https://liberty.ioCurated star highlights with fresh energy.en-usMon, 24 Aug 2026 18:27:29 -0400https://liberty.io/laplace-transform-of-the-product-of-two-functions.htmlhttps://liberty.io/laplace-transform-of-the-product-of-two-functions.htmlWhat is the Laplace Transform of the product of two functions? Specifically, my function is sin(5t)*cos(5t), but I'd like to know a general principle if it's available. What's the easy way to com......Mon, 24 Aug 2026 22:22:07 -0400https://liberty.io/serre-duality-in-derived-category.htmlhttps://liberty.io/serre-duality-in-derived-category.htmlLet $X$ be a smooth projective variety over a field $k$, and $\omega_X$ its canonical bundle. Denote by $D^b(X):=D^b(\mathbf{Coh})$ the bounded derived category of coherent sheaves on $X$. Theorem......Mon, 24 Aug 2026 22:15:10 -0400https://liberty.io/get-the-equation-of-a-circle-when-given-3-points.htmlhttps://liberty.io/get-the-equation-of-a-circle-when-given-3-points.htmlGet the equation of a circle through the points $(1,1), (2,4), (5,3) $. I can solve this by simply drawing it, but is there a way of solving it (easily) without having to draw?...Mon, 24 Aug 2026 22:12:19 -0400https://liberty.io/recurrence-relations-question.htmlhttps://liberty.io/recurrence-relations-question.htmlA vending machine dispensing books of stamps accepts only one-dollar coins, $1 bills, and $5 bills. a) Find a recurrence relation for the number of ways to deposit $n$ dollars in the vending machi......Mon, 24 Aug 2026 22:09:17 -0400https://liberty.io/why-is-it-always-necessary-that-a-second-order-differential-equation-h.htmlhttps://liberty.io/why-is-it-always-necessary-that-a-second-order-differential-equation-h.htmlI'm an AL student , and i have been recently introduced to the concept of Homogeneous Second Order Linear Differential Equations.I could understand how the Characteristic Equation forms and why it ......Mon, 24 Aug 2026 22:03:50 -0400https://liberty.io/find-a-singular-value-decomposition-of-the-matrix.htmlhttps://liberty.io/find-a-singular-value-decomposition-of-the-matrix.htmlFina a singular value decomposition for $$A=\begin{bmatrix} -2 & 2 \\ -1 & 1 \\ 2 & -2\end{bmatrix}.$$ Find a (full svd)singular value decomposition for the matrix $A$. My working......Mon, 24 Aug 2026 22:02:40 -0400https://liberty.io/tranform-region-of-type-1-into-a-region-of-type-2.htmlhttps://liberty.io/tranform-region-of-type-1-into-a-region-of-type-2.htmlWe have the integral of type 1 : $\int_0^1 \left (\int_{e^x}^e f(x,y)dy\right )dx$ and we want to transform it into an integral of type 2. The region of type 1 is $D_1=\{(x,y) \mid 0\leq x\leq 1,......Mon, 24 Aug 2026 21:57:08 -0400https://liberty.io/if-a-plane-is-divided-by-n-lines-then-it-is-possible-to-color-the-regi.htmlhttps://liberty.io/if-a-plane-is-divided-by-n-lines-then-it-is-possible-to-color-the-regi.htmlI am self-studying Discrete Mathematics, and there is the following exercise. (in Portuguese) A plane is divided by many lines. Show that it is possible to color the regions formed with only two ...Mon, 24 Aug 2026 21:51:30 -0400https://liberty.io/how-do-you-prove-the-domain-of-a-function.htmlhttps://liberty.io/how-do-you-prove-the-domain-of-a-function.htmlEdit: Man, I actually had a bachelor's completed when I asked this dumb question, like even more than the terry tao dumb thing. Don't judge me! But to be fair even when I was in calculus we're always ...Mon, 24 Aug 2026 21:42:47 -0400https://liberty.io/how-to-simplify-frac-1-sqrt-3-3-1-frac-2-sqrt-3-3-1-to-sqrt-3-3.htmlhttps://liberty.io/how-to-simplify-frac-1-sqrt-3-3-1-frac-2-sqrt-3-3-1-to-sqrt-3-3.html$$\frac{1}{\sqrt[3]{3}-1} - \frac{2}{\sqrt[3]{3}+1}$$ I have simplified above to: $$\frac{3-\sqrt[3]{3}}{(\sqrt[3]{3}+1)(\sqrt[3]{3}-1)}$$ What is equal to: $$\frac{3-\sqrt[3]{3}}{\sqrt[3]{9}-1}$$ ...Mon, 24 Aug 2026 21:42:45 -0400https://liberty.io/how-is-it-that-tan-a-b-frac-tan-a-tan-b-1-tan-a-tan-b-for-all-angles-e.htmlhttps://liberty.io/how-is-it-that-tan-a-b-frac-tan-a-tan-b-1-tan-a-tan-b-for-all-angles-e.htmlHow is it that $$\tan(A +B) = \frac{\tan A + \tan B}{1-\tan A\tan B}$$ for any value of $A$, $B$? I have doubts about this since we arrive at this by dividing the numerator and denominator of $$\......Mon, 24 Aug 2026 21:40:15 -0400https://liberty.io/derivative-of-x-x-without-logarithmic-differentiation.htmlhttps://liberty.io/derivative-of-x-x-without-logarithmic-differentiation.htmlWith logarithmic differentiation, it is quite simple to compute the derivative of $x^x$: $$y=x^x$$ $$\ln {y} =x \ln{x}$$ $$\frac {1}{y} \frac {dy}{dx} = \ln{x} +1$$ $$\frac {dy}{dx} =\l......Mon, 24 Aug 2026 21:39:43 -0400https://liberty.io/true-definition-of-the-real-numbers.htmlhttps://liberty.io/true-definition-of-the-real-numbers.htmlI've found lots of resources that say this is a real number if it's not rational, but what is a real number, really? I mean what is the definition of a real number? If nothing else, anyone know o......Mon, 24 Aug 2026 21:22:14 -0400https://liberty.io/number-of-ways-to-draw-a-pair-in-a-poker-deck.htmlhttps://liberty.io/number-of-ways-to-draw-a-pair-in-a-poker-deck.htmlI am confused over calculating the number of ways in which I can select a pair out of a deck of 52 cards. This is how I go about solving the problem: Following the definition of a pair in card ga......Mon, 24 Aug 2026 21:19:07 -0400https://liberty.io/mathematical-operator-to-flip-a-vector.htmlhttps://liberty.io/mathematical-operator-to-flip-a-vector.htmlIs there a mathematical operator which flips a vector from left to right (or up to down). Say \begin{align} a = [1~ 2~ 3]\quad\text{and}\quad b = [3~ 2~ 1] \end{align} I'd like to have \begin{align......Mon, 24 Aug 2026 21:16:13 -0400https://liberty.io/finding-best-response-function.htmlhttps://liberty.io/finding-best-response-function.htmlConsider two players Player 1 choose $x_1$ Player 2 choose $x_2$ $x_1, x_2 \in [0,b]$ and $b>0$ The payoffs for players 1 and 2 are identical and equal $$Z=u(x_1+x_2)+v(2b-x_1-x_2)$$ $u$......Mon, 24 Aug 2026 21:08:01 -0400https://liberty.io/lambert-w-product-log-function.htmlhttps://liberty.io/lambert-w-product-log-function.htmlI would like to solve this equation: $$n \cdot 2^n = 15000$$ And according to WolframAlpha $$n=\frac{W(15000\log(2))}{\log(2)}, \text{ where }\log\text{ is ln}$$ Which shows that I need to use ......Mon, 24 Aug 2026 21:07:51 -0400https://liberty.io/what-does-expected-value-of-sum-of-two-discrete-random-variables-mean.htmlhttps://liberty.io/what-does-expected-value-of-sum-of-two-discrete-random-variables-mean.htmlI am confused with summing two random variables. Suppose $X$ and $Y$ are two random variables denoting how much is gained from each two games. If two games are played together, we can gain $E[X] + ......Mon, 24 Aug 2026 21:04:32 -0400https://liberty.io/taylor-series-third-order-approximation.htmlhttps://liberty.io/taylor-series-third-order-approximation.htmlThere has been this question that had been bothering me for a while and I could not find a satisfying answer on the internet or any of the books even though it is not very complex. i) Its because ......Mon, 24 Aug 2026 20:57:50 -0400https://liberty.io/even-digit-series-2-4-6-8-20-duplicate.htmlhttps://liberty.io/even-digit-series-2-4-6-8-20-duplicate.htmlI have a series of numbers whose Nth term is a number whose all of the digits are even. The series is of course divergent , but i am interested in finding a formula to find the Nth term of this ser......Mon, 24 Aug 2026 20:57:11 -0400https://liberty.io/calculating-an-ellipse-given-the-orbital-eccentricity-and-a-vertex.htmlhttps://liberty.io/calculating-an-ellipse-given-the-orbital-eccentricity-and-a-vertex.htmlI know that the formula for Eccentricity is e = c/a where c is the distance from the center to a focus and a is the distance from that focus to a vertex. I know the distance from the center of the ...Mon, 24 Aug 2026 20:50:15 -0400https://liberty.io/lie-algebra-of-lorentz-group-o-1-3.htmlhttps://liberty.io/lie-algebra-of-lorentz-group-o-1-3.htmlLet $$ O(1,3)=\{A\in GL_4(\mathbb R):A^TgA=g\} $$ where $g$ is the diagonal matrix with $1$ on the first diagonal entry, and $-1$ on the other diagonal entries. I want to show that the Lie algebra ...Mon, 24 Aug 2026 20:13:31 -0400https://liberty.io/construct-an-a4-paper.htmlhttps://liberty.io/construct-an-a4-paper.htmlThe A4 European paper format is designed such that if you cut in halve the longest side you would obtain two papers where the ratio of the longest side of each is the same as the original paper. I......Mon, 24 Aug 2026 20:07:15 -0400https://liberty.io/what-is-a-compact-element.htmlhttps://liberty.io/what-is-a-compact-element.htmlI was reading the definition of an Algebraic Lattice: "An algebraic lattice is a complete lattice L, such that every element x of L is the supremum of the compact elements below x". Then I looked f......Mon, 24 Aug 2026 20:02:41 -0400https://liberty.io/given-a-3-times-3-matrix-a-with-det-a-frac-1-8-find-det-3a-det-6a-1.htmlhttps://liberty.io/given-a-3-times-3-matrix-a-with-det-a-frac-1-8-find-det-3a-det-6a-1.htmlHow do I go about solving this? The $\det(3A)$ is not simply $3\cdot\frac{1}{8}$ correct?...Mon, 24 Aug 2026 20:01:45 -0400https://liberty.io/on-the-commutative-properties-of-subtraction-and-division.htmlhttps://liberty.io/on-the-commutative-properties-of-subtraction-and-division.htmlCan someone explain to me why subtraction and division is both commutative? The reason I believe that they are commutative are as follows: $$ 3-2 = 3+(-2) $$ $$ 3+(-2) = (-2)+(3) $$ I've read a w......Mon, 24 Aug 2026 19:55:39 -0400https://liberty.io/how-to-handle-optimization-problems-when-optimization-variable-is-matr.htmlhttps://liberty.io/how-to-handle-optimization-problems-when-optimization-variable-is-matr.htmlSuppose we have the following optimization problem $$ \min_{0\preceq M \preceq I} y^TMy $$ where $y \in \mathbb{R}^n$ and $M \in \mathbb{R}^{n \times n}$ is a positive semi-definite matrix. Notice......Mon, 24 Aug 2026 19:53:02 -0400https://liberty.io/what-functions-can-be-represented-as-power-series.htmlhttps://liberty.io/what-functions-can-be-represented-as-power-series.htmlHow do we know if a particular function can be represented as a power series? And once we have come up with a power series representation, how does one figure out its radius of convergence ?...Mon, 24 Aug 2026 19:43:11 -0400https://liberty.io/find-fourth-side-of-quadrilateral-given-three-sides-and-two-angles.htmlhttps://liberty.io/find-fourth-side-of-quadrilateral-given-three-sides-and-two-angles.htmlTo completely determine a quadrilateral, you have to have five independent pieces of information, of sides and angles. If you have five data (all outer sides and a diagonal), finding the angles is ......Mon, 24 Aug 2026 19:41:20 -0400https://liberty.io/relation-between-inradius-and-exradii-of-a-triangle-where-did-my-proof.htmlhttps://liberty.io/relation-between-inradius-and-exradii-of-a-triangle-where-did-my-proof.htmlI am given a triangle $\triangle ABC$ with side lengths $AB=c$, $AC=b$, $BC=a$. $r$ is the radius of the incircle, and $r_a,r_b,r_c$ are the radii of the excircles. The incircle divides the sides o......Mon, 24 Aug 2026 19:36:54 -0400https://liberty.io/how-to-find-correlation-coefficient-given-joint-distribution.htmlhttps://liberty.io/how-to-find-correlation-coefficient-given-joint-distribution.htmlRandom variables $X$ and $Y$ follow a joint distribution $$f(x, y) = \left\{ \begin{array}{ll} 2,& 0 < x \leq y < 1,\\ 0,&\text{otherwise}\end{array}\right. $$ Determine the correlat......Mon, 24 Aug 2026 19:22:03 -0400https://liberty.io/a-tourist-in-france-wants-to-visit-12-different-cities-find-the-probab.htmlhttps://liberty.io/a-tourist-in-france-wants-to-visit-12-different-cities-find-the-probab.htmlThe Question : A tourist in France wants to visit 12 different cities. If the route is randomly selected, what is the probability that she will visit the cities in alphabetical order? I thought ...Mon, 24 Aug 2026 19:20:16 -0400https://liberty.io/how-to-simplify-the-integration.htmlhttps://liberty.io/how-to-simplify-the-integration.htmlIf any integration is in form $$\int \frac{1}{1+x^2}dx$$ it easily follows the $\tan^{-1}x$ but How to simplify if we have $$\int \frac{1}{(1+x^2)^2}dx$$...Mon, 24 Aug 2026 19:17:33 -0400https://liberty.io/constructing-a-matrix-such-its-square-is-not-039-0-039-but-its-cube-is.htmlhttps://liberty.io/constructing-a-matrix-such-its-square-is-not-039-0-039-but-its-cube-is.htmli have been asked to construct a matrix A such that $A^2$ is not equal to '0' but, $A^3=0$. how should i proceed. i can only understand that all the eigenvalues for A , $A^2$ and $A^3$ will be 'z......Mon, 24 Aug 2026 19:16:35 -0400https://liberty.io/group-theory-notation.htmlhttps://liberty.io/group-theory-notation.htmlWhat does the notation $(G,.)$ mean in group theory? I have seen in places that $.$ implies the binary operation multiplication on group $G$. But then, why do we show an abelian group as $(G, +)$? ......Mon, 24 Aug 2026 19:10:42 -0400https://liberty.io/usage-of-the-word-formal-ly.htmlhttps://liberty.io/usage-of-the-word-formal-ly.htmlThis is weird. To me, in mathematical contexts, "formally" means something like "rigorously", i.e. the opposite of informally/heuristically. And yet, I very often read papers very the word seems t......Mon, 24 Aug 2026 18:59:47 -0400https://liberty.io/dfa-union-operation-how-to.htmlhttps://liberty.io/dfa-union-operation-how-to.htmlI'm currently looking at deterministic finite automata, and learning how to combine two DFAs using AND or OR. I think I understand how to construct the INTERSECTION (AND) of two DFAs, but I'm at a ......Mon, 24 Aug 2026 18:49:10 -0400https://liberty.io/which-of-the-following-statements-are-true-and-which-are-false.htmlhttps://liberty.io/which-of-the-following-statements-are-true-and-which-are-false.htmlLet $f$ and $g$ be continuous functions such that $f(x) ≤ g(x)$ for all $x ∈ [0, 1]$. Determine which of the following statements are true and which are false: $$ \begin{align} (a) & {}\quad \......Mon, 24 Aug 2026 18:44:39 -0400https://liberty.io/straightforward-logical-techniques-for-solving-matrices.htmlhttps://liberty.io/straightforward-logical-techniques-for-solving-matrices.htmlI'm studying for a Linear Algebra exam, and having difficulty with solving matrices in a timely manner. Our professor provided us with several sample exams to practice with, which begin with a matr......Mon, 24 Aug 2026 18:35:38 -0400https://liberty.io/determining-the-area-of-a-heart-curve-2d.htmlhttps://liberty.io/determining-the-area-of-a-heart-curve-2d.htmlI found out the area of a heart but I am not sure whether it is correct. Can someone please check whether what I have done makes any sense at all. The equation of a heart is shown below: I first ...Mon, 24 Aug 2026 18:27:39 -0400https://liberty.io/chromatic-number-of-a-graph-obtained-from-petersen-s-graph.htmlhttps://liberty.io/chromatic-number-of-a-graph-obtained-from-petersen-s-graph.htmlI am studying on Chromatic numbers of graphs and I came across Petersen’s graph. Petersen’s graph which is an undirected graph with 10 vertices and 15 edges. And it has the chromatic number 3, mea......Mon, 24 Aug 2026 18:16:21 -0400https://liberty.io/determining-the-derivative-of-e-5x-tan-2x.htmlhttps://liberty.io/determining-the-derivative-of-e-5x-tan-2x.htmlDetermine the derivative of $e^{5x}\tan(2x)$ I have no idea, what is this question mean- but what I can do is $$\begin{align*}y&=e^{5x}\\ \frac{dy}{dx}&=e^{5x}(5)\\ \frac{dy}{dx}&=5......Mon, 24 Aug 2026 18:07:38 -0400https://liberty.io/how-to-calculate-e-x-with-a-standard-calculator.htmlhttps://liberty.io/how-to-calculate-e-x-with-a-standard-calculator.htmlIs there a simple method for calculating the $e^x$ ($x\in\mathbb{R}$) with a basic add/subtract/multiply/divide calculator that converges in reasonable time, preferably without having to memorize ...Mon, 24 Aug 2026 18:05:46 -0400https://liberty.io/why-is-this-a-valid-proof-for-the-harmonic-series.htmlhttps://liberty.io/why-is-this-a-valid-proof-for-the-harmonic-series.htmlThe other day, I encountered the Harmonic series which I though is an interesting concept. There were many ways to prove that it's divergent and the one I really liked was rather simple but did the......Mon, 24 Aug 2026 18:03:57 -0400https://liberty.io/number-of-possible-permutations-of-three-pairs-of-socks-one-blue-one-b.htmlhttps://liberty.io/number-of-possible-permutations-of-three-pairs-of-socks-one-blue-one-b.htmlThere are 3 colours of socks; blue, black and white and each colour has two pairs of socks and we need to distribute them to six people and the left leg socks are distinguishable from right leg soc......Mon, 24 Aug 2026 17:59:09 -0400https://liberty.io/finding-the-major-and-minor-axis-vertices-for-an-ellipse-given-two-con.htmlhttps://liberty.io/finding-the-major-and-minor-axis-vertices-for-an-ellipse-given-two-con.htmlI've been googling, searching forums and looking in my old algebra/trig books to try to understand how to find the end points to the major and minor axis of an ellipse given the end points of two ...Mon, 24 Aug 2026 17:54:24 -0400https://liberty.io/how-to-calculate-this-limit-without-l-039-hopital-039-s-rule.htmlhttps://liberty.io/how-to-calculate-this-limit-without-l-039-hopital-039-s-rule.htmlI know that in the very small values of $x$ $$(1+x)^n ≈1+ nx$$ and I can prove it using Taylor series. But I wanted to prove it without any smell of derivative. So... In order to calculate the ...Mon, 24 Aug 2026 17:46:53 -0400https://liberty.io/principia-mathematica-and-axioms.htmlhttps://liberty.io/principia-mathematica-and-axioms.htmlI have been trying to understand how mathematicians achieve the truth through axiomatic systems. Surely, if one searches a bit more, it will get to know how axiomatic systems work, generally. After......Mon, 24 Aug 2026 17:41:49 -0400https://liberty.io/calculate-expected-value-in-dice-game.htmlhttps://liberty.io/calculate-expected-value-in-dice-game.htmlWe play a game in 2 stages: In stage one, we throw a dice until we get number 6, Let N represent the number of times played until we got 6 for the first time. In stage two, we throw N dices (each one ...Mon, 24 Aug 2026 17:37:42 -0400https://liberty.io/what-is-the-expected-area-of-a-polygon-whose-vertices-lie-on-a-circle.htmlhttps://liberty.io/what-is-the-expected-area-of-a-polygon-whose-vertices-lie-on-a-circle.htmlI came across a nice problem that I would like to share. Problem: What is expected value of the area of an $n$-gon whose vertices lie on a circle of radius $r$? The vertices are uniformly distribu......Mon, 24 Aug 2026 17:36:53 -0400https://liberty.io/differentiate-log-gamma-function.htmlhttps://liberty.io/differentiate-log-gamma-function.htmlI am working with the log of negative binomial distribution $NegBin(r,p)$. I need to differentiate the following with respect to $r$ such that at the end, I am NOT left with $r$ in factorial form. ......Mon, 24 Aug 2026 17:30:04 -0400https://liberty.io/writing-out-some-propositions-using-disjunctions-and-conjunctions.htmlhttps://liberty.io/writing-out-some-propositions-using-disjunctions-and-conjunctions.htmlSuppose that the universe of discourse of the atomic formula $P(x,y)$ is $\{1, 2, 3\}$. Write out the following propositions using disjunctions and conjunctions. a) $\exists x \forall y \ P(x,y)$ ...Mon, 24 Aug 2026 17:27:47 -0400https://liberty.io/calculate-the-sum-using-fourier-series-of-big-cos-frac-x-2-big.htmlhttps://liberty.io/calculate-the-sum-using-fourier-series-of-big-cos-frac-x-2-big.htmlI have been given task to evaluate the sum $\dfrac{(-1)^{n-1}}{4n^2-1}$ using Fourier series for function $|\cos{\frac{x}{2}}|$. I used the interval $(-\pi, +\pi)$ for evaluation of the sum and I n......Mon, 24 Aug 2026 17:19:17 -0400https://liberty.io/gradient-of-2-norm-squared.htmlhttps://liberty.io/gradient-of-2-norm-squared.htmlCould someone please provide a proof for why the gradient of the squared $2$-norm of $x$ is equal to $2x$? $$\nabla\|x\|_2^2 = 2x$$...Mon, 24 Aug 2026 17:02:39 -0400https://liberty.io/calculate-sin9-closed.htmlhttps://liberty.io/calculate-sin9-closed.htmlI found this question in the olympiad book. But I could not find the solution. The question is to calculate the following real number: $$\sin{9°}$$...Mon, 24 Aug 2026 16:55:08 -0400https://liberty.io/heaviside-step-function-squared.htmlhttps://liberty.io/heaviside-step-function-squared.htmlI have a question about the Heaviside step function $\theta(\xi)$, defined by $$\theta(\xi):=\begin{cases} 1, & \xi\geq0\\ 0, & \xi<0 \end{cases}$$ I need to evaluate the square of the ...Mon, 24 Aug 2026 16:47:44 -0400https://liberty.io/sard-039-s-theorem-for-algebraic-varieties.htmlhttps://liberty.io/sard-039-s-theorem-for-algebraic-varieties.html(One version of) Sard's theorem states that: Theorem (Sard): Given $M$ and $N$ smooth manifolds of dimensions $m$ and $n$ respectively, and a smooth map $f:M\to N$, then the set of singular valu......Mon, 24 Aug 2026 16:40:48 -0400https://liberty.io/the-equation-of-a-line-reflected-about-another-line.htmlhttps://liberty.io/the-equation-of-a-line-reflected-about-another-line.htmlI need to find the equation of this reflected line:...Mon, 24 Aug 2026 16:35:39 -0400https://liberty.io/find-the-parameters-for-which-the-matrix.htmlhttps://liberty.io/find-the-parameters-for-which-the-matrix.html$\left(\begin{matrix} 1 & 2 & -1 \\ 3a & 2 & b \\ 1 & 2& 1 \end{matrix}\right)$ For example I have such a matrix. Can you, please explain how can I find for what values of ......Mon, 24 Aug 2026 16:34:26 -0400https://liberty.io/what-does-it-mean-when-all-the-values-of-a-row-in-a-matrix-are-0.htmlhttps://liberty.io/what-does-it-mean-when-all-the-values-of-a-row-in-a-matrix-are-0.htmlI'm fairly new to linear algebra and I'm trying to make sense of what I'm being taught in class. I'm a little confused as to what happens when all the values of a matrix's row are equal to $0$: doe......Mon, 24 Aug 2026 16:33:59 -0400https://liberty.io/questions-tagged-ramanujan-summation.htmlhttps://liberty.io/questions-tagged-ramanujan-summation.htmlQ&A for people studying math at any level and professionals in related fields...Mon, 24 Aug 2026 16:14:19 -0400https://liberty.io/fixed-point-iteration-method-with-parameters.htmlhttps://liberty.io/fixed-point-iteration-method-with-parameters.htmlWe want to approach the number $\alpha =\sqrt[3]{2}$. The function $f(x) = x^{3} - 2$ has $\alpha$ as a root. Now take a function $g$ so that $\alpha$ is a fixed point, $g(\alpha) = \alpha$. Use $ ......Mon, 24 Aug 2026 16:08:52 -0400https://liberty.io/what-is-the-peano-definition-of-subtraction.htmlhttps://liberty.io/what-is-the-peano-definition-of-subtraction.htmlI came up with this: (S.1) $a - a = 0$ (S.2) $a - b = S(a - S(b))$ This seems to work. At least for $a$$\ge$$b$. Is this the correct or most efficient formulation? Also, does there happen to b......Mon, 24 Aug 2026 16:00:26 -0400https://liberty.io/law-of-iterated-expectations-example.htmlhttps://liberty.io/law-of-iterated-expectations-example.htmlI have a question that I hope can be shown using the LIE. There is a urn which contains 3 marbles (2 white and 1 black). The person who gets to pick the black marble gets to win $100 whereas the ...Mon, 24 Aug 2026 15:41:45 -0400https://liberty.io/why-does-there-seem-to-be-so-much-error-in-the-laws-of-sines-and-cosin.htmlhttps://liberty.io/why-does-there-seem-to-be-so-much-error-in-the-laws-of-sines-and-cosin.htmlI've been computing the angles of a triangle with sides a = 17, b = 6 and c = 15 using the law of cosines to find the first angle and then the law of sines to find the other 2. I follow the convent......Mon, 24 Aug 2026 15:15:09 -0400https://liberty.io/how-should-i-write-down-the-alternating-group-a-3-closed.htmlhttps://liberty.io/how-should-i-write-down-the-alternating-group-a-3-closed.htmlI didn't understand how to write down the alternating group A3. Is this the group consisting of only the even permutations? Also, what familiar group is this isomorphic to?...Mon, 24 Aug 2026 15:14:13 -0400https://liberty.io/recommended-maths-book-for-beginner-to-study-in-computer-science-close.htmlhttps://liberty.io/recommended-maths-book-for-beginner-to-study-in-computer-science-close.htmlI am going to study computer science next year. I am afraid I can't handle the mathematics in the university because I only know some basic mathematics, such as set theory, simple probability, simple ...Mon, 24 Aug 2026 15:03:45 -0400https://liberty.io/ti-83-plus-is-it-possible-to-find-the-eigenvalues-on-this-calculator.htmlhttps://liberty.io/ti-83-plus-is-it-possible-to-find-the-eigenvalues-on-this-calculator.htmlI searched the Internet and can't seem to find a source on how to find the eigenvalues on a TI-83 Plus calculator. Can anyone give me a hand?...Mon, 24 Aug 2026 15:00:16 -0400https://liberty.io/rules-for-modulus-and-multiplication.htmlhttps://liberty.io/rules-for-modulus-and-multiplication.htmlMy question is rather simple as I'm interested about modulus and multiplication, specifically whether it holds that $(a*b)\,mod\,n=(a\,mod\,n)*(b\,mod\,n)$?...Mon, 24 Aug 2026 14:57:39 -0400https://liberty.io/what-does-test-function-mean.htmlhttps://liberty.io/what-does-test-function-mean.htmlI am trying to learn weak derivatives. In that, we call $\mathbb{C}^{\infty}_{c}$ functions as test functions and we use these functions in weak derivatives. I want to understand why these are called ...Mon, 24 Aug 2026 14:54:43 -0400https://liberty.io/derive-fourier-transform-for-summation-of-shifted-dirac-delta-function.htmlhttps://liberty.io/derive-fourier-transform-for-summation-of-shifted-dirac-delta-function.htmlHow to derive this Fourier transform: $$ F\{ \sum_{n=-\infty}^{\infty} \delta (t- nT) \text{ }\} =\omega_o \sum_{n=-\infty}^{\infty} \delta (\omega - n \omega_o)$$ where: $$ \omega_o = 2 \pi / T $$...Mon, 24 Aug 2026 14:43:56 -0400https://liberty.io/proof-for-how-9999999-1-duplicate.htmlhttps://liberty.io/proof-for-how-9999999-1-duplicate.htmlI was just curious for the proof/theorem for a very close decimal (so when you keep adding a decimal) to equal the next integer such as .999999999999....=1. Thank you....Mon, 24 Aug 2026 14:43:23 -0400https://liberty.io/alternating-series-determine-if-it-converges-absolutely-conditionally-.htmlhttps://liberty.io/alternating-series-determine-if-it-converges-absolutely-conditionally-.htmlI am trying to determine to whether the series $\sum _{n=0}^{\infty }\:\left(-1\right)^n\frac{1}{\sqrt{n}\left(ln\left(n\right)^{2021}\right)}$ is conditionally convergent, absolutely convergent, or ...Mon, 24 Aug 2026 14:25:19 -0400https://liberty.io/intuitive-understanding-of-the-first-isomorphism-theorem.htmlhttps://liberty.io/intuitive-understanding-of-the-first-isomorphism-theorem.htmlI've been reading some things about an intuitive understanding of the first isomorphism theorem, and there's just one more part that I do not understand. Specifically, the first isomorphism theorem ...Mon, 24 Aug 2026 14:07:51 -0400https://liberty.io/what-does-formal-mean.htmlhttps://liberty.io/what-does-formal-mean.htmlI know the definition of formal power series, power series and polynomials. But what does the adjective "formal" mean? In google English dictionary, does it mean "9. Of or relating to linguistic or ...Mon, 24 Aug 2026 13:55:16 -0400https://liberty.io/convert-complex-number-to-polar-coordinates-round-2.htmlhttps://liberty.io/convert-complex-number-to-polar-coordinates-round-2.htmlThis is follow up problem to this. I post new question related to previous question since all of the problem were not solved. I have new questions / problems that came after i closed the previous p......Mon, 24 Aug 2026 13:28:25 -0400https://liberty.io/what-is-2-1-1-duplicate.htmlhttps://liberty.io/what-is-2-1-1-duplicate.html$2-1+1$; a fairly straightforward question, but I (well, not me, but Henry Reich) found something strange. Most people would evaluate it as $2+(-1)+1 = 2$; however, this goes against the famed, and ...Mon, 24 Aug 2026 13:27:25 -0400https://liberty.io/polar-form-modulus-argument.htmlhttps://liberty.io/polar-form-modulus-argument.htmlQuestions: a) Express $-4i$ in polar form b) Hence solve the equation $z^2 =-4i$ I've attempted the question and was only able to find $z=4$ using the form $r(\cos x+i\sin x)=z$. Could someone ...Mon, 24 Aug 2026 13:14:46 -0400https://liberty.io/what-is-the-structural-hierarchy-in-mathematics.htmlhttps://liberty.io/what-is-the-structural-hierarchy-in-mathematics.htmlBy structural hierarchy, I mean the mental concept in which things are 'done' in mathematics. At the top, you have mathematics itself, which is a collection of systems, like arithmetic, algebra, ge......Mon, 24 Aug 2026 13:07:54 -0400https://liberty.io/why-is-slope-rise-run.htmlhttps://liberty.io/why-is-slope-rise-run.htmlWhat makes slope rise over run? What makes the standard equation for a line use a slope of rise over run as opposed to run over rise? What would the standard equation of a line look like if m was ...Mon, 24 Aug 2026 13:06:28 -0400https://liberty.io/square-root-of-x-3.htmlhttps://liberty.io/square-root-of-x-3.htmlI understand the concept behind the expression $\sqrt{x^2} = |x|$. So, then why is the square root of $x^3$ NOT equal to $|x|\sqrt{x}$? Specifically, I can write $\sqrt{x^3}$ as $\sqrt{x^2\time......Mon, 24 Aug 2026 13:02:05 -0400https://liberty.io/prove-by-induction-involving-calculus-closed.htmlhttps://liberty.io/prove-by-induction-involving-calculus-closed.htmlProve by induction. Let $$f(x)={ e }^{ { x }^{ 2 } }$$ Then for every $n\in \mathbb{N}$ there exists a polynomial ${ P }_{ n }(x)$ of degree $n$ such that $$\frac { { d }^{ n } }{ { dx }^{ n } }......Mon, 24 Aug 2026 12:57:40 -0400https://liberty.io/prove-that-there-are-infinitely-many-n-in-mathbb-n-such-that-6n-1-6n-1.htmlhttps://liberty.io/prove-that-there-are-infinitely-many-n-in-mathbb-n-such-that-6n-1-6n-1.htmlLet m be a integer. Then if $6n+1$ is a composite number we have that $\operatorname{lcd}(6n+1,m)$ is not just $1$, because then $6n+1$ would be prime. Also this is for $6n-1$. Now I must find all ......Mon, 24 Aug 2026 12:55:38 -0400https://liberty.io/super-perfect-numbers.htmlhttps://liberty.io/super-perfect-numbers.htmlA super-perfect number is a number with $\sigma(\sigma (n))=2n$. How can I prove that every even super perfect number is from the form $n=2^k$ when $2^{k+1}-1$ is prime. I tried every way please ......Mon, 24 Aug 2026 12:55:33 -0400https://liberty.io/what-really-is-an-indeterminate-form.htmlhttps://liberty.io/what-really-is-an-indeterminate-form.htmlWe can apply l’Hôpital’s Rule to the indeterminate quotients $ \dfrac{0}{0} $ and $ \dfrac{\infty}{\infty} $, but why can’t we directly apply it to the indeterminate difference $ \infty - \infty $ ......Mon, 24 Aug 2026 12:55:18 -0400https://liberty.io/how-to-parameterize-an-ellipse.htmlhttps://liberty.io/how-to-parameterize-an-ellipse.htmlI need to parameterize the ellipse $\frac{x^2}{2}+y^2=2$, so this is how I proceed: I know that $a=2$ and $b=1$ (where $a$ and $b$ are the axis of the ellipse), so I parameterize as: \begin{cases}......Mon, 24 Aug 2026 12:55:07 -0400https://liberty.io/prove-that-a-planar-graph-has-four-coloring.htmlhttps://liberty.io/prove-that-a-planar-graph-has-four-coloring.htmlThere is a theorem which says that every planar graph can be colored with five colors. It can also be colored with four colors. How can I prove that any planar graph with max degree of $4$, has a f......Mon, 24 Aug 2026 12:51:12 -0400https://liberty.io/how-many-non-isomorphic-trees-with-four-vertices-are-there-what-are-th.htmlhttps://liberty.io/how-many-non-isomorphic-trees-with-four-vertices-are-there-what-are-th.htmlAs of now I have 4 assuming I got this right. I apologize for formatting because I have no idea how I'd display this in text but here I go. a b c d are my points. - means there is a connection betw......Mon, 24 Aug 2026 12:48:10 -0400https://liberty.io/circle-equation-transformations.htmlhttps://liberty.io/circle-equation-transformations.htmlIf we have a circle of the form $x^2 + y^2 = 4$ it has the middle point $(0,0)$ and radius $2$. If we want to move it $5$ to the right, the equation becomes $(x-5)^2 + y^2 = 4.$ What happens if we ......Mon, 24 Aug 2026 12:45:46 -0400https://liberty.io/what-is-the-difference-between-a-limit-and-derivative.htmlhttps://liberty.io/what-is-the-difference-between-a-limit-and-derivative.htmlWhen you get the derivative of a point, arn't you just getting the limit at that point? I'm not quite sure why they need to be named differently when they seem to be doing the same thing....Mon, 24 Aug 2026 12:40:56 -0400https://liberty.io/how-to-determine-if-a-given-function-is-a-valid-cdf-pmf-or-pdf.htmlhttps://liberty.io/how-to-determine-if-a-given-function-is-a-valid-cdf-pmf-or-pdf.htmlStudying for a statistics exam. I have come across this problem: and it presents to me some important and extremely basic questions (I have a LONG way to go before I'm prepared for this exam). Can ...Mon, 24 Aug 2026 12:25:21 -0400https://liberty.io/how-to-get-polynomial-equation-from-points.htmlhttps://liberty.io/how-to-get-polynomial-equation-from-points.htmlHow can i get polynomial equation from points? I have data in excel and i want to get polynomial equation from it. I can get polynomial equation using excel using trendline but only to order of 6. I ...Mon, 24 Aug 2026 12:22:37 -0400https://liberty.io/how-to-draw-a-graph-f-x-y-in-x-y-plane.htmlhttps://liberty.io/how-to-draw-a-graph-f-x-y-in-x-y-plane.htmlIs there possible way to draw function $\vec{f}: R^2 \to R^2 $ such as like $\vec{f}(x,y) = \sin x+ \sin y, y+\sin x\ $ and other $\vec{f}(x,y)$ in $xy$-corordinate instead of $xyz$ plane. Is it o......Mon, 24 Aug 2026 12:20:55 -0400https://liberty.io/incomparable-elements-in-a-poset.htmlhttps://liberty.io/incomparable-elements-in-a-poset.htmlThe problem I am working on is, Find two incomparable elements in these posets. a) $(P(\{0,1,2\}),⊆)$ b) $(\{1,2,4,6,8\},|)$ For a, I said that $R \subseteq p(\{0,1,2,3\}) \times p(\{0,1,2,3\})$, ...Mon, 24 Aug 2026 12:07:26 -0400https://liberty.io/factoring-a-4th-degree-trinomial.htmlhttps://liberty.io/factoring-a-4th-degree-trinomial.htmlI am trying to factor $3x^4-8x^3+16$, but I have no idea how to even start. I put into Wolfram Alpha, and it said that the answer was $(x-2)^2 (3 x^2+4 x+4)$. How would you factor something like th......Mon, 24 Aug 2026 11:55:26 -0400https://liberty.io/what-do-free-variable-and-leading-variables-mean.htmlhttps://liberty.io/what-do-free-variable-and-leading-variables-mean.htmlWhat do the leading variables and free variables in a matrix mean? I have the system below and am trying to understand which are which. I searched a lot for this, please help me ! $$w + x + y + z =......Mon, 24 Aug 2026 11:55:13 -0400https://liberty.io/subtraction-by-addition.htmlhttps://liberty.io/subtraction-by-addition.htmlCan anyone explain to me "subtraction by addition" in a visual way? The steps say: Take the "complement" of the number we are subtracting Add it to to the number we are subtracting from Disc......Mon, 24 Aug 2026 11:54:39 -0400https://liberty.io/verifying-trigonometric-identities-2-cos-2x-1-frac-1-tan-2x-1-tan-2x.htmlhttps://liberty.io/verifying-trigonometric-identities-2-cos-2x-1-frac-1-tan-2x-1-tan-2x.htmlVerify that $$ 2\cos^2x-1 = \frac{1-\tan^2x}{1+\tan^2x}$$...Mon, 24 Aug 2026 11:50:14 -0400https://liberty.io/angles-in-rhombus-when-an-equilateral-triangle-is-inscribed-in-it-clos.htmlhttps://liberty.io/angles-in-rhombus-when-an-equilateral-triangle-is-inscribed-in-it-clos.htmlWhen one inscribs an equilateral triangle in a rhombus, all the corners are multiples of 30 degrees. I can see this, but I can't proof it. Question: How can I proof that the angle ADC is 120 degr......Mon, 24 Aug 2026 11:43:06 -0400https://liberty.io/proving-3-n-geq-3n-using-mathematical-induction.htmlhttps://liberty.io/proving-3-n-geq-3n-using-mathematical-induction.htmlSo I have to prove that $3^n$ is greater than or equal to $3n$ using induction. The base case is a not a problem, but I can't seem to figure out where to go for $(n-1)$. I've tried saying: $$3^n=3\......Mon, 24 Aug 2026 11:27:52 -0400