https://liberty.ioCurated star highlights with fresh energy.en-usThu, 13 Aug 2026 15:37:59 -0400https://liberty.io/what-is-the-geometric-interpretation-of-the-transpose.htmlhttps://liberty.io/what-is-the-geometric-interpretation-of-the-transpose.htmlI can follow the definition of the transpose algebraically, i.e. as a reflection of a matrix across its diagonal, or in terms of dual spaces, but I lack any sort of geometric understanding of the ...Thu, 13 Aug 2026 19:32:00 -0400https://liberty.io/negative-area-below-x-axis-and-above-f-x.htmlhttps://liberty.io/negative-area-below-x-axis-and-above-f-x.htmlI was finding the area below the x-axis and above $y = x^2 - 4x$. My outcome was a negative number (fair enough, it's under the x-axis), but wolframalpha for instance gives me a positive number (I ......Thu, 13 Aug 2026 19:22:41 -0400https://liberty.io/a-mod-b-when-b-is-greater-than-a.htmlhttps://liberty.io/a-mod-b-when-b-is-greater-than-a.htmlWhy 6 % 10 = 6 As fa as I know 10 % 6 = 4, but I don't know how to handle the first one Thanks!...Thu, 13 Aug 2026 19:21:23 -0400https://liberty.io/permutation-test-for-one-way-anova.htmlhttps://liberty.io/permutation-test-for-one-way-anova.htmlI want to know what steps to do the permutation test (and the R/SAS code) for one-way ANOVA? Would you give me the specific steps? Thanks, Buddies!...Thu, 13 Aug 2026 19:16:41 -0400https://liberty.io/riemann-rearrangement-theorem.htmlhttps://liberty.io/riemann-rearrangement-theorem.htmlThe Riemann rearrangement theorem states that if $\sum\limits_{n=0} ^{+ \infty} a_n$ is conditionally convergent and $M \in \mathbb{R}$ then there exists a permutation $ \sigma (n) $ such that $\sum\...Thu, 13 Aug 2026 19:15:56 -0400https://liberty.io/symmetric-matrix-multiplication.htmlhttps://liberty.io/symmetric-matrix-multiplication.htmlLet $A$ and $B$ be symmetric matrices. Prove: $AB=BA$ $AB$ is a symmetric matrix As for 1. due to the axiom $(AB)^T=B^T A^T$ so $AB=BA$ As for 2. I did not find any axiom that can support the cla......Thu, 13 Aug 2026 19:01:14 -0400https://liberty.io/multiply-same-trig-functions.htmlhttps://liberty.io/multiply-same-trig-functions.htmlI apologize if this is a duplicate of another question in advance. I did not find any in the search. Does $\sin(x)\sin(y)=\sin(xy)$? I am wondering because I need to know if $\sin^2(44)=\sin(44)\si......Thu, 13 Aug 2026 18:53:55 -0400https://liberty.io/dimensions-of-singular-value-decomposition-matrices.htmlhttps://liberty.io/dimensions-of-singular-value-decomposition-matrices.htmlI have seen the Singular Value Decomposition (SVD) represented in two different ways and wanted to know a) if they're both correct and b) what their relationship is. In the first version a matrix ......Thu, 13 Aug 2026 18:52:36 -0400https://liberty.io/cardinality-of-a-power-set-is-given-be-2-n-why.htmlhttps://liberty.io/cardinality-of-a-power-set-is-given-be-2-n-why.htmlCardinality of power set of $A$ is determined by $2^n$, where n is the number of elements in Set $A$. How does this formula work? I was taught that there are $n$ elements in Set $A$ and each eleme......Thu, 13 Aug 2026 18:48:08 -0400https://liberty.io/how-to-find-vertices-of-a-rectangle-when-center-coordinates-and-angle-.htmlhttps://liberty.io/how-to-find-vertices-of-a-rectangle-when-center-coordinates-and-angle-.htmlNeed to find z1, z2 z3 and z4 when zc and r are given. Been trying to solve this seemingly simple geometry problem. Can't wrap my head around it. It's a rectangle with center coordinates given and......Thu, 13 Aug 2026 18:44:17 -0400https://liberty.io/card-drawing-with-replacement.htmlhttps://liberty.io/card-drawing-with-replacement.htmlIf 6 cards are drawn from a deck of cards with replacement. What is the probability that the outcome is a club at least once. I thought it would be 13/52. But I was wrong....Thu, 13 Aug 2026 18:36:44 -0400https://liberty.io/how-to-correctly-compate-f-n-and-g-n-when-working-through-o-n-notation.htmlhttps://liberty.io/how-to-correctly-compate-f-n-and-g-n-when-working-through-o-n-notation.htmlGoing through theory, missing the idea, need a bit of help. So, the initial state is: $$f(n) = O(g(n))$$ Assume that $f$ and $g$ are both nondecreasing and always bigger than 1. And, from my ...Thu, 13 Aug 2026 18:30:47 -0400https://liberty.io/calculate-the-area-of-the-crescent.htmlhttps://liberty.io/calculate-the-area-of-the-crescent.htmlI found this problem on a thread on Stack overflow where it was posted as "job interview question". Unfortunately I cannot find the question. But I saved the picture and just cannot figure it out. ......Thu, 13 Aug 2026 18:29:51 -0400https://liberty.io/how-to-find-the-closed-form-formula-for-this-recurrence-relation.htmlhttps://liberty.io/how-to-find-the-closed-form-formula-for-this-recurrence-relation.html$ x_{0} = 5 $ $ x_{n} = 2x_{n-1} + 9(5^{n-1})$ I have computed: $x_{0} = 5, x_{1} = 19, x_{2} = 83, x_{3} = 391, x_{4} = 1907$, but cannot see any pattern for the general $n^{th}$ term....Thu, 13 Aug 2026 18:21:13 -0400https://liberty.io/can-matrices-do-cross-product-with-vectors.htmlhttps://liberty.io/can-matrices-do-cross-product-with-vectors.htmlthis is a practical problem raised from my chemistry research, apologize if somewhere I used incorrect words. I'm dealing with the derivatives of a vector with respect to another vector. After some ...Thu, 13 Aug 2026 18:16:04 -0400https://liberty.io/why-does-a-monotonic-function-always-have-a-positive-rate-of-change.htmlhttps://liberty.io/why-does-a-monotonic-function-always-have-a-positive-rate-of-change.htmlBased on my book The rate of change of $f(u)$ as $u$ changes can be measured by looking at the change in $f$ between two values of $u$, divided by the change in $u$: $$\frac{Δf}{Δu} = \frac{f(......Thu, 13 Aug 2026 18:14:43 -0400https://liberty.io/this-statement-is-false-duplicate.htmlhttps://liberty.io/this-statement-is-false-duplicate.htmlIn propositional logic, a proposition is a statement that is either true or false, but not both. In a text I am reading and in many others, "this statement is false" is not considered a proposition......Thu, 13 Aug 2026 18:09:17 -0400https://liberty.io/the-sum-of-the-first-n-squares-1-4-9-cdots-n-2-is-frac-n-n-1-2n-1-6-du.htmlhttps://liberty.io/the-sum-of-the-first-n-squares-1-4-9-cdots-n-2-is-frac-n-n-1-2n-1-6-du.htmlProve that the sum of the first $n$ squares $(1 + 4 + 9 + \cdots + n^2)$ is $\frac{n(n+1)(2n+1)}{6}$. Can someone pls help and provide a solution for this and if possible explain the question...Thu, 13 Aug 2026 18:01:19 -0400https://liberty.io/pencils-of-circles.htmlhttps://liberty.io/pencils-of-circles.htmlI want to solve an exercise regarding the connection between the projective model of möbius geometry and pencils of circles. Consider the projective model of two-dimensional Möbius geometry P($R^{......Thu, 13 Aug 2026 17:59:32 -0400https://liberty.io/maximize-area-of-a-triangle-with-fixed-perimeter.htmlhttps://liberty.io/maximize-area-of-a-triangle-with-fixed-perimeter.htmlIf perimeter of a triangle is $2d$, what is the length of sides so the triangle has maximal area? I found some solution using circle and angles, but I think I have to use derivatives. I need ......Thu, 13 Aug 2026 17:55:43 -0400https://liberty.io/calculating-the-length-of-a-tangent-drawn-to-a-circle-from-a-named-poi.htmlhttps://liberty.io/calculating-the-length-of-a-tangent-drawn-to-a-circle-from-a-named-poi.htmlMy book (New Tertiary Mathematics Volume 1 Part 1, by C Plumpton and P S W Macilwaine) describes a method for calculating the length of a tangent to a circle from the point $(x_{1}, y_{1})$ outside......Thu, 13 Aug 2026 17:42:06 -0400https://liberty.io/applications-of-the-isomorphism-theorems.htmlhttps://liberty.io/applications-of-the-isomorphism-theorems.htmlIn my study of groups, rings, modules etc, I've seen the three isomorphism theorems stated and proved many times. I use the first one ( $G/\ker \phi \cong \operatorname{im} \phi$ ) very often, but ......Thu, 13 Aug 2026 17:26:43 -0400https://liberty.io/using-logic-laws-to-verify-the-logical-equivalences.htmlhttps://liberty.io/using-logic-laws-to-verify-the-logical-equivalences.htmlI need to use theorem 2.1.1 in order to verify the following: ∼((∼p ∧ q) ∨ (∼ p ∧ ∼q)) ∨ (p ∧ q) ≡ p I applied the Double Negative Law to get ((p ∧ q) ∨ (∼p ∧ ∼q)) ∨ (p ∧ q) I noticed we have (......Thu, 13 Aug 2026 17:24:42 -0400https://liberty.io/how-many-irrational-numbers.htmlhttps://liberty.io/how-many-irrational-numbers.htmlHow many irrational numbers can exist between two rational numbers ? As there are infinite numbers between two rational numbers and also there are infinite rational numbers between two rational num......Thu, 13 Aug 2026 17:23:45 -0400https://liberty.io/congruency-in-bow-tie-triangles.htmlhttps://liberty.io/congruency-in-bow-tie-triangles.htmlWe've just started congruency in my class, and we've stumbled across a question which goes like this: Prove that ∆AOB $\equiv$ ∆COD I drew up a diagram which portrays exactly the same informati......Thu, 13 Aug 2026 17:21:53 -0400https://liberty.io/trace-of-an-inverse-inequality-text-tr-a-1-ge-n-2-text-tr-a-1.htmlhttps://liberty.io/trace-of-an-inverse-inequality-text-tr-a-1-ge-n-2-text-tr-a-1.htmlLet $A \in \mathbb{R}^n$ be a positive definite matrix. Then, it is well known that \begin{align} \text{Tr}(A^{-1}) \ge n^2 \text{Tr}(A)^{-1} \end{align} The proof follows by using the fact that ......Thu, 13 Aug 2026 17:18:49 -0400https://liberty.io/ratio-of-circumference-to-diameter-on-a-sphere.htmlhttps://liberty.io/ratio-of-circumference-to-diameter-on-a-sphere.htmlI was listening to an audiobook of Einstein when they started discussing spherical geometry and how Pi was no longer the ratio of a circle's circumference to its diameter, so I set out to find the ......Thu, 13 Aug 2026 17:14:27 -0400https://liberty.io/how-to-define-being-inside-of-something-in-the-context-of-topology.htmlhttps://liberty.io/how-to-define-being-inside-of-something-in-the-context-of-topology.htmlI'm a Psychologist and Neuroscientist with interest in math and I just started reading about Topology. I have to say it's not easy to grasp the concepts without a practical example, so I'm trying to ...Thu, 13 Aug 2026 17:09:41 -0400https://liberty.io/function-domain-of-cubic-root-duplicate.htmlhttps://liberty.io/function-domain-of-cubic-root-duplicate.htmlDoes the domain of the function $y=\sqrt[3]{x^3+1}$ include $x<-1$? If yes, why is Mathematica and Wolfram Alpha not plotting that part of the function?...Thu, 13 Aug 2026 17:05:52 -0400https://liberty.io/why-are-horizontal-transformations-of-functions-reversed.htmlhttps://liberty.io/why-are-horizontal-transformations-of-functions-reversed.htmlWhile studying graph transformations I came across horizontal and vertical scale and translations of functions. I understand the ideas below. $f(x+a)$ - grouped with x, horizontal translation, inv......Thu, 13 Aug 2026 16:55:29 -0400https://liberty.io/how-to-convert-a-fraction-into-ternary.htmlhttps://liberty.io/how-to-convert-a-fraction-into-ternary.htmlIn a ternary system how is $\dfrac{1}{2}=0.\bar1$ ,$\dfrac{1}{3}=0.1=0.0\bar2$??, etc. In general, how does one write the ternary expression for a given fraction?...Thu, 13 Aug 2026 16:50:45 -0400https://liberty.io/is-there-a-symbol-or-abbreviation-for-coefficient-of-x.htmlhttps://liberty.io/is-there-a-symbol-or-abbreviation-for-coefficient-of-x.htmlDoing some binomial expansions with algebra where I need to equate different coefficients together but don't know what to write: [Coefficient of $x^3$] = _ $k=+1.5$...Thu, 13 Aug 2026 16:49:02 -0400https://liberty.io/why-is-this-the-basis-for-the-indiscrete-topology.htmlhttps://liberty.io/why-is-this-the-basis-for-the-indiscrete-topology.htmlIndiscrete topology only has $\{\varnothing, X\}$ where $X$ is the entire space. I know this is probably a dumb question. Why is the basis (according to wikipedia) for this topology just $\{X\}$ i......Thu, 13 Aug 2026 16:32:31 -0400https://liberty.io/questions-tagged-complex-integration.htmlhttps://liberty.io/questions-tagged-complex-integration.htmlQ&A for people studying math at any level and professionals in related fields...Thu, 13 Aug 2026 16:28:34 -0400https://liberty.io/probability-that-exactly-2-balls-are-white.htmlhttps://liberty.io/probability-that-exactly-2-balls-are-white.htmlWhat I did. I let $X$ be number of withdraws before $x$ white balls. We can call our succes in this case to be gettin a white ball and probability is of course $p = \frac{3}{6} = \frac{1}{2}$. So ......Thu, 13 Aug 2026 16:27:46 -0400https://liberty.io/if-x-2-5x-4-y-2-y-1-2y-for-all-real-values-of-y-then-what-interval-doe.htmlhttps://liberty.io/if-x-2-5x-4-y-2-y-1-2y-for-all-real-values-of-y-then-what-interval-doe.htmlOkay so I'm stuck on this one for a long time : If $(x^2-5x+4)(y^2+y+1)<2y$ for all real values of $y$ then what interval does $x$ belong to? My Try: Since $y$ can take any real value, I tr......Thu, 13 Aug 2026 16:06:22 -0400https://liberty.io/how-to-find-center-of-a-circle-from-only-an-arbitary-arc-of-that-circl.htmlhttps://liberty.io/how-to-find-center-of-a-circle-from-only-an-arbitary-arc-of-that-circl.htmlHow to find the center of a circle with given an arbitrary arc. we only have the arc nothing else. Is there any known equation or way to complete the circle....Thu, 13 Aug 2026 15:58:19 -0400https://liberty.io/proof-that-the-derivative-of-a-constant-is-zero.htmlhttps://liberty.io/proof-that-the-derivative-of-a-constant-is-zero.htmlI know that the derivative of a constant is zero, but the only proof that I can find is: given that $f(x) = {x}^{0}$, $$ f'(x) = \lim_{h\to 0} \frac {f(x+h) - f(x)}{h} $$ $$ f'(x) = \lim_{h\to 0} \......Thu, 13 Aug 2026 15:52:21 -0400https://liberty.io/difficulty-understanding-what-the-n-1-part-refers-to-in-a-basic-arithm.htmlhttps://liberty.io/difficulty-understanding-what-the-n-1-part-refers-to-in-a-basic-arithm.htmlI am 28 and learned no maths in school. I'm going back through and learning on my own time. I am up to arithmetic sequences. The general formula (according to Khan Academy) for a recursive arithmetic ...Thu, 13 Aug 2026 15:48:59 -0400https://liberty.io/integration-by-parts-twice.htmlhttps://liberty.io/integration-by-parts-twice.htmlUse the substition $u=4x-3$ to find $\int\frac{4x}{4x-3}dx$, giving your answer in terms of $x$ $$u=4x-3$$ $$4x=u+3$$ $$\frac{du}{dx}=4$$ $$dx=\frac{1}{4}du$$ $$\int\frac{4x}{4x-3}dx$$ $$\Rightarro......Thu, 13 Aug 2026 15:45:45 -0400https://liberty.io/matrix-product-notation.htmlhttps://liberty.io/matrix-product-notation.htmlMy lecturer has used some notation that I've never seen before: it is a (matrix) product symbol with a left-to-right arrow over the top. Does anybody know what this means? Thanks in advance. Edit......Thu, 13 Aug 2026 15:39:50 -0400https://liberty.io/understanding-the-rigorous-definition-of-continuity.htmlhttps://liberty.io/understanding-the-rigorous-definition-of-continuity.htmlI have an intuitive understanding of continuity, however, I’m struggling to understand the rigorous definition. I understand that a function is continuous when you can get infinitely close to each ...Thu, 13 Aug 2026 15:32:55 -0400https://liberty.io/domain-of-rational-functions.htmlhttps://liberty.io/domain-of-rational-functions.htmlI have the following definition for rational functions: "Let $p(x), q(x)$ be polynomials and define $D=\{x\in \mathbb{R} : q(x) \neq 0 \}$ Then, the function $$f:D \rightarrow \mathbb{R}$$ $$f......Thu, 13 Aug 2026 15:23:56 -0400https://liberty.io/what-are-the-components-in-the-ito-process.htmlhttps://liberty.io/what-are-the-components-in-the-ito-process.htmlWe know that if W(t) is a Brownian Motion, then the Ito Process is given by, X(t)= X(0) + $\int_0^t \Delta(u)dW(u)$ + $\int_0^t \Phi(u)du$ Could someone explain briefly what the two components a......Thu, 13 Aug 2026 15:23:33 -0400https://liberty.io/what-is-undecidability.htmlhttps://liberty.io/what-is-undecidability.htmlWhat does it mean that some problem is undecidable? For instance the halting problem. Does it mean that humans can never invent a new technique that always decides whether a turing machine will h......Thu, 13 Aug 2026 15:19:41 -0400https://liberty.io/finding-moment-of-inertia-of-a-cone-why-can-039-t-i-integrate-like-thi.htmlhttps://liberty.io/finding-moment-of-inertia-of-a-cone-why-can-039-t-i-integrate-like-thi.htmlI have a cone (vertex at $(x,y,z)=0$) with height $H$ and radius $a$ (radius of the base). Here is a picture: I want to find the moment of inertia around the $z$-axis using calculus. I know the in......Thu, 13 Aug 2026 15:15:04 -0400https://liberty.io/equivalent-systems-of-linear-equation.htmlhttps://liberty.io/equivalent-systems-of-linear-equation.htmlI've just begun to re-learn linear algebra because is so important, the book that I chose is naturally the Hoffman's for a lot of reason. Well, In the first chapter I'm stuck with the following, ...Thu, 13 Aug 2026 15:13:56 -0400https://liberty.io/real-valued-analytic-function-defined-on-a-connected-set-is-constant.htmlhttps://liberty.io/real-valued-analytic-function-defined-on-a-connected-set-is-constant.htmlLet $G$ be a connected set and $f : G \rightarrow \mathbb{C}$ a real valued analytic function. Prove that $f$ is constant. My idea to prove the result is to prove a subset $A \neq \varnothing$ of......Thu, 13 Aug 2026 15:13:38 -0400https://liberty.io/four-non-coplanar-points-in-mathbb-r-3-determine-an-unique-sphere.htmlhttps://liberty.io/four-non-coplanar-points-in-mathbb-r-3-determine-an-unique-sphere.htmlShow that four non-coplanar points in $\mathbb{R}^3$ determine an unique sphere. I have no idea how to solve this exercise. Thank you for your help....Thu, 13 Aug 2026 15:07:00 -0400https://liberty.io/finding-the-value-of-k-duplicate.htmlhttps://liberty.io/finding-the-value-of-k-duplicate.html$$π1:( 1+k , -3 , 6 )$$ $$π2: ( 1 , 5+k , 3k)$$ Does anyone know the values of k that would make these parallel and perpendicular, ive been trying for hours but nothing seems to work The original ...Thu, 13 Aug 2026 15:05:55 -0400https://liberty.io/no-mode-or-all-mode.htmlhttps://liberty.io/no-mode-or-all-mode.htmlLet's say I have a data set:- $${1,2,3,4,5}$$ I decide to find the mode though it would not be an appropriate measure of central tendency. But still........ Now this is an exceptional case and for ...Thu, 13 Aug 2026 15:05:45 -0400https://liberty.io/is-a-set-contained-in-its-cartesian-product-with-itself.htmlhttps://liberty.io/is-a-set-contained-in-its-cartesian-product-with-itself.htmlGiven the Cartesian product $X = A \times A$, is $A \subset X$?...Thu, 13 Aug 2026 14:38:22 -0400https://liberty.io/find-the-derivative-dy-dx-given-some-value-for-x.htmlhttps://liberty.io/find-the-derivative-dy-dx-given-some-value-for-x.htmlI know how to find the derivative of a function by itself given a function. I can use multiple rules to determine the derivative. However in a situation where I must find the derivative when the x......Thu, 13 Aug 2026 14:10:51 -0400https://liberty.io/how-to-prove-if-something-is-a-linear-transformation.htmlhttps://liberty.io/how-to-prove-if-something-is-a-linear-transformation.html(I think) My textbook says something is a linear transformation if $L(ax) = aL(x)$ $L(x+y) = L(x) + L(y)$ $L(x) = A(x)$ But a lot of sites I've been on haven't proved these 3 things, so I just w......Thu, 13 Aug 2026 13:57:13 -0400https://liberty.io/proof-by-lewis-carroll-that-an-urn-cannot-contain-two-balls-duplicate.htmlhttps://liberty.io/proof-by-lewis-carroll-that-an-urn-cannot-contain-two-balls-duplicate.htmlI am self-learning probability theory and solving some problems from the book Probability and Random processes by Grimmett and Stirzaker. Exercise 1.4.4 poses a nice problem. I would like someone to ...Thu, 13 Aug 2026 13:54:13 -0400https://liberty.io/time-complexity-of-a-modulo-operation.htmlhttps://liberty.io/time-complexity-of-a-modulo-operation.htmlI am trying to prove that if $p$ is a decimal number having $m$ digits, then $p \bmod q$ can be performed in time $O(m)$ (at least theoretically), if $q$ is a prime number. How do I go about this? ......Thu, 13 Aug 2026 13:28:46 -0400https://liberty.io/subcoalgebra-generated-by-an-element.htmlhttps://liberty.io/subcoalgebra-generated-by-an-element.htmlLet $(A, \Delta, \epsilon)$ be a coalgebra and $f\in A$. What is the subcoalgebra generated by $f$ like? For example, if $A$ is the dual of the quaternions $\mathbb{H}$ (which is $\mathbb{R}$-alge......Thu, 13 Aug 2026 13:27:27 -0400https://liberty.io/a-basic-property-of-slowly-varying-functions.htmlhttps://liberty.io/a-basic-property-of-slowly-varying-functions.htmlIt seems that a fundamental property of a slowly varying function is that for all $\delta > 0$ $$ \lim_{x\to \infty} L(x) \, x^{-\delta} = 0.$$ How to prove this? The only book I found (that was......Thu, 13 Aug 2026 13:26:12 -0400https://liberty.io/weierstrass-m-test-proof.htmlhttps://liberty.io/weierstrass-m-test-proof.htmlLet (X,d) be a metric space. For each n $\epsilon$ N let $g_n$:X$\rightarrow$R be a continuous function. Let ($a_n$) be a sequence of positive real numbers such that the series $\sum_{n=1}^\infty a......Thu, 13 Aug 2026 13:24:53 -0400https://liberty.io/what-exactly-is-a-0-form.htmlhttps://liberty.io/what-exactly-is-a-0-form.htmlFrom what I understand, a k-form in the real numbers is essentially a mapping $\mathbb{R^k} \rightarrow \mathbb{R}$, but I can't seem to find a corresponding definition for a "0-form". Wikipedia se......Thu, 13 Aug 2026 13:24:21 -0400https://liberty.io/how-to-find-an-eulerian-circuit-in-a-complicated-graph.htmlhttps://liberty.io/how-to-find-an-eulerian-circuit-in-a-complicated-graph.htmlConsider the following graph G: (a) Give a decomposition of G into cycles. (b) Find an Eulerian circuit in G. This is a very complicated graph and each time I am trying to find the solution I am ...Thu, 13 Aug 2026 13:24:08 -0400https://liberty.io/how-do-i-convert-the-distance-between-two-lat-long-points-into-feet-me.htmlhttps://liberty.io/how-do-i-convert-the-distance-between-two-lat-long-points-into-feet-me.htmlI've been reading around the net and everything I find is really confusing. I just need a formula that will get me 95% there. I have a tool that outputs the distance between two lat/long points. P......Thu, 13 Aug 2026 13:18:52 -0400https://liberty.io/interpretation-of-log-differences.htmlhttps://liberty.io/interpretation-of-log-differences.htmlI have a very simple question. I am confused about the interpretation of log differences. Here a simple example: $$\log(2)-\log(1)=.3010$$ With my present understanding, I would interpret the resul......Thu, 13 Aug 2026 13:17:30 -0400https://liberty.io/how-to-rigorously-find-range-of-a-function.htmlhttps://liberty.io/how-to-rigorously-find-range-of-a-function.htmlWhat is a rigorous method/mechanism to find the range of a function $f(x)$? Is it acceptable to find $f^{-1}(x)$ and then its domain? I understand that $f^{-1}(x)$ does not always exist (at least not ...Thu, 13 Aug 2026 13:14:24 -0400https://liberty.io/determine-the-derivative-of-the-reciprocal-function-only-with-the-prod.htmlhttps://liberty.io/determine-the-derivative-of-the-reciprocal-function-only-with-the-prod.htmlFollowing is given: $( \alpha )$ the derivative of a inverse function: $$\left(\frac{1}{f}\right)' = -\frac{f'}{f^2}$$ $( \beta )$ The the product rule: $$({f}*g) = ({f'}*{g})+({f}*{g'})$$ Follo......Thu, 13 Aug 2026 13:04:56 -0400https://liberty.io/how-to-calculate-csc-2-85-in-calculator.htmlhttps://liberty.io/how-to-calculate-csc-2-85-in-calculator.htmlIn my calculator (TI-84), there are only $sin, cos,$ and $tan$ commands (and inverse sin, inverse cos, inverse tan). I had a question that was as follows: Calculate $csc(2.85)$ in which I was perm......Thu, 13 Aug 2026 12:55:56 -0400https://liberty.io/find-the-amplitude-of-the-steady-state-solution.htmlhttps://liberty.io/find-the-amplitude-of-the-steady-state-solution.htmlHeave motion of a point energy absorber in the form of a buoy due t incident wave of frequency $w$ and amplitude $a$ is described by $(m+m_a)y'' + (c_h + c_e)y' +ky = F_i(t)$ $F_i(t)=aF(w)\cos(wt......Thu, 13 Aug 2026 12:46:38 -0400https://liberty.io/how-many-proper-nontrivial-subgroups-do-d5-have.htmlhttps://liberty.io/how-many-proper-nontrivial-subgroups-do-d5-have.htmlDo I have to find out every element in D5 and draw a table to find out subgroups? I know how to find out every single element in D5, but can't think of how to find proper nontrivial subgroups...Thu, 13 Aug 2026 12:46:19 -0400https://liberty.io/conic-section-equation-of-section-plane-in-3d.htmlhttps://liberty.io/conic-section-equation-of-section-plane-in-3d.htmlAll of us know general equation for any conic section: $$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$$ If we'll change of parameters of this equation we get different type of conic section. If we imagine ...Thu, 13 Aug 2026 12:41:04 -0400https://liberty.io/is-a-temperature-change-in-celsius-larger-than-a-temperature-change-in.htmlhttps://liberty.io/is-a-temperature-change-in-celsius-larger-than-a-temperature-change-in.htmlIs a temperature change in Celsius larger than a temperature change in Fahrenheit? The teacher offers this second way of thinking about the question. If the temperature increases by 1 degree Ce......Thu, 13 Aug 2026 12:40:47 -0400https://liberty.io/simple-trig-equations-why-is-it-wrong-to-cancel-trig-terms.htmlhttps://liberty.io/simple-trig-equations-why-is-it-wrong-to-cancel-trig-terms.htmlIn the following problem, I first did it using a cancellation of $sin^2\theta$, working shown below, which gave the wrong answer. Having looked at the question again, I saw it could be solved by ...Thu, 13 Aug 2026 12:40:30 -0400https://liberty.io/specific-strings-that-cannot-be-found-in-pi.htmlhttps://liberty.io/specific-strings-that-cannot-be-found-in-pi.htmlConsidering that the decimals of PI aren't like prime numbers (random), they are pseudo-random (can be calculated through a formula or an infinite series, yet it is calculable). Keeping this in min......Thu, 13 Aug 2026 12:38:24 -0400https://liberty.io/finding-the-average-velocity-and-instantaneous-velocity.htmlhttps://liberty.io/finding-the-average-velocity-and-instantaneous-velocity.htmlA water balloon is launched upward with an initial velocity of $40 ft/s$. the height h at time $t$ is given $h(t)= 16t^2+40t$ a. find the avg. velocity of the balloon between $t=1$ and $t=2$ secon......Thu, 13 Aug 2026 12:37:32 -0400https://liberty.io/how-to-find-the-common-base-of-terms-in-an-expression.htmlhttps://liberty.io/how-to-find-the-common-base-of-terms-in-an-expression.htmlI'm teaching myself basic algebra from a book and am stuck on a question. In the current section it is about expressing numbers as powers of the same base. So $9$ maybe expressed as $3^2$. Another......Thu, 13 Aug 2026 12:32:40 -0400https://liberty.io/does-the-existence-of-a-free-variable-mean-there-are-infinitely-many-s.htmlhttps://liberty.io/does-the-existence-of-a-free-variable-mean-there-are-infinitely-many-s.htmllets say I have an augmented matrix $$\begin{bmatrix} 2&0&3\\ 0&1&0\\ 2&1&3 \end{bmatrix}$$ The question is: Is the matrix consistent with a unique solution, inconsistent,......Thu, 13 Aug 2026 12:32:11 -0400https://liberty.io/are-all-symmetric-matrices-invertible.htmlhttps://liberty.io/are-all-symmetric-matrices-invertible.htmlIs any symmetric matrix ​​invertible? I'm trying to prove this theoretical question, but I don't know what I need to do. I apologize for the simple question, but I'm in doubt and need clarification....Thu, 13 Aug 2026 12:01:44 -0400https://liberty.io/continuous-uniformly.htmlhttps://liberty.io/continuous-uniformly.htmlI need help to demonstrate the following or if you find a function I would appreciate it, I have already looked for many functions to see if any one fulfills it but none works for me and I can't th......Thu, 13 Aug 2026 11:47:05 -0400https://liberty.io/finding-mean-and-standard-deviation-of-normal-distribution-given-2-poi.htmlhttps://liberty.io/finding-mean-and-standard-deviation-of-normal-distribution-given-2-poi.htmlIn general, how do do you calculate the mean and standard deviation of a normal distribution given 2 values on the distribution with their respective probabilities? For Example: Suppose that the a......Thu, 13 Aug 2026 11:31:35 -0400https://liberty.io/comparing-the-growth-rates.htmlhttps://liberty.io/comparing-the-growth-rates.htmlHow can I go about comparing the growth rate of the following functions? $$\sqrt n,\quad 10^n,\quad n^{1.5},\quad 2^{\sqrt{\log n}},\quad n^{5/3}.$$ I am looking for a more generic answer on how ......Thu, 13 Aug 2026 11:26:49 -0400https://liberty.io/what-is-the-math-behind-the-game-spot-it.htmlhttps://liberty.io/what-is-the-math-behind-the-game-spot-it.htmlI just purchased the game Spot It. As per this site, the structure of the game is as follows: Game has 55 round playing cards. Each card has eight randomly placed symbols. There are a total of 50 ...Thu, 13 Aug 2026 11:26:06 -0400https://liberty.io/set-notation-inclusion-and-proper-inclusion.htmlhttps://liberty.io/set-notation-inclusion-and-proper-inclusion.htmlBy definition: If$ \ A$ and $B\ $are sets and every element of$ \ A\ $ is also an element of $ \ B \ $then we can say $ \ A\ $is a subset of $ \ B, \ $and denote this by $ \ A\subset B\ $or $ \ A\...Thu, 13 Aug 2026 11:18:26 -0400https://liberty.io/first-order-logic-vs-second-order-logic.htmlhttps://liberty.io/first-order-logic-vs-second-order-logic.htmlWikipedia describes the first-order vs. second-order logic as follows: First-order logic uses only variables that range over individuals (elements of the domain of discourse); second-order logic......Thu, 13 Aug 2026 11:18:17 -0400https://liberty.io/the-coefficient-of-x-100-in-the-expansion-of-1-x-3.htmlhttps://liberty.io/the-coefficient-of-x-100-in-the-expansion-of-1-x-3.htmlPlease solve it and tell me the technique so that I can solve it in examination in multiple choice questions....Thu, 13 Aug 2026 11:17:31 -0400https://liberty.io/unnecessary-condition-for-alternating-series-test.htmlhttps://liberty.io/unnecessary-condition-for-alternating-series-test.htmlGiven a series $$ \sum^\infty_{n=0} (-1)^n b_n$$ the Alternating Series test states that if $b_n$ is convergent if 1) $b_n$ is positive 2) $b_n$ is monotonically decreasing 3) $\lim_{n\to\infty}......Thu, 13 Aug 2026 11:16:21 -0400https://liberty.io/calculate-an-integral-with-riemann-sum.htmlhttps://liberty.io/calculate-an-integral-with-riemann-sum.htmlWe know that Riemann sum gives us the following formula for a function $f\in C^1$: $$\lim_{n\to \infty}\frac 1n\sum_{k=0}^n f\left(\frac kn\right)=\int_0^1f(x) dx.$$ I am looking for an example......Thu, 13 Aug 2026 11:09:59 -0400https://liberty.io/deriving-sin-ix-i-sinh-x.htmlhttps://liberty.io/deriving-sin-ix-i-sinh-x.htmlDerive $\sin{ix}=i\sinh{x}$ from $(5)$. What is $\sin{i}$? $$\cos{x}=\frac{1}{2}\left(e^{ix}+e^{-ix}\right)\quad\text{and}\quad\sin{x}=\frac{1}{2i}\left(e^{ix}-e^{-ix}\right)\tag 5$$ We have $$\......Thu, 13 Aug 2026 11:02:33 -0400https://liberty.io/what-are-the-uses-of-euler-039-s-number-e.htmlhttps://liberty.io/what-are-the-uses-of-euler-039-s-number-e.htmlPeople make such a big deal of the number $e$. I do not get why it is so important, other than the fact that $\ln(x)=\log_e(x)$. People say it is used all over mathematics and such, but they never ......Thu, 13 Aug 2026 11:01:38 -0400https://liberty.io/user-jialin-zhang-mathematics-stack-exchange.htmlhttps://liberty.io/user-jialin-zhang-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Thu, 13 Aug 2026 10:56:01 -0400https://liberty.io/find-the-domain-of-the-given-function.htmlhttps://liberty.io/find-the-domain-of-the-given-function.htmlI have the function $f(x)=\cos^{-1}(\frac{1}{2\cos(x)})$ and I have to find its domain. What I know is that the domain of $\cos^{-1}(x)$ is $[-1,1]$ so I think that $\frac{1}{2\cos(x)}$ should be at ...Thu, 13 Aug 2026 10:55:27 -0400https://liberty.io/what-is-the-time-complexity-of-euclid-039-s-algorithm-upper-bound-lowe.htmlhttps://liberty.io/what-is-the-time-complexity-of-euclid-039-s-algorithm-upper-bound-lowe.htmlI looked it up online in many sites but none give a clear answer. They all give a lot of complicated mathematical stuff which is not only hard for me to grasp but also irrelevant as I simply want t......Thu, 13 Aug 2026 10:53:21 -0400https://liberty.io/how-to-prove-that-n-frac-1-3-is-not-a-polynomial.htmlhttps://liberty.io/how-to-prove-that-n-frac-1-3-is-not-a-polynomial.htmlI'm reading Barbeau's Polynomials, there's an exercise: How to prove that $n^{\frac{1}{3}}$ is not a polynomial? I've made this question and with the first answer as an example, I guess I should ...Thu, 13 Aug 2026 10:48:38 -0400https://liberty.io/solve-for-a-and-b-of-this-piecewise-function-such-that-the-function-f-.htmlhttps://liberty.io/solve-for-a-and-b-of-this-piecewise-function-such-that-the-function-f-.htmlConsider the function defined by \begin{align*} f(x) = \begin{cases} ax^{2} + x - b, & x < 2\\ ax+b, & 2\leq x \leq 5\\ 2ax - 7, & x > 5. \end{cases} \end{align*} I was able to ......Thu, 13 Aug 2026 10:48:13 -0400https://liberty.io/expectation-of-absolute-difference-of-two-uniform-random-variables.htmlhttps://liberty.io/expectation-of-absolute-difference-of-two-uniform-random-variables.htmlLet $X \sim U(0, 1)$ and $Y \sim U(0, 2)$ be independent random variables. Find $\mathbb{E}[|X - Y|]$. I found $f_{X}(x) = 1$ and $f_{Y}(y) = 1/2$, and I took an integral to find $P(Y > X) = 3......Thu, 13 Aug 2026 10:45:45 -0400https://liberty.io/determine-the-fourth-roots-of-16-duplicate.htmlhttps://liberty.io/determine-the-fourth-roots-of-16-duplicate.htmlDetermine the fourth roots of -16 in the form $x +iy$ where $x$ and $y$ are not trigonometric functions. I do not even know what they really want from me in this question. My initial thought wass......Thu, 13 Aug 2026 10:45:44 -0400https://liberty.io/a-lot-of-confusion-in-the-polynomial-remainder-theorem.htmlhttps://liberty.io/a-lot-of-confusion-in-the-polynomial-remainder-theorem.htmlLately I've been reading about Polynomial Remainder Theorem from various sources, mainly from the wikipedea article, this post and some high school books. Wikipedea says that if we divide a polynom......Thu, 13 Aug 2026 10:43:38 -0400https://liberty.io/lie-group-and-lie-algebra-homomorphisms.htmlhttps://liberty.io/lie-group-and-lie-algebra-homomorphisms.htmlI am reading Brian C Hall's Lie Groups, Lie Algebras, and Representations and trying to do one of the exercises about showing that isomorphic matrix Lie groups give rise to isomorphic Lie Algebras.......Thu, 13 Aug 2026 10:33:00 -0400https://liberty.io/find-closest-power-of-2-to-a-specific-number.htmlhttps://liberty.io/find-closest-power-of-2-to-a-specific-number.htmlThis question is weird and it's not a homework question. I couldn't come up with anything substantial, so sorry if you think that I should have posted my tried methods (I have tried but most of the......Thu, 13 Aug 2026 10:31:51 -0400https://liberty.io/you-can-039-t-find-an-equation-for-the-cross-sectional-shape-when-give.htmlhttps://liberty.io/you-can-039-t-find-an-equation-for-the-cross-sectional-shape-when-give.htmlWhy not? I'm trying to design a calculus I problem. I want them to eventually prove that the volume of a cone (oblique or even with irregular base) is $\frac{\pi r^2 h}{3}$, I also want them to know ...Thu, 13 Aug 2026 10:18:42 -0400https://liberty.io/is-a-omega-spectrum-a-connective-one.htmlhttps://liberty.io/is-a-omega-spectrum-a-connective-one.htmlI can't find this result anywhere, but it seems pretty straightforward. I want to avoid silly mistakes, but I can't see any fault. I'd love to receive some feedback Let $X$ be a $\Omega$-spectrum ......Thu, 13 Aug 2026 10:18:21 -0400https://liberty.io/what-are-we-really-doing-when-we-integrate-a-complex-exponential.htmlhttps://liberty.io/what-are-we-really-doing-when-we-integrate-a-complex-exponential.htmlFirstly, I have read this related question from this site, but it does not answer what I am asking here. What I like to know is why $$\int_{-n}^n e^{ix}dx \ne 0$$ for any finite $n \in \mathbb N$ ......Thu, 13 Aug 2026 10:04:47 -0400