug 2026 10:44:27 -0400https://liberty.io/algebra-trick-for-dividing-polynomials.htmlhttps://liberty.io/algebra-trick-for-dividing-polynomials.htmlI saw that "once have a root, we can divide the polynomials" and somehow $\frac{3n^3 - 39n^2 + 360n + 20}{3(n-n_1)} = n^2 + (n_1 - 13)n + (n_1^2 - 13n_1 + 120)$ I'm noticing a pattern that there's......Wed, 19 Aug 2026 10:37:53 -0400https://liberty.io/how-does-one-calculate-all-orbits-of-an-ode-system.htmlhttps://liberty.io/how-does-one-calculate-all-orbits-of-an-ode-system.htmlI have the following ODE system and have been tasked to find all of it's orbits. The system is $$x' = y(x^2 + 1)$$ $$y' = x(x^2 + 1)$$ From here is where I'm lost. I found one definition of an orbit ...Wed, 19 Aug 2026 10:35:15 -0400https://liberty.io/how-to-calculate-curvature-of-earth-per-surface-kilometer.htmlhttps://liberty.io/how-to-calculate-curvature-of-earth-per-surface-kilometer.htmlI was watching a video regarding Flat Earthers giving curvatures of Earth that sounded way too big, so I decided to calculate how much the surface of Earth would drop with respect to a line perpend......Wed, 19 Aug 2026 10:35:12 -0400