https://liberty.ioCurated star highlights with fresh energy.en-usFri, 04 Sep 2026 13:30:32 -0400https://liberty.io/is-a-constant-function-periodic-closed.htmlhttps://liberty.io/is-a-constant-function-periodic-closed.htmlCan we regard a constant function "$f(x)=\text{constant}$" to be a periodic function? If yes, what is its period?...Fri, 04 Sep 2026 17:00:39 -0400https://liberty.io/approximating-the-cosine-by-taylor-polynomial.htmlhttps://liberty.io/approximating-the-cosine-by-taylor-polynomial.htmlLet $f:=\cos(x)$ I'm asked to find for which values of $x$ we can be sure the 4th degree Taylor polynomial will give an error lesser than $\frac{1}{1000}$. Now, $\cos(x)=1-\frac{x^2}{2}+\frac{x^4}......Fri, 04 Sep 2026 16:39:37 -0400https://liberty.io/complete-probability-spaces.htmlhttps://liberty.io/complete-probability-spaces.htmlI am taking a course on Stochastic Analysis, and the introduction of the course covers Measure Theory\ Probability Theory. I am very new to this area of abstract maths and am trying to get to grips......Fri, 04 Sep 2026 16:36:07 -0400https://liberty.io/finite-sum-of-cosecant-squared.htmlhttps://liberty.io/finite-sum-of-cosecant-squared.htmlI need help with that sum $$\sum_{k=1}^N \csc^2\left(\frac{k\pi}{2N}\right)$$ - it is given in many tables as a standard result, but I can't figure it out. I know how to do $\sin^2$ and $\cos^2$, ......Fri, 04 Sep 2026 16:31:15 -0400https://liberty.io/positive-and-negative-integer-that-is-congruent-to-0-mod-5-and-incongr.htmlhttps://liberty.io/positive-and-negative-integer-that-is-congruent-to-0-mod-5-and-incongr.htmlI'm kind of confused by this because I thought 0 mod 5 = 0, and 0 mod 6 = 0 as well. So what's an integer that is congruent to one but not the other?...Fri, 04 Sep 2026 16:02:43 -0400https://liberty.io/proofs-of-the-properties-of-jacobi-symbol.htmlhttps://liberty.io/proofs-of-the-properties-of-jacobi-symbol.htmlThe definition and properties of Jacobi symbol are stated in this article. I don't have a textbook handy containing the proofs of the following properties of Jacobi symbol. It seems to me that not ......Fri, 04 Sep 2026 15:50:29 -0400https://liberty.io/integral-int-0-1-dx-frac-ln-x-ln-2-1-x-ln-1-x-x.htmlhttps://liberty.io/integral-int-0-1-dx-frac-ln-x-ln-2-1-x-ln-1-x-x.htmlI am trying to calculate $$ I:=\int_0^1 dx \frac{\ln x \ln^2(1-x)\ln(1+x)}{x}$$ Note, the closed form is beautiful (yes beautiful) and is given by $$ I=−\frac{3}{8}\zeta_2\zeta_3 -\frac{2}{3}\zeta_......Fri, 04 Sep 2026 15:47:45 -0400https://liberty.io/differentiate-y-e-ax-3.htmlhttps://liberty.io/differentiate-y-e-ax-3.html$y=e^{ax^3}$ What in a problem indicates I should use the laws of logarithm? Here is how I differentiated the function: $\ln(y)=\ln{e^{ax^3}}$ $\ln(y)=ax^3\ln(e)$ $\frac{dy}{dx}\cdot\frac{1}{......Fri, 04 Sep 2026 15:40:47 -0400https://liberty.io/last-digit-on-3-100-duplicate.htmlhttps://liberty.io/last-digit-on-3-100-duplicate.htmlHow to find the last digit on $3^{100}$? Is there any proper method to solve such questions without calculator of course?...Fri, 04 Sep 2026 15:33:45 -0400https://liberty.io/does-zero-curl-imply-a-conservative-field.htmlhttps://liberty.io/does-zero-curl-imply-a-conservative-field.htmlA field that is conservative must have a curl of zero everywhere. However, I was wondering whether the opposite holds for functions continuous everywhere: if the curl is zero, is the field conserva......Fri, 04 Sep 2026 15:33:27 -0400https://liberty.io/evaluate-6-log-8-4-without-using-a-calculator.htmlhttps://liberty.io/evaluate-6-log-8-4-without-using-a-calculator.htmlI am to evaluate $6\log_8(4)$ without a calculator. The answer is provided as 4 but I cannot see how to arrive at 4. Ignoring the 6 at the beginning of the expression, $log_8(4)$ can be written as ......Fri, 04 Sep 2026 15:32:05 -0400https://liberty.io/solve-simple-xor-equation.htmlhttps://liberty.io/solve-simple-xor-equation.htmlI have to solve few simple equation like this: $$ 117 = 86 \oplus x $$ I don't know how to move x to one side and calculate its value. How to do it?...Fri, 04 Sep 2026 15:30:59 -0400https://liberty.io/what-is-the-039-implicit-function-theorem-039.htmlhttps://liberty.io/what-is-the-039-implicit-function-theorem-039.htmlPlease give me an intuitive explanation of 'implicit function theorem'. I read some bits and pieces of information from some textbook, but they look too confusing, especially I do not understand wh......Fri, 04 Sep 2026 15:18:22 -0400https://liberty.io/what-does-the-derivative-of-area-with-respect-to-length-signify.htmlhttps://liberty.io/what-does-the-derivative-of-area-with-respect-to-length-signify.htmlSuppose that we have a square sheet of edge length $L$. Its area $A=L^2$. Differentiating $A$ w.r.t. L, we get $$\dfrac{dA}{dL}=2L$$ I do understand what it means to differentiate, graphicall......Fri, 04 Sep 2026 15:17:45 -0400https://liberty.io/why-does-grapher-a-mac-application-plot-the-3d-inequality-x-1-using-sp.htmlhttps://liberty.io/why-does-grapher-a-mac-application-plot-the-3d-inequality-x-1-using-sp.htmlGrapher (sometimes called Grapher.app) is a mathematics graphing application that comes bundled with Mac computers (see https://en.wikipedia.org/wiki/Grapher). I was recently playing around with G......Fri, 04 Sep 2026 15:16:21 -0400https://liberty.io/what-is-a-combined-function-and-what-factors-affect-its-domain.htmlhttps://liberty.io/what-is-a-combined-function-and-what-factors-affect-its-domain.htmlCan someone please explain to me what a combined function is and how it is different from a composite function. From my understanding a combined function would be $(f+g)(x) = f(x)+g(x)$ and a com......Fri, 04 Sep 2026 15:03:41 -0400https://liberty.io/example-of-a-subgroup-that-is-not-normal.htmlhttps://liberty.io/example-of-a-subgroup-that-is-not-normal.htmlCan you kindly provide an example of a subgroup that is not normal? I have been told many times that, for coset multiplication to be defined, the subgroup must be normal. I have seen the proof and ...Fri, 04 Sep 2026 15:02:29 -0400https://liberty.io/how-to-find-the-dual-cone.htmlhttps://liberty.io/how-to-find-the-dual-cone.htmlThe similar question is here , but I do not see the desired answer. Assume a cone $K=\{(x,y) |\ x+y=0\}$, find the dual cone of $K$. The definition of dual cone is here: $K^*=\{y|x^{T}y\geq0, \fora......Fri, 04 Sep 2026 14:56:32 -0400https://liberty.io/square-root-of-a-binary-number-and-beyond.htmlhttps://liberty.io/square-root-of-a-binary-number-and-beyond.htmlhow to find (efficiently) binary expansion of a square root of a number which is given as a binary number. Is there any general method to find the n-nary expansion of m-th root of a number which is ...Fri, 04 Sep 2026 14:56:32 -0400https://liberty.io/what-is-the-best-book-to-learn-probability.htmlhttps://liberty.io/what-is-the-best-book-to-learn-probability.htmlQuestion is quite straight... I'm not very good in this subject but need to understand at a good level....Fri, 04 Sep 2026 14:55:33 -0400https://liberty.io/multivariable-chain-rule-formula-doesn-039-t-make-sense-to-me.htmlhttps://liberty.io/multivariable-chain-rule-formula-doesn-039-t-make-sense-to-me.htmlConsider $z=f(x(t),y(t))$, then its chain rule formula is: $\frac{dz}{dt} = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt}$ What I cannot make sense of, a......Fri, 04 Sep 2026 14:46:48 -0400https://liberty.io/do-matrices-ab-and-ba-have-the-same-minimal-and-characteristic-polynom.htmlhttps://liberty.io/do-matrices-ab-and-ba-have-the-same-minimal-and-characteristic-polynom.htmlLet $ A, B $ be two square matrices of order $n$. Do $ AB $ and $ BA $ have same minimal and characteristic polynomials? I have a proof only if $ A$ or $ B $ is invertible. Is it true for all cases?...Fri, 04 Sep 2026 14:29:30 -0400https://liberty.io/definition-of-measure-preserving-why-inverse-image.htmlhttps://liberty.io/definition-of-measure-preserving-why-inverse-image.htmlIn the definition of measure-preserving dynamical system, the crucial equation is $$ \mu \left(T^{-1} \left(A\right)\right) = \mu\left(A\right) . $$ Why is it not the seemingly more natural $......Fri, 04 Sep 2026 14:13:40 -0400https://liberty.io/realification-and-complexification-of-vector-spaces.htmlhttps://liberty.io/realification-and-complexification-of-vector-spaces.htmlI am interested in a good comprehensive resource on realification and complexification of vector spaces over the reals or complexes (and the interplay of these structures on the 'same' space in gen......Fri, 04 Sep 2026 14:06:30 -0400https://liberty.io/finding-derivative-of-frac-1-sqrt-x-2-using-only-the-definition-of-the.htmlhttps://liberty.io/finding-derivative-of-frac-1-sqrt-x-2-using-only-the-definition-of-the.htmlI need to find $\frac{d}{dx}(\frac{1}{\sqrt{x+2}})$ only using the basic definition of the derivative $f'(x)=\lim_{h \to 0}\frac{f(x+h)-f(x)}{h}$, but I'm not having any luck with the algebra. Any ......Fri, 04 Sep 2026 14:00:56 -0400https://liberty.io/probability-of-getting-3-cards-in-the-same-suit-from-a-deck.htmlhttps://liberty.io/probability-of-getting-3-cards-in-the-same-suit-from-a-deck.htmlWhen three cards are randomly selected at a time from a standard deck of 52 playing cards, what is the probability that all of these three cards are in the same suit (heart, diamond, spade, o......Fri, 04 Sep 2026 13:56:34 -0400https://liberty.io/how-do-you-find-an-angle-between-two-points-on-the-edge-of-a-circle.htmlhttps://liberty.io/how-do-you-find-an-angle-between-two-points-on-the-edge-of-a-circle.htmlI have two points on the circumference of circle, and I also know the center of the circle. I want to calculate the angle between those two points which are on the circumference of circle. Is this ...Fri, 04 Sep 2026 13:55:51 -0400https://liberty.io/definition-of-convex-function.htmlhttps://liberty.io/definition-of-convex-function.htmlSpivak's book states first this definition of Convex Function: Definition 1: A function $f$ is convex on an interval, if for all $a$ and $b$ in the interval, the line segment joining $(a, f(a))......Fri, 04 Sep 2026 13:52:40 -0400https://liberty.io/user-andrew-stacey-mathematics-stack-exchange.htmlhttps://liberty.io/user-andrew-stacey-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Fri, 04 Sep 2026 13:52:28 -0400https://liberty.io/finding-n-value-basic-statistics.htmlhttps://liberty.io/finding-n-value-basic-statistics.htmlI have some practice homework questions. I did the first one I will go over the steps please tell me If I am doing it right. a) As mentioned earlier, it is claimed that 70% of households in Ontar......Fri, 04 Sep 2026 13:49:06 -0400https://liberty.io/given-q-and-cos-q-pi-to-be-rational-find-all-possible-values-of-cos-q-.htmlhttps://liberty.io/given-q-and-cos-q-pi-to-be-rational-find-all-possible-values-of-cos-q-.htmlQuestion: Given $q$ and $\cos(q\pi)$ to be rational, find all possible values of $\cos(q\pi)$ After some trial and error I think the possible values are $\{0,\pm 1,\pm 1/2 \}$. I could show that $......Fri, 04 Sep 2026 13:49:04 -0400https://liberty.io/proving-that-a-point-is-a-midpoint.htmlhttps://liberty.io/proving-that-a-point-is-a-midpoint.htmlI'm trying to solve this using elementary geometry. Let $\triangle ABC$ be a triangle and consider the squares $CBFG$ and $ACDE$. Draw $\overline{EF}$ and let $M$ be its midpoint. Draw the perpendi......Fri, 04 Sep 2026 13:39:51 -0400https://liberty.io/conditions-for-cyclic-quotient-group.htmlhttps://liberty.io/conditions-for-cyclic-quotient-group.htmlLet $G$ be an arbitrary finite group and $H$ a normal subgroup. What are some good conditions on $H$ that make the quotient $G/H$ cyclic? I want to avoid any further restriction on $G$....Fri, 04 Sep 2026 13:39:08 -0400https://liberty.io/modulo-question.htmlhttps://liberty.io/modulo-question.htmlI thought you could only mod positive numbers but then I saw this and became confused...How does this even work ? how can you have negatives? $$\begin{align*} -8 &\equiv 7 \pmod{5}\\ 2 &\...Fri, 04 Sep 2026 13:37:50 -0400https://liberty.io/inscribed-rhombus.htmlhttps://liberty.io/inscribed-rhombus.htmlCan a rhombus that is NOT a square be inscribed in a circle? A quadrilaterals opposite angles must add up to 180 in order to be inscribed in a circle, but a rhombuses opposite angles are equal and ......Fri, 04 Sep 2026 13:37:43 -0400https://liberty.io/what-is-the-symbol-for-undefined.htmlhttps://liberty.io/what-is-the-symbol-for-undefined.htmlJust how there is a symbol for no real solutions (the empty set) and there is DNE for does not exist, what symbol or abbreviation would you use for undefined?...Fri, 04 Sep 2026 13:36:38 -0400https://liberty.io/puzzle-where-did-the-extra-dollar-come-from-duplicate.htmlhttps://liberty.io/puzzle-where-did-the-extra-dollar-come-from-duplicate.htmlPossible Duplicate: Riddle (simple arithmetic problem/illusion) I have this small story (sorry about my bad English) : One man (M) took 25 dollar from man (A), and another 25 dollar from man (......Fri, 04 Sep 2026 13:24:15 -0400https://liberty.io/unclear-about-what-the-diagonal-relation-means.htmlhttps://liberty.io/unclear-about-what-the-diagonal-relation-means.htmlI am reading through Skornjakov's Elements of Lattice Theory (1977), and am having trouble understanding what is meant by a diagonal relation. I am also unclear about how the diagonal differs from ......Fri, 04 Sep 2026 13:15:30 -0400https://liberty.io/solve-the-ode-y-039-039-y-039-y-2.htmlhttps://liberty.io/solve-the-ode-y-039-039-y-039-y-2.htmlSolve the ODE $$ y''+ y' = y^2$$ I tried to solve, but I don't know any algorithm except for numerical methods....Fri, 04 Sep 2026 13:07:39 -0400https://liberty.io/how-to-solve-x-5-2x-6-1.htmlhttps://liberty.io/how-to-solve-x-5-2x-6-1.html$|x-5|=|2x+6|-1$. The answer is $0$ or $-12$, but how would I solve it by algebraically solving it as opposed to sketching a graph? $|x-5|=|2x+6|-1\\ (|x-5|)^2=(|2x+6|-1)^2\\ ...\\ 9x^4+204x^3+1......Fri, 04 Sep 2026 13:03:27 -0400https://liberty.io/derivative-of-an-exponentially-weighted-moving-average.htmlhttps://liberty.io/derivative-of-an-exponentially-weighted-moving-average.htmlIt has been a while since my university math courses, so let me apologize right off the bat... I'm using GSL to perform non-linear regression analysis and am mostly happy with the outcome, however......Fri, 04 Sep 2026 13:02:34 -0400https://liberty.io/what-do-trivial-and-non-trivial-solution-of-homogeneous-equations-mean.htmlhttps://liberty.io/what-do-trivial-and-non-trivial-solution-of-homogeneous-equations-mean.htmlSuppose I have system of 3 equations $$a_1x+b_1y+c_1z=0$$ $$a_2x+b_2y+c_2z=0$$ $$a_3x+b_3y+c_3z=0$$ and cofficient matrix $A=\begin{equation} \begin{pmatrix} a_1 & b_1 & c_1 \\ a_2 & b_......Fri, 04 Sep 2026 13:02:04 -0400https://liberty.io/find-the-maximum-displacement-given-trajectory.htmlhttps://liberty.io/find-the-maximum-displacement-given-trajectory.htmlI'm trying to find the maximum displacement of a object given the trajectory. The trajectory is given by the equation: $$p(t)=-11.79t^2+25.9t+4.35$$ Looking around online, the closest article I've ......Fri, 04 Sep 2026 12:57:12 -0400https://liberty.io/why-negating-universal-quantifier-gives-existential-quantifier.htmlhttps://liberty.io/why-negating-universal-quantifier-gives-existential-quantifier.htmlNegating a universal quantifier gives the existential quantifier, and vice versa: $\neg \forall x = \exists x \neg \\ \neg \exists x = \forall x \neg $ Why is this, and is there a proof for it (i......Fri, 04 Sep 2026 12:51:02 -0400https://liberty.io/range-two-039-s-complement.htmlhttps://liberty.io/range-two-039-s-complement.htmlThe most used formula's to calculate the range of numbers in a two's complement system are + $2^{n-1}-1$ for the highest number and $-2^{n-1}$ for the lowest number. The problem is that this only ......Fri, 04 Sep 2026 12:45:17 -0400https://liberty.io/dual-of-a-positive-semidefinite-cone.htmlhttps://liberty.io/dual-of-a-positive-semidefinite-cone.htmlThe PSD cone is the set of all positive semidefinite matrices. The dual is the set of all matrices $A$ such that tr($A^T X$) $\geq 0$ for all positive semidefinite matrices $X$. How to prove that......Fri, 04 Sep 2026 12:44:36 -0400https://liberty.io/finding-velocity-from-a-table.htmlhttps://liberty.io/finding-velocity-from-a-table.htmlI have a homework question I am seeking an alternative solution to. Basically, the question is... "The table provided below shows the position of a particle S, at several times, t. as the particle......Fri, 04 Sep 2026 12:42:03 -0400https://liberty.io/find-a-particular-solution-for-the-differential-equation-by-the-method.htmlhttps://liberty.io/find-a-particular-solution-for-the-differential-equation-by-the-method.htmlFind a particular solution for the differential equation by the method of undetermined coefficients. $$2y'' - 16y' + 32y = -e^{4x}$$ Also, find the general solution of this equation. The step......Fri, 04 Sep 2026 12:41:25 -0400https://liberty.io/how-well-does-probability-work-in-the-real-world.htmlhttps://liberty.io/how-well-does-probability-work-in-the-real-world.htmlWhen I was first introduced to probability at the age of about 13, me and my classmates always used to ridicule the subject in a most immature manner. We used to take a deck of well shuffled cards ......Fri, 04 Sep 2026 12:38:41 -0400https://liberty.io/what-is-the-centroid-of-a-hollow-spherical-cap.htmlhttps://liberty.io/what-is-the-centroid-of-a-hollow-spherical-cap.htmlI have a unit hollow sphere which I cut along a diameter to generate two equivalent hollow hemispheres. I place one of these hemispheres on an (x,y) plane, letting it rest on the circular planar f......Fri, 04 Sep 2026 12:32:21 -0400https://liberty.io/symbol-for-the-shape-of-a-matrix.htmlhttps://liberty.io/symbol-for-the-shape-of-a-matrix.htmlI'm writing a technical document and need to describe some mathematics involving the shape of a matrix. I realize that I can write $X_{m\times n}$ and someone can simply see the dimensions. I'm ...Fri, 04 Sep 2026 12:31:07 -0400https://liberty.io/example-of-strictly-concave-functions.htmlhttps://liberty.io/example-of-strictly-concave-functions.htmlI'm learning concave/convex functions. Wikipedia article provides us with example functions of concave or convex. However, it does not provide strictly concave function. Do you have some examples of ...Fri, 04 Sep 2026 12:15:14 -0400https://liberty.io/what-does-the-d-represent-in-the-equation-of-a-plane-ax-by-cz-d-0.htmlhttps://liberty.io/what-does-the-d-represent-in-the-equation-of-a-plane-ax-by-cz-d-0.htmlLook at the following problem: Consider the plane $\alpha$ defined by $x-2y+z+3=0$ and $A(0;0;2)$. Write an equation of the plane that is parallel to $\alpha$ and goes through A. My book ...Fri, 04 Sep 2026 12:06:06 -0400https://liberty.io/questions-about-relations-on-directed-and-possibly-cyclical-graphs.htmlhttps://liberty.io/questions-about-relations-on-directed-and-possibly-cyclical-graphs.htmlLet $(P, E)$ be a directed graph with $P$ being the set of nodes and $E \subset P^2$ the set of directed edges. It is well-known that the transitive closure of $E$ is a preorder $\leq$, which is more ...Fri, 04 Sep 2026 11:58:21 -0400https://liberty.io/finding-a-b-c-d-in-a-quartic-expression.htmlhttps://liberty.io/finding-a-b-c-d-in-a-quartic-expression.htmlLet $p(x)=x^4+ax^3+bx^2+cx+d$ where a,b,c,d are constants. If $p(1)=10$, $p(2)=20$, $p(3)=30$, compute $\frac {p(12)+p(-8)}{10}$. I have tried so far. \begin{align} a+b+c+d=&9\\8a+4b+2c+d=&amp......Fri, 04 Sep 2026 11:51:15 -0400https://liberty.io/finding-x-intercept-of-a-cubic-equation.htmlhttps://liberty.io/finding-x-intercept-of-a-cubic-equation.htmlWhat is the $x$ intercept of $y=(x-2)(x^2+25) $? To find $x$ intercept:$ 0=(x-2)(x^2+25) $ I tried $ 0=(x-2)(x+5)(x+5)$ in which the $X$ intercepts are $2,-5$ and $-5$. Is this correct?...Fri, 04 Sep 2026 11:45:29 -0400https://liberty.io/proving-the-trig-identity-sin-20-circ-cos-65-circ-cos-20-circ-sin-65-c.htmlhttps://liberty.io/proving-the-trig-identity-sin-20-circ-cos-65-circ-cos-20-circ-sin-65-c.htmlUsing the trigonometric identities I have to prove that: $$ \sin(20^\circ)\cos(65^\circ)-\cos(20^\circ)\sin(65^\circ)=-\frac{1}{\sqrt{2}} $$ I solved $$ \sin(20^\circ)\cos(65^\circ) = \frac{\s......Fri, 04 Sep 2026 11:36:06 -0400https://liberty.io/prime-divisor-of-3n-2-proof.htmlhttps://liberty.io/prime-divisor-of-3n-2-proof.htmlI have to prove that any number of the form $3n+2$ has a prime factor of the form $3m+2$. Ive started the proof I tried saying by the division algorithm the prime factor is either the form 3m,3m+1......Fri, 04 Sep 2026 11:32:36 -0400https://liberty.io/what-is-the-algebraic-closure-of-mathbb-q.htmlhttps://liberty.io/what-is-the-algebraic-closure-of-mathbb-q.htmlGiven the rational numbers $\mathbb{Q}$, I am wondering what is the algebraic closure of $\mathbb{Q}$, i.e., I need to find a field extension $\mathbb{K}/ \mathbb{Q}, $ such that for an arbitrary ...Fri, 04 Sep 2026 11:15:22 -0400https://liberty.io/why-homology-is-homotopy-invariant.htmlhttps://liberty.io/why-homology-is-homotopy-invariant.htmlI've red Hatcher's proof of homotopy invariance for singular homology. I understood how the proof works and the reason why we construct the prism operator (i.e. its a chain contraction). However, t......Fri, 04 Sep 2026 11:04:23 -0400https://liberty.io/double-integral-change-of-variables-to-polar-coordinates.htmlhttps://liberty.io/double-integral-change-of-variables-to-polar-coordinates.htmlQuick question on Polar Coordinates. When evaluating the double integral and changing variables, I'm not sure if the limits are correct. The question is as follows: Evaluate $$\int\!\!\!\int_D......Fri, 04 Sep 2026 10:32:13 -0400https://liberty.io/mean-of-gamma-distribution.htmlhttps://liberty.io/mean-of-gamma-distribution.htmlSo I was trying to prove the mean result of gamma distribution which is $\frac{\alpha}{\lambda}$. My attempt, $E(X)=\int_{0}^{\infty }x f(x)dx$ $=\int_{0}^{\infty } \frac{\lambda^{\alpha}}{\Gam......Fri, 04 Sep 2026 10:26:02 -0400https://liberty.io/visual-intuition-for-the-definition-of-asymptotically-equivalent.htmlhttps://liberty.io/visual-intuition-for-the-definition-of-asymptotically-equivalent.htmlI'm trying to intuitively grasp the following definition: The real-valued functions $f$ and $g$ are asymptotically equivalent as $x \to \infty$ if $$\lim_{x \to \infty} \dfrac{f(x)}{g(x)}=1.$$ We ......Fri, 04 Sep 2026 10:19:51 -0400https://liberty.io/what-is-a-common-symbol-for-angles-and-what-are-semantic-differences.htmlhttps://liberty.io/what-is-a-common-symbol-for-angles-and-what-are-semantic-differences.htmlI've just seen that LaTeX offers at least three symbols for denoting angles: $\text{\sphericalangle}:\sphericalangle ABC \\\text{\measuredangle}:\measuredangle ABC\\\text{\angle}: \angle ABC$ I t......Fri, 04 Sep 2026 10:18:24 -0400https://liberty.io/how-to-simplify-the-boolean-expression-a-039-bc-ab-039-c-039-a-039-b-0.htmlhttps://liberty.io/how-to-simplify-the-boolean-expression-a-039-bc-ab-039-c-039-a-039-b-0.htmlHow to simplify this Boolean expression? $$A'BC + AB'C' + A'B'C' + AB'C + AB$$ I have solved it using Kmap and found the answer to be $$A + BC + B'C'$$ I tried simplifying it using the rules but o......Fri, 04 Sep 2026 10:16:33 -0400https://liberty.io/the-need-for-the-gram-schmidt-process.htmlhttps://liberty.io/the-need-for-the-gram-schmidt-process.htmlAs far as I understood Gram–Schmidt orthogonalization starts with a set of linearly independent vectors and produces a set of mutually orthonormal vectors that spans the same space that starting ve......Fri, 04 Sep 2026 10:14:51 -0400https://liberty.io/what-is-the-purpose-of-jordan-canonical-form.htmlhttps://liberty.io/what-is-the-purpose-of-jordan-canonical-form.htmlI don't claim at all to be an expert on this topic. In many (advanced) linear algebra textbooks for undergraduates, I usually find something about the "Jordan Canonical Form" of a matrix. What is......Fri, 04 Sep 2026 10:03:05 -0400https://liberty.io/could-you-please-explain-why-the-remainder-when-the-square-of-49-is-di.htmlhttps://liberty.io/could-you-please-explain-why-the-remainder-when-the-square-of-49-is-di.htmlHere is the original question, What is the remainder when the square of 49 is divided by the square root of 49? According to UKMT, Because 49 = 7×7, it follows both that $\sqrt 49$ = 7 and that......Fri, 04 Sep 2026 10:00:10 -0400https://liberty.io/if-2-n-1-is-prime-why-must-n-be-a-power-of-2.htmlhttps://liberty.io/if-2-n-1-is-prime-why-must-n-be-a-power-of-2.htmlA little bird told me that if $2^n+1$ is prime, then $n$ is a power of $2$. I tend not to trust talking birds, so I'm trying to verify that statement independently. Suppose $n$ is not a power of $......Fri, 04 Sep 2026 10:00:05 -0400https://liberty.io/how-many-4-letter-words-can-we-make-from-bananas.htmlhttps://liberty.io/how-many-4-letter-words-can-we-make-from-bananas.htmlAs the title suggests, I want to know the value of possible permutations without repetition given the word BANANAS BA1N1A2N2A3S so for example to word A1N1A2S is the A3N1A2S and we want to exclu......Fri, 04 Sep 2026 09:57:37 -0400https://liberty.io/row-reduced-2-times3-echelon-matrices.htmlhttps://liberty.io/row-reduced-2-times3-echelon-matrices.htmlSuppose $R$ and $R'$ are two $2\times3$ row reduced echelon matrices if $RX=0$ and $R'X$ have axactly same solutions then prove that $R=R'$. My try: Let $x,y,z$ be the same solution of $R$ and......Fri, 04 Sep 2026 09:48:14 -0400https://liberty.io/pure-sub-module-in-m.htmlhttps://liberty.io/pure-sub-module-in-m.htmlWhat is the difference between a pure module $P$ and a pure module $P$ in a module $M$ ? I have a slight intuition though, what is the relationship between purity and solvability of equations....Fri, 04 Sep 2026 09:47:58 -0400https://liberty.io/help-me-understand-a-3d-graph.htmlhttps://liberty.io/help-me-understand-a-3d-graph.htmlI've just seen this graph and while it's isn't the first 3d graph I've seen, as a math "noob" I never thought how these graphs are plotted. I can draw 2d graphs on paper by marking the input and ou......Fri, 04 Sep 2026 09:43:25 -0400https://liberty.io/how-to-calculate-the-amplitude-and-phase-of-such-a-complex-expression.htmlhttps://liberty.io/how-to-calculate-the-amplitude-and-phase-of-such-a-complex-expression.htmlI got an expression like this: $a*\sin(x)+b*\cos(x)+i*c*\sin(x)+i*d*\cos(x)$. I want to calculate the amplitude and the phase of it. Can someone give me some hints? Thx very much!...Fri, 04 Sep 2026 09:42:49 -0400https://liberty.io/the-expression-sqrt-13-3-sqrt-frac-23-3-sqrt-13-3-sqrt-frac-23-3-is-wh.htmlhttps://liberty.io/the-expression-sqrt-13-3-sqrt-frac-23-3-sqrt-13-3-sqrt-frac-23-3-is-wh.html(a) A natural number, (b) A rational number but not a natural number, (c) An irrational number not exceeding 6, (d) An irrational number exceeding 6. Please help with this, i can't manage to simpl......Fri, 04 Sep 2026 09:42:20 -0400https://liberty.io/what-do-you-call-a-3d-annulus-cylindrical-shell-and-parts-of-it.htmlhttps://liberty.io/what-do-you-call-a-3d-annulus-cylindrical-shell-and-parts-of-it.htmlTake a cylinder and punch a centered circular hole through it--like a washer. What is that shape called? In 2D it's an annulus, but what's the name of its 3D equivalent? Also, what is this shape c......Fri, 04 Sep 2026 09:36:59 -0400https://liberty.io/how-to-calculate-the-distance-between-two-lines-drawn-at-separate-angl.htmlhttps://liberty.io/how-to-calculate-the-distance-between-two-lines-drawn-at-separate-angl.htmlI don't know much about math, so forgive me if this is an obvious / unnecessary question. Please feel free to point me towards any resource that might enlighten me. However I need to know this for ......Fri, 04 Sep 2026 09:22:26 -0400https://liberty.io/how-to-find-specific-percentage-in-a-normal-distribution.htmlhttps://liberty.io/how-to-find-specific-percentage-in-a-normal-distribution.htmlHere is the exercise question I'm trying to answer: A certain fast food outlet produces hamburgers with weights that are normally distributed with mean $\mu = 990g$ and standard deviation $\s......Fri, 04 Sep 2026 09:08:05 -0400https://liberty.io/traditional-marriage-gs-female-pessimality.htmlhttps://liberty.io/traditional-marriage-gs-female-pessimality.htmlIn the context of stable matchings I am currently trying to understand the Gale-Shapely Algorithm (traditional marriage algorithm). I have proved it s correctness and I have shown that its male o......Fri, 04 Sep 2026 09:06:25 -0400https://liberty.io/why-does-imread-generate-3-images-closed.htmlhttps://liberty.io/why-does-imread-generate-3-images-closed.htmlThe MATLAB help for imread says: A = imread(filename, fmt) The return value A is an array containing the image data. If the file contains a grayscale image, A is an M-by-N array. If the f......Fri, 04 Sep 2026 08:39:34 -0400https://liberty.io/how-do-i-find-percentiles-of-data-sets-even-vs-odd.htmlhttps://liberty.io/how-do-i-find-percentiles-of-data-sets-even-vs-odd.htmlGiven the following data set with an even number of values: $100, 100, 105, 113, 129, 132, 146, 152, 176, 200$ The value representing the 30th percentile, using the formula n(p/100) where n = sam......Fri, 04 Sep 2026 08:37:48 -0400https://liberty.io/finding-the-height-given-the-angle-of-elevation-and-depression.htmlhttps://liberty.io/finding-the-height-given-the-angle-of-elevation-and-depression.htmlPlease, I need help for this problem. I'm a little confused about it :( From a point A 10ft. above the water the angle of elevation of the top of a lighthouse is 46 degrees and the angle of depr......Fri, 04 Sep 2026 08:32:04 -0400https://liberty.io/find-n-of-geometric-sequence-closed.htmlhttps://liberty.io/find-n-of-geometric-sequence-closed.htmlI need to find the $n$ of a geometric sequence, this formula is to find the $n$th term, $n$th = $ar ^{n-1}$; so I need $n$ to be on the left instead of $n$th, I have all the other variables except......Fri, 04 Sep 2026 08:26:10 -0400https://liberty.io/why-is-ln-cos-x-equal-to-ln-sec-x.htmlhttps://liberty.io/why-is-ln-cos-x-equal-to-ln-sec-x.htmlWhy does the value $-\ln(|\cos(x)|)$ become $\ln(|\sec(x)|)?$ I was doing an integral and I got my final answer as that, but I don't understand how you can just send the negative sign inside and m......Fri, 04 Sep 2026 08:23:00 -0400https://liberty.io/find-the-coordinates-of-a-point-on-a-circle.htmlhttps://liberty.io/find-the-coordinates-of-a-point-on-a-circle.htmlI have a circle like so $r$, with angle $\theta$ to the $y$-axis"> Given a rotation θ and a radius r, how do I find the coordinate (x,y)? Keep in mind, this rotation could be anywhere between 0 ......Fri, 04 Sep 2026 08:16:14 -0400https://liberty.io/equation-of-a-line-as-a-determinant-equation.htmlhttps://liberty.io/equation-of-a-line-as-a-determinant-equation.htmlI've been looking for an answer in the forums and Google but couldn't find one. Given two different points in the plane, $(x_1,\,y_1)$ and $(x_2,\,y_2)$, we know that the equation of that line can......Fri, 04 Sep 2026 08:13:11 -0400https://liberty.io/how-to-express-transformations-of-exponential-functions-stretches-and-.htmlhttps://liberty.io/how-to-express-transformations-of-exponential-functions-stretches-and-.htmlWhen referring to the transformations of exponential functions, how do you refer to vertical "stretches" and "compressions" in terms of "a dilation of $z$ from the y-axis o......Fri, 04 Sep 2026 08:10:17 -0400https://liberty.io/how-to-calculate-the-moment-of-inertia-of-a-uniform-disk-about-an-axis.htmlhttps://liberty.io/how-to-calculate-the-moment-of-inertia-of-a-uniform-disk-about-an-axis.htmlConsider a uniform disk of mass m and radius R, and let's calculate the moment of inertia about an axis passing through a diameter of the disk. One easy way is to use the perpendicular axis theorem: ...Fri, 04 Sep 2026 08:01:57 -0400https://liberty.io/isomorphism-vs-equality-of-graphs.htmlhttps://liberty.io/isomorphism-vs-equality-of-graphs.htmlI have just started studying graph theory and having trouble with understanding the difference b/w isomorphism and equality of two graphs.According what I have studied so far, I am able to conclude......Fri, 04 Sep 2026 07:52:11 -0400https://liberty.io/what-is-a-co-prime.htmlhttps://liberty.io/what-is-a-co-prime.htmlI've never encountered this question in any of my math classes and it just shows up randomly in my comsci class with no further info about it. I've wiki'ed it, but can't even understand that. Could ...Fri, 04 Sep 2026 07:50:42 -0400https://liberty.io/basic-equivalence-class-discrete-math.htmlhttps://liberty.io/basic-equivalence-class-discrete-math.htmlI read through the textbook definition of the equivalence class, but still cannot clearly understand what an equivalence class is. Does anyone have a good example with a definition that can hit m......Fri, 04 Sep 2026 07:47:03 -0400https://liberty.io/what-is-the-outward-normal-vector-at-the-end-points-of-the-one-dimensi.htmlhttps://liberty.io/what-is-the-outward-normal-vector-at-the-end-points-of-the-one-dimensi.htmlIn the one-dimensional space, suppose the interval $[a, b]$ exists on the real line with boundary points $a$ and $b$. What is the outward normal vector at the point $a$ and at the point $b$? The an......Fri, 04 Sep 2026 07:43:35 -0400https://liberty.io/are-the-eigenvectors-of-a-real-symmetric-matrix-always-an-orthonormal-.htmlhttps://liberty.io/are-the-eigenvectors-of-a-real-symmetric-matrix-always-an-orthonormal-.htmlI was reading the wikipedia page for symmetric matrices, and I noticed this part: a real n×n matrix A is symmetric if and only if there is an orthonormal basis of Rn consisting of eigenvectors f......Fri, 04 Sep 2026 07:30:59 -0400https://liberty.io/how-to-pronounce-tableaux-closed.htmlhttps://liberty.io/how-to-pronounce-tableaux-closed.htmlHow do you pronounce Young tableaux? Does it sound just like its singular form?...Fri, 04 Sep 2026 07:23:53 -0400https://liberty.io/finding-cosets-and-index-in-d8.htmlhttps://liberty.io/finding-cosets-and-index-in-d8.htmla). Find all the left and right cosets of $<τ>$ in $D_8$. $<τ>$ is reflection and I know that there are two reflections about a diagonal. So I am wondering how to represent this idea a......Fri, 04 Sep 2026 07:10:48 -0400https://liberty.io/show-that-the-sum-of-2n-1-consecutive-integers-is-divisible-by-2n-1.htmlhttps://liberty.io/show-that-the-sum-of-2n-1-consecutive-integers-is-divisible-by-2n-1.htmlBase case: $n=1$. Picking $2n+1$ random numbers 5,6,7 we get $5+6+7=18$. So, $2(1)+1=3$ which indeed does divide 18. The base case holds. Let $n=k>=1$ and let $2k+1$ be true. We want to show $2......Fri, 04 Sep 2026 07:06:12 -0400https://liberty.io/zassenhaus-formula-exponents.htmlhttps://liberty.io/zassenhaus-formula-exponents.htmlfound Zassenhaus formula for noncommutative $X,Y$ $$e^{t(X+Y)}= e^{tX}~ e^{tY} ~e^{-\frac{t^2}{2} [X,Y]} ~ e^{\frac{t^3}{6}(2[Y,[X,Y]]+ [X,[X,Y]] )} ~ e^{\frac{-t^4}{24}([[[X,Y],X],X] + 3[[[X,Y],X......Fri, 04 Sep 2026 07:05:15 -0400https://liberty.io/right-triangular-pyramid.htmlhttps://liberty.io/right-triangular-pyramid.htmlI am somehow confused with this problem. It is said that this is a right triangular pyramid, but it has an equilateral triangle base? I am confused. How can I find VA? It doesn't look like it has a......Fri, 04 Sep 2026 06:53:20 -0400https://liberty.io/strangest-notation-closed.htmlhttps://liberty.io/strangest-notation-closed.htmlWhile this may be a fruitless pursuit of anecdotes, I still ask: what is the strangest (or most blatantly wrong (at least in the eyes of common notation)) mathematical notation you have ever seen?...Fri, 04 Sep 2026 06:53:19 -0400https://liberty.io/calculators-using-taylor-polynomials.htmlhttps://liberty.io/calculators-using-taylor-polynomials.htmlI've always heard that calculators (TI-84's and the like) use Taylor polynomials to approximate trigonometric/exponential/etc functions. Do any of you know this for a fact?...Fri, 04 Sep 2026 06:48:47 -0400