https://liberty.ioCurated star highlights with fresh energy.en-usMon, 17 Aug 2026 08:12:48 -0400https://liberty.io/user-oxodo-mathematics-stack-exchange.htmlhttps://liberty.io/user-oxodo-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Mon, 17 Aug 2026 12:11:31 -0400https://liberty.io/determining-if-a-statement-is-tautology-contingent-or-contradiction.htmlhttps://liberty.io/determining-if-a-statement-is-tautology-contingent-or-contradiction.htmlSay we are given the following statement A ≡ B where A is some logic statement and B is some logic statement. Determine if it is tautology, contingent or contradiction. I know the definition of ......Mon, 17 Aug 2026 12:06:36 -0400https://liberty.io/how-do-you-find-a-derivative-of-y-1-cos-x-sin-x.htmlhttps://liberty.io/how-do-you-find-a-derivative-of-y-1-cos-x-sin-x.htmlThe question asks to find the derivative of the function $1-\cos(x)\sin(x)$, and I thought maybe using some derivative rules I could, but I don't know where to start....Mon, 17 Aug 2026 11:58:06 -0400https://liberty.io/why-do-equations-work-and-how-do-they-relate-to-each-other.htmlhttps://liberty.io/why-do-equations-work-and-how-do-they-relate-to-each-other.htmlOk, so I understand that an equation is something like 15 = 15 , and that the only criteria as far as I can tell for it being an equation is that both sides are equal to each other. I have a few ...Mon, 17 Aug 2026 11:54:07 -0400https://liberty.io/what-is-the-radial-direction.htmlhttps://liberty.io/what-is-the-radial-direction.htmlI'm currently in an electrostatics course; and wanting to rip my hair out. The question says: A sphere of radius 𝑎 is polarized such that the polarization at 𝒓 within the sphere is given by 𝑷 = ......Mon, 17 Aug 2026 11:45:04 -0400https://liberty.io/determine-which-positive-integers-have-exactly-36-positive-divisors.htmlhttps://liberty.io/determine-which-positive-integers-have-exactly-36-positive-divisors.htmlDetermine which positive integers have exactly 36 positive divisors. Let $n=p_1^a{_2}p_1^a{_2}...p_s^a{_s}$. Then $\tau(n)= \prod_{j=1}^{s} a_j+1$. We need $\tau(n)=36$. So we list the factor pair......Mon, 17 Aug 2026 11:36:15 -0400https://liberty.io/in-the-given-figure-pqrs-is-a-parallelogram.htmlhttps://liberty.io/in-the-given-figure-pqrs-is-a-parallelogram.htmlIn the given figure, $PQRS$ is a parallelogram. $PQ$ is produced to $L$ so that $QL=PQ$. The line $SL$ cuts $QR$ at $O$. Prove that: $\triangle PQS=2\triangle ROL$. Attempt $$PL=LQ$$ $$ Q \ \tex......Mon, 17 Aug 2026 11:31:57 -0400https://liberty.io/show-that-any-non-cyclic-group-of-order-9-is-isomorphic-to-mathbb-z-3-.htmlhttps://liberty.io/show-that-any-non-cyclic-group-of-order-9-is-isomorphic-to-mathbb-z-3-.htmlShow that any non-cyclic group of order $9$ is isomorphic to $\mathbb{Z}_3 \times \mathbb{Z}_3$ The basic idea to show this is to take any non-cyclic group $G$ of order $9$ and apply the theorem b......Mon, 17 Aug 2026 11:30:25 -0400https://liberty.io/symbol-for-finite.htmlhttps://liberty.io/symbol-for-finite.htmlI understand there is a symbol for infinite. Is there one for finite? I searched and found there is none. How is finite represented symbolically?...Mon, 17 Aug 2026 11:21:30 -0400https://liberty.io/completeness-can-be-omitted-from-banach-fixed-point-theorem.htmlhttps://liberty.io/completeness-can-be-omitted-from-banach-fixed-point-theorem.htmlIn Kreyszig's Functional Analysis, page no. 303, exercise no. 3 says that completeness cannot be omitted from Banach's Fixed Point Theorem. But if we take $f(x)=x^2$ from an incomplete metric space......Mon, 17 Aug 2026 11:19:53 -0400https://liberty.io/is-there-a-way-to-write-the-convolution-form-of-a-derivative.htmlhttps://liberty.io/is-there-a-way-to-write-the-convolution-form-of-a-derivative.htmlIn image filtering and computer simulations, derivatives can be applied to a image by performing a convolution of the image/data with an appropriate kernel matrix. For example, in 1D this could be ......Mon, 17 Aug 2026 11:16:05 -0400https://liberty.io/derivative-of-ln-x-e-duplicate.htmlhttps://liberty.io/derivative-of-ln-x-e-duplicate.htmlIn Randall Munroe's What If, he says that "if you want to be mean to first-year calculus students, you can ask them to take the derivative of $(lnx)^e$" He says, as I would expect, that the result ......Mon, 17 Aug 2026 11:03:37 -0400https://liberty.io/markov-process-vs-markov-chain-vs-random-process-vs-stochastic-process.htmlhttps://liberty.io/markov-process-vs-markov-chain-vs-random-process-vs-stochastic-process.htmlI'm trying to understand each of the above terms, and I'm having a lot of trouble deciphering the difference between them. According to Wikipeda: A Markov chain is a memoryless, random process. A ...Mon, 17 Aug 2026 11:03:10 -0400https://liberty.io/infinite-algebraic-extension-of-mathbb-q.htmlhttps://liberty.io/infinite-algebraic-extension-of-mathbb-q.htmlI have this problem in a exercise list: "Prove that $K=\mathbb{Q}(2^{\frac{1}{2}},2^{\frac{1}{3}}, 2^{\frac{1}{4}}, \ldots)$ is an algebraic extension, but not a finite extension of $\mathbb{Q}$."......Mon, 17 Aug 2026 11:02:28 -0400https://liberty.io/what-is-the-formula-for-the-birthday-problem.htmlhttps://liberty.io/what-is-the-formula-for-the-birthday-problem.htmlWhat is the formula for the birthday problem? I can't seem to find JUST the formula. I don't care how it works at this point, I just want to know what to do. Can someone please just tell me? Most ...Mon, 17 Aug 2026 10:57:23 -0400https://liberty.io/plotting-the-complex-exponential-function-and-it-039-s-period-by-hand.htmlhttps://liberty.io/plotting-the-complex-exponential-function-and-it-039-s-period-by-hand.htmlI'm reading a bit about the exponential function $e_k: \mathbb{R} \longrightarrow \mathbb{C}$ given by $$e_{k}: x \mapsto e^{i k x}=\cos k x+i \sin k x$$ The period is given by $2\pi/|k|$. I do no......Mon, 17 Aug 2026 10:56:13 -0400https://liberty.io/how-to-solve-trigonometric-equation-sin-x-x-cdot-cos-x-0.htmlhttps://liberty.io/how-to-solve-trigonometric-equation-sin-x-x-cdot-cos-x-0.htmlI'm facing the problem of solving $$\sin(x)+x \cdot \cos(x)=0$$ using $$\tan(x)=\sin(x)/\cos(x)$$ I end at $$x+\tan(x)=0$$ on the other hand, I also tried $\cos(x)= \pm \sqrt{1-\sin......Mon, 17 Aug 2026 10:52:47 -0400https://liberty.io/lotus-law-of-the-unconscious-statistician-for-non-one-to-one-functions.htmlhttps://liberty.io/lotus-law-of-the-unconscious-statistician-for-non-one-to-one-functions.htmlMy textbook, Introduction to Probability by Blitzstein and Hwang, defines LOTUS (law of the unconscious statistician) as follows: Theorem 4.5.1 (LOTUS). If $X$ is a discrete r.v. and $g$ is a func......Mon, 17 Aug 2026 10:49:23 -0400https://liberty.io/find-a-quintic-polynomial-function-given-roots-and-y-intercept.htmlhttps://liberty.io/find-a-quintic-polynomial-function-given-roots-and-y-intercept.htmlI need to find an expression that satisfies the qualifying conditions for a quintic polynomial. $f(0)=3$ and $f(-2)=f(\frac{1}{2})=f(1)=0$. With this information, I found that the zeros are $2, ......Mon, 17 Aug 2026 10:43:38 -0400https://liberty.io/finding-the-slope-of-a-tangent-line-to-polar-curve.htmlhttps://liberty.io/finding-the-slope-of-a-tangent-line-to-polar-curve.htmlQuestion: Find the slope of the tangent line to the graph of $r = e^\theta - 4$ at $\theta = \frac{\pi}{4}$. $$x = r\cos \theta = (e^\theta - 4)\cos\theta$$ $$y = r\sin \theta = (e^\theta - 4)\sin\...Mon, 17 Aug 2026 10:43:06 -0400https://liberty.io/area-of-a-triangle-adjacent-to-two-similar-triangles.htmlhttps://liberty.io/area-of-a-triangle-adjacent-to-two-similar-triangles.htmlThis GMAT problem states: In the figure above $AD = 4$, $AB = 3$ and $CD = 9$. What is the area of triangle $AEC$ The solution states: to find the base we need to see that triangles $AEB$ and ......Mon, 17 Aug 2026 10:39:49 -0400https://liberty.io/if-squaring-a-number-means-multiplying-that-number-with-itself-then-sh.htmlhttps://liberty.io/if-squaring-a-number-means-multiplying-that-number-with-itself-then-sh.htmlIf squaring a number means multiplying that number with itself then shouldn't taking square root of a number mean to divide a number by itself? For example the square of $2$ is $2^2=2 \cdot 2=4 ......Mon, 17 Aug 2026 10:24:13 -0400https://liberty.io/finding-all-solutions-to-tan-5x-9-tan-x-0.htmlhttps://liberty.io/finding-all-solutions-to-tan-5x-9-tan-x-0.htmlI am stuck when it comes to finding the end value of a trig function. I have the following question: $$ \tan^5x - 9\tan{x} = 0 $$ I worked the problem and got: $$ \tan x = 0\\ \tan^4x-9 = 0\\ x ......Mon, 17 Aug 2026 10:22:35 -0400https://liberty.io/when-to-use-total-probability-rule-and-bayes-039-theorem.htmlhttps://liberty.io/when-to-use-total-probability-rule-and-bayes-039-theorem.htmlConsider the following information about travelers on vacation: $40$% check work email, $30$% use a cell phone, $25$% bring a laptop with them, $23$% both check work email and use a cell phone, and......Mon, 17 Aug 2026 10:11:41 -0400https://liberty.io/prove-casting-out-nines-of-an-integer-is-equivalent-to-that-integer-mo.htmlhttps://liberty.io/prove-casting-out-nines-of-an-integer-is-equivalent-to-that-integer-mo.htmlLet $s(x)$ be an abstraction for casting out nines of integer $x$. For all integers $x$, prove $s(x) \equiv x$ mod $9$ I'm not asking for an answer more of a way to attack this problem. Can't thin......Mon, 17 Aug 2026 09:54:10 -0400https://liberty.io/is-my-general-formula-for-polynomial-multiplication-right.htmlhttps://liberty.io/is-my-general-formula-for-polynomial-multiplication-right.htmlIs the following equation for polynomial multiplication correct? $ p(x)=p_{0} + \dots + p_{m}x^m \\ q(x)=q_{0} + \dots + q_{n}x^n \\ $ If $m \geq n$ then $i := m$. If $n \gt m$ then $i := n$. $ ......Mon, 17 Aug 2026 09:51:53 -0400https://liberty.io/use-the-alternative-form-of-the-derivative-to-find-the-derivative-at-x.htmlhttps://liberty.io/use-the-alternative-form-of-the-derivative-to-find-the-derivative-at-x.htmlUse the alternative form of the derivative to find the derivative at $x=c$ if $f(x)=x^3+4x$ and $c=2$. I keep getting stuck with the answer being $0$, no matter how I try to solve it. If someone ......Mon, 17 Aug 2026 09:47:59 -0400https://liberty.io/a-matrix-to-the-power-of-zero-gives-identity-matrix-even-if-it-doesn-0.htmlhttps://liberty.io/a-matrix-to-the-power-of-zero-gives-identity-matrix-even-if-it-doesn-0.htmlIf one matrix whose determinant is equal to 0 which means it doesn't have an inverse. Then how is possible to find the value of the matrix to the power of 0 equal to identity matrix when multiplyin......Mon, 17 Aug 2026 09:41:11 -0400https://liberty.io/magnitude-and-direction-of-a-vector-5-6.htmlhttps://liberty.io/magnitude-and-direction-of-a-vector-5-6.htmlFind the magnitude and direction of the vector $<-5,6>$ I found the magnitude: $||v||=\sqrt{(-5)^2+6^2}=\sqrt{25+36}=\sqrt{61}$ In the direction this is what I did: $\theta=\tan^{-1}(\frac{6}......Mon, 17 Aug 2026 09:38:03 -0400https://liberty.io/when-do-you-use-the-quadratic-formula.htmlhttps://liberty.io/when-do-you-use-the-quadratic-formula.htmlI am revising for a mathematics exam and am looking over simultaneous equations. I was curious as to when I use the quadratic formula and when I don't? I realize there are multiple ways to solve a ...Mon, 17 Aug 2026 09:36:13 -0400https://liberty.io/how-to-find-lim-limits-x-to-0-frac-sin4x-sin2x.htmlhttps://liberty.io/how-to-find-lim-limits-x-to-0-frac-sin4x-sin2x.htmlHow to find the limit $$\lim\limits_{x \to 0}\dfrac{\sin4x}{\sin2x}\,?$$ Should I do $$\lim\limits_{x \to 0}\dfrac{\dfrac{\sin4x}{4x}}{\dfrac{\sin2x}{2x}}\,?$$ This doesn't seem to look right, c......Mon, 17 Aug 2026 09:30:59 -0400https://liberty.io/deriving-second-order-transfer-function-from-spring-mass-damper-system.htmlhttps://liberty.io/deriving-second-order-transfer-function-from-spring-mass-damper-system.htmlI am having a hard time understanding how a differential equation based on a spring mass damper system $$ m\ddot{x} + b\dot{x} + kx = 0$$ can be described as an second order transfer function for an ...Mon, 17 Aug 2026 09:26:27 -0400https://liberty.io/degeneracy-in-simplex-algorithm.htmlhttps://liberty.io/degeneracy-in-simplex-algorithm.htmlAccording to my understanding, Degeneracy in a linear optimization problem, occurs when the same extreme point of a bounded feasible region $X$ can be represented by more than one basis, that is not ...Mon, 17 Aug 2026 09:16:04 -0400https://liberty.io/find-corresponding-number-in-a-range-by-index.htmlhttps://liberty.io/find-corresponding-number-in-a-range-by-index.htmlThere's a array: array(1,2,3,...12, 1,2,3,...9,9,10,...12, 1,2,3,...12, 1,2,3,...12, 1,2,3,...6,6,7,8,...12, 1,2,3,3,4,...12, ...); All numbers are in the range 1-12, one number maybe occur twi......Mon, 17 Aug 2026 09:15:43 -0400https://liberty.io/least-common-multiple-in-euclidean-algorithm.htmlhttps://liberty.io/least-common-multiple-in-euclidean-algorithm.htmlI want to prove that in last step of Euclidean algorithm we have lcm representation (by last step I mean the step with zero representation as $0 = x * E_0 + y * E_1$, where we apply euclidean algo......Mon, 17 Aug 2026 09:09:41 -0400https://liberty.io/how-do-you-integrate-e-x-2.htmlhttps://liberty.io/how-do-you-integrate-e-x-2.htmlI know that $\int{\frac{1}{x}}dx$ is simply $\ln{(x)}+c$ (-which is clearly unrelated to the problem but I just thought I would share anyway) but I am not sure how to approach $e^{x{^2}}$. Perhaps a ...Mon, 17 Aug 2026 09:09:03 -0400https://liberty.io/prove-using-rolle-039-s-theorem-that-an-equation-has-exactly-one-real-.htmlhttps://liberty.io/prove-using-rolle-039-s-theorem-that-an-equation-has-exactly-one-real-.htmlProve that the equation $x^7+x^5+x^3+1=0$ has exactly one real solution. You should use Rolle’s Theorem at some point in the proof. Since $f(x) = x^7+x^5+x^3+1$ is a polynomial then it is continu......Mon, 17 Aug 2026 09:08:18 -0400https://liberty.io/cauchy-binet-formula-proof-intuition-determinants.htmlhttps://liberty.io/cauchy-binet-formula-proof-intuition-determinants.htmlI am having trouble proving the Cauchy-Binet Theorem. I jotted down how far I got in the proof, but I just find myself stuck. Any guidance would be greatly appreciated! I understand that $\begin......Mon, 17 Aug 2026 09:06:24 -0400https://liberty.io/why-is-the-berry-paradox-a-paradox-at-all.htmlhttps://liberty.io/why-is-the-berry-paradox-a-paradox-at-all.htmlThe Berry Paradox arises from first assigning every combination of 12 english words to an integer and then asking for "the smallest positive integer not definable in fewer than twelve words". This ...Mon, 17 Aug 2026 09:05:05 -0400https://liberty.io/express-in-interval-notation-where-f-is-differentiable-graph.htmlhttps://liberty.io/express-in-interval-notation-where-f-is-differentiable-graph.htmlThe graph of the function is cut off in the vertical direction because the function escapes to negative infinity. I know [2,-1)U(-1,1)U(1,3)U(3,4], but the system is telling me that my second set is ...Mon, 17 Aug 2026 09:04:13 -0400https://liberty.io/number-of-positive-real-roots-in-equation-x-6-x-1-0-closed.htmlhttps://liberty.io/number-of-positive-real-roots-in-equation-x-6-x-1-0-closed.htmlHow can I find the number of positive real roots in the equation $$x^6 - x - 1 =0$$ This question is given in my analysis book and I have no idea how to start...Mon, 17 Aug 2026 09:01:08 -0400https://liberty.io/finding-the-gcd-of-three-numbers.htmlhttps://liberty.io/finding-the-gcd-of-three-numbers.htmlI have found an algorithm called the Binary GCD/Stein Algorithm which takes two non-negative integers and finds the greatest common factor of the two. What I am trying to do is take three numbers ......Mon, 17 Aug 2026 08:58:12 -0400https://liberty.io/finding-eigenvalues-of-block-2x2-matrix.htmlhttps://liberty.io/finding-eigenvalues-of-block-2x2-matrix.htmlIn my studies of probability theory, I have recently come across the following eigenvalue problem as part of my research: Let $ \Sigma $ be a given positive definite $ d \times d $ matrix, and l......Mon, 17 Aug 2026 08:57:01 -0400https://liberty.io/word-problem-involving-time-and-square-meters.htmlhttps://liberty.io/word-problem-involving-time-and-square-meters.htmlBetty paints twice as fast as Dan. Working together Dan and Betty can paint, $2,400$ square feet in $4$ hours. Another employee sue, joined their painting team. Working together, Dan, Betty and......Mon, 17 Aug 2026 08:55:39 -0400https://liberty.io/finding-an-expression-for-the-general-term-of-a-taylor-series.htmlhttps://liberty.io/finding-an-expression-for-the-general-term-of-a-taylor-series.htmlI am working on a homework problem that asks the following: "Find an expression for the general term of each of the series below. Use $n$ as your index, and pick your general term so that the sum ......Mon, 17 Aug 2026 08:43:07 -0400https://liberty.io/show-the-function-is-decreasing-in-an-interval.htmlhttps://liberty.io/show-the-function-is-decreasing-in-an-interval.htmlLet $s>1, s\in\mathbb{R}$, and let $f$ be a function defined by $$f(x)=\frac{ln(x)}{x^s}, x>0$$ Determine the monotone intervals of $f$. I note that $f(x)$'s domain is $(0,\infty)$. I then......Mon, 17 Aug 2026 08:42:43 -0400https://liberty.io/calculate-limit-of-sequence-a-n-1-3-1-a-n.htmlhttps://liberty.io/calculate-limit-of-sequence-a-n-1-3-1-a-n.htmlI have been given the following problem: $$ a_1 = 1, a_{n+1} = 3-\frac{1}{a_n}\\ \text{ Find: }\lim\limits_{n \to \infty} a_n $$ I have already proved that the function is always increasing and bou......Mon, 17 Aug 2026 08:35:21 -0400https://liberty.io/limit-of-the-geometric-sequence.htmlhttps://liberty.io/limit-of-the-geometric-sequence.htmlI have the following homework question: For what values of $r$ does $\lim_{n\rightarrow \infty} r^n$ exist? What does it converge to? Correct me if I am wrong, but if $r$ is greater than or equa......Mon, 17 Aug 2026 08:27:36 -0400https://liberty.io/calculate-regular-or-equilateral-triangle-altitude-with-radius-only-po.htmlhttps://liberty.io/calculate-regular-or-equilateral-triangle-altitude-with-radius-only-po.htmlI need to calculate the altitude of a regular triangle (equilateral) but i only have the radius (polygon radius) available (http://www.mathopenref.com/polygonradius.html). I have been searching for ...Mon, 17 Aug 2026 08:23:48 -0400https://liberty.io/can-a-sum-of-a-finite-number-of-exponentially-growing-numbers-be-calcu.htmlhttps://liberty.io/can-a-sum-of-a-finite-number-of-exponentially-growing-numbers-be-calcu.htmlThis is a question based on the exponential growth pattern of a game mechanic in World of Warcraft: The Heart of Azeroth item in the game has a level that can be increased by gathering Azerite, wit......Mon, 17 Aug 2026 08:20:04 -0400https://liberty.io/idempotent-matrix.htmlhttps://liberty.io/idempotent-matrix.htmlI have a question: Let A be an idempotent matrix. Show that I-A is idempotent Show that I+A is nonsingular and (I+A)^(-1) = I- ${1\over 2}$A I solved 1: (I - A)^2 = (I - A)(I - A) = I^2 - ......Mon, 17 Aug 2026 08:19:56 -0400https://liberty.io/how-to-find-the-area-of-a-quarter-circle-given-only-the-perimeter-of-a.htmlhttps://liberty.io/how-to-find-the-area-of-a-quarter-circle-given-only-the-perimeter-of-a.htmlI have been looking for a formula for these types of problems but I still haven't gotten an answer. The question you guys can use can be found here. All I need is the formula but if you want you can ...Mon, 17 Aug 2026 08:01:06 -0400https://liberty.io/how-to-find-the-value-of-this-summation-equation.htmlhttps://liberty.io/how-to-find-the-value-of-this-summation-equation.htmlThe question is: $$\sum_{i=1}^n (i^2+3i+4)$$ I get that $$\sum_{i=1}^n i^2 = \frac{n(n+1)(n+2)}{6}$$ and $$3\sum_{i=1}^n i = \frac{3n(n+1)}{2}$$ so one would get I'll call this form1: $$\frac......Mon, 17 Aug 2026 07:50:59 -0400https://liberty.io/shadow-prices-in-linear-programming.htmlhttps://liberty.io/shadow-prices-in-linear-programming.htmlI am quite confused about the meaning of shadow price from explanations on the internet. It can be understood as the value of a change in revenue if the constraint is relaxed, or how much you woul......Mon, 17 Aug 2026 07:49:05 -0400https://liberty.io/finding-the-span-of-3-vectors.htmlhttps://liberty.io/finding-the-span-of-3-vectors.htmlLet $V = \mathbb R^3$, a vector space over the reals. Find the span $W$ of $\{(1, 2, 1), (3, −1, −4), (0, 7, 7)\}$ in the form $\{(x, y, z) ∈ V \mid ax + by + cz = 0\}$ for some $a, b, c$. Find a b......Mon, 17 Aug 2026 07:46:55 -0400https://liberty.io/inverse-of-the-curl.htmlhttps://liberty.io/inverse-of-the-curl.htmlGiven $\nabla \cdot u = 0$ and $w = \nabla \times u$ equation 9 of http://projecteuclid.org/euclid.cmp/1103941230 has the identity $u = - \nabla \times (\nabla^{-1} w)$. What is $\nabla^{-1}$ exac......Mon, 17 Aug 2026 07:42:37 -0400https://liberty.io/equation-of-hyperbola-with-eccentricity-frac32.htmlhttps://liberty.io/equation-of-hyperbola-with-eccentricity-frac32.htmlA hyperbola whose transverse axis is along the major axis of the conic $\frac{2x^2}{3}+\frac{y^2}{4}=4$ and has vertices on the foci of this conic. If the eccentricity of the hyperbola is $\frac32$......Mon, 17 Aug 2026 07:40:12 -0400https://liberty.io/how-to-estimate-a-definite-integral-using-taylor-series.htmlhttps://liberty.io/how-to-estimate-a-definite-integral-using-taylor-series.htmlI have a problem that asks Evaluate to five decimal places using the taylor series for the definite integral. $$\int_{1}^{2} \frac{e^x}{x} dx$$ I don't get how to do this, given that the minute ......Mon, 17 Aug 2026 07:27:43 -0400https://liberty.io/distribution-and-absolute-value.htmlhttps://liberty.io/distribution-and-absolute-value.htmlI have a question about distribution and absolute values. I was solving a problem and was wondering if it would be okay to distribute a number into an absolute value with two terms. For example $3|......Mon, 17 Aug 2026 07:17:00 -0400https://liberty.io/what-is-the-definition-of-a-relation.htmlhttps://liberty.io/what-is-the-definition-of-a-relation.htmlLet $S$ be a set. An equivalence relation on $S$ is a subset $R\subseteq S\times S$ satisfying that $(s,s)\in R$ for all $s\in S$, $(s,t)\in R \implies (t,s)\in R$, $(s,t)\in R\; \land\; (t,u)\in R\...Mon, 17 Aug 2026 07:10:47 -0400https://liberty.io/should-we-use-parentheses-or-brackets-around-functions.htmlhttps://liberty.io/should-we-use-parentheses-or-brackets-around-functions.htmlShould we use parentheses or brackets around functions? Is there a widely accepted rule/style? For example, which of the following is the preferred way of writing formulas? $(f(x) + g(x)) * h(x)$......Mon, 17 Aug 2026 07:09:44 -0400https://liberty.io/weak-convergence-of-product-of-weakly-and-strongly-convergent-l-2-sequ.htmlhttps://liberty.io/weak-convergence-of-product-of-weakly-and-strongly-convergent-l-2-sequ.htmlthere is one question bothering me for quite a while now. Let $a_{n},b_{n}\in L^{2}:a_{n}\stackrel{L^{2}}{\rightharpoonup} a\in L^{2} $ weakly $ b_{n}\stackrel{L^{2}}{\rightarrow} b \in L^{2}$ str......Mon, 17 Aug 2026 06:55:50 -0400https://liberty.io/help-with-converting-the-derivative-of-cos-3-x.htmlhttps://liberty.io/help-with-converting-the-derivative-of-cos-3-x.htmlI needed to find the derivative of $\cos^3(x)$ and find it in a list of given derivatives, but I could not find $-3\sin(x)\cos^2(x)$, so I waited until I did more problems and the only one left was......Mon, 17 Aug 2026 06:44:55 -0400https://liberty.io/linear-algebra-notation-a-dagger.htmlhttps://liberty.io/linear-algebra-notation-a-dagger.htmlI'm a little confused as to some linear algebra notation and would appreciate some clarification. There exists an $n\times n$ diagonal matrix $Q$ where the entries along the diagonal are the eigen......Mon, 17 Aug 2026 06:38:18 -0400https://liberty.io/projecting-a-surface-segment-of-a-cone-onto-a-2d-plane.htmlhttps://liberty.io/projecting-a-surface-segment-of-a-cone-onto-a-2d-plane.htmlFirstly, I'd like to apologise - I do not know the correct terms for what I am asking. Assume that the top/bottom of the highlighted portion there is actually aligned with the base. To help expla......Mon, 17 Aug 2026 06:32:29 -0400https://liberty.io/extensions-and-kazhdan-039-s-property-t.htmlhttps://liberty.io/extensions-and-kazhdan-039-s-property-t.htmlIs Kazhdan's property (T) stable under extensions? i.e. if $G$ is an extension of a group with property (T) by a group with property (T), does it follow that $G$ has property (T)?...Mon, 17 Aug 2026 06:27:57 -0400https://liberty.io/the-langlands-program-for-beginners.htmlhttps://liberty.io/the-langlands-program-for-beginners.htmlAssuming that a person has taken standard undergraduate math courses (algebra, analysis, point-set topology), what other things must a person know before they can understand the Langlands program a......Mon, 17 Aug 2026 06:10:34 -0400https://liberty.io/proof-for-combination-formula-n-choose-k.htmlhttps://liberty.io/proof-for-combination-formula-n-choose-k.htmlI have been looking at this problem for a long time. Can anyone prove the combination formula using factorials N choose K? In case anyone does not know how to list all combinations in a set, you ......Mon, 17 Aug 2026 06:03:03 -0400https://liberty.io/understanding-the-ideal-generated-by-a-polynomial.htmlhttps://liberty.io/understanding-the-ideal-generated-by-a-polynomial.htmlSo my class on ring theory recently began, and I'm having a bit of trouble understanding ideals that are generated by polynomials. For an arbitrary ring, I know the definition of such an ideal, tha......Mon, 17 Aug 2026 06:00:50 -0400https://liberty.io/finding-the-inverse-of-a-number-under-a-certain-modulus.htmlhttps://liberty.io/finding-the-inverse-of-a-number-under-a-certain-modulus.htmlHow does one get the inverse of 7 modulo 11? I know the answer is supposed to be 8, but have no idea how to reach or calculate that figure. Likewise, I have the same problem finding the inverse o......Mon, 17 Aug 2026 05:49:54 -0400https://liberty.io/rational-numbers.htmlhttps://liberty.io/rational-numbers.htmlIt is a rule that: A rational number can be expressed in $\frac{a}{b}$ form and an irrational number cannot be expressed in such form. It is even said that (in order to promote the $\frac{a}{b}$......Mon, 17 Aug 2026 05:43:10 -0400https://liberty.io/what-is-the-integral-of-e-cos-x.htmlhttps://liberty.io/what-is-the-integral-of-e-cos-x.htmlQuestion: Find out $\displaystyle{\int e^{\cos x}~dx}$. My Attempt: Let $\cos x = y$. Hence $-\sin x\ dx = dy$ or $$dx = \displaystyle{\frac{-dy}{\sin x}=\frac{-dy}{\sqrt{1-\cos^2x}}=\frac{-dy}{\...Mon, 17 Aug 2026 05:28:54 -0400https://liberty.io/pontryagin-classes-of-a-product-manifold.htmlhttps://liberty.io/pontryagin-classes-of-a-product-manifold.htmlI'm imagining there's a way to relate the pontryagin classes of $T(M\times N)$ to the pontryagin classes of $M$ and those of $N$, but I haven't been able to find a helpful reference. Could someone ...Mon, 17 Aug 2026 05:21:37 -0400https://liberty.io/the-hat-function-closed.htmlhttps://liberty.io/the-hat-function-closed.htmlA hat function in 2D Cartesian is defined as $$f(x)= \begin{cases} x, & x\geq 0\\ -x, & x<0 \end{cases}$$ Could any one help me how the hat function is defined in 3D Cartesian. Note, ......Mon, 17 Aug 2026 05:12:40 -0400https://liberty.io/finding-the-scale-factor-and-rotation-angle-of-a-matrix.htmlhttps://liberty.io/finding-the-scale-factor-and-rotation-angle-of-a-matrix.htmlI was given the following question without the material appearing first in the book (I am learning independently). I would gladly look into the matter, but I am not sure where to look - because I d......Mon, 17 Aug 2026 05:09:06 -0400https://liberty.io/calculate-the-value-of-x-y-and-z-coordinates.htmlhttps://liberty.io/calculate-the-value-of-x-y-and-z-coordinates.htmlSystem of the equations: Equation 1: $$-360 = -6x + x^2 - 40y + y^2 - 20z + z^2$$ Equation 2: $$-600 = -40x + x^2 - 30y + y^2 - 100z + z^2$$ Equation 3: $$59.85 = -10x + x^2 - 4y + y^2 - 10z ......Mon, 17 Aug 2026 04:44:00 -0400https://liberty.io/how-to-solve-ln-e-2.htmlhttps://liberty.io/how-to-solve-ln-e-2.htmlI know that $$\ln e^2=2$$ But what about this? $$(\ln e)^2$$ A calculator gave 1. I'm really confused....Mon, 17 Aug 2026 04:33:12 -0400https://liberty.io/why-was-o-gabber-not-an-author-of-the-1983-039-s-faisceaux-pervers.htmlhttps://liberty.io/why-was-o-gabber-not-an-author-of-the-1983-039-s-faisceaux-pervers.htmlEssentially the title. I noticed that in the 2018 re-printed version of Faisceaux Pervers, O. Gabber is an author (making the work BBDG) as opposed to the classic Faisceaux Pervers (BBD) of 1983 in......Mon, 17 Aug 2026 04:19:44 -0400https://liberty.io/b-splines-of-degree-1-2-and-3.htmlhttps://liberty.io/b-splines-of-degree-1-2-and-3.htmlThe basis functions for linear B-splines are: \begin{equation*} \begin{aligned} & B_0(u) = (1-u)\\ & B_1(u) = u. \end{aligned} \end{equation*} For quadratic B-plines: \b......Mon, 17 Aug 2026 04:04:18 -0400https://liberty.io/finding-delta-algebraically-for-a-given-epsilon.htmlhttps://liberty.io/finding-delta-algebraically-for-a-given-epsilon.htmlFor the limit $$\lim_{x\to 5}\sqrt{x-1}=2$$ find a $\delta>0$ that works for $\epsilon=1$. In another words, find a $\delta>0$ such that for all $x$, $$0<|x-5|<\delta \implies |\sqrt......Mon, 17 Aug 2026 04:01:57 -0400https://liberty.io/square-with-perpendicular-lines-drawn-prove-that-they-are-equal.htmlhttps://liberty.io/square-with-perpendicular-lines-drawn-prove-that-they-are-equal.htmlABCD is a square. C' is a point on BA and B' is a point on AD such that BB' and CC' are perpendicular. Show that BB' = CC' I don't know where to start. Use similarities?...Mon, 17 Aug 2026 04:01:51 -0400https://liberty.io/verification-of-stokes-039-theorem-for-a-paraboloid-and-a-given-field.htmlhttps://liberty.io/verification-of-stokes-039-theorem-for-a-paraboloid-and-a-given-field.htmlI've been trying different methods but keep getting the same answer for the following question: Verify that Stokes’ Theorem is true for the given vector field F and surface S. $14.\; \vec F(x, y,......Mon, 17 Aug 2026 03:56:50 -0400https://liberty.io/technique-for-simplifying-e-g-sqrt-8-4-sqrt-3-to-sqrt-6-sqrt-2.htmlhttps://liberty.io/technique-for-simplifying-e-g-sqrt-8-4-sqrt-3-to-sqrt-6-sqrt-2.htmlHow to find the square root of an irrational expression, to simplify that root. e.g.: $$ \sqrt{ 8 - 4\sqrt{3} } = \sqrt{6} - \sqrt{2} $$ Easy to verify: \begin{align} (\sqrt{6} - \sqrt{2})^2 = 6......Mon, 17 Aug 2026 03:53:27 -0400https://liberty.io/how-do-i-simplify-a-fraction-with-a-square-root-on-the-numerator-close.htmlhttps://liberty.io/how-do-i-simplify-a-fraction-with-a-square-root-on-the-numerator-close.htmlHow do I simplify this fraction : $$\frac{(3x^2+4x)\sqrt {x+1}}{2(x+1)^2}$$ The final result is $\frac{3x^2+4x} {2(x+1)^{3/2}}$....Mon, 17 Aug 2026 03:44:39 -0400https://liberty.io/sum-of-the-series-sum-n-1-infty-n-2e-n.htmlhttps://liberty.io/sum-of-the-series-sum-n-1-infty-n-2e-n.htmlThe question is to find out the sum of the series $$\sum_{n=1}^\infty n^2 e^{-n}$$ I tried to bring the summation in some form of telescoping series but failed. I then tried approximating the sum ......Mon, 17 Aug 2026 03:42:53 -0400https://liberty.io/show-that-9-5-sqrt-3-2-sqrt-2-is-not-a-square-in-mathbb-q-sqrt-2-sqrt-.htmlhttps://liberty.io/show-that-9-5-sqrt-3-2-sqrt-2-is-not-a-square-in-mathbb-q-sqrt-2-sqrt-.htmlShow that $(9-5\sqrt 3)(2-\sqrt 2) $ is not a square in $\mathbb Q (\sqrt 2, \sqrt 3) $; equivalently, prove that $\mathbb Q (\sqrt 2, \sqrt 3, u)$ is of degree $2$ over $\mathbb Q (\sqrt 2, \sqrt ......Mon, 17 Aug 2026 03:41:00 -0400https://liberty.io/find-a-system-of-linear-equations-for-the-given-solution-set.htmlhttps://liberty.io/find-a-system-of-linear-equations-for-the-given-solution-set.html$$\{ (-1, 1, -2, 2) + \alpha(1, 2, -3, 4) + \beta(1, -2, 3, -4) \mid \alpha, \beta \in \mathbb{R}\}$$ The given solution set suggests the system of equations will have 2 pivots and 2 free variable......Mon, 17 Aug 2026 03:39:22 -0400https://liberty.io/what-039-s-the-difference-between-direction-sense-and-orientation.htmlhttps://liberty.io/what-039-s-the-difference-between-direction-sense-and-orientation.htmlI'm trying to understand the difference between the sense, orientation, and direction of a vector. According to this, sense is specified by two points on a line parallel to a vector. Orientation is ...Mon, 17 Aug 2026 03:24:52 -0400https://liberty.io/evaluate-the-integral-int-cot-x-cos-x-dx.htmlhttps://liberty.io/evaluate-the-integral-int-cot-x-cos-x-dx.htmlI am asked to integrate $$ \int \cot(x) \cos(x) \ dx $$ How ca I do that? Using the trig substitution $$ \cos^2(x) = \frac{1}{2} \left( 1+\cos(2x) \right) $$ doesn't get me any further. Thank ......Mon, 17 Aug 2026 03:18:17 -0400https://liberty.io/expectation-and-variance-of-the-pareto-distribution.htmlhttps://liberty.io/expectation-and-variance-of-the-pareto-distribution.htmlGiven the distribution funciton of the r.v. $X$ for $\alpha, \beta >0$ $$ F(x) = 1-\Big( \frac{\beta}{\beta +x}\Big)^{\alpha} $$ for $x \geq 0$ and $0$ elsewhere What is the expectation and vari......Mon, 17 Aug 2026 03:15:25 -0400https://liberty.io/how-to-solve-x-2-ln-x-0.htmlhttps://liberty.io/how-to-solve-x-2-ln-x-0.htmlI was just investigating $y = f(x) = e^{-x^2}$ and then went ahead to plot $x=f(y), y=-f(x), and x=-f(y)$, and what I got was interest rounded square shape, and I think we can calculate this area u......Mon, 17 Aug 2026 03:14:36 -0400https://liberty.io/help-finding-coordinates-of-centre-and-radius-of-circle.htmlhttps://liberty.io/help-finding-coordinates-of-centre-and-radius-of-circle.htmlI'm having trouble with some questions relating to finding the centre coordinates and radius of a circle when given the equation. I understand how to find it in the form: (x-p)^2 + (y-q)^2 = r^2 I.e ...Mon, 17 Aug 2026 02:54:33 -0400https://liberty.io/create-context-free-grammar-for-w-mid-w-is-odd-with-a-0-in-the-middle.htmlhttps://liberty.io/create-context-free-grammar-for-w-mid-w-is-odd-with-a-0-in-the-middle.htmlI need to find a CFG where the word length $|w|$ is odd. Plus there must be a $0$ in the middle. In a previous exercise I had to specify a CFG only for odd word length. I chose the following: $G ......Mon, 17 Aug 2026 02:47:11 -0400https://liberty.io/to-prove-cayley-hamilton-theorem-why-can-039-t-we-substitute-a-for-lam.htmlhttps://liberty.io/to-prove-cayley-hamilton-theorem-why-can-039-t-we-substitute-a-for-lam.htmlCayley Hamilton Theorem states that if $A$ is an $n \times n$ matrix over the field $F$ then $p(A) = 0$. We note that $p(\lambda) = \det(\lambda I - A)$. Hence, why can't we just substitute $\lamb......Mon, 17 Aug 2026 02:41:39 -0400https://liberty.io/find-the-second-derivative-d-2-y-over-d-x-2-in-terms-of-t-when-x-3-2-t.htmlhttps://liberty.io/find-the-second-derivative-d-2-y-over-d-x-2-in-terms-of-t-when-x-3-2-t.htmlThis is my attempt: $\eqalign{ & x = 3 - 2{t^2} \cr & y = {1 \over t} \cr & {{dx} \over {dt}} = - 4t \cr & {{dy} \over {dt}} = - {t^{ - 2}} = {{ - 1} \over {{t^2}}}......Mon, 17 Aug 2026 02:40:18 -0400https://liberty.io/expansion-of-cube-of-four-terms-closed.htmlhttps://liberty.io/expansion-of-cube-of-four-terms-closed.htmlHow do I get to know the pattern that appears when we open the cube of four terms ? For example, how do I get to know this pattern? $(a+b+c+d)^3=\sum a^3 $ +6 $\sum abc$ + 3$\sum b a^2$...Mon, 17 Aug 2026 02:36:46 -0400https://liberty.io/inverse-of-a-singular-matrix.htmlhttps://liberty.io/inverse-of-a-singular-matrix.htmlMy question is why we have to restrict over selves to find inverse for non singular matrix? We can define inverse of a square matrix as follows: A matrix $B$ is said to be inverse of $A$ if $BA =......Mon, 17 Aug 2026 02:36:04 -0400https://liberty.io/why-are-representations-of-quivers-such-a-big-deal.htmlhttps://liberty.io/why-are-representations-of-quivers-such-a-big-deal.htmlQuivers are directed graphs where loops and multi-arrows are allowed. And we can talk about representations of quivers by assigning each vertex a vector space and each arrow a homomorphism. Moreover, ...Mon, 17 Aug 2026 02:24:57 -0400https://liberty.io/how-to-solve-an-equation-of-the-form-ax-2-by-2-cx-dy-e-0.htmlhttps://liberty.io/how-to-solve-an-equation-of-the-form-ax-2-by-2-cx-dy-e-0.htmlI am trying to find out how to solve $ax^2 - by^2 + cx - dy + e = 0$ to get integer solutions, failing this the rational solutions. Thanks!...Mon, 17 Aug 2026 02:24:31 -0400https://liberty.io/mathematics-behind-this-card-trick.htmlhttps://liberty.io/mathematics-behind-this-card-trick.htmlSuppose I have $21$ playing cards. I distribute them in $3$ columns and tell you to choose mentally a card. Then just indicate in which column the card is. I pick up one of the columns which doe......Mon, 17 Aug 2026 02:04:55 -0400