https://liberty.ioCurated star highlights with fresh energy.en-usThu, 27 Aug 2026 01:37:21 -0400https://liberty.io/how-to-express-other-logical-operations-via-pierce-039-s-arrow.htmlhttps://liberty.io/how-to-express-other-logical-operations-via-pierce-039-s-arrow.htmlx↑y, x⇒y, and x⇔y. So I have really given my best, but all I could do is express the conjunction, disjunction, negation, and impilcation....Thu, 27 Aug 2026 05:30:42 -0400https://liberty.io/prove-that-the-square-of-an-integer-a-is-congruent-to-0-or-1-modulo-4.htmlhttps://liberty.io/prove-that-the-square-of-an-integer-a-is-congruent-to-0-or-1-modulo-4.htmlI'm currently trying to go through Artin's Algebra textbook via a Harvard Extension Course (available on YouTube). I wanted to make sure that my proof was solid for this question. Prove that the ...Thu, 27 Aug 2026 05:29:24 -0400https://liberty.io/why-we-need-to-know-how-to-solve-a-quadratic.htmlhttps://liberty.io/why-we-need-to-know-how-to-solve-a-quadratic.htmlFive years ago I was tutoring orphans in a local hospital. One of them asked me the following question when I tried to ask him to solve a quadratic: Why do I need how to solve a quadratic? I am ......Thu, 27 Aug 2026 05:27:08 -0400https://liberty.io/does-an-increasing-sequence-of-reals-converge-if-the-difference-of-con.htmlhttps://liberty.io/does-an-increasing-sequence-of-reals-converge-if-the-difference-of-con.htmlIf $a_n$ is a sequence such that $$a_1 \leq a_2 \leq a_3 \leq \dotsb$$ and has the property that $a_{n+1}-a_n \to 0$, then can we conclude that $a_n$ is convergent? I know that without the con......Thu, 27 Aug 2026 05:22:46 -0400https://liberty.io/examples-of-groups-in-the-real-world.htmlhttps://liberty.io/examples-of-groups-in-the-real-world.htmlI'm looking for some examples of groups in the real world to show students in a liberal arts math course. For example the Rubik's cube. Keep in mind these students have only a college algebra backg......Thu, 27 Aug 2026 05:19:53 -0400https://liberty.io/taylor-series-expansion-about-x-3.htmlhttps://liberty.io/taylor-series-expansion-about-x-3.htmlFind the Taylor series expansion of $f(x)=\sqrt{6x-x^2} $ about $x = 3$. Looking through my notes, I think this could be solved with the binomial theorem. I am no longer sure binomial theorem is u......Thu, 27 Aug 2026 05:05:36 -0400https://liberty.io/find-x-location-using-3-known-x-y-location-using-trilateration.htmlhttps://liberty.io/find-x-location-using-3-known-x-y-location-using-trilateration.htmlI post this question in stackoverflow here and was advised it was best suited for here. I am trying to understand the maths behind trilateration, we have 3 access points $(\text{AP's: } 1,2,3)$ an......Thu, 27 Aug 2026 05:04:38 -0400https://liberty.io/compound-angle-formula.htmlhttps://liberty.io/compound-angle-formula.htmlI understand how to use the compound angle formula when solving $\sin(\pi/12)$. However I dont understand how I can use a compound angle formula to show $$\arctan(3)-\arctan(1/2)=\pi/4$$ Thankyou ......Thu, 27 Aug 2026 04:59:20 -0400https://liberty.io/define-the-subgroup-of-m2-r.htmlhttps://liberty.io/define-the-subgroup-of-m2-r.htmlFind a subgroup of $M_2(R)$, the group of $2 \times 2$ matrices with real entries under addition, called $H$, if for all $A,B$ in $M_2(R)$, $A-B$ is an element of subgroup $H$ implies that $\operat......Thu, 27 Aug 2026 04:55:49 -0400https://liberty.io/how-many-sides-can-a-polygon-have-before-it-is-considered-a-circle-clo.htmlhttps://liberty.io/how-many-sides-can-a-polygon-have-before-it-is-considered-a-circle-clo.htmlGood day, my family had a dinner discussion about polygons and how many sides a polygon has in relation to the angle measurement you'll get when you measure an "arc" encompassing a "side" of the po......Thu, 27 Aug 2026 04:48:50 -0400https://liberty.io/a-rectangle-is-scissors-congruent-to-a-square-of-the-same-area.htmlhttps://liberty.io/a-rectangle-is-scissors-congruent-to-a-square-of-the-same-area.htmlCan somebody explain me step 3, in the following link - here...Thu, 27 Aug 2026 04:48:36 -0400https://liberty.io/verifying-eigenvalues.htmlhttps://liberty.io/verifying-eigenvalues.htmlHow would you check whether eigenvalues $\lambda_1=8$, $\lambda_2=3$, $\lambda_3=-1$ belong to a matrix? $$ \begin{matrix} 7 & 1 & 1\\ 3 & 1 & 2 \\ 1......Thu, 27 Aug 2026 04:46:08 -0400https://liberty.io/formula-for-skewed-bell-curve.htmlhttps://liberty.io/formula-for-skewed-bell-curve.htmlI can plot a bell curve using the formula: $$ Y = \frac{1}{\sqrt{2\pi S^2} } e^{ -\frac{(X-A)^2}{(2S)^2} } $$ where A=mean and S=standard deviation. This is obviously on an X,Y plot. I want to add a ...Thu, 27 Aug 2026 04:38:20 -0400https://liberty.io/orthogonality-of-sine-and-cosine-integrals.htmlhttps://liberty.io/orthogonality-of-sine-and-cosine-integrals.htmlHow to prove that $$ \int_{t_0}^{t_0+T} \sin(m\omega t)\sin(n\omega t)\,\mathrm{d}t$$ will equal to $0$ when $m\ne n$ and $\frac{T}{2}$ when $m=n\ne 0$? Besides $$ \int_{t_0}^{t_0+T} \cos(m\omega ......Thu, 27 Aug 2026 04:30:39 -0400https://liberty.io/what-are-the-5-simplest-lambda-calculus-expresions.htmlhttps://liberty.io/what-are-the-5-simplest-lambda-calculus-expresions.htmlI'm struggling to learn lambda Calculus. I think what might really help is to see the simplest functions that you can create in lambda calculus and how they might be combined to make more complex ...Thu, 27 Aug 2026 04:23:03 -0400https://liberty.io/how-to-obtain-3-parameters-from-a-linearized-model.htmlhttps://liberty.io/how-to-obtain-3-parameters-from-a-linearized-model.htmlI have the following model: $$ q_e=\dfrac{K_s\ {C_e}^\beta}{1+a_s\ {C_e}^\beta} $$ Its linearized form is: $$ \beta\ln(C_e)=-\ln\left(\dfrac{K_s}{q_e}\right)+\ln a_s $$ The known values are $C_......Thu, 27 Aug 2026 04:20:12 -0400https://liberty.io/total-derivative-vs-chain-rule.htmlhttps://liberty.io/total-derivative-vs-chain-rule.htmlThe total derivative as I understand it would be a linear map along the lines on the wikipedia page. https://en.wikipedia.org/wiki/Total_derivative However in some instances it looks alot like a c......Thu, 27 Aug 2026 04:20:11 -0400https://liberty.io/parameterized-hexagonal-spiral-equation-s.htmlhttps://liberty.io/parameterized-hexagonal-spiral-equation-s.htmlI need to design a fully parameterized hexagonal spiral, where I can choose/change the distance between turns, the inner and outer radii etc. I understand that it will be somehow related to the ...Thu, 27 Aug 2026 04:15:28 -0400https://liberty.io/characterization-of-faithfully-flat-homomorphisms.htmlhttps://liberty.io/characterization-of-faithfully-flat-homomorphisms.htmlLet $A \to B$ be a homomorphism of commutative rings. Why are the following conditions equivalent? $A \to B$ is faithfully flat. $A \to B$ is injective, flat and $B/A$ is a flat $A$-module. This ......Thu, 27 Aug 2026 03:59:30 -0400https://liberty.io/how-do-you-take-the-conjugate-of-a-real-matrix.htmlhttps://liberty.io/how-do-you-take-the-conjugate-of-a-real-matrix.htmlI've only seen resources pointing to how to take the conjugate of a complex matrix but how do you take the conjugate of a real matrix? Let's take this matrix as an example: 0 1 1 0...Thu, 27 Aug 2026 03:40:18 -0400https://liberty.io/standard-deviation-with-exponential-distribution.htmlhttps://liberty.io/standard-deviation-with-exponential-distribution.htmlLet x denote the distance that an animal moves from its birth site to the first territorial vacancy it encounters. Suppose that x has an exponential distribution with parameter lambda = 0.01386. a......Thu, 27 Aug 2026 03:35:35 -0400https://liberty.io/using-unit-circle-to-explain-cos-0-1-and-sin-90-1.htmlhttps://liberty.io/using-unit-circle-to-explain-cos-0-1-and-sin-90-1.htmlWe have been taught $\cos(0) = 1$ and $\sin(90) = 1$. But, how do I visualize these angles on the unit circle?...Thu, 27 Aug 2026 03:20:36 -0400https://liberty.io/what-does-1-13-times-faster-mean.htmlhttps://liberty.io/what-does-1-13-times-faster-mean.htmlI'm reading High Performance JavaScript, and I think the graphs in one chapter are just plain wrong. Here is one on Google Books. The y axis is "Times faster", and it runs from -1.5 to +4.0. Now, I ...Thu, 27 Aug 2026 03:19:06 -0400https://liberty.io/subset-of-a-regular-language.htmlhttps://liberty.io/subset-of-a-regular-language.htmlI need to show that a subset of a regular language is regular or not. I think it may not be regular but I could not find a counter example. Do you have any simple example to prove that? Thanks in ...Thu, 27 Aug 2026 03:17:19 -0400https://liberty.io/modulus-of-continuity-properties-and-uniform-continuity.htmlhttps://liberty.io/modulus-of-continuity-properties-and-uniform-continuity.htmlLet $f:[a,b]\rightarrow \mathbb{R}$ bounded and $\omega(f,r)=\sup\{|f(x)-f(y)| \colon x,y \in [a,b], \ |x-y|<r\}$ (called modulus of continuity of $f$ EDIT note: the original question is in ...Thu, 27 Aug 2026 03:11:01 -0400https://liberty.io/0-order-logic-construction-sequence.htmlhttps://liberty.io/0-order-logic-construction-sequence.htmlI am asked to exhibit a construction sequence to show whether the following is a well-formed formula; Let S = {A, B, C, D} $\alpha = ((A\land C) \rightarrow (((\lnot A) \land (\lnot C)) \lor B)))......Thu, 27 Aug 2026 03:01:56 -0400https://liberty.io/what-does-039-039-and-039-039-mean-in-algebra.htmlhttps://liberty.io/what-does-039-039-and-039-039-mean-in-algebra.htmlIn the book I'm reading it says about ratios and proportions, If $a:b::c:d;$ then $ad=bc$, and $\frac{a}{b}=\frac{c}{d}$ What do ':' and '::' mean? Also, it is not stating the above as a defini......Thu, 27 Aug 2026 02:59:27 -0400https://liberty.io/definition-of-monotone-increasing-sets.htmlhttps://liberty.io/definition-of-monotone-increasing-sets.htmlI've started reading All of Statistics by Larry Wasserman with the hope of gaining understanding of machine learning fundamentals. The author gives the following definition in the first chapter: ......Thu, 27 Aug 2026 02:46:48 -0400https://liberty.io/square-roots-of-complex-number-in-exponential-form.htmlhttps://liberty.io/square-roots-of-complex-number-in-exponential-form.htmlThe complex number $z$ is defined by $z=\frac{9\sqrt{3}+9i}{\sqrt{3}-i}$. Find the square roots of $z$, giving your answers in the form $re^{i\theta}$.where $r>0$ and $-\pi < \theta \leq \pi$......Thu, 27 Aug 2026 02:46:07 -0400https://liberty.io/how-do-i-find-the-orthogonal-basis-for-this-plane.htmlhttps://liberty.io/how-do-i-find-the-orthogonal-basis-for-this-plane.htmlQuestion: $P$ is a plane through the origin given by $x + y + 2z = 0$. Find an orthogonal basis v1, v2 ∈ $P$. My answer: I'm assuming the question asks for two vectors that span this plane ......Thu, 27 Aug 2026 02:29:10 -0400https://liberty.io/why-is-the-pdf-of-the-sum-of-two-continuous-random-variables-the-convo.htmlhttps://liberty.io/why-is-the-pdf-of-the-sum-of-two-continuous-random-variables-the-convo.htmlI have taken a probability course last year but we didn't cover that notion. I do know the steps in the discrete case. finding the support of $X + Y$... calculating the the probability of each el......Thu, 27 Aug 2026 02:18:45 -0400https://liberty.io/how-to-show-i-1-i-duplicate.htmlhttps://liberty.io/how-to-show-i-1-i-duplicate.htmlHow can I show $i^{-1} = -i$, where $i$ is the imaginary unit? Here's what I've tried: $i^{-1} = (-1)^{-1/2} = \dots ?$...Thu, 27 Aug 2026 01:58:20 -0400https://liberty.io/how-to-find-local-maximum-and-minimum-of-function-f-n-x-n-sin-x-at-x-0.htmlhttps://liberty.io/how-to-find-local-maximum-and-minimum-of-function-f-n-x-n-sin-x-at-x-0.htmlHow to find local maximum and minimum of function $f_{n} = x^{n} \sin x$ at $x=0$? Where $n≥2$. I tried to find local Maxima or minima by finding the critical points, but I'm getting no critical p......Thu, 27 Aug 2026 01:54:40 -0400https://liberty.io/is-it-possible-to-have-a-function-differentiable-but-not-continuous-in.htmlhttps://liberty.io/is-it-possible-to-have-a-function-differentiable-but-not-continuous-in.htmlIs there any possible function that is not continuous but differentiable in a given interval. It sounds non-logical to me since differentiation is a special limit function in itself therefore non-...Thu, 27 Aug 2026 01:48:58 -0400https://liberty.io/re-learning-math-from-scratch-closed.htmlhttps://liberty.io/re-learning-math-from-scratch-closed.htmlTwo or three years ago I decided to re-learn math from scratch, I did some research on what to read, made a list of books. I started from a Sawyer, it opened my eyes a little bit on the fact that m......Thu, 27 Aug 2026 01:46:18 -0400https://liberty.io/the-pepperoni-pizza-problem.htmlhttps://liberty.io/the-pepperoni-pizza-problem.htmlThis problem arose in a different context at work, but I have translated it to pizza. Suppose you have a circular pizza of radius $R$. Upon this disc, $n$ pepperoni will be distributed completely ...Thu, 27 Aug 2026 01:27:41 -0400https://liberty.io/what-are-the-odds-of-rolling-a-1-and-a-20-in-two-d20-dice.htmlhttps://liberty.io/what-are-the-odds-of-rolling-a-1-and-a-20-in-two-d20-dice.htmlSo, one of these days a friend of mine rolled two d20 dice and got a $1$ in one of them and a $20$ in the other. I was thinking, what are the odds of this happening?...Thu, 27 Aug 2026 01:27:00 -0400https://liberty.io/unique-solution-for-ode-y-039-sqrt-1-y-2.htmlhttps://liberty.io/unique-solution-for-ode-y-039-sqrt-1-y-2.htmlI was given to solve the next ode: $y' = {\sqrt{1-y^2}} $ I found its solution: $y=sin(x+c)$ Now, I'm given that $y(0)=0$ and asked to show the only solution is $y=sin(x)$ in the region $(-\infty,\...Thu, 27 Aug 2026 01:23:54 -0400https://liberty.io/what-is-the-probability-that-a-five-card-poker-hand-has-four-aces.htmlhttps://liberty.io/what-is-the-probability-that-a-five-card-poker-hand-has-four-aces.htmlWhat is the probability that a five-card poker hand has four ACES? When I was solving the above stated problem, I got confused while trying different methods : Assume a normal $52$ deck of cards. ...Thu, 27 Aug 2026 01:13:10 -0400https://liberty.io/how-to-prove-that-a-function-is-differentiable-everywhere.htmlhttps://liberty.io/how-to-prove-that-a-function-is-differentiable-everywhere.htmlThe function I come up with is: $f(x)= -5x$ and I think this function can be differentiated everywhere because the domain is $\mathbb R$ right? $f'(x) = -5$ which is constant, but my question i......Thu, 27 Aug 2026 01:12:54 -0400https://liberty.io/partial-differential-equation-with-a-flux-term.htmlhttps://liberty.io/partial-differential-equation-with-a-flux-term.htmlRecall how we derived all our equations: Take an interval $[a,b]$ and consider $$\dfrac{\mathrm d}{\mathrm dt}\int_a^b(\text{quantity) d}x=\big[\text{Flux}\big]_a-\big[{\rm Flux}\big]_b.$$ For our ...Thu, 27 Aug 2026 01:08:33 -0400https://liberty.io/find-side-bc-of-a-triangle-given-ab-ac-and-a-relation-between-angle-a-.htmlhttps://liberty.io/find-side-bc-of-a-triangle-given-ab-ac-and-a-relation-between-angle-a-.htmlA question from my class: In triangle $ABC$, $3\angle A+2\angle B=180$ and $AB=10, AC=4$. So question is, what all can we comment on side $BC$. Can we find its exact length? I have a crude ...Thu, 27 Aug 2026 01:07:39 -0400https://liberty.io/tessellating-the-sphere.htmlhttps://liberty.io/tessellating-the-sphere.htmlIt is a famous result that the plane can be tessellated by regular triangles, squares, and hexagons. Which regular polygons can tessellate the sphere?...Thu, 27 Aug 2026 01:06:48 -0400https://liberty.io/spherical-cow-calculus-1-closed.htmlhttps://liberty.io/spherical-cow-calculus-1-closed.htmlCan someone please help me with the 3 calculus one word problems below, I've done all three of them several times but can never seem to arrive at the correct answer. This is greatly frustrating bec......Thu, 27 Aug 2026 01:01:48 -0400https://liberty.io/which-is-asymptotically-larger-lg-lg-n-or-lg-lg-n.htmlhttps://liberty.io/which-is-asymptotically-larger-lg-lg-n-or-lg-lg-n.htmlThis definition is extracted from "Introduction to Algorithm, 2nd Edition". The iterated logarithm function We use the notation $\lg^* n$ (read "log star of $n$") to denote the ...Thu, 27 Aug 2026 01:00:43 -0400https://liberty.io/how-are-the-tangential-and-normal-components-of-the-acceleration-vecto.htmlhttps://liberty.io/how-are-the-tangential-and-normal-components-of-the-acceleration-vecto.htmlLet acceleration be $$\overrightarrow a= a_T \overrightarrow T + a_N \overrightarrow N$$ where $\overrightarrow T$ is the unit tangent component and $\overrightarrow N$ is the unit normal component......Thu, 27 Aug 2026 00:52:53 -0400https://liberty.io/category-theory-homset-preserves-limits.htmlhttps://liberty.io/category-theory-homset-preserves-limits.htmledit I updated my question at the end, I think the claim may be false? Let $(L,\lambda)$ be a limit cone of a diagram $D$ in a category. For any object $X$ it is said that the hom functor $......Thu, 27 Aug 2026 00:48:06 -0400https://liberty.io/how-is-fubini-039-s-theorem-used-in-the-following-proof.htmlhttps://liberty.io/how-is-fubini-039-s-theorem-used-in-the-following-proof.htmlI'm having trouble to understand exactly how we are using Fubini's theorem in the following proof involving the distribution function, since it newer explicitly involves an integral with product me......Thu, 27 Aug 2026 00:47:58 -0400https://liberty.io/how-to-proof-that-phi-1-1-phi-1-for-standard-normal-distribution.htmlhttps://liberty.io/how-to-proof-that-phi-1-1-phi-1-for-standard-normal-distribution.htmlI have been studying normal distribution and frequently I see that $$\Phi(1)=1-\Phi(-1)$$ where $\Phi(x)=\int_{-\infty}^{x}\phi(u)du$ is the cdf of a standard normal variable. How one can prove eq......Thu, 27 Aug 2026 00:41:24 -0400https://liberty.io/what-is-equipotent-relation.htmlhttps://liberty.io/what-is-equipotent-relation.htmlIn Royden's book Real Analysis, page 13, he writes that "We call two sets A and b equipotent provided there is a one-to-one mapping $f$ from A onto B Equipotence defines a equivalence relation amon......Thu, 27 Aug 2026 00:17:23 -0400https://liberty.io/what-do-you-call-the-point-where-two-lines-meet.htmlhttps://liberty.io/what-do-you-call-the-point-where-two-lines-meet.htmlThis is from a third grader. His example is the point where the hands on the clock meet. It's not pivot. Or "if you start with a dot and make two lines go out from it, on straight up and one to the......Thu, 27 Aug 2026 00:16:37 -0400https://liberty.io/is-the-theory-of-dual-numbers-strong-enough-to-develop-real-analysis-a.htmlhttps://liberty.io/is-the-theory-of-dual-numbers-strong-enough-to-develop-real-analysis-a.htmlI've been interested in non-standard analysis recently. I was reading up on it and noticed the following interesting comment on the Wikipedia page about hyperreal numbers, right after giving an exa......Wed, 26 Aug 2026 23:55:08 -0400https://liberty.io/why-should-i-care-about-adjoint-functors.htmlhttps://liberty.io/why-should-i-care-about-adjoint-functors.htmlI am comfortable with the definition of adjoint functors. I have done a few exercises proving that certain pairs of functors are adjoint (tensor and hom, sheafification and forgetful, direct image ......Wed, 26 Aug 2026 23:50:41 -0400https://liberty.io/rank-of-an-augmented-matrix.htmlhttps://liberty.io/rank-of-an-augmented-matrix.htmlCan the rank of coeffecient matrix be greater than augmented matrix? Also,what is the condition for an inconsistent set of linear equations?...Wed, 26 Aug 2026 23:49:32 -0400https://liberty.io/the-use-of-gershgorin-circle-theorem.htmlhttps://liberty.io/the-use-of-gershgorin-circle-theorem.htmlIn order to estimate the eigenvalues of a real symmetric $n\times n$ matrix, I intend to use the Gershgorin Circle Theorem. Unfortunately, the examples one might find on the internet are a bit conf......Wed, 26 Aug 2026 23:42:17 -0400https://liberty.io/having-trouble-understanding-meaning-of-a-finite-dimensional-vector-sp.htmlhttps://liberty.io/having-trouble-understanding-meaning-of-a-finite-dimensional-vector-sp.htmlThe definition I have is the following: A vector space V is said to be finite-dimensional if there is a finite set of vectors in V that spans V and is said to be infinite-dimensional if no such ......Wed, 26 Aug 2026 23:31:47 -0400https://liberty.io/when-does-a-system-of-equations-have-infinite-unique-and-no-solutions.htmlhttps://liberty.io/when-does-a-system-of-equations-have-infinite-unique-and-no-solutions.htmlCan someone please tell me what a matrix looks like when there is infinite solutions, unique solution and no solutions. I have been searching the internet and I cannot find a straightforward answer......Wed, 26 Aug 2026 23:18:04 -0400https://liberty.io/polar-form-of-frac-1-z.htmlhttps://liberty.io/polar-form-of-frac-1-z.htmlWrite the following in polar form:$$\frac{1}{z}$$ $$\frac{1}{z}=\frac{1}{x+yi}$$ $$\frac{1}{x+yi}\cdot \frac{x-yi}{x-yi}=\frac{x-yi}{x^2+y^2}=\frac{x-yi}{r^2}$$ Because: $$x-yi=rcis(-\frac{\pi}......Wed, 26 Aug 2026 23:15:19 -0400https://liberty.io/how-to-integrate-sqrt-x-2-y-2-1-the-easy-way.htmlhttps://liberty.io/how-to-integrate-sqrt-x-2-y-2-1-the-easy-way.htmlI know you can change to polar coordinates but then you still have to integrate $\sqrt{1+r^2}$ which is still non-trivial. I remember there being some trigonometric substitution, possibly hyperbo......Wed, 26 Aug 2026 23:10:26 -0400https://liberty.io/bound-for-the-fourier-transform.htmlhttps://liberty.io/bound-for-the-fourier-transform.htmlCan someone come up with a proof for this bound of the Fourier transformation? Let $f$ be $s$ times continuously differentiable, with compact support, then there exists a real constant $c$ such th......Wed, 26 Aug 2026 22:52:12 -0400https://liberty.io/for-what-value-of-constant-a-is-function-continuous.htmlhttps://liberty.io/for-what-value-of-constant-a-is-function-continuous.htmlI know there is a similar question. I had a read through it and it didn't help me so I'm posting this one. The question is for what value(s) of the constant $a\in \mathbb R$ is $$f_a(x) = \left\{ \...Wed, 26 Aug 2026 22:50:48 -0400https://liberty.io/prove-this-identity-cos-x-sec-x-cos-x-sin-2x-closed.htmlhttps://liberty.io/prove-this-identity-cos-x-sec-x-cos-x-sin-2x-closed.htmlI have been trying to prove this identity for about 25 minutes and I can't figure it out. Can you help me please? $$\cos x(\sec x-\cos x)=\sin^2x$$...Wed, 26 Aug 2026 22:44:49 -0400https://liberty.io/find-the-least-significant-digit-of-2-3a-where-a-10-100.htmlhttps://liberty.io/find-the-least-significant-digit-of-2-3a-where-a-10-100.htmlCan anyone please tell me - What is the least significant digit in $2^{3A}$? How do we define least significant digit?...Wed, 26 Aug 2026 22:43:13 -0400https://liberty.io/unimodular-matrix-definition.htmlhttps://liberty.io/unimodular-matrix-definition.htmlI'm a bit confused. Based on Wikipedia: In mathematics, a unimodular matrix M is a square integer matrix having determinant +1, 0 or −1. Equivalently, it is an integer matrix that is invertibl......Wed, 26 Aug 2026 22:38:40 -0400https://liberty.io/choosing-randomly-integers-from-1-to-10.htmlhttps://liberty.io/choosing-randomly-integers-from-1-to-10.htmlFrom Question 5 the practice book of the GRE math subject test: Sofia and Tess will each randomly choose one of the $10$ integers from $1$ to $10$. What is the probability that neither integer cho......Wed, 26 Aug 2026 22:29:10 -0400https://liberty.io/rotate-a-shape-360-degrees-on-one-axis-and-not-be-the-same-as-the-star.htmlhttps://liberty.io/rotate-a-shape-360-degrees-on-one-axis-and-not-be-the-same-as-the-star.htmlAre there shapes in nth dimensions (in euclidian geometry), that can be rotated 360 degrees on one axis and have more than one point not in the same place? Alternatively, in non-euclidian geometry,......Wed, 26 Aug 2026 22:23:44 -0400https://liberty.io/summation-of-1-k-duplicate.htmlhttps://liberty.io/summation-of-1-k-duplicate.htmlWhat is summation of 1/k where n ranges from 1 to n. I need the general formula for the summation. I know the series tends to infinity when k tends to infinity . But upto n terms there must be a ...Wed, 26 Aug 2026 22:19:19 -0400https://liberty.io/sobolev-norm-in-the-definition-of-sobolev-spaces.htmlhttps://liberty.io/sobolev-norm-in-the-definition-of-sobolev-spaces.htmlI've seen the Sobolev space defined as: The Sobolev space $H^k(\Omega)$ is the set of all functions $u \in L_2(\Omega)$ for which the weak derivative $\partial^\alpha u \in L_2(\Omega)$ for all $|\...Wed, 26 Aug 2026 22:17:58 -0400https://liberty.io/conflicting-definition-of-the-hessian-matrix-does-the-order-of-the-par.htmlhttps://liberty.io/conflicting-definition-of-the-hessian-matrix-does-the-order-of-the-par.htmlOn Wikipedia, the Hessian matrix is defined as, https://en.wikipedia.org/wiki/Hessian_matrix $ {\displaystyle \mathbf {H} ={\begin{bmatrix}{\dfrac {\partial ^{2}f}{\partial x_{1}^{2}}}&{\dfrac {\...Wed, 26 Aug 2026 22:12:18 -0400https://liberty.io/how-to-visualize-triple-integrals.htmlhttps://liberty.io/how-to-visualize-triple-integrals.htmlI am struggling a lot with triple integrals. I can evaluate them, but I find it extremely difficult to write the triple integral. I cannot visualize them. An example question: $\iiint_E6xy\;dV$, w......Wed, 26 Aug 2026 22:00:07 -0400https://liberty.io/orthogonal-projection-onto-a-vector-with-matrix-transformation.htmlhttps://liberty.io/orthogonal-projection-onto-a-vector-with-matrix-transformation.htmlSo I have this question here which says: $a)$ Find the standard matrix of the linear operator $T:R^2\rightarrow R^2$ given by the orthogonal projection onto the vector $(1,-2)$. $b)$ Given the li......Wed, 26 Aug 2026 21:57:07 -0400https://liberty.io/how-to-express-rounding-down-up-to-nearest-multiplum-of-a-number.htmlhttps://liberty.io/how-to-express-rounding-down-up-to-nearest-multiplum-of-a-number.htmlI've written an equation in a programming language which I need to express using maths: slots = (elements_count - (elements_count % slot_size) + slot_size) / slot_size In Excel, you can also wr......Wed, 26 Aug 2026 21:48:52 -0400https://liberty.io/domain-and-range-of-f-x.htmlhttps://liberty.io/domain-and-range-of-f-x.htmlFind the domain and range of Function? $$F(x) = \frac{1}{\sqrt{25-x^2}}$$ Domanin f(x) has real value , if $$25-x^2\geqslant0$$ $$-x^2\geqslant-25$$ $$x^2\le25$$ $$-5\le x\le5$$ $$D_f= [-5,5]$$ ......Wed, 26 Aug 2026 21:46:40 -0400https://liberty.io/angle-between-edges-in-a-square-pyramid.htmlhttps://liberty.io/angle-between-edges-in-a-square-pyramid.htmlHow do you calculate the missing angle? I couldn't figure it out. This is not homework, just a math practice question that I made up to test myself....Wed, 26 Aug 2026 21:35:01 -0400https://liberty.io/derivative-of-1-1-x-2.htmlhttps://liberty.io/derivative-of-1-1-x-2.html$ \frac{\partial f}{\partial x}\dfrac{1}{(1-x)^2} = \dfrac{0 *(1-x)^2 - (-1)(1)}{(1-x)^2*(1-x)} $ But: $ \frac{\partial f}{\partial x}\dfrac{1}{(1-x)^2} \neq \dfrac{0 *(x^2-2x+1) - (2x-2)(1)}{(x^2-......Wed, 26 Aug 2026 21:33:22 -0400https://liberty.io/mercer-039-s-condition-and-rkhss.htmlhttps://liberty.io/mercer-039-s-condition-and-rkhss.htmlIn general when considering an RKHS $\mathcal{H}$ over a set $X$ one has a kernel function $k:X\times X\rightarrow\mathbb{C}$. It is then not too hard to show that its reproducing property implies a ...Wed, 26 Aug 2026 21:31:20 -0400https://liberty.io/instantaneous-rate-of-change-and-average-rate-of-change.htmlhttps://liberty.io/instantaneous-rate-of-change-and-average-rate-of-change.htmlI'm currently studying for my calculus AP exam and I'm currently stuck on a question. Unless otherwise specified, the domain of a function $f$ is assumed to be the set of all real numbers $x$ for ......Wed, 26 Aug 2026 21:27:58 -0400https://liberty.io/which-of-the-following-sets-of-vectors-are-linearly-independent.htmlhttps://liberty.io/which-of-the-following-sets-of-vectors-are-linearly-independent.htmlWhich of the following set of vectors are linearly independent? I'm a bit confused. I think the answer(s) would be A, C, and D. I'm unsure of how to actually figure out if each one is or isn't LI/LD...Wed, 26 Aug 2026 21:20:54 -0400https://liberty.io/applying-mean-value-theorem-to-formula.htmlhttps://liberty.io/applying-mean-value-theorem-to-formula.htmlAs I understand it, the mean value theorem is where $${f}'(c)=\frac{f(b)-f(a)}{b-a}$$ if $f$ is continuous on the open interval (a, b) and differentiable on the closed interval [a,b]. A proble......Wed, 26 Aug 2026 21:13:13 -0400https://liberty.io/dimension-of-solution-set-for-ax-b-but-the-solution-set-isn-039-t-a-ve.htmlhttps://liberty.io/dimension-of-solution-set-for-ax-b-but-the-solution-set-isn-039-t-a-ve.htmlThe solution set of a system of $n$ linear equations written as a matrix equation $AX=b$,$b\neq0$ cannot have zero vector in it. This means that the solution set cannot be a vector space and talking ...Wed, 26 Aug 2026 21:12:01 -0400https://liberty.io/assuming-a-n-is-a-convergent-sequence-prove-that-the-lim-inf-of-a-n-1-.htmlhttps://liberty.io/assuming-a-n-is-a-convergent-sequence-prove-that-the-lim-inf-of-a-n-1-.htmlI'm aware that you have to use the definition of a limit of a sequence, which is: $\lim\limits_{n \to \infty} a_n = L $ if for every $E > 0$, there is an $N$ such that if $n >N $then $|a_n -......Wed, 26 Aug 2026 21:04:07 -0400https://liberty.io/why-can-a-matrix-without-a-full-rank-not-be-invertible.htmlhttps://liberty.io/why-can-a-matrix-without-a-full-rank-not-be-invertible.htmlI know you could just say because the $\det = 0$. But during the introduction of determinants the professor said, obviously if two columns of the matrix are linearly dependent the matrix can't be ...Wed, 26 Aug 2026 20:58:06 -0400https://liberty.io/proof-of-a-2-ab-b-2-0-a-b-both-are-not-zero-both-are-real-duplicate.htmlhttps://liberty.io/proof-of-a-2-ab-b-2-0-a-b-both-are-not-zero-both-are-real-duplicate.htmlShow that for any reals a,b, both of which are not zero, it holds true that a^2 + ab + b^2 >0...Wed, 26 Aug 2026 20:55:07 -0400https://liberty.io/what-is-the-difference-between-average-and-expected-value.htmlhttps://liberty.io/what-is-the-difference-between-average-and-expected-value.htmlQuestion : What is the difference between Average and Expected value? I have been going through the definition of expected value on Wikipedia beneath all that jargon it seems that the expected val......Wed, 26 Aug 2026 20:54:15 -0400https://liberty.io/prove-that-x-is-rational.htmlhttps://liberty.io/prove-that-x-is-rational.htmlLet $x$ be a real number with the properties that $x^3+x$ and $x^5+x$ are rational. Prove that $x$ is rational. Denote $a=x^3+x$; $b=x^5+x$. We can multiply and add them together until we get the d......Wed, 26 Aug 2026 20:36:45 -0400https://liberty.io/differentiating-with-respect-to-1-x.htmlhttps://liberty.io/differentiating-with-respect-to-1-x.htmlI am fairly sure this is a silly question, but a Google search was insufficient to find a satisfactory answer. If I differentiate some function of $x$ with respect to $1-x$, what do I get compare......Wed, 26 Aug 2026 20:36:15 -0400https://liberty.io/minimal-sufficient-statistics-for-uniform-distribution-on-theta-theta.htmlhttps://liberty.io/minimal-sufficient-statistics-for-uniform-distribution-on-theta-theta.htmlLet $X_1,\dots,X_n$ be a sample from uniform distribution on $(-\theta,\theta)$ with parameter $\theta>0$. It is easy to show that $T(X) = (X_{(1)},X_{(n)})$ is a sufficient statistic for $\the......Wed, 26 Aug 2026 20:34:22 -0400https://liberty.io/partial-differential-equation-u-t-3u-xx.htmlhttps://liberty.io/partial-differential-equation-u-t-3u-xx.htmlSolve the partial differential equation $$u_t=3u_{xx}, u(0,t)=0, u(x,0)=\cos{x}\sin{5x}$$ Attempt: Using separation of variables, let $u(x,t)=f(x)g(t)$, so $$f(x)g'(t)=3f''(x)g(t)$$ $$\frac{g'(t)}{......Wed, 26 Aug 2026 20:26:57 -0400https://liberty.io/is-the-rank-of-a-matrix-the-same-of-its-transpose-if-yes-how-can-i-pro.htmlhttps://liberty.io/is-the-rank-of-a-matrix-the-same-of-its-transpose-if-yes-how-can-i-pro.htmlI am auditing a Linear Algebra class, and today we were taught about the rank of a matrix. The definition was given from the row point of view: "The rank of a matrix A is the number of non-zero ...Wed, 26 Aug 2026 20:26:33 -0400https://liberty.io/how-to-use-parametric-equations-for-line-integral.htmlhttps://liberty.io/how-to-use-parametric-equations-for-line-integral.htmlThe question is to find the work done in moving a particle in a force field: $$\overrightarrow F = 3x^2i+(2xy-y)j+3k$$ along the straight line from (0, 0, 0) to (2, 1, 3) So, work done $=\int \...Wed, 26 Aug 2026 20:22:25 -0400https://liberty.io/what-039-s-the-difference-between-theorem-lemma-and-corollary.htmlhttps://liberty.io/what-039-s-the-difference-between-theorem-lemma-and-corollary.htmlCan anybody explain me what is the basic difference between theorem, lemma and corollary? We have been using it for a long time but I never paid any attention. I am just curious to know....Wed, 26 Aug 2026 20:19:07 -0400https://liberty.io/proving-and-distributive-law-using-boolean-algebra.htmlhttps://liberty.io/proving-and-distributive-law-using-boolean-algebra.html$$ X(Y+Z) = (XY) + (XZ) $$ I can’t seem to derive the proper steps to prove this equation using Boolean axioms. The hint I’ve been given is using demorgans laws proofs but I still can’t seem to fi......Wed, 26 Aug 2026 20:17:02 -0400https://liberty.io/finding-contrapositive-of-a-universal-statement.htmlhttps://liberty.io/finding-contrapositive-of-a-universal-statement.htmlFor all $x$, if $x^2$ is even, then $x$ is even. The contrapositive to this statement is: For all $x$, if $x$ is odd, then $x^2$ is odd. Why do we ignore the "For all $x$" and not say ......Wed, 26 Aug 2026 20:16:58 -0400https://liberty.io/soft-question-definition-of-the-real-plane.htmlhttps://liberty.io/soft-question-definition-of-the-real-plane.htmlI have only seen the real plane to be defined as $\mathbb R \times \mathbb R$. In Euclidean geometry, the origin does not play any central role, but in this set theoretic definition, it does. Is t......Wed, 26 Aug 2026 20:13:26 -0400https://liberty.io/how-to-prepare-for-integral-calculus-calculus-2.htmlhttps://liberty.io/how-to-prepare-for-integral-calculus-calculus-2.htmlI'm majoring in computer engineering and I have Calculus 2 coming up this semester. From what I understand, Calculus 2 is the most difficult math class in the engineering path. Over the summer, I......Wed, 26 Aug 2026 20:10:41 -0400https://liberty.io/probability-of-drawing-a-flush-from-a-standard-deck-of-cards.htmlhttps://liberty.io/probability-of-drawing-a-flush-from-a-standard-deck-of-cards.htmlThe following is the problem put to me: A 5-card poker hand is dealt from a well shuffled regular 52-card playing card deck. Find the probability that the hand is a Flush (5 nonconsecutive cards e......Wed, 26 Aug 2026 19:54:52 -0400https://liberty.io/how-to-calculate-the-height-of-a-cone-at-particular-volume.htmlhttps://liberty.io/how-to-calculate-the-height-of-a-cone-at-particular-volume.htmlThe equation for the volume of a cone is $\frac{1}{3}\pi r^2h$ starting from the base. However, I wanna calculate the height of the cone for a particular volume from the tip of the cone. Could you ......Wed, 26 Aug 2026 19:31:46 -0400https://liberty.io/with-the-help-of-chinese-remainder-theorem-show-that-x-144-equiv-1-pmo.htmlhttps://liberty.io/with-the-help-of-chinese-remainder-theorem-show-that-x-144-equiv-1-pmo.htmlWith the help of Chinese remainder theorem show that $ x^{144} \equiv 1 \pmod{323}$ for all $x$ relatively prime to 323. The problem with me is that I used to use CRT when $x$ is raised to a power......Wed, 26 Aug 2026 19:25:38 -0400https://liberty.io/what-does-curly-curved-less-than-sign-succcurlyeq-mean.htmlhttps://liberty.io/what-does-curly-curved-less-than-sign-succcurlyeq-mean.htmlI am reading Boyd & Vandenberghe's Convex Optimization. The authors use curved greater than or equal to (\succcurlyeq) $$f(x^*) \succcurlyeq \alpha$$ and curved less than or equal to (\preccu......Wed, 26 Aug 2026 19:22:58 -0400https://liberty.io/is-it-possible-to-reverse-a-gradient-vec-nabla-operation.htmlhttps://liberty.io/is-it-possible-to-reverse-a-gradient-vec-nabla-operation.htmlIn calculus, the antiderivative (indefinite integral) can be considered as the reverse operation of a derivative. A gradient yields a vector. Is there a similar way of reversing gradient, as you d......Wed, 26 Aug 2026 19:15:48 -0400