https://liberty.ioCurated star highlights with fresh energy.en-usTue, 25 Aug 2026 03:01:13 -0400https://liberty.io/continuity-essential-supremum-and-supremum.htmlhttps://liberty.io/continuity-essential-supremum-and-supremum.htmlTake $f: \mathbb{R}\to \mathbb{R}$ be continuous, then w.r.t. Lebesgue measure $L$, we have $$\operatorname{ess sup} |f|=\sup |f|$$ w.r.t. $L$. I have been provided a proof in a Measure theory co......Tue, 25 Aug 2026 06:54:15 -0400https://liberty.io/runge-kutta-and-butcher-table.htmlhttps://liberty.io/runge-kutta-and-butcher-table.htmlIn the Wikipedia article on Runge-Kutta methods, there is a notation explained using a Butcher table with a $c_{i}$ vector (nodes), a $b_{i}$ vector (weights) and a runge-kutta matrix $a_{ij}$. My ...Tue, 25 Aug 2026 06:52:22 -0400https://liberty.io/when-rolling-two-dice-which-of-the-following-events-are-independent-of.htmlhttps://liberty.io/when-rolling-two-dice-which-of-the-following-events-are-independent-of.htmlWhen rolling two dice, which of the following events are independent of the event that the first die is 4: (a) The second is 2 (b) The sum is 6 (c) The sum is 7 (d) The sum is even I thought only o......Tue, 25 Aug 2026 06:49:59 -0400https://liberty.io/proof-of-polya-039-s-theorem-probability-theory.htmlhttps://liberty.io/proof-of-polya-039-s-theorem-probability-theory.htmlI know a theorem called Polya's theorem: $X_n \rightarrow X$ in distribution as $n\rightarrow \infty$ is equivalent to $\sup_{x} | F_n(x) -F(x)| \rightarrow 0$ as $n \rightarrow \infty$, where $F_n......Tue, 25 Aug 2026 06:47:51 -0400https://liberty.io/two-system-of-linear-equation-have-same-solution-if.htmlhttps://liberty.io/two-system-of-linear-equation-have-same-solution-if.htmlTwo system of linear equation have same solution if and only if each equation in each system is a linear combination of the equations in other system? I didn't get it what are they trying to say?...Tue, 25 Aug 2026 06:43:40 -0400https://liberty.io/find-a-formula-for-cos-5x-in-terms-of-sin-x-and-cos-x.htmlhttps://liberty.io/find-a-formula-for-cos-5x-in-terms-of-sin-x-and-cos-x.htmlI was asked to find a formula for $\cos(5x)$ in terms of $\sin(x)$ and $\cos(x)$. I tried to use the formula $\cos(5x) + i\sin(5x) = (\cos(x)+i\sin(x))^5 $ and what I get is $16i\sin^5(x) - 2......Tue, 25 Aug 2026 06:42:13 -0400https://liberty.io/how-do-i-find-a-closed-form-of-the-characteristic-function-of-gamma-di.htmlhttps://liberty.io/how-do-i-find-a-closed-form-of-the-characteristic-function-of-gamma-di.htmlIt’s written in wikipedia that $(1-i \beta t)^{-\alpha}$ is the characteristic function of the Gamma distribution $Gamma(\alpha,\beta)$. I tried to prove this but I failed. So I googled it for a pr......Tue, 25 Aug 2026 06:36:52 -0400https://liberty.io/understanding-the-definition-of-a-local-minimizer.htmlhttps://liberty.io/understanding-the-definition-of-a-local-minimizer.htmlWhat does the following notations mean? As far as I understood, Suppose that $f$ is a function which takes $n$ real values and spits out one real value where those input real values are ...Tue, 25 Aug 2026 06:31:05 -0400https://liberty.io/what-039-s-the-different-among-the-concepts-probability-possibility-an.htmlhttps://liberty.io/what-039-s-the-different-among-the-concepts-probability-possibility-an.htmlCan you please explain the difference among the concepts Probability, Possibility and Belief through some simple examples?...Tue, 25 Aug 2026 06:29:25 -0400https://liberty.io/how-to-find-the-radius-of-convergence-and-the-interval-of-convergence-.htmlhttps://liberty.io/how-to-find-the-radius-of-convergence-and-the-interval-of-convergence-.htmlFor this question, I'm stuck on finding the radius of convergence and interval of convergence for the power series. Here is what I have so far. Can anyone please help me out? Find the radius of ...Tue, 25 Aug 2026 06:22:11 -0400https://liberty.io/what-is-the-period-and-phase-shift-of-f-x-3-sin-4-left-x-frac-pi-4-rig.htmlhttps://liberty.io/what-is-the-period-and-phase-shift-of-f-x-3-sin-4-left-x-frac-pi-4-rig.htmlFor the function what is the period and the phase shift respectively are:? $$f(x) = 3\sin 4\left(x +\frac{\pi}{4}\right) - 1$$ My attempt: It is wrong though. $y = a\sin b(x\pm h) \pm k$ $a$ : ...Tue, 25 Aug 2026 06:14:58 -0400https://liberty.io/dividing-each-term-by-the-highest-degree-in-the-denominator-of-a-limit.htmlhttps://liberty.io/dividing-each-term-by-the-highest-degree-in-the-denominator-of-a-limit.htmlLimits that involve infinity are not a problem for me, but there is an idea that is used in rational limits that I never really understood why. The so called "rule" says that given a rational ...Tue, 25 Aug 2026 06:07:27 -0400https://liberty.io/what-is-wrong-with-my-approach-to-solving-x-log25-25-log-x-10.htmlhttps://liberty.io/what-is-wrong-with-my-approach-to-solving-x-log25-25-log-x-10.htmlFound this equation on the web: $$x^{\log25} + 25^{\log x} = 10$$ The person solved by substitution and got $x = \sqrt{10}$ which satisfies the equation. I tried different ways after following the ......Tue, 25 Aug 2026 06:04:23 -0400https://liberty.io/how-do-you-calculate-the-modulo-of-a-high-raised-number.htmlhttps://liberty.io/how-do-you-calculate-the-modulo-of-a-high-raised-number.htmlI need some help with this problem: $$439^{233} \mod 713$$ I can't calculate $439^{223}$ since it's a very big number, there must be a way to do this. Thanks....Tue, 25 Aug 2026 06:01:44 -0400https://liberty.io/what-is-the-difference-between-finite-difference-methods-finite-elemen.htmlhttps://liberty.io/what-is-the-difference-between-finite-difference-methods-finite-elemen.htmlCan you help me explain the basic difference between FDM, FEM and FVM? What is the best method and why? Advantage and disadvantage of them?...Tue, 25 Aug 2026 06:01:39 -0400https://liberty.io/getting-the-inverse-of-a-lower-upper-triangular-matrix.htmlhttps://liberty.io/getting-the-inverse-of-a-lower-upper-triangular-matrix.htmlFor a lower triangular matrix, the inverse of itself should be easy to find because that's the idea of the LU decomposition, am I right? For many of the lower or upper triangular matrices, often I ......Tue, 25 Aug 2026 05:51:51 -0400https://liberty.io/fat-cantor-set.htmlhttps://liberty.io/fat-cantor-set.htmlSo the standard Cantor set has an outer measure equal to $0$, but how can you construct a "fat" Cantor set with a positive outer measure? I was told that it is even possible to produce one with an ......Tue, 25 Aug 2026 05:50:55 -0400https://liberty.io/difference-between-nonpositive-and-negative-numbers.htmlhttps://liberty.io/difference-between-nonpositive-and-negative-numbers.htmlI am wondering if there is any difference between non-positive and negative numbers? I think that negative numbers mean "negative real numbers" and "Non-positive numbers" are negative real numbe......Tue, 25 Aug 2026 05:48:26 -0400https://liberty.io/continuous-rotation-formula-of-turning-particle.htmlhttps://liberty.io/continuous-rotation-formula-of-turning-particle.htmlContext I am writing a "flight" trajectory program in Mathematica that ignores all physics. I model a turning aircraft (pitching at full deflection) as a parametrised circle. I have been able to ...Tue, 25 Aug 2026 05:40:50 -0400https://liberty.io/how-to-use-lemniscate-sine-and-lemniscate-cosine-elliptic-integrals.htmlhttps://liberty.io/how-to-use-lemniscate-sine-and-lemniscate-cosine-elliptic-integrals.htmlI’ve been reading up on lemniscate sine and cosine functions and found that they are essentially the lemniscate analogues of circular sine and cosine functions. The following wiki page elaborates a......Tue, 25 Aug 2026 05:30:25 -0400https://liberty.io/what-does-multiple-edges-mean-in-simple-graph-definition.htmlhttps://liberty.io/what-does-multiple-edges-mean-in-simple-graph-definition.htmlFrom Wolfram Alpha : A simple graph, also called a strict graph, is an unweighted, undirected graph containing no graph loops or multiple edges. Unless stated otherwise, the unqualified term "...Tue, 25 Aug 2026 05:28:03 -0400https://liberty.io/the-shortest-distance-between-two-parallel-lines.htmlhttps://liberty.io/the-shortest-distance-between-two-parallel-lines.htmlI was working on a set of problems involving finding the shortest distance between two skew lines, which was fine, but then parallel lines showed up. In essence this should be much easier to solve ......Tue, 25 Aug 2026 05:05:26 -0400https://liberty.io/how-to-find-the-analytical-expression-for-the-supporting-hyperplane.htmlhttps://liberty.io/how-to-find-the-analytical-expression-for-the-supporting-hyperplane.htmlIn Stephen Boyd & Lieven Vandenberghe's Convex Optimization book we have following set $$\{x\in \mathbb R^2_+ \mid x_1x_2\geq1\}$$ Now the analytical expression for the supporting hyperplane......Tue, 25 Aug 2026 05:03:36 -0400https://liberty.io/when-to-use-washers-method-and-when-to-use-shells-method.htmlhttps://liberty.io/when-to-use-washers-method-and-when-to-use-shells-method.htmlI know that I can use either washers method or shells method. I have a few questions which I have googled but I did not receive much luck. How do I decide in which case which method is easier and......Tue, 25 Aug 2026 05:01:43 -0400https://liberty.io/what-is-foundations-of-mathematics-and-why-isn-039-t-it-taught-more-of.htmlhttps://liberty.io/what-is-foundations-of-mathematics-and-why-isn-039-t-it-taught-more-of.htmlI have just come across "Foundations of Mathematics". Wikipedia describes it as Foundations of mathematics can be conceived as the study of the basic mathematical concepts (number, geometrical ...Tue, 25 Aug 2026 04:58:25 -0400https://liberty.io/what-exactly-is-a-tensor.htmlhttps://liberty.io/what-exactly-is-a-tensor.htmlI've repeatedly read things that reference tensors, and despite reading the wiki page and other answers here on stackexchange I still don't know what a tensor is. I'm fine with hearing things in the ...Tue, 25 Aug 2026 04:55:16 -0400https://liberty.io/how-to-calculate-the-norm-of-an-ideal.htmlhttps://liberty.io/how-to-calculate-the-norm-of-an-ideal.htmlWould someone please help explain how to calculate the norm of an ideal? I can't find a source that explains this clearly. For example, I know that the norm $N_\mathbb{Q(\sqrt10)}(\langle2,\sqrt10\......Tue, 25 Aug 2026 04:37:30 -0400https://liberty.io/help-with-using-euler-s-formula-to-prove-that-cos-2-theta-frac-cos-2-t.htmlhttps://liberty.io/help-with-using-euler-s-formula-to-prove-that-cos-2-theta-frac-cos-2-t.htmlI have to use Euler's Formula to prove that: $$\cos^2(\theta) = \frac{\cos(2\theta)+1}{2}.$$ I have managed to prove this using trigonometric identities but I'm not sure how to use Euler's Formul......Tue, 25 Aug 2026 04:33:12 -0400https://liberty.io/question-about-an-isometric-immersion.htmlhttps://liberty.io/question-about-an-isometric-immersion.htmlThis is the question: Let $M,N$ be Riemannian manifolds, such that the inclusion $i:M \to N$ is a isometric immersion. Give a example where the inequality $d_M > d_N$ may occur. I thought ab......Tue, 25 Aug 2026 04:30:57 -0400https://liberty.io/complex-number-solutions-for-a-circle.htmlhttps://liberty.io/complex-number-solutions-for-a-circle.htmlI have a circle of radius 5 with its center at the origin represented as $X^2+Y^2=25$. I get that it has a solution for all values ranging from $-5$ to $+5$. My question is what does it mean when ......Tue, 25 Aug 2026 04:20:28 -0400https://liberty.io/is-there-a-pythagorean-triple-such-that-all-angles-of-the-correspondin.htmlhttps://liberty.io/is-there-a-pythagorean-triple-such-that-all-angles-of-the-correspondin.htmlObviously, the most interesting pythagorean triple $(a, b, c)$ would be one for which the corresponding triangle (with integer side lengths $a, b, c$) has angles 90°, 60° and 30° ($\frac{\pi}{2}, \......Tue, 25 Aug 2026 04:11:44 -0400https://liberty.io/what-is-the-output-of-a-function-which-is-not-defined-for-input-0-for-.htmlhttps://liberty.io/what-is-the-output-of-a-function-which-is-not-defined-for-input-0-for-.htmlMy teacher told me that every function has to be able to assign an output to every input we give to this function. What is than an output for a function which is not defined in zero? Nothing? Or do ...Tue, 25 Aug 2026 04:02:37 -0400https://liberty.io/hatcher-and-house-with-two-rooms.htmlhttps://liberty.io/hatcher-and-house-with-two-rooms.htmlOn page 4 of Hatcher's "Algebraic Topology" he constructs the "house with two rooms" space. He claims that there is some neighborhood containing this space that is homeomorphic to the unit ball (in $\...Tue, 25 Aug 2026 03:55:17 -0400https://liberty.io/proof-verification-for-proof-by-contraposition.htmlhttps://liberty.io/proof-verification-for-proof-by-contraposition.htmlprove using contraposition that if a and b are integers: $a^2-4b-2\neq0$ So the contrapositive: $a^2-4b-2=0$ implies that at least one of either a or b are not integers Working from the above, I go......Tue, 25 Aug 2026 03:50:14 -0400https://liberty.io/how-to-calculate-15-without-using-calculator.htmlhttps://liberty.io/how-to-calculate-15-without-using-calculator.htmlI am joining a maths competition and recently I am preparing for it. I came across a question that asks me to fill the blank of a number: 1_0767436_000 And this number is the product of $15!= 15\...Tue, 25 Aug 2026 03:44:10 -0400https://liberty.io/why-is-y-sqrt-x-4-a-function-and-y-sqrt-4-x-2-should-be-a-circle.htmlhttps://liberty.io/why-is-y-sqrt-x-4-a-function-and-y-sqrt-4-x-2-should-be-a-circle.htmlSo why is it a function, even though for example $x = 8$; you'll have $y = +2$ and $y = -2$. It'll fail the vertical line test. But every textbook considers it as a function. Did I misunderstand ...Tue, 25 Aug 2026 03:28:43 -0400https://liberty.io/what-is-the-intuition-behind-the-terminology-surrounding-baire-039-s-t.htmlhttps://liberty.io/what-is-the-intuition-behind-the-terminology-surrounding-baire-039-s-t.htmlBaire's theorem is considered fundamental for functional analysis. It could be simply stated as In complete metric spaces (e.g.) countable intersections of dense open sets are dense. But in ......Tue, 25 Aug 2026 03:28:17 -0400https://liberty.io/clarification-on-the-augmented-filtration.htmlhttps://liberty.io/clarification-on-the-augmented-filtration.htmlConsider the following definition. Definition. Let $\left(\Omega,\mathcal{F},\mathbb{P}\right)$ be a probability space and $W$ a Brownian motion. Let $\mathcal{F}^W_t=\sigma\left(\left\{W_s\mid ......Tue, 25 Aug 2026 03:26:54 -0400https://liberty.io/how-is-the-sum-of-simplices-defined.htmlhttps://liberty.io/how-is-the-sum-of-simplices-defined.htmlI have recently started to learn about simplices, simplicial complexes and simplicial homology. I understand that for some set of points $\{ p_0, \ldots, p_k \} \subset \mathbb{R}^n$ a simplex $\si......Tue, 25 Aug 2026 03:19:55 -0400https://liberty.io/linear-algebra-find-a-quadratic-function-so-that-it-fits-these-data.htmlhttps://liberty.io/linear-algebra-find-a-quadratic-function-so-that-it-fits-these-data.htmlFit a quadratic function of the form $f(t)=c_0+c_1t+c_2t^2$ to the data points (0, 0), (1, -9), (2, -2), (3, -19), using least squares. I'd like to know how to use least squares to do this. Thanks....Tue, 25 Aug 2026 03:10:22 -0400https://liberty.io/what-is-basis-of-a-group.htmlhttps://liberty.io/what-is-basis-of-a-group.htmlI have an assignment, that beside other things, asks about a basis of a group, more precisely, it have been asked to "give an example of a basis for $T_d$" (tetrahedron group). But I been on all ...Tue, 25 Aug 2026 03:04:22 -0400https://liberty.io/solving-coupled-second-order-odes-via-laplace-transforms-function-theo.htmlhttps://liberty.io/solving-coupled-second-order-odes-via-laplace-transforms-function-theo.htmlI have used Laplace transforms to transform a system of 2 coupled second order ODEs into 2 simultaneous equations. 1st ode: $$\frac{3d^2y}{dt^2}+\frac{dy}{dx}=0$$ 2nd ode: $$\frac{5d^2y}{dx^2}-......Tue, 25 Aug 2026 03:01:25 -0400https://liberty.io/how-to-rotate-a-line-segment-around-one-of-the-end-points.htmlhttps://liberty.io/how-to-rotate-a-line-segment-around-one-of-the-end-points.htmlI am given x1, y1, x2, y2 and θ. How can I find x3 and y3? By the way, there can be another line segment on the other side of AB (as if the line was rotated counter-clockwise). How to find that too?...Tue, 25 Aug 2026 03:00:26 -0400https://liberty.io/explicit-trapezoidal-rule.htmlhttps://liberty.io/explicit-trapezoidal-rule.htmlHi I am studying 'Explicit Methods Based on Quadrature'. I found that 'Heun's method may refer to the improved or modified Euler's method (that is, the explicit trapezoidal rule)'. See here for more ...Tue, 25 Aug 2026 03:00:19 -0400https://liberty.io/why-does-the-shortcut-for-finding-inverse-of-2x2-matrices-work-duplica.htmlhttps://liberty.io/why-does-the-shortcut-for-finding-inverse-of-2x2-matrices-work-duplica.htmlThere is a short-cut for finding the inverse of 2x2 matrices Can someone please explain why it works ? because I can't find any proof ?...Tue, 25 Aug 2026 02:48:03 -0400https://liberty.io/simplifying-square-of-integral-in-general.htmlhttps://liberty.io/simplifying-square-of-integral-in-general.htmlFor a real-valued function $f=f(x)$, over the real variable $x$, with the following integral $$ \left[ \int_{a}^{b} f(x)dx \right]^{2}, $$ is there a known general method/approach to handle this ......Tue, 25 Aug 2026 02:47:01 -0400https://liberty.io/finding-the-next-rational-number.htmlhttps://liberty.io/finding-the-next-rational-number.htmlA rational number is one that can be written as $a/b$ where $a$ and $b$ are integers, $b\gt0$ ($a$ can take care of negative rationals), and I suppose $\gcd(a,b) = 1$. Given some $n\in\mathbb{Q}$ ......Tue, 25 Aug 2026 02:46:40 -0400https://liberty.io/algebra-of-limits-what-nbd-means.htmlhttps://liberty.io/algebra-of-limits-what-nbd-means.html9. If $f(x)\le g(x)$ for every $x$ in the NBD of $a$, then $$\lim_{x\to a}f(x)\le\lim_{x\to a}g(x).$$ NBD stands for what?...Tue, 25 Aug 2026 02:44:40 -0400https://liberty.io/how-to-prove-a-complex-limit-with-epsilon-delta-definition.htmlhttps://liberty.io/how-to-prove-a-complex-limit-with-epsilon-delta-definition.htmlI have $$\lim_{z \to i} \frac {iz^3-1}{z+i}=0$$ To prove this I am trying to use the epsilon-delta definition. By saying that for any $\delta >0$ and any $\varepsilon >0$ then: $$0<|z-i|<\...Tue, 25 Aug 2026 02:44:14 -0400https://liberty.io/formula-for-odds-of-succeeding-at-least-one-1-4096-roll-out-of-n-attem.htmlhttps://liberty.io/formula-for-odds-of-succeeding-at-least-one-1-4096-roll-out-of-n-attem.htmlI'm trying to calculate the "expected" success/failure rate for getting a shiny pokemon after n encounters, which for my purposes has a flat 1/4096 chance of occuring. To my understanding......Tue, 25 Aug 2026 02:39:50 -0400https://liberty.io/equilibrium-solutions-of-differential-equations.htmlhttps://liberty.io/equilibrium-solutions-of-differential-equations.htmlFind the equilibrium solutions of the following differential equation: $$\dfrac{dy}{dt} = \dfrac{(t^2 - 1)(y^2 - 2)}{(y^2 -4)}$$ I'm not sure how to go about doing this since t appears explicitly......Tue, 25 Aug 2026 02:12:07 -0400https://liberty.io/a-function-that-satisfies-the-intermediate-value-theorem-and-takes-eac.htmlhttps://liberty.io/a-function-that-satisfies-the-intermediate-value-theorem-and-takes-eac.htmlI'm having a confusion over the veracity of the statement that a function that satisfies the Intermediate Value Theorem and takes each value only finitely many times is continuous. I've seen from a ...Tue, 25 Aug 2026 02:10:02 -0400https://liberty.io/how-to-use-matlab-for-plotting-functions-that-contain-summations.htmlhttps://liberty.io/how-to-use-matlab-for-plotting-functions-that-contain-summations.htmlI am having a terrible time trying to figure out how to plot this function in matlab: $$\frac{1}{\pi} + \frac{1}{2}\sin(4t) - \frac{2}{\pi} \sum\limits_{k=2,4,6,8}\frac{\cos(4kt)}{k^2-1}$$ I am not ...Tue, 25 Aug 2026 02:09:08 -0400https://liberty.io/formula-for-combinations-with-replacement.htmlhttps://liberty.io/formula-for-combinations-with-replacement.htmlI understand how combinations and permutations work (without replacement). I also see why a permutation of $n$ elements ordered $k$ at a time (with replacement) is equal to $n^{k}$. Through some br......Tue, 25 Aug 2026 02:05:23 -0400https://liberty.io/divisibility-by-41-and-5-digit-number.htmlhttps://liberty.io/divisibility-by-41-and-5-digit-number.htmlHow to prove that if a $5$-digit number is divisible by $41$,then all the numbers generated from it by cyclic shift are also divisible by $41$...Tue, 25 Aug 2026 02:04:10 -0400https://liberty.io/probability-of-getting-a-full-house.htmlhttps://liberty.io/probability-of-getting-a-full-house.htmlIf five cards are selected at random from a standard 52 card deck, what is the probability of getting a full house. This is what I am thinking. $(52*\binom{4}{3}*\binom{4}{2})/_{52}C_5$ Is that r......Tue, 25 Aug 2026 02:01:40 -0400https://liberty.io/how-can-i-find-the-equation-for-a-reverse-exponential-curve-based-on-t.htmlhttps://liberty.io/how-can-i-find-the-equation-for-a-reverse-exponential-curve-based-on-t.htmlI have a curve such that as $x$ approaches infinity, $y$ approaches $2$, and as $x$ approaches negative infinity, $y$ approaches infinity. I know it contains the following approximate points: $$(......Tue, 25 Aug 2026 01:58:51 -0400https://liberty.io/why-is-frac-15-sqrt-4-125-sqrt-4-5-15-sqrt-5-and-not-15-sqrt-4-25.htmlhttps://liberty.io/why-is-frac-15-sqrt-4-125-sqrt-4-5-15-sqrt-5-and-not-15-sqrt-4-25.htmlI have an expression I am to simplify: $$\frac{15\sqrt[4]{125}}{\sqrt[4]{5}}$$ I arrived at $15\sqrt[4]{25}$. My textbook tells me that the answer is in fact $15\sqrt{5}$. Here is my thought proc......Tue, 25 Aug 2026 01:56:56 -0400https://liberty.io/mathematical-toys.htmlhttps://liberty.io/mathematical-toys.htmlAnybody know of "serious" mathematical ornaments or toys like the Gömböc, etc? Already have a rubix and abacus (that's more of a tool though)....Tue, 25 Aug 2026 01:53:55 -0400https://liberty.io/appropriate-notation-equiv-versus.htmlhttps://liberty.io/appropriate-notation-equiv-versus.htmlWith respect to assignments/definitions, when is it appropriate to use $\equiv$ as in $$M \equiv \max\{b_1, b_2, \dots, b_n\}$$ which I encountered in my analysis textbook as opposed to the "c......Tue, 25 Aug 2026 01:53:37 -0400https://liberty.io/how-to-prove-x-p-i-implies-x-p-n-inp-n-1.htmlhttps://liberty.io/how-to-prove-x-p-i-implies-x-p-n-inp-n-1.htmlHow to prove $[x,p]=i$ $\implies$ $[x,p^n]=inp^{n-1}$? I can do this using $p=i\frac{d}{dx}$, but my book hasn't introduced this yet so is there another proof without using this ? These are just l......Tue, 25 Aug 2026 01:51:26 -0400https://liberty.io/is-an-equilateral-triangle-the-same-as-an-equiangular-triangle-in-any-.htmlhttps://liberty.io/is-an-equilateral-triangle-the-same-as-an-equiangular-triangle-in-any-.htmlI have heard of both equilateral triangles and equiangular triangles. (For example, this sporcle quiz lists both.) Are these always equivalent, regardless of geometry? I know they are the same in ...Tue, 25 Aug 2026 01:49:56 -0400https://liberty.io/logic-coin-game.htmlhttps://liberty.io/logic-coin-game.html2 people are playing a game with coins. First, 10 coins are put in a circular form. Then, the players take turns removing 1 coin or 2 coins that are together at each turn. The player who takes the ......Tue, 25 Aug 2026 01:36:26 -0400https://liberty.io/how-to-integrate-x-1-x-1.htmlhttps://liberty.io/how-to-integrate-x-1-x-1.htmlI want to calculate the integral $$\int\frac{x-1}{x+1}\,\mathrm{d}x.$$ I have tried solving it by differentiating the denominator and substituting it, but I didn't get it. How else can I solve it?...Tue, 25 Aug 2026 01:33:03 -0400https://liberty.io/user-mathma-mathematics-stack-exchange.htmlhttps://liberty.io/user-mathma-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Tue, 25 Aug 2026 01:31:34 -0400https://liberty.io/how-to-calculate-subspace-of-a-set-of-solutions-of-matrix-ax-b.htmlhttps://liberty.io/how-to-calculate-subspace-of-a-set-of-solutions-of-matrix-ax-b.htmlI am looking through some old linear algebra exam papers. However i do not understand how to calculate whether a set of solutions is within a certain subspace R. This is the problem in question: I......Tue, 25 Aug 2026 01:26:14 -0400https://liberty.io/integrate-gaussian-function.htmlhttps://liberty.io/integrate-gaussian-function.htmlI am trying to integrate Gaussian distribution from -m to m to find parametar A. I have done this so far: $\int_{-m}^{m}\frac{A}{\sqrt(2\pi)\sigma}e^{-(x-m)^{2}/(2\sigma^{2})}dx=1$ after $u=\fra......Tue, 25 Aug 2026 01:25:11 -0400https://liberty.io/lfloor-25x-2-4-rfloor-13x-4-3.htmlhttps://liberty.io/lfloor-25x-2-4-rfloor-13x-4-3.htmlThis problem is designated for those who have the basic knowledge of floor brackets I know the answer will be 6. But tell me how....Tue, 25 Aug 2026 01:23:17 -0400https://liberty.io/mathcad-variable-undefined-even-though-i-just-defined-it.htmlhttps://liberty.io/mathcad-variable-undefined-even-though-i-just-defined-it.htmlHoping to run into some mathcadders: I have a worksheet which includes the following except (and error) Any help?...Tue, 25 Aug 2026 01:20:57 -0400https://liberty.io/when-is-an-upper-triangular-matrix-invertible.htmlhttps://liberty.io/when-is-an-upper-triangular-matrix-invertible.htmlI did the following (the exercise assumes $2\times 2$ matrices): $$\begin{pmatrix} {a}&{b}\\ {0}&{d} \end{pmatrix} \begin{pmatrix} {A}&{B}\\ {C}&{D} \end{pmatrix} = \begin{pmatri......Tue, 25 Aug 2026 01:19:20 -0400https://liberty.io/receding-horizon-vs-finite-horizon-in-planning.htmlhttps://liberty.io/receding-horizon-vs-finite-horizon-in-planning.htmlSo, I am reading a paper on TSP. In one of the papers it is mentioned: We plan using a fixed-horizon approach, alternating between replanning and execution until the elapsed time $t$ exceeds the ...Tue, 25 Aug 2026 01:16:35 -0400https://liberty.io/on-solvibility-of-heat-equation-with-time-space-reversed.htmlhttps://liberty.io/on-solvibility-of-heat-equation-with-time-space-reversed.htmlConsider the heat equation $$u_t=u_{xx}, (t,x)\in \mathbb{R}^2$$ with initial condition on $x=0$: \begin{cases} u(0,t)=1+\sin t\\ u_x(0,t)=\sin\left(\dfrac{\pi}{4}+t\right) \end{cases} and with per......Tue, 25 Aug 2026 01:10:48 -0400https://liberty.io/calculus-dx-on-top-of-fraction.htmlhttps://liberty.io/calculus-dx-on-top-of-fraction.htmlI'm studying for a final, and I haven't seen any mention of any problem of this form in class or in my homework. I can't figure out how to go about solving this problem: $$\int^{e^6}_{1}{\frac{dx}......Tue, 25 Aug 2026 01:09:25 -0400https://liberty.io/characteristics-and-mantessa.htmlhttps://liberty.io/characteristics-and-mantessa.htmlI've just heard about these terms. Could someone elaborate on what's their use is? And plus could you explain it using a few examples?...Tue, 25 Aug 2026 01:05:46 -0400https://liberty.io/value-of-cdf-at-infinity.htmlhttps://liberty.io/value-of-cdf-at-infinity.htmlI had a question, given a continuous CDF F, is $F(+\infty)$ essentially equal to 1?...Tue, 25 Aug 2026 01:02:08 -0400https://liberty.io/how-to-convert-numbers-with-related-bases-quickly.htmlhttps://liberty.io/how-to-convert-numbers-with-related-bases-quickly.htmlLet: $a = (1011011)_2 = (1123)_4$ There two ways to solve it: Convert the number in base 2 to base 10 then to base $4$. Consider $4 = 2^2$ and group each of two numbers in base $2$ to one in ba......Tue, 25 Aug 2026 01:00:15 -0400https://liberty.io/diagonalizing-quadratic-forms-linear-algebra.htmlhttps://liberty.io/diagonalizing-quadratic-forms-linear-algebra.htmlI have a question that reads: Diagonalize the quadratic form $A(x,y) = 3x^2 -12xy + 7y^2$ by completing the square. What is diagonalization? Is that when I should find the eigenvector matrix, ......Tue, 25 Aug 2026 00:48:47 -0400https://liberty.io/does-negative-infinity-squared-positive-infinity.htmlhttps://liberty.io/does-negative-infinity-squared-positive-infinity.htmlI googled this question and saw this answer but I wasn't satisfied....Tue, 25 Aug 2026 00:43:41 -0400https://liberty.io/probability-of-there-being-a-bullet-in-the-chamber.htmlhttps://liberty.io/probability-of-there-being-a-bullet-in-the-chamber.htmlSuppose we have three $6$-chambered guns: The first has no bullets. The second has one bullet. The third has two bullets in consecutive chambers. The cylinder advances automatically as the trigg......Tue, 25 Aug 2026 00:42:05 -0400https://liberty.io/what-base-is-roman-numerals.htmlhttps://liberty.io/what-base-is-roman-numerals.htmlWhat is the base for Roman Numerals? It starts off with unary then goes back and forth between multiples of 5 and 10....Tue, 25 Aug 2026 00:39:08 -0400https://liberty.io/questions-tagged-binomial-coefficients.htmlhttps://liberty.io/questions-tagged-binomial-coefficients.htmlQ&A for people studying math at any level and professionals in related fields...Tue, 25 Aug 2026 00:36:40 -0400https://liberty.io/using-implicit-differentiation-with-a-fraction.htmlhttps://liberty.io/using-implicit-differentiation-with-a-fraction.htmlHow do I solve this? What steps? I have been beating my head into the wall all evening. $$ x^2 + y^2 = \frac{x}{y} + 4 $$...Tue, 25 Aug 2026 00:25:30 -0400https://liberty.io/exponential-growth-and-decay-compound-interest.htmlhttps://liberty.io/exponential-growth-and-decay-compound-interest.htmlThis is the question: "If you want to have $\$75,000$ after $35$ years in your account that pays $12\%$ annual interest compounded quarterly, how much should you put in as your original investment?......Tue, 25 Aug 2026 00:17:52 -0400https://liberty.io/stability-of-fixed-points-for-ode-range.htmlhttps://liberty.io/stability-of-fixed-points-for-ode-range.htmlGiven the ordinary differential equation $$\frac{dx}{dt} = x^3 - 2x^2 + (1- \mu)x,$$ where $\mu$ is a parameter, I want to find an expression for the fixed point(s) $x^*$ as a function of $\mu$; ......Tue, 25 Aug 2026 00:16:34 -0400https://liberty.io/three-mutually-tangent-circles-have-centers-at-given-distances-from-ea.htmlhttps://liberty.io/three-mutually-tangent-circles-have-centers-at-given-distances-from-ea.htmlThree circles of different radii are tangent to each other externally. The distance between their centers are $9\ cm$, $8\ cm$, and $11\ cm$. Find the radius of each circle. Find the area in betw......Tue, 25 Aug 2026 00:07:41 -0400https://liberty.io/calculate-width-and-height-of-a-rectangle-given-its-diagonal-and-ratio.htmlhttps://liberty.io/calculate-width-and-height-of-a-rectangle-given-its-diagonal-and-ratio.htmlWell, I know, it's easy. We did it in class some time ago and I forgot it, I'm stupid because I can't figure it out: E.g. I have a 32" TV with 16:9 ratio and I want to know its width and height. ......Tue, 25 Aug 2026 00:06:57 -0400https://liberty.io/difference-between-indefinite-and-definite-integration-int-f-x-dx-int-.htmlhttps://liberty.io/difference-between-indefinite-and-definite-integration-int-f-x-dx-int-.html$\int f(x) \;dx = \int^x_0 f(x) \; dx$ I sometimes get the impression that there's meant to be a basic and important difference between definite and indefinite integration. But when I consider sim......Mon, 24 Aug 2026 23:54:35 -0400https://liberty.io/meaning-of-c-in-the-linear-equation-ax-by-c.htmlhttps://liberty.io/meaning-of-c-in-the-linear-equation-ax-by-c.htmlSo I found: $ax+by=c$ What exactly does $c$ stand for? What does it do and why is $c$ important? I googled everywhere but couldn't find an answer other than $c$ being a constant....Mon, 24 Aug 2026 23:39:24 -0400https://liberty.io/is-the-game-2048-always-solveable.htmlhttps://liberty.io/is-the-game-2048-always-solveable.htmlGames got me on math. I always want to play best. I don't know how to answer my question. My question is : How to show that the game 2048 is (always) solvable>? Is there any method other than......Mon, 24 Aug 2026 23:37:39 -0400https://liberty.io/what-is-the-predual-of-l-1.htmlhttps://liberty.io/what-is-the-predual-of-l-1.htmlIs there a nice characterization of the predual of $L^1$? So, what does the space $X$ look like, such that $X^*=L^1$, where the star denotes the dual of a Banach space. How do you start to find such ...Mon, 24 Aug 2026 23:25:21 -0400https://liberty.io/inductive-proof-using-fitch-software.htmlhttps://liberty.io/inductive-proof-using-fitch-software.htmlI am trying to prove the integer square root theorem $\forall x: \mathbb{N}, \exists y : \mathbb{N}((y^2 \leq x) \land (x < (y+1)^2))$ for $\lfloor \sqrt{x} \rfloor$. In words: for any natural ...Mon, 24 Aug 2026 23:17:56 -0400https://liberty.io/how-to-simulate-monty-hall-problem-in-computer.htmlhttps://liberty.io/how-to-simulate-monty-hall-problem-in-computer.htmlSuppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1 [but the door is not opened], and the host,......Mon, 24 Aug 2026 23:13:48 -0400https://liberty.io/finding-the-complementary-function-of-a-second-order-linear-differenti.htmlhttps://liberty.io/finding-the-complementary-function-of-a-second-order-linear-differenti.htmlI have a question stating Find the complementary function of $\frac{d^2y}{dx^2} + 4\frac{dy}{dx} + 40y = x$ So I solved the auxillary equation, $ \lambda^2 + 4\lambda +40 = 0$, which gives $\......Mon, 24 Aug 2026 23:13:45 -0400https://liberty.io/distribution-of-the-difference-of-two-normal-random-variables.htmlhttps://liberty.io/distribution-of-the-difference-of-two-normal-random-variables.htmlIf $U$ and $V$ are independent identically distributed standard normal, what is the distribution of their difference? I will present my answer here. I am hoping to know if I am right or wrong. U......Mon, 24 Aug 2026 23:08:24 -0400https://liberty.io/correct-way-to-calculate-numeric-derivative-in-discrete-time.htmlhttps://liberty.io/correct-way-to-calculate-numeric-derivative-in-discrete-time.htmlGiven a set of discrete measurements in time $x_t, t \in \{0,\Delta t, 2\Delta t,\ldots,T-\Delta t,T\}$, what is the correct way to compute the discrete derivative $\dot x_t$. Is it more correct to......Mon, 24 Aug 2026 23:07:10 -0400https://liberty.io/given-two-events-a-and-b-what-is-the-probability-that-at-most-one-will.htmlhttps://liberty.io/given-two-events-a-and-b-what-is-the-probability-that-at-most-one-will.htmlI know this is a very basic question, but my mathematics is coming out a bit wonky. Assume Events A, B are independent. Define: $Pr(A) = 1/6$ $Pr(B) = 1/4$ Let Event C = "at most one event out ......Mon, 24 Aug 2026 22:56:16 -0400https://liberty.io/what-is-the-correct-inter-arrival-time-distribution-in-a-poisson-proce.htmlhttps://liberty.io/what-is-the-correct-inter-arrival-time-distribution-in-a-poisson-proce.htmlGiven a Poisson process (e.g. radioactive decay) with rate $\lambda$, then the expression $\exp(-\lambda t)$ is the probability of observing no counts in time interval $t$. This can be interpreted (...Mon, 24 Aug 2026 22:41:36 -0400https://liberty.io/understanding-of-the-formal-and-intuitive-definition-of-a-limit.htmlhttps://liberty.io/understanding-of-the-formal-and-intuitive-definition-of-a-limit.htmlThe intuitive definition for $\lim\limits_{x \to a} f(x) = L$ is the value of $f ( x )$ can be made arbitrarily close to $L$ by making $x$ sufficiently close, but not equal to, $a$ . I can easily ...Mon, 24 Aug 2026 22:36:20 -0400https://liberty.io/polynomials-as-vector-spaces.htmlhttps://liberty.io/polynomials-as-vector-spaces.htmlCan someone please explain how polynomials are vector spaces? I don't understand this at all. Vectors are straight, so how could a polynomial be a vector. Is it just that the coefficients are vector ...Mon, 24 Aug 2026 22:34:50 -0400https://liberty.io/how-can-i-find-the-horizontal-asymptotes-of-the-function-y-frac-x-3-sq.htmlhttps://liberty.io/how-can-i-find-the-horizontal-asymptotes-of-the-function-y-frac-x-3-sq.htmlIt is not a homework question. I ask this question because it was on my test and I genuinely want to understand. I have a very minimal knowledge on the subject of limits. The question is: How can I......Mon, 24 Aug 2026 22:30:52 -0400