https://liberty.ioCurated star highlights with fresh energy.en-usWed, 12 Aug 2026 01:17:27 -0400https://liberty.io/show-that-for-all-real-numbers-a-and-b-ab-le-1-2-a-2-b-2-duplicate.htmlhttps://liberty.io/show-that-for-all-real-numbers-a-and-b-ab-le-1-2-a-2-b-2-duplicate.htmlso as in the title, I have the following theorem to prove. Theorem Show that for all $a$, $b\in \mathbb R$, that the following inequality holds, $\begin{equation} ab \leq \frac{1}{2}(a^2 + b^2) ......Wed, 12 Aug 2026 05:15:22 -0400https://liberty.io/using-square-area-finding-quadrilateral-area.htmlhttps://liberty.io/using-square-area-finding-quadrilateral-area.htmlArea of square ABCD is 169 and that of square EFGD is 49. Find area of quadrilateral FBCG I am stuck and just thinking which way can be helpful for me finding this area of quadrilateral FBCG. Inst......Wed, 12 Aug 2026 05:11:23 -0400https://liberty.io/difference-between-factorization-theorem-and-fischer-neymann-theorem-f.htmlhttps://liberty.io/difference-between-factorization-theorem-and-fischer-neymann-theorem-f.htmlDifference between Factorization theorem and Fischer-Neymann theorem for t to be sufficient estimator of $\theta$ Factorization theorem says for parameter t to be sufficient statistic of $\theta$,......Wed, 12 Aug 2026 05:09:15 -0400https://liberty.io/where-039-s-the-error-in-this-2-1-fake-proof-duplicate.htmlhttps://liberty.io/where-039-s-the-error-in-this-2-1-fake-proof-duplicate.htmlI'm reading Spivak's Calculus: 2 What's wrong with the following "proof"? Let $x=y$. Then $$x^2=xy\tag{1}$$ $$x^2-y^2=xy-y^2\tag{2}$$ $$(x+y)(x-y)=y(x-y)\tag{3}$$ $$x+y=y\tag{4}$$ $$2y=y......Wed, 12 Aug 2026 04:57:55 -0400https://liberty.io/total-injective-relations-are-always-functions.htmlhttps://liberty.io/total-injective-relations-are-always-functions.htmlI was watching the following video, and around 6:12, he says "Whenever there is a total injective relation, there is always a total injective function" as well. https://ocw.mit.edu/courses/electr......Wed, 12 Aug 2026 04:56:35 -0400https://liberty.io/is-there-a-way-to-solve-sin-x-x.htmlhttps://liberty.io/is-there-a-way-to-solve-sin-x-x.htmlNote: Question was originally to solve it algebraically, though I've decided to change it to analytically due to the comments and answers. When trying to solve $\sin(x)=x$, the obvious first solut......Wed, 12 Aug 2026 04:53:31 -0400https://liberty.io/euler-and-the-factorial-function.htmlhttps://liberty.io/euler-and-the-factorial-function.htmlI recently purchased H. M. Edwards' book entitled The Riemann Zeta Function. In the early pages of the volume, concerning the factorial function $\Gamma$, Edwards notes that "Euler observed that $\...Wed, 12 Aug 2026 04:51:05 -0400https://liberty.io/asymptote-of-a-line.htmlhttps://liberty.io/asymptote-of-a-line.htmlIf a rational function is defined as $f(x) = \frac{p(x)} {q(x)}$ for any two polynomials $p(x)$ and $q(x)$; given $p(x) = 2x+1$ (polynomial of degree 1) and $q(x) = 3$ (polynomial of degree zero), ......Wed, 12 Aug 2026 04:50:22 -0400https://liberty.io/properties-of-triangles-that-have-3-equal-120-degree-angles-around-an-.htmlhttps://liberty.io/properties-of-triangles-that-have-3-equal-120-degree-angles-around-an-.htmlWhere is the point in a triangle, that when connected to the three vertices forms equal angles of 120 degrees? Does this point exist in all triangles? For a triangle with side lengths of 2,4,5, this ...Wed, 12 Aug 2026 04:40:22 -0400https://liberty.io/projection-of-ellipsoid.htmlhttps://liberty.io/projection-of-ellipsoid.htmlFind the projections of the ellipsoid $$ x^2 + y^2 + z^2 -xy -1 = 0$$ on the cordinates plan I have no idea how to do this. I couldn't find much on google to help me with it too. Thanks in adva......Wed, 12 Aug 2026 04:23:59 -0400https://liberty.io/what-is-a-covering-set-of-a-sierpinski-number-what-does-it-do.htmlhttps://liberty.io/what-is-a-covering-set-of-a-sierpinski-number-what-does-it-do.htmlRecently a new prime number has been discovered, which eliminates one of the six remaining candidates for the smallest Sierpinski numbers. So I was reading the wikipedia article about the Sierpinski ...Wed, 12 Aug 2026 04:16:36 -0400https://liberty.io/when-given-the-dimensions-of-a-cylinder-a-x-b-x-c-what-does-each-lette.htmlhttps://liberty.io/when-given-the-dimensions-of-a-cylinder-a-x-b-x-c-what-does-each-lette.htmlWhen looking at fire extinguishers online, the dimensions are always listed in the above format (A x B x C). What does each letter represent? Example: 3" x 5.5" x 14.9" I assume the third (C) is ...Wed, 12 Aug 2026 04:12:59 -0400https://liberty.io/choose-a-coefficient-so-that-the-system-of-equations-has-exactly-one-s.htmlhttps://liberty.io/choose-a-coefficient-so-that-the-system-of-equations-has-exactly-one-s.htmlChoose the value of $a$ from $\{-4, 88\}$ so that the system of equations has exactly one solution: $$2x +20y+3z=1$$ $$2x +2y+41z=2$$ $$ax -22y-44z=-3$$ I tried solving the system for $a=-4$ and ......Wed, 12 Aug 2026 04:12:31 -0400https://liberty.io/why-does-trapezoidal-rule-have-potential-error-greater-than-midpoint.htmlhttps://liberty.io/why-does-trapezoidal-rule-have-potential-error-greater-than-midpoint.htmlI can approximate the area beneath a curve using the Midpoint and Trapezoidal methods, with errors such that: $Error_m \leq \frac{k(b-a)^3}{24n^2}$ and $Error_T \leq \frac{k(b-a)^3}{12n^2}$. Does......Wed, 12 Aug 2026 04:07:36 -0400https://liberty.io/a-place-to-learn-about-math-etymology.htmlhttps://liberty.io/a-place-to-learn-about-math-etymology.htmlI was recently wondering where the word `kernel' comes from in mathematics. I am sure the internet must know. I did manage to find http://www.pballew.net/etyindex.html#k which contains the ori......Wed, 12 Aug 2026 03:58:08 -0400https://liberty.io/integer-coefficients-of-cubic-equation-imply-integer-roots.htmlhttps://liberty.io/integer-coefficients-of-cubic-equation-imply-integer-roots.htmlProblem: Let $a,b,c$ be three integers for which the sum $ \frac{ab}{c}+ \frac{ac}{b}+ \frac{bc}{a}$ is integer. Prove that each of the three numbers $ \frac{ab}{c}, \quad \frac{ac}{b},\quad \frac......Wed, 12 Aug 2026 03:55:33 -0400https://liberty.io/in-classical-geometry-why-is-a-line-considered-to-be-parallel-to-itsel.htmlhttps://liberty.io/in-classical-geometry-why-is-a-line-considered-to-be-parallel-to-itsel.htmlA definition in classical geometry (for example, Birkhoff's formulation, but I suppose it could be all of them) is that a line is always considered to be parallel to itself. I understand this is pr......Wed, 12 Aug 2026 03:54:20 -0400https://liberty.io/find-difference-between-a-positive-and-a-negative-number.htmlhttps://liberty.io/find-difference-between-a-positive-and-a-negative-number.htmlI am trying to work out the percentage difference between two numbers (one of which is negative), NOT PERCENTAGE CHANGE. I have done quite a bit of reading, but there doesn't appear to be a definit......Wed, 12 Aug 2026 03:47:31 -0400https://liberty.io/prove-the-median-of-a-uniform-distributions-is-frac-1-2-a-b.htmlhttps://liberty.io/prove-the-median-of-a-uniform-distributions-is-frac-1-2-a-b.htmlLet X~U(a,b) with a and b in the real line, such that b>a with X's pdf given by $$f_X(x)=\frac{1}{b-a}\mathbb{1}(a<x< b)$$ Show the median of X's distribution is given by $$m=\frac{1}{2}(b+a)......Wed, 12 Aug 2026 03:33:43 -0400https://liberty.io/getting-a-hermite-polynomial-expansion-of-gaussian-with-given-variance.htmlhttps://liberty.io/getting-a-hermite-polynomial-expansion-of-gaussian-with-given-variance.htmlI am trying to find an expansion of centered Gaussian - $\frac{1}{\sqrt{2\pi}\sigma}\exp({-\frac{x^2}{2\sigma^2})}$ in terms of Hermite polynomials. Namely to calculate $a_n=\frac{1}{\sqrt{2\pi}\......Wed, 12 Aug 2026 03:32:26 -0400https://liberty.io/parallelogram-constructed-through-medians.htmlhttps://liberty.io/parallelogram-constructed-through-medians.htmlBdmo In $\Delta ABC$, Medians AD and CF intersect at P.Let Q be any point on AC.Construct QM and QN parallel to AD and CF respectively.Now the line joining M and N intersects CF and T and AD at ......Wed, 12 Aug 2026 03:06:59 -0400https://liberty.io/what-is-the-practical-difference-between-a-differential-and-a-derivati.htmlhttps://liberty.io/what-is-the-practical-difference-between-a-differential-and-a-derivati.htmlI ask because, as a first-year calculus student, I am running into the fact that I didn't quite get this down when understanding the derivative: So, a derivative is the rate of change of a functio......Wed, 12 Aug 2026 03:06:25 -0400https://liberty.io/sum-of-perfect-cubes-formula-proof-without-induction-duplicate.htmlhttps://liberty.io/sum-of-perfect-cubes-formula-proof-without-induction-duplicate.htmlis there any proof for the sum of cubes except induction supposition? there are some proofs using induction in below page Proving $1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$ using ...Wed, 12 Aug 2026 03:03:40 -0400https://liberty.io/integral-of-d-2y-dx-2-dy.htmlhttps://liberty.io/integral-of-d-2y-dx-2-dy.htmlI saw this as a step in a calculation and it was confusing. The left cancels down to: $$\int \frac{d^2y}{dx^2}dy$$ But surely the answer is $\frac{d^2y}{dx^2}y$; instead, $\frac {d^2}{dx^2}$ is being ...Wed, 12 Aug 2026 03:03:29 -0400https://liberty.io/union-and-difference-of-convex-set.htmlhttps://liberty.io/union-and-difference-of-convex-set.htmlsuppose X,Y are two convex sets x1, x2 in X and y1, y2 in Y defn of X and Y being convex: tx1+(1-t)x2 in X ty1+(1-t)y2 in Y it is clear that: 1) X+Y is convex. 2) X intersection Y is convex 3) ...Wed, 12 Aug 2026 02:37:20 -0400https://liberty.io/integrate-e-2x-sin-x.htmlhttps://liberty.io/integrate-e-2x-sin-x.htmlFirst of all, sorry for my poor English. Can I integrate the following using integration by parts, because I've tried a lot and it didn't work. (And sorry if it's too easy for you, guys! I'm such a......Wed, 12 Aug 2026 02:31:59 -0400https://liberty.io/how-to-prove-sum-frac-1-n-1-1-n-is-divergent.htmlhttps://liberty.io/how-to-prove-sum-frac-1-n-1-1-n-is-divergent.htmlWe know that $\sum \frac{1}{n^p}$ is convergent for $p>1$. However the series $\sum {\frac{1}{n^{1+1/n}}}$ is apparently divergent since $1+1/n$ tends to 1 as $n$ tends to infinity. But how to p......Wed, 12 Aug 2026 02:25:59 -0400https://liberty.io/complex-analysis-trigonometric-conjugate.htmlhttps://liberty.io/complex-analysis-trigonometric-conjugate.htmlI'm having trouble with this assignment: Show that $ \overline {\cos(z)} = \cos(\bar z)$, where $\bar z$ represents conjugate of $z$. I know that the two are equal, but how to mathematically show......Wed, 12 Aug 2026 02:25:08 -0400https://liberty.io/graph-of-negative-exponential-function.htmlhttps://liberty.io/graph-of-negative-exponential-function.htmlIs it possible to graph a function $Y = (-2)^x$ where $x \geq 1$ It is written in my text that exponential functions are only defined for positive bases, but why not negative bases having restricti......Wed, 12 Aug 2026 02:20:16 -0400https://liberty.io/limit-approaching-infinity-of-sine-function.htmlhttps://liberty.io/limit-approaching-infinity-of-sine-function.htmlI'd like to ask a question which I have been reflecting on for some time now. What is the limit of: $f(x) = \sin(x)$ as $x$ tends to infinity? As we know, the function has a definite value for each ...Wed, 12 Aug 2026 02:17:33 -0400https://liberty.io/what-039-s-an-isomorphism.htmlhttps://liberty.io/what-039-s-an-isomorphism.htmlI'm familiar with the definition (inverses and bijections, preserving operations) in the context of groups and vector spaces, the hoeomorphism of topological spaces, and have some feeling for the ...Wed, 12 Aug 2026 02:13:33 -0400https://liberty.io/why-is-the-chain-rule-applied-to-derivatives-of-trigonometric-function.htmlhttps://liberty.io/why-is-the-chain-rule-applied-to-derivatives-of-trigonometric-function.htmlI'm having trouble to understand why is the Chain rule applied to trigonometric functions, like: $$\frac{d}{dx}\cos 2x=[2x]'*[\cos 2x]'=-2 \sin 2x$$ Why isn't it like in other variable derivatives?......Wed, 12 Aug 2026 02:10:18 -0400https://liberty.io/reflexive-graph-meaning-of-the-reflection.htmlhttps://liberty.io/reflexive-graph-meaning-of-the-reflection.htmlHere on the page 1 there is a definition of reflexive graph. I need an intuition how it works the morphism $e:X_0\to X_1.$ What is it and to what edge in $X_1$ it sends a vertex from $X_0$?...Wed, 12 Aug 2026 01:57:46 -0400https://liberty.io/leibniz-convergence-test.htmlhttps://liberty.io/leibniz-convergence-test.htmlI want to prove if $\sum\limits_{n=2}^\infty (-1)^n \frac{1}{n!}$ is convergent or not. I can prove this by the $ratio \ test$. Isn't this series an alternating series with $a_n = (-1)^n$ and $b_......Wed, 12 Aug 2026 01:56:17 -0400https://liberty.io/hadamard-039-s-inequality-proof.htmlhttps://liberty.io/hadamard-039-s-inequality-proof.htmlI have the following inequality to prove. With $A \in M_n(R)$ show that: $$ (\det(A))^2 \leq \prod_{i=1}^n\left( \sum_{k=1}^n A_{k,i}^2\right) $$ What I already have: I found out that: $$G(v_1,......Wed, 12 Aug 2026 01:55:24 -0400https://liberty.io/elliptical-version-of-pythagoras-theorem.htmlhttps://liberty.io/elliptical-version-of-pythagoras-theorem.htmlConsider any right triangle $\triangle ABC$. We focus on one side, $AC$, and we take the midpoint $E$ of this side. Then, we draw the circle with center in $E$ and passing by $A,C$. If we take the ...Wed, 12 Aug 2026 01:52:00 -0400https://liberty.io/continuous-mapping-on-a-compact-metric-space-is-uniformly-continuous.htmlhttps://liberty.io/continuous-mapping-on-a-compact-metric-space-is-uniformly-continuous.htmlI am struggling with this question: Prove or give a counterexample: If $f : X \to Y$ is a continuous mapping from a compact metric space $X$, then $f$ is uniformly continuous on $X$. Thanks fo......Wed, 12 Aug 2026 01:49:58 -0400https://liberty.io/how-do-we-find-the-value-of-x-where-the-tangent-line-to-f-x-is-horizon.htmlhttps://liberty.io/how-do-we-find-the-value-of-x-where-the-tangent-line-to-f-x-is-horizon.htmlConsider the function $f(x)=9x^2 +7x$ The derivative is $f'(x)=18x+7$ The slope of the tangent to the graph of $f(x)$ at $x=2$ is $43$ The equation for the tangent line at $x=2$ is $y=43x-36$ I fo......Wed, 12 Aug 2026 01:19:24 -0400https://liberty.io/uniform-convergence-problem-of-sequences-of-functions.htmlhttps://liberty.io/uniform-convergence-problem-of-sequences-of-functions.htmlI have been trying to solve this problem for hours... At this point, I believe that there is a mistake in the problem, but I assume that I am in the one who is wrong. May you help me? Here is the ...Wed, 12 Aug 2026 01:16:01 -0400https://liberty.io/algorithm-of-tree-partition.htmlhttps://liberty.io/algorithm-of-tree-partition.htmlI am looking for algorithm which allows to do following: Let $T$ is some tree, where each vertex $V_i$ has its cost $c_i$. I am looking for algorithm which allows to split $T$ (by removing two ed......Wed, 12 Aug 2026 01:09:29 -0400https://liberty.io/why-is-the-distance-between-two-circles-spheres-that-don-039-t-interse.htmlhttps://liberty.io/why-is-the-distance-between-two-circles-spheres-that-don-039-t-interse.htmlFrom GRE 0568 From MathematicsGRE.Com: I'm guessing the idea applies to circles also? Is there a way to prove this besides the following non-elegant way? Form a line between centers $C_1$ and $C......Wed, 12 Aug 2026 01:09:25 -0400https://liberty.io/why-are-the-two-probabilities-in-the-monty-hall-problem-dependent-on-e.htmlhttps://liberty.io/why-are-the-two-probabilities-in-the-monty-hall-problem-dependent-on-e.htmlLike most people, one of the first things I did after ringing in the new year was get into a discussion about the Monty Hall problem. Past discussions typically amounted to the other person saying,......Wed, 12 Aug 2026 00:50:25 -0400https://liberty.io/get-rid-of-the-square-roots-of-the-denominator-dfrac-1-sqrt-7-2-sqrt-5.htmlhttps://liberty.io/get-rid-of-the-square-roots-of-the-denominator-dfrac-1-sqrt-7-2-sqrt-5.htmlHow to get rid of the square roots of the denominator: $\dfrac{1}{\sqrt{7}-2\sqrt{5}+\sqrt{3}}$? I squared the whole denominator, but that didn't help. Also I searched for a propriety or identity ......Wed, 12 Aug 2026 00:47:41 -0400https://liberty.io/rudin-039-s-principles-of-mathematical-analysis-or-apostol-039-s-mathe.htmlhttps://liberty.io/rudin-039-s-principles-of-mathematical-analysis-or-apostol-039-s-mathe.htmlI am in high school and have no access to a professor or anyone. I previously used Calculus Volume I by Tom Apostol and Spivak's Calculus (for the differential calculus bit). I can choose between ...Wed, 12 Aug 2026 00:46:57 -0400https://liberty.io/diagonally-dominant-matrix-by-rows-and-or-by-columns.htmlhttps://liberty.io/diagonally-dominant-matrix-by-rows-and-or-by-columns.htmlConsider a binary square matrix ${\bf B} \in \{0,1\}^{N \times N}$, with $b_{i,i} = 0$ (this can be, for example, the adjacency matrix of a directed graph). Let's define $r_i = \sum_{j=1}^N b_{i,j......Wed, 12 Aug 2026 00:43:05 -0400https://liberty.io/what-does-p-to-q-mean-in-propositional-logic.htmlhttps://liberty.io/what-does-p-to-q-mean-in-propositional-logic.htmlI have a question that is focussed on a more language oriented side of propositional logic. In such a way that I have no clue what the question means. I want to know what the "-q to -p" part means......Wed, 12 Aug 2026 00:38:04 -0400https://liberty.io/find-all-irreducible-monic-polynomials-in-mathbb-z-2-x-with-degree-equ.htmlhttps://liberty.io/find-all-irreducible-monic-polynomials-in-mathbb-z-2-x-with-degree-equ.htmlFind all irreducible monic polynomials in $\mathbb{Z}/(2)[x]$ with degree equal or less than $5$. This is what I tried: It's evident that $x,x+1$ are irreducible. Then, use these to find all redu......Wed, 12 Aug 2026 00:18:18 -0400https://liberty.io/integrating-the-area-of-a-quarter-circle.htmlhttps://liberty.io/integrating-the-area-of-a-quarter-circle.html$A=\int_0^r\sqrt{r^2-x^2}\,dx$ is the area of a quarter-circle. Show why, with a graph and thin rectangles. Calculate this integral by substituting $x=r\sin\theta$ and $\,dx = r\cos\theta\,d\theta$......Wed, 12 Aug 2026 00:16:12 -0400https://liberty.io/integrate-int-frac-1-sqrt-x-2-1-dx.htmlhttps://liberty.io/integrate-int-frac-1-sqrt-x-2-1-dx.htmlI am solving an ODE $$y^2 \, dx + \left(x\sqrt{y^2 - x^2} - xy\right)dy=0$$ I let $y=xu$ and did some algebra, and I ended up with this $$\ln(x)+C=\int \frac{1}{\sqrt{u^2 - 1}} \, du - \int \fra......Wed, 12 Aug 2026 00:15:16 -0400https://liberty.io/compounded-interest-differential-equation.htmlhttps://liberty.io/compounded-interest-differential-equation.htmlYou borrow $8000 to buy a car. The lender charges an annual rate of 10% compounded continuously. You make payments of k dollars per year continuously. A) write a differential equation describing the ...Wed, 12 Aug 2026 00:13:53 -0400https://liberty.io/proof-by-induction-involving-factorials.htmlhttps://liberty.io/proof-by-induction-involving-factorials.htmlMy "factorial" abilities are a slightly rusty and although I know of a few simplifications such as: $(n+1)\,n! = (n+1)!$, I'm stuck I have to prove by induction that: $$\sum_{i=1}^n\frac{i-1}{i!}......Wed, 12 Aug 2026 00:11:42 -0400https://liberty.io/how-is-the-dot-product-of-two-cartesian-unit-vectors-equal-to-the-kroe.htmlhttps://liberty.io/how-is-the-dot-product-of-two-cartesian-unit-vectors-equal-to-the-kroe.htmlIn my lectures, the relationship $\underline{\hat{e_k}} \cdot \underline{\hat{ e_j}}$ is given to be equal to $\delta_{kj}$ - how is this so? From my understanding this is equal to $ \left( \begin{...Wed, 12 Aug 2026 00:01:28 -0400https://liberty.io/example-of-non-compact-set-for-rationals.htmlhttps://liberty.io/example-of-non-compact-set-for-rationals.htmlthere is an example from lectures which i do not understand. given a closed set [0,1] , for the below set K, how is there not a finite subcover on K? why is the set K=[0,1] $\bigcap \mathbb{Q}$ ......Tue, 11 Aug 2026 23:48:43 -0400https://liberty.io/evaluating-int-0-infty-frac-arctan-x-2-ln-1-x-4-x-x-8-x-4-1-mathrm-dx.htmlhttps://liberty.io/evaluating-int-0-infty-frac-arctan-x-2-ln-1-x-4-x-x-8-x-4-1-mathrm-dx.htmlSo, my friend has challenged me to solve the following integral $\displaystyle \tag*{} I=\int_0^\infty \frac{\arctan (x^2) \ln(1+x^4)}{x(x^8+x^4+1)} \mathrm dx$ I started by doing $x^2=t$, so $\...Tue, 11 Aug 2026 23:44:16 -0400https://liberty.io/unequal-circles-within-circle-with-least-possible-radius.htmlhttps://liberty.io/unequal-circles-within-circle-with-least-possible-radius.htmlIt is the classical will-my-cables-fit-within-the-tube-problem which lead me to the interest of circle packing. So basically, I have 3 circles where r = 3 and 1 circle where r = 7 and I am trying to ...Tue, 11 Aug 2026 23:43:12 -0400https://liberty.io/how-do-i-solve-this-analytically-3-x-9x.htmlhttps://liberty.io/how-do-i-solve-this-analytically-3-x-9x.htmlOne of my friends ask me how to solve this equation analytically $3^x=9x$. Looking at it I guess 3 is the answer and I also plot a graph of line $9x$ and the curve $3^x$, they intersect at 3. But,......Tue, 11 Aug 2026 23:33:36 -0400https://liberty.io/is-there-a-general-solution-to-the-water-bucket-logic-problem-duplicat.htmlhttps://liberty.io/is-there-a-general-solution-to-the-water-bucket-logic-problem-duplicat.htmlHaving an infinite supply of water and two containers, one for 3 liters and one for 5 liters, how would you measure 4 liters? Each step in the solution can be one of three things: Fill up a contai......Tue, 11 Aug 2026 23:29:51 -0400https://liberty.io/how-to-understand-conjugate-points-on-a-riemannian-manifold.htmlhttps://liberty.io/how-to-understand-conjugate-points-on-a-riemannian-manifold.htmlI'm having trouble grasping what it means for two points to be conjugate on a Riemannian manifold. Could someone provide a geometric or intuitive explanation for this? For clarification: given a ...Tue, 11 Aug 2026 23:18:12 -0400https://liberty.io/trying-to-understand-polar-coordinate-vectors.htmlhttps://liberty.io/trying-to-understand-polar-coordinate-vectors.htmlI'm trying to understand what the unit polar coordinate vectors (I'll denote them $\hat r$ and $\hat \phi$) are and if they form a basis for $\Bbb R^2$. So from what I understand, $\hat r$ points ...Tue, 11 Aug 2026 23:03:47 -0400https://liberty.io/how-to-find-one-endpoint-with-the-other-end-point-and-a-point-1-3-away.htmlhttps://liberty.io/how-to-find-one-endpoint-with-the-other-end-point-and-a-point-1-3-away.htmlIf you had one endpoint and a point 1/3 of the way away from that endpoint. How would you find the other endpoint?...Tue, 11 Aug 2026 22:56:38 -0400https://liberty.io/how-to-find-the-area-bounded-by-two-curves.htmlhttps://liberty.io/how-to-find-the-area-bounded-by-two-curves.htmlfind the area bounded by the curves $y = x^2$ and $y = 5x-6$. First method I tried was to find where the two curves meet, which is at $x = 2, x =3$. Then integrated $x^2$ with limits from $0$ t......Tue, 11 Aug 2026 22:53:02 -0400https://liberty.io/proving-an-inequality-using-the-cauchy-schwarz-inequality-with-a-given.htmlhttps://liberty.io/proving-an-inequality-using-the-cauchy-schwarz-inequality-with-a-given.htmlit is given that $x^2 + 2y^2 + 3z^2 = \frac{1}{126}$ (all variables are real numbers) And i need to prove that it is always the case that: $7x + 10y + 9z \leq 1$ and what I've been able to come up ......Tue, 11 Aug 2026 22:52:29 -0400https://liberty.io/what-039-s-the-least-positive-integer-that-is-not-a-factor-of-25-and-i.htmlhttps://liberty.io/what-039-s-the-least-positive-integer-that-is-not-a-factor-of-25-and-i.htmlIt's more of a question of what's the definition of factor. To my understanding, for a number 25, 1,5,25 would be the factor because 1*25=25and 5*5=25. It is a question from GRE official guide book......Tue, 11 Aug 2026 22:40:53 -0400https://liberty.io/how-to-find-the-determinant-of-a-matrix-using-the-pa-lu-method.htmlhttps://liberty.io/how-to-find-the-determinant-of-a-matrix-using-the-pa-lu-method.htmlI am currently trying to write a code in matlab for a course I'm taking. My program needs to implement the PA = LU method to solve a matrix and compute it's determinant. I am not allowed to use the......Tue, 11 Aug 2026 22:11:57 -0400https://liberty.io/conjugates-of-complex-numbers.htmlhttps://liberty.io/conjugates-of-complex-numbers.htmlThis is from a textbook: $w=i\frac{1-z}{1+z}$ where $z$ is also a complex number Then $$\mathrm{Im}(w)=\frac{w-\bar w}{2i}$$ Then $$\mathrm{Im}(w)=\frac 1 {2i} \left(i\frac{1-z}{1+z}+i......Tue, 11 Aug 2026 22:11:08 -0400https://liberty.io/how-do-i-simplify-the-expression-a-1-b-1-1-closed.htmlhttps://liberty.io/how-do-i-simplify-the-expression-a-1-b-1-1-closed.htmlHow do I simplify the expression ... $$(a^{-1} + b^{-1}) ^{-1}$$ ?...Tue, 11 Aug 2026 22:07:01 -0400https://liberty.io/composite-trapezoidal-rule-and-simpson-rule-question.htmlhttps://liberty.io/composite-trapezoidal-rule-and-simpson-rule-question.htmlWe have the following integral $$\int_{1}^{3} \frac x{x^2+4}\; dx$$ and $n=6$. I have to approximate it using composite trapezoidal rule and then composite simpson rule. Using trapezoidal rule: $$\...Tue, 11 Aug 2026 22:06:12 -0400https://liberty.io/circle-puzzle-geometry.htmlhttps://liberty.io/circle-puzzle-geometry.htmlTwo friends are playing a game. One friend stands in the middle of a circle radius 100m. His objective is to leave the circle. He may take one step at a time, distance 1m, in any direction. However......Tue, 11 Aug 2026 22:05:41 -0400https://liberty.io/chain-rule-partial-derivatives-dw-dt.htmlhttps://liberty.io/chain-rule-partial-derivatives-dw-dt.htmlExpress dw/dt as function of t, if w = xy, x = cos t, y = sen t, z = t My first step is to sketch the tree. w - x - t w - y - t w - z - t dw/dx = y dw/dy = x dw/dz = 1 Then: dx/dt = -sen......Tue, 11 Aug 2026 22:00:38 -0400https://liberty.io/writing-a-general-formula-for-nth-derivative-of-cos-ax.htmlhttps://liberty.io/writing-a-general-formula-for-nth-derivative-of-cos-ax.htmlSo as stated in the title I have to create a general formula for nth derivative of $f(x) = \cos(ax)$ and proving it with induction. I get $f(x) = \cos(ax)$ $f'(x) = -a\sin(ax)$ $f''(x) = -a^2\......Tue, 11 Aug 2026 21:55:20 -0400https://liberty.io/multiplying-reciprocal-fractions-with-exponents.htmlhttps://liberty.io/multiplying-reciprocal-fractions-with-exponents.htmlThe question is as follows: If we multiply a fraction with its reciprocal then we get 1. So by this logic this should be equal to 1^(a/a+b). But, the solutions I have been provided with states the ...Tue, 11 Aug 2026 21:54:59 -0400https://liberty.io/what-039-s-the-difference-between-early-transcendentals-and-late-trans.htmlhttps://liberty.io/what-039-s-the-difference-between-early-transcendentals-and-late-trans.htmlAnton, Bivens, and Davis have written calculus books with late transcendentals and Stewart has a calculus book with early transcendentals. What's this all about? edit 1: (Both terms show up in ......Tue, 11 Aug 2026 21:45:10 -0400https://liberty.io/what-is-the-base-of-log-x-duplicate.htmlhttps://liberty.io/what-is-the-base-of-log-x-duplicate.htmlI've seen "$\log x$" being used in some papers (and by Wolfram|Alpha), and I was confused because so far I have only ever seen the $\log$ used with a base ( so e.g. $\log_y x$). Am I correct that ......Tue, 11 Aug 2026 21:43:37 -0400https://liberty.io/why-does-the-mandelbrot-shape-show-up-in-other-fractals.htmlhttps://liberty.io/why-does-the-mandelbrot-shape-show-up-in-other-fractals.htmlIn the pictures below, the Collatz map fractal includes parts resembling the Mandelbrot set. Why? Do other fractals do so? The Mandelbrot set From Wikimedia Commons Part of the Collatz map fract......Tue, 11 Aug 2026 21:41:21 -0400https://liberty.io/how-many-subsets-does-the-set-1-2-3-n-have-that-contain-no-three-conse.htmlhttps://liberty.io/how-many-subsets-does-the-set-1-2-3-n-have-that-contain-no-three-conse.htmlQ: How many subsets does the set $\{1,2,3,...n\}$ have that contain no three consecutive integers? Find a recurrence. I don’t think I understand the solution. According to the solution if I take......Tue, 11 Aug 2026 21:38:43 -0400https://liberty.io/cyclic-and-circular-permutation-are-they-different-concepts.htmlhttps://liberty.io/cyclic-and-circular-permutation-are-they-different-concepts.html"Cyclic" and "Circular" permutations - are these two different concepts? I have been reading about permutation and encountering them in many places. What are the definitions of them, in simple Engl......Tue, 11 Aug 2026 21:37:26 -0400https://liberty.io/good-books-on-complex-numbers.htmlhttps://liberty.io/good-books-on-complex-numbers.htmlI searched on this site but I found posts asking for books on complex analysis which I guess is not what I want. I want a book on problems concerning with roots of unity, argand plane, exponentia......Tue, 11 Aug 2026 21:36:52 -0400https://liberty.io/if-we-have-a-n-times-n-and-ax-xa-a-how-do-we-show-that-det-a-0.htmlhttps://liberty.io/if-we-have-a-n-times-n-and-ax-xa-a-how-do-we-show-that-det-a-0.htmlIf we have $A_{n \times n}$ and $AX-XA=A$, how do we show that $\det(A)=0$? I've tried taking the trace on both sides to get $$tr(AX)-tr(XA)=tr(A)$$ $$tr(AX)-tr(AX)=tr(A)$$ $$0=tr(A)$$ However, that ...Tue, 11 Aug 2026 21:07:46 -0400https://liberty.io/calculate-the-divergence-of-the-polar-coordinate-vector-field-partial-.htmlhttps://liberty.io/calculate-the-divergence-of-the-polar-coordinate-vector-field-partial-.htmlI have to solve this problem: $v=\partial_\phi$ on $M=\mathbb{R}^2\backslash{0}$ where the components of $v$ are in polar coordinates. Calculate the divergence of $v$. Even with the help of comme......Tue, 11 Aug 2026 21:05:21 -0400https://liberty.io/probability-of-guessing-m-m-colors-correctly.htmlhttps://liberty.io/probability-of-guessing-m-m-colors-correctly.htmlSuppose there are 5 M and M beans of different colors. You randomly choose 3. What is the probability that you guess 2 out of the 3 numbers correctly? Apparently the answer is $12/125$, but I don't......Tue, 11 Aug 2026 21:05:21 -0400https://liberty.io/infinite-countable-union-of-finite-sets.htmlhttps://liberty.io/infinite-countable-union-of-finite-sets.htmlIf $A_n$ is finite $\forall n\in \mathbb N,\;$ then countable infinite union of these sets is infinite, right?...Tue, 11 Aug 2026 21:00:46 -0400https://liberty.io/definition-of-bounded-set-in-mathbb-r-n.htmlhttps://liberty.io/definition-of-bounded-set-in-mathbb-r-n.htmlI have a question regarding the definition of a bounded set in $\mathbb{R}^n$ here: http://mathworld.wolfram.com/BoundedSet.html Specifically, the definition in the link says that "A set in $\math......Tue, 11 Aug 2026 20:58:16 -0400https://liberty.io/what-is-wrong-with-my-derivation-of-the-exponential-form-of-sinh-x.htmlhttps://liberty.io/what-is-wrong-with-my-derivation-of-the-exponential-form-of-sinh-x.htmlI know the exponential form of $\sin(x)=\dfrac{e^{ix}-e^{-ix}}{2i}$. I wish to derive the exponential form of $\sinh(x)=\dfrac{e^x-e^{-x}}{2}$, and I am stuck. I try to replace $\sin(x)$ with $\sin......Tue, 11 Aug 2026 20:56:38 -0400https://liberty.io/using-matlab-to-find-the-max-elements-and-their-positions-closed.htmlhttps://liberty.io/using-matlab-to-find-the-max-elements-and-their-positions-closed.htmlMATLAB documentation says "[C,I] = max(...) finds the indices of the maximum values of A, and returns them in output vector I." But that does not seem to work. See below. X = [2 8 4; 7 3 9]; ......Tue, 11 Aug 2026 20:52:26 -0400https://liberty.io/how-many-points-are-necessary-to-determine-a-cube-in-3-dimensions.htmlhttps://liberty.io/how-many-points-are-necessary-to-determine-a-cube-in-3-dimensions.htmlIf you have 2 points in 3D, there is only one line passing through them, but infinite planes can pass through them. If you have 3 points, only one plane can pass through all 3, but infinitely many......Tue, 11 Aug 2026 20:47:44 -0400https://liberty.io/if-s-is-a-multiple-of-both-a-and-b-then-s-is-a-multiple-of-operatornam.htmlhttps://liberty.io/if-s-is-a-multiple-of-both-a-and-b-then-s-is-a-multiple-of-operatornam.htmlI came across this as I was doing work for one of my classes. We just use this property, proved presumably in number theory, which we didn't need to take. Could someone help me? Prove that if $a$, ......Tue, 11 Aug 2026 20:47:32 -0400https://liberty.io/denomination-of-cards-probability.htmlhttps://liberty.io/denomination-of-cards-probability.htmlWe have a condensed deck of 40 cards containing only the denominations from Ace through 10. We draw four cards uniformly at random (without replacement). What is the probability that we draw four ...Tue, 11 Aug 2026 20:44:49 -0400https://liberty.io/finding-product-of-maxterms-form.htmlhttps://liberty.io/finding-product-of-maxterms-form.htmlI need help to resolve this problem, i have the following boolean function: [(A.!C)+!(A.!C)].!(A.!B) The Truth table is: (please see this LINK TO wolframalpha for more detail) Then the Sum-of-...Tue, 11 Aug 2026 20:36:08 -0400https://liberty.io/is-there-an-english-translation-of-diophantus-039-s-arithmetica-availa.htmlhttps://liberty.io/is-there-an-english-translation-of-diophantus-039-s-arithmetica-availa.htmlIt should be in the public domain (obviously), so I'd thought I could find the English text on the web somewhere. Apparently not?...Tue, 11 Aug 2026 20:35:58 -0400https://liberty.io/is-it-possible-to-have-a-spherical-object-with-only-hexagonal-faces.htmlhttps://liberty.io/is-it-possible-to-have-a-spherical-object-with-only-hexagonal-faces.htmlIf so, what would be the most efficient algorithm for generating spheres with different number of hexagonal faces at whatever interval required to make them fit uniformly or how might you calculate......Tue, 11 Aug 2026 20:31:28 -0400https://liberty.io/system-of-equations-has-no-solution.htmlhttps://liberty.io/system-of-equations-has-no-solution.htmlIf the system of linear equations, \begin{cases} x &+ ay &+ z &= 3\\ x &+ 2y &+ 2z &= 6\\ x &+ 5y &+ 3z &= b\\ \end{cases} has no solution, then: $a......Tue, 11 Aug 2026 20:29:53 -0400https://liberty.io/why-does-1-0-5-equal-2-duplicate.htmlhttps://liberty.io/why-does-1-0-5-equal-2-duplicate.htmlI would like to know, in $1/0.5 =2$ why does $1/0.5$ yield $2$. I want to know a conceptual explanation, not merely a method how to get the answer. If you are able to explain this in example u......Tue, 11 Aug 2026 20:18:18 -0400https://liberty.io/the-caret-symbol-means-exponentiation-informally-in-math-why-not-a-sym.htmlhttps://liberty.io/the-caret-symbol-means-exponentiation-informally-in-math-why-not-a-sym.htmlPlus, minus, multiply, divide, and exponentiation all have symbols in math (+, -, *, /, ^ ) . But why isn't there the missing log symbol too? Here's how it would work: 4 ^ 5 = 1024 (as is standard......Tue, 11 Aug 2026 20:17:02 -0400https://liberty.io/how-to-evaluate-the-gradient-of-a-function-at-a-point.htmlhttps://liberty.io/how-to-evaluate-the-gradient-of-a-function-at-a-point.htmlI have a problem where I am to create a function in terms of $x$ and $y$ and compute the gradient at the point $(1,1)$. I computed the gradient but in order to evaluate it at the given point do I j......Tue, 11 Aug 2026 20:14:47 -0400https://liberty.io/how-to-determine-the-periods-of-a-periodic-function.htmlhttps://liberty.io/how-to-determine-the-periods-of-a-periodic-function.htmlI am aware of the other similar questions but was not able to figure out what I want to know from those question thus posting it here. Given a periodic function $f(x)=sin(x)$, Why is the period of......Tue, 11 Aug 2026 20:04:19 -0400https://liberty.io/matrix-columns-vs-rows.htmlhttps://liberty.io/matrix-columns-vs-rows.htmlI am confused over the geometric meaning of columns and rows in a matrix. My understanding from 3Blue1Brown's Linear Algebra series was that the elements of a matrix column are the coordinates whi......Tue, 11 Aug 2026 20:03:52 -0400https://liberty.io/angle-measure-angles-in-standard-position-is-it-a-useless-term.htmlhttps://liberty.io/angle-measure-angles-in-standard-position-is-it-a-useless-term.htmlI have a pre calculus test next week, and I have been going over the chapters to gain a deeper understanding; however, I find it difficult, or at least I find the concept of "Angles in Standard Pos......Tue, 11 Aug 2026 20:00:20 -0400https://liberty.io/what-is-the-meaning-of-mean-field.htmlhttps://liberty.io/what-is-the-meaning-of-mean-field.htmlIn lots of Bayesian papers, people use variational approximation. In lots of them they call it "mean-field variational approximation". Does anyone know what is the meaning of mean-field in this co......Tue, 11 Aug 2026 19:54:52 -0400https://liberty.io/orthogonal-position-and-velocity-vectors.htmlhttps://liberty.io/orthogonal-position-and-velocity-vectors.htmlIs it true that if the position and velocity vectors of a moving particle are always perpendicular the path of the particle is on a sphere? If so how do I prove it? Geometrically I believe it makes ...Tue, 11 Aug 2026 19:44:25 -0400https://liberty.io/lower-bound-on-tail-probabilities.htmlhttps://liberty.io/lower-bound-on-tail-probabilities.htmlInequalities such as Markov's and Chebyshev’s provide upper bounds on tail probabilities. Are there similar inequalities that give lower bounds in the form $P(X \geq \alpha)>\theta$?...Tue, 11 Aug 2026 19:44:08 -0400