https://liberty.ioCurated star highlights with fresh energy.en-usMon, 24 Aug 2026 00:25:27 -0400https://liberty.io/defining-a-principal-square-root-in-the-complex-space-wolfram-case.htmlhttps://liberty.io/defining-a-principal-square-root-in-the-complex-space-wolfram-case.htmlIn Wikipedia(https://en.wikipedia.org/wiki/Square_root), we define the square root as In mathematics, a square root of a number $a$ is a number y such that $y^2 = a$; in other words, a number $y$ ......Mon, 24 Aug 2026 04:16:13 -0400https://liberty.io/probability-to-get-all-6-sides-of-dice.htmlhttps://liberty.io/probability-to-get-all-6-sides-of-dice.htmlBy rolling a dice 7 times, what is the probability that we could get all the sides to come up at least once? My attempt: Let the first role be any number. 2nd roll: 5/6 chance to get different nu......Mon, 24 Aug 2026 04:07:13 -0400https://liberty.io/what-is-the-exact-definition-of-an-injective-function.htmlhttps://liberty.io/what-is-the-exact-definition-of-an-injective-function.htmlAm I right to believe that a function is injective, if some elements of the first set are mapped to some elements of the second set? It is also possible to 4 elements of the first set, are mapped ......Mon, 24 Aug 2026 04:02:46 -0400https://liberty.io/changing-modulus-in-modular-arithmetic.htmlhttps://liberty.io/changing-modulus-in-modular-arithmetic.htmlIs it true that $$a\equiv b\pmod{m}\implies\frac{a}{n}\equiv\frac{b}{n} \pmod{\frac{m}{n}},$$ where $a, b, m, n, \frac{a}{n}, \frac{b}{n}, \frac{m}{n}\in\mathbb{N}$? If so, how do I prove it?...Mon, 24 Aug 2026 03:54:34 -0400https://liberty.io/n-4-4-n-is-a-not-a-prime-duplicate.htmlhttps://liberty.io/n-4-4-n-is-a-not-a-prime-duplicate.htmlProve that $n^4 + 4^n$ is not a prime for all $n > 1$ and $n \in \mathbb{N}$. This question appeared in the undergrad entrance exam of the Indian Statistical institute. When $n$ is even the p......Mon, 24 Aug 2026 03:52:58 -0400https://liberty.io/appearance-of-formal-derivative-in-algebra.htmlhttps://liberty.io/appearance-of-formal-derivative-in-algebra.htmlWhen studying polynomials, I know it is useful to introduce the concept of a formal derivative. For example, over a field, a polynomial has no repeated roots iff it and its formal derivative are co......Mon, 24 Aug 2026 03:51:05 -0400https://liberty.io/laplace-transform-of-tf-t.htmlhttps://liberty.io/laplace-transform-of-tf-t.htmlQ. prove that $\mathfrak{L}\{tf(x)\}=-\frac{d\mathfrak{L}\{f(x)\}}{ds}$ where the notation used is standard one. Attempt I tried what would seem obvious way to start: $$\mathfrak{L}\{tf(x)\}=\int_0^\...Mon, 24 Aug 2026 03:47:27 -0400https://liberty.io/probability-of-winning-fantasy-5-lotto.htmlhttps://liberty.io/probability-of-winning-fantasy-5-lotto.htmlIn Fantasy 5, you choose 5 numbers, each 1-39. You win if you choose match all 5 numbers, but you can also win if you match 4, 3, or even 2 of the numbers. The lotto website lists the probabiliti......Mon, 24 Aug 2026 03:35:44 -0400https://liberty.io/polygons-with-a-unique-triangulation.htmlhttps://liberty.io/polygons-with-a-unique-triangulation.htmlFor each n > 3, find a polygon with n vertices that has a unique triangulation. I want to say that you can somehow build these polygons by continuously adding triangles somehow, but I'm not sure....Mon, 24 Aug 2026 03:26:51 -0400https://liberty.io/writing-an-expression-for-the-volume-of-a-spherical-shell.htmlhttps://liberty.io/writing-an-expression-for-the-volume-of-a-spherical-shell.htmlThe question is as follows: Write an expression for the volume of the spherical shell formed between two concentric spheres, the inner one of radius $r$, the outer one of radius $R$. Factor your ...Mon, 24 Aug 2026 03:26:28 -0400https://liberty.io/sum-of-all-the-digits-from-1-to-10000-closed.htmlhttps://liberty.io/sum-of-all-the-digits-from-1-to-10000-closed.htmlI need to add up all the digits in the numbers $1$ to $10000$. How would I do that without using a calculator? I don't get it one bit. We are doing something in our math class about this....Mon, 24 Aug 2026 03:22:29 -0400https://liberty.io/the-meaning-of-i-duplicate.htmlhttps://liberty.io/the-meaning-of-i-duplicate.htmlCan someone kind explain the mathematical quantity i to me (which is the square root of -1)? Just to be clear, I'm not actually trying to understand i per se, I'm just trying to understand how it ......Mon, 24 Aug 2026 03:20:56 -0400https://liberty.io/how-do-i-evaluate-the-antiderivative-of-e-cos-x.htmlhttps://liberty.io/how-do-i-evaluate-the-antiderivative-of-e-cos-x.htmlFunctions that do not have an elementary antiderivative can be evaluated by generating a Taylor series, provided the function is infinitely differentiable and uniformly convergent in its domain. Ho......Mon, 24 Aug 2026 03:13:30 -0400https://liberty.io/standard-deviation-probability-of-a-poission-distribution.htmlhttps://liberty.io/standard-deviation-probability-of-a-poission-distribution.htmlJust looking for a bit of help on the topic. The question is basically this: Number of drivers who travel between a particular origin and destination during a designated time period has a Poisson ...Mon, 24 Aug 2026 03:11:28 -0400https://liberty.io/how-many-different-color-combinations-red-black-black-red-red-black-ar.htmlhttps://liberty.io/how-many-different-color-combinations-red-black-black-red-red-black-ar.htmlI have regular deck of playing cards and I want to find out how many color combinations there are in the deck. The cards fall into two categories: red and black. Here are the givens. Total $52$ ite......Mon, 24 Aug 2026 03:11:20 -0400https://liberty.io/total-of-non-zero-value.htmlhttps://liberty.io/total-of-non-zero-value.htmlI want to write a math formula that represents the average of non zero value e.g. if I have 4 numbers, one of them is zero, then the sum of the numbers will be divided by 3. Is it correct to say: ......Mon, 24 Aug 2026 03:07:25 -0400https://liberty.io/finding-the-equation-for-a-sine-wave-given-only-two-points.htmlhttps://liberty.io/finding-the-equation-for-a-sine-wave-given-only-two-points.htmlGiven the co-ordinates of only two points A (x1,y1) and B (x2,y2) on a graph where the amplitude is y2-y1, what is the equation for the sine curve? The distance from A to B is half the period....Mon, 24 Aug 2026 03:05:29 -0400https://liberty.io/solve-x-2-2-y-2021-for-x-y-in-mathbb-n.htmlhttps://liberty.io/solve-x-2-2-y-2021-for-x-y-in-mathbb-n.htmlI have this equation to solve $x^2-2^y=2021, x,y \in N$ I was thinking of seeing it as a Diophantine equation, but it doesn't seem very logical...Mon, 24 Aug 2026 02:56:23 -0400https://liberty.io/inverse-laplace-transform-of-s-s-1.htmlhttps://liberty.io/inverse-laplace-transform-of-s-s-1.htmlWhat is the inverse laplace transform of $\frac{s}{s+1}$? My work was: $$ X(s)=\frac{s}{s+1}\\ X(s)=s\frac{1}{s+1}\\ x(t)=\frac{d}{dt}e^{-t}=-e^{-t} $$ My only issue is that when I check my answe......Mon, 24 Aug 2026 02:52:43 -0400https://liberty.io/definition-of-an-affine-set.htmlhttps://liberty.io/definition-of-an-affine-set.htmlSome resource says that a set $A \subset V$ is affine if $\forall x,y \in A, t \in F$, $tx+(1-t)y \in A$ while others say that $A$ is affine if $\forall x_1,\dotsb,x_n \in A, a_1, \dotsb a_n, \in ......Mon, 24 Aug 2026 02:41:01 -0400https://liberty.io/why-is-a-set-of-orthonormal-vectors-linearly-independent.htmlhttps://liberty.io/why-is-a-set-of-orthonormal-vectors-linearly-independent.htmlA set of orthonormal vectors is linearly independent. Since I wouldn't memorize any such facts that is mentioned in my course book, what is the reason behind it and why are they linearly independ......Mon, 24 Aug 2026 02:33:57 -0400https://liberty.io/how-to-represent-not-an-empty-set.htmlhttps://liberty.io/how-to-represent-not-an-empty-set.htmlI'm writing a academic paper and need to represent "A is not the empty set". What is usual way for professional mathematicians? My idea is: $|A| > 0$ However, using the emptyset $\emptyset$ ......Mon, 24 Aug 2026 02:33:28 -0400https://liberty.io/what-do-these-symbols-mean-bigcap-bigcup-bigwedge-bigvee.htmlhttps://liberty.io/what-do-these-symbols-mean-bigcap-bigcup-bigwedge-bigvee.htmlI know that some of these symbols are used in set theory like $A \cup B$, but that's not what I'm talking about. I have seen those symbols used in a way similar to $\Sigma$ summation and $\Pi$ prod......Mon, 24 Aug 2026 02:33:23 -0400https://liberty.io/power-series-expansion-of-f-x-1-1-x-around-x-0-and-x-1.htmlhttps://liberty.io/power-series-expansion-of-f-x-1-1-x-around-x-0-and-x-1.htmlFor the power series expansion of the function $f(x)$ I worked out the at $x=0$ the power series expansion is $$1(x-0)^n$$ and at $x=-1$ the power series expansion is $$\left(\frac{1}{2^n}+1\right)......Mon, 24 Aug 2026 02:22:15 -0400https://liberty.io/cuboid-not-the-correct-name-for-3d-rectangle.htmlhttps://liberty.io/cuboid-not-the-correct-name-for-3d-rectangle.htmlI have always thought the best name of the 3d equivalent of a rectangle was "cuboid". I am talking about the 3d shape with 6 rectangular faces shown below. However, when looking up the name of this ...Mon, 24 Aug 2026 02:18:09 -0400https://liberty.io/what-will-be-the-count-of-4cardstraight-in-a-poker-game.htmlhttps://liberty.io/what-will-be-the-count-of-4cardstraight-in-a-poker-game.htmlLet's modify the poker hand(fake): "4cardstraight" -- i.e. a straight, but with only 4 cards in a row instead of 5. Rules of the game: A hand is counted as a "4cardstraight" if it includes 4 c......Mon, 24 Aug 2026 02:17:09 -0400https://liberty.io/inverse-of-an-absolute-value.htmlhttps://liberty.io/inverse-of-an-absolute-value.htmlSuppose $f(x) = e^{-|x|}$ then it follows: $$y = e^{-|x|} $$ $$x = e^{-|y|} $$ $$ \log(x) = -|y| $$ $$-\log(x) = |y| $$ Im assuming the equation for the inverse depends on the range of $x$, and the......Mon, 24 Aug 2026 02:13:16 -0400https://liberty.io/can-anyone-recommend-a-good-book-on-quaternions.htmlhttps://liberty.io/can-anyone-recommend-a-good-book-on-quaternions.htmlAlthough I have used them at work, I don't understand quaternions. How were they discovered intuitively? How do they work? Can anyone recommend a good book on quaternions?...Mon, 24 Aug 2026 02:10:03 -0400https://liberty.io/integration-by-substitution-why-do-we-change-the-limits.htmlhttps://liberty.io/integration-by-substitution-why-do-we-change-the-limits.htmlI've highlighted the part I don't understand in red. Why do we change the limits of integration here? What difference does it make? Source of Quotation: Calculus: Early Transcendentals, 7th Edition, ...Mon, 24 Aug 2026 02:05:43 -0400https://liberty.io/partial-derivative-of-an-integral-function.htmlhttps://liberty.io/partial-derivative-of-an-integral-function.htmlGiven the function $f(x,y) = \int^x_y g(t) dt $, $g(t)$ continuous for all $t$. How to evaluate the partial derivatives of $f$ ? It is correct to do like this (by FTC) ?: $f_x$ = $g(x)x'$. $f_y......Mon, 24 Aug 2026 02:01:24 -0400https://liberty.io/what-do-m-and-d-mean-in-the-formula-f-a-b-c-d-sigma-m-9-10-12-d-3-5-6-.htmlhttps://liberty.io/what-do-m-and-d-mean-in-the-formula-f-a-b-c-d-sigma-m-9-10-12-d-3-5-6-.htmlWhat do "$m$" and "$d$" mean here? I know that "$\Sigma$" means sum of products but I do not know what "$m$" and "$d$" mean. Please help....Mon, 24 Aug 2026 01:50:34 -0400https://liberty.io/why-isn-039-t-fourier-series-taught-in-calculus-2-closed.htmlhttps://liberty.io/why-isn-039-t-fourier-series-taught-in-calculus-2-closed.htmlI am self taught and have read the book "Essential Calculus ETF" by Larson and Hosteller from cover to cover and have since been evaluating the more difficult problems in calculus. During my ventur......Mon, 24 Aug 2026 01:42:13 -0400https://liberty.io/dimension-of-eigenspace.htmlhttps://liberty.io/dimension-of-eigenspace.htmlGiven a matrix A $ A = \begin{bmatrix} 5 & 4 & -1 \\ 4 & 5 & -1 \\ -4 & -4 & 2 \\ \end{bmatrix} $ I have to find out if A is diagonalizable or not. Also I have to ...Mon, 24 Aug 2026 01:37:09 -0400https://liberty.io/the-derivatives-of-the-logarithm-of-a-moment-generating-function.htmlhttps://liberty.io/the-derivatives-of-the-logarithm-of-a-moment-generating-function.htmlLet $M_{X}(t)$ be an mgf of $X$. Show that the first derivative of $\ln M_{X}(t)$ at $t=0$ is $\mathbb{E}[X]$ and the second derivative of $\ln M_{X}(t)$ at $t=0$ is $\text{Var}[X]$ I'm not enti......Mon, 24 Aug 2026 01:08:22 -0400https://liberty.io/sigma-algebra-is-it-an-algebra-field-or-something-else.htmlhttps://liberty.io/sigma-algebra-is-it-an-algebra-field-or-something-else.htmlThe Wikipedia page for $\sigma$-algebra says this set is called a "sigma-algebra" by some, and called a "sigma-field" by others. I'm writing a paper on measure theory, where the topic of sigma-alge......Mon, 24 Aug 2026 00:55:32 -0400https://liberty.io/what-is-the-best-algorithm-to-find-the-inverse-of-matrix-a.htmlhttps://liberty.io/what-is-the-best-algorithm-to-find-the-inverse-of-matrix-a.htmlI'm going to build a C-library so I can do linear algebra at embedded systems. It's most for machine learning. https://github.com/DanielMartensson/EmbeddedAlgebra/ Anyway, I need to compute invers......Mon, 24 Aug 2026 00:54:54 -0400https://liberty.io/difference-of-concave-functions.htmlhttps://liberty.io/difference-of-concave-functions.htmlSuppose that there are two concave functions $f_1(x)$ and $f_2(x)$ defined on $x\geq0$. In addition, the functions are positive, smooth, bounded ($|f_2|\leq b_2,|f_1|\leq b_1$ such that $b_2 = b_1......Mon, 24 Aug 2026 00:53:46 -0400https://liberty.io/exactly-how-does-the-equation-n-n-1-2-determine-the-number-of-pairs-of.htmlhttps://liberty.io/exactly-how-does-the-equation-n-n-1-2-determine-the-number-of-pairs-of.htmlI understand the purpose for division. Multiplication is simply repeated addition and division is simply repeated subtraction. So lets say we have 4 total items. Plugging 4 into the equation we get......Mon, 24 Aug 2026 00:49:59 -0400https://liberty.io/9-by-9-matrix-finding-the-determinant.htmlhttps://liberty.io/9-by-9-matrix-finding-the-determinant.htmlCan it be done analytically? I have a system I need to solve, but would need to take a determinant of a 9 by 9 matrix. Is it worth the effort, or is there a limit (in rank) above which it's not pos......Mon, 24 Aug 2026 00:46:04 -0400https://liberty.io/how-to-solve-for-p-in-this-equation-closed.htmlhttps://liberty.io/how-to-solve-for-p-in-this-equation-closed.htmlHow do I solve for p in the following equation: $50,000=\frac{25000p}{100-p}$ ? Thanks!!...Mon, 24 Aug 2026 00:45:20 -0400https://liberty.io/three-numbers-that-multiply-to-ten-it-gets-harder.htmlhttps://liberty.io/three-numbers-that-multiply-to-ten-it-gets-harder.htmlI am working on a problem that involves the following: One must find three numbers, integers and/or decimals, that multiply to ten. Here's the catch: You must use all integers, $0-9$, in the an......Mon, 24 Aug 2026 00:41:17 -0400https://liberty.io/parabolic-elements-of-sl-2-mathbb-r.htmlhttps://liberty.io/parabolic-elements-of-sl-2-mathbb-r.htmlThe elements of $SL_2(\mathbb{R})$ are classified in elliptic, parabolic and hyperbolic elements. It seems to me that elliptic elements preserve an ellipse and hyperbolic elements preserve a hyper......Mon, 24 Aug 2026 00:39:20 -0400https://liberty.io/how-does-false-false-imply-true-duplicate.htmlhttps://liberty.io/how-does-false-false-imply-true-duplicate.htmlSuppose I have $p, q$ as variables and lets say $p$ is false and $q$ is false, then why and how is $p \implies q$ true?...Mon, 24 Aug 2026 00:25:18 -0400https://liberty.io/using-poisson-distribution.htmlhttps://liberty.io/using-poisson-distribution.htmlCustomers arrive at a checkout counter in a department score according to a Poisson distribution at an average of seven per hour. During a given hour, what are the probabilities that: a.) no more ......Mon, 24 Aug 2026 00:12:41 -0400https://liberty.io/h-is-normal-subgroup-of-g-iff-ghg-1-h-for-every-g-of-g.htmlhttps://liberty.io/h-is-normal-subgroup-of-g-iff-ghg-1-h-for-every-g-of-g.htmlGiven the following definition of normal subgroup A subgroup $H$ of a group $G$ is said to be normal if, for every $g\in G$: $$gH=Hg$$ I've tried to show that $H\mathrel{\unlhd} G$ if and only ......Mon, 24 Aug 2026 00:06:42 -0400https://liberty.io/find-the-equation-of-the-diameter-of-the-circle-having-equation-x-2-y-.htmlhttps://liberty.io/find-the-equation-of-the-diameter-of-the-circle-having-equation-x-2-y-.htmlFind the equation of the diameter of the circle having equation $x^2+y^2-6x+2y-8=0$. My Work: $$x^2+y^2-6x+2y-8=0$$ Comparing the above equation with $x^2+y^2+2gx+2fy+c=0$, we get: $$g=-3$$, $$f=1......Sun, 23 Aug 2026 23:40:59 -0400https://liberty.io/proof-percentage-change-in-area-if-side-of-a-two-dimensional-figure-is.htmlhttps://liberty.io/proof-percentage-change-in-area-if-side-of-a-two-dimensional-figure-is.htmlIf each of side of a rectangle or any two dimensional shape is increased by $x\%$, its area is increased by $\left(\dfrac{x^2}{100}+2x\right)\%$ Source: careerbless.com I am trying to gen......Sun, 23 Aug 2026 23:35:14 -0400https://liberty.io/how-was-the-quadratic-formula-created-duplicate.htmlhttps://liberty.io/how-was-the-quadratic-formula-created-duplicate.htmlI have tested the quadratic formula and I have found that it works, yet I am curious as to how it was created. Can anybody please tell me one of the ways that it was created?...Sun, 23 Aug 2026 23:30:14 -0400https://liberty.io/3-times-3-matrices-are-similar-if-and-only-if-they-have-the-same-chara.htmlhttps://liberty.io/3-times-3-matrices-are-similar-if-and-only-if-they-have-the-same-chara.htmlI want to prove: $B$ is similar to $A \Leftrightarrow m_A(x) = m_B(x)$ and $P_A(x) = P_B(x)$, where $m,P$ are the minimal and characteristic polynomial, respectively. "$\Rightarrow$" Let $A$ t......Sun, 23 Aug 2026 23:16:24 -0400https://liberty.io/what-is-the-moment-of-inertia-of-a-disc-about-an-axis-through-its-diam.htmlhttps://liberty.io/what-is-the-moment-of-inertia-of-a-disc-about-an-axis-through-its-diam.htmlWhat is the moment of inertia of a dics of mass $m$ and radius $r$ about an axis through its diameter (like when spinning a coin)? My textbook says $$I = \int r^2\, dm = \int_{-r}^r(x^2 \rho \sqrt......Sun, 23 Aug 2026 23:15:19 -0400https://liberty.io/finding-square-roots-of-sqrt-3-3i.htmlhttps://liberty.io/finding-square-roots-of-sqrt-3-3i.htmlI was reading an example, where it is calculating the square roots of $\sqrt 3 +3i$. $w=\sqrt 3 +3i=2\sqrt 3\left(\frac{1}{2}+\frac{1}{2}\sqrt3i\right)\\=2\sqrt 3(\cos\frac{\pi}{3}+i\sin\frac{\pi}......Sun, 23 Aug 2026 23:04:10 -0400https://liberty.io/related-rates-volume-of-spherical-balloon.htmlhttps://liberty.io/related-rates-volume-of-spherical-balloon.htmlThe question I was given was: The volume of a spherical balloon is increasing at a constant rate of $0.78$ inches per minute. At the instant when the radius is $3.20$ inches, the radius is increas......Sun, 23 Aug 2026 23:02:53 -0400https://liberty.io/why-is-the-axis-of-symmetry-of-a-parabola-b-over-2a-and-not-b-over-2a.htmlhttps://liberty.io/why-is-the-axis-of-symmetry-of-a-parabola-b-over-2a-and-not-b-over-2a.htmlI'm working on a lesson plan for my students regarding completing the square for a parabola, and I've done the following: $$\begin{align}ax^2+bx+c &= a\left(x^2+{b\over a}x\right) + c \\ &am......Sun, 23 Aug 2026 23:01:04 -0400https://liberty.io/the-symbol-for-translation-transformation-or-conversion.htmlhttps://liberty.io/the-symbol-for-translation-transformation-or-conversion.htmlWhat is the symbol for demonstrating syntactic conversion (transformation or translation)? For example, I want to show a calculation sequence, from $ \neg ( A \wedge B ) $ to $ \neg A \vee \neg B ......Sun, 23 Aug 2026 22:51:12 -0400https://liberty.io/constructing-a-cochain-complex-out-of-a-chain-complex.htmlhttps://liberty.io/constructing-a-cochain-complex-out-of-a-chain-complex.htmlLet $(C,\partial)$ be a chain complex where $C_i$ is an $R$-module ($R$ is a given ring) , we can always construct a cochain complex out of the chain complex $(C,\partial)$ in the following way: We ...Sun, 23 Aug 2026 22:22:06 -0400https://liberty.io/any-simple-example-of-lebesgue-integration.htmlhttps://liberty.io/any-simple-example-of-lebesgue-integration.htmlI saw several conceptual explanations regarding Lebesgue Integration, but can I see few practical examples that require Lebesgue Integration? What I need is just a toy case of Lebesgue Integration.......Sun, 23 Aug 2026 22:06:34 -0400https://liberty.io/levi-civita-symbol-from-4-to-3-indices.htmlhttps://liberty.io/levi-civita-symbol-from-4-to-3-indices.htmlIn Four-dimensional space, the Levi-Civita symbol is defined as: $$ \varepsilon_{ijkl } =$$ \begin{cases} +1 & \text{if }(i,j,k,l) \text{ is an even permutation of } (1,2,3,4) \\ -1 & \tex......Sun, 23 Aug 2026 21:46:08 -0400https://liberty.io/angles-of-a-known-3d-vector.htmlhttps://liberty.io/angles-of-a-known-3d-vector.htmlI have a 3d vector r known by its coordinates rx, ry, rz. How can calculate angles Theta and Phi ?...Sun, 23 Aug 2026 21:45:58 -0400https://liberty.io/what-does-k-in-mathbb-z-in-the-general-solutions-of-trigonometric-equa.htmlhttps://liberty.io/what-does-k-in-mathbb-z-in-the-general-solutions-of-trigonometric-equa.htmlI wish to understand the meaning of the term $k\in\mathbb{Z}$, in solving trigonometric equations, for example, it is written $\theta=2k\pi+\frac{\pi}{2}$, for all $k\in\mathbb{Z}$....Sun, 23 Aug 2026 21:41:53 -0400https://liberty.io/counting-5-letter-words-that-include-at-least-one-vowel.htmlhttps://liberty.io/counting-5-letter-words-that-include-at-least-one-vowel.htmlHow many sequences are there of five lowercase letters that includes at least one vowel (a, e, i, o, u)? What is erroneous about the following solution? There are five choices for the certain vow......Sun, 23 Aug 2026 21:38:51 -0400https://liberty.io/entire-extension-of-the-fouier-transform-of-a-smooth-compactly-support.htmlhttps://liberty.io/entire-extension-of-the-fouier-transform-of-a-smooth-compactly-support.htmlThe statement which is not very clear is this: if $\phi \in C^{\infty}_0$ then its Fourier transform extends to a complex-analytic function. Of course the candidate extension is $\hat{\phi}(z)=\i......Sun, 23 Aug 2026 21:36:29 -0400https://liberty.io/find-the-angle-between-two-planes-using-their-normal-vectors.htmlhttps://liberty.io/find-the-angle-between-two-planes-using-their-normal-vectors.htmlThe angle between two intersecting planes is defined to be the angle between their normal vectors. Find the angle between the planes $x – 2y + z = 0$ and $2x + 3y – 2z = 0$. Find the parametric ...Sun, 23 Aug 2026 21:35:17 -0400https://liberty.io/finding-the-largest-possible-delta-in-a-epsilon-delta-limit.htmlhttps://liberty.io/finding-the-largest-possible-delta-in-a-epsilon-delta-limit.html$$\lim_{x\to0}\frac{x^2+2x}{x^2-3x} = - \frac{2}{3}$$ Given that $\epsilon = 0.01$, what is the largest possible $\delta$ according to the epsilon-delta definition of the limit? Here's what I'v......Sun, 23 Aug 2026 21:33:32 -0400https://liberty.io/the-product-of-any-number-and-zero-is-zero.htmlhttps://liberty.io/the-product-of-any-number-and-zero-is-zero.htmlI don't understand why this proof is valid. The proof just swaps the left and right hand side of the equality. Is this a valid method of proof for equalities? If so, what's the logical intuition be......Sun, 23 Aug 2026 21:31:23 -0400https://liberty.io/can-i-conjugate-a-complex-number-sqrt-a-ib.htmlhttps://liberty.io/can-i-conjugate-a-complex-number-sqrt-a-ib.htmlCan I conjugate a complex number: $\sqrt{a+ib}$ ? Actually my maths school teacher says and argues with each and every student that we can't conjugate $\sqrt{a+ib}$ to $\sqrt{a-ib}$ because acc......Sun, 23 Aug 2026 21:28:04 -0400https://liberty.io/measuring-angles-in-the-beltrami-klein-model-of-hyperbolic-geometry.htmlhttps://liberty.io/measuring-angles-in-the-beltrami-klein-model-of-hyperbolic-geometry.htmlI am learning bits of hyperbolic geometry and the wikipedia page gives two such standard models for it ; the Beltrami Klein (BK) model and the Poincare (P) disk model. Now as I understand it hyper......Sun, 23 Aug 2026 21:14:44 -0400https://liberty.io/how-do-i-rotate-a-vector-90-degrees-in-a-random-direction.htmlhttps://liberty.io/how-do-i-rotate-a-vector-90-degrees-in-a-random-direction.htmlI'm building a tree generator and I'm at the point where I want to have sub branches branch off at right angles off the current branch, in random directions. I have a 3D vector defining the directi......Sun, 23 Aug 2026 21:14:35 -0400https://liberty.io/how-is-this-true-bookworm-puzzle.htmlhttps://liberty.io/how-is-this-true-bookworm-puzzle.htmlThis is from Eugene Northrop's book Riddles in Mathematics. Why is the answer 1 inch. Iit should be three. What logic am I missing here?...Sun, 23 Aug 2026 21:07:11 -0400https://liberty.io/finding-all-left-inverse-of-a-matrix-b.htmlhttps://liberty.io/finding-all-left-inverse-of-a-matrix-b.htmlLet $$ \mathbf{B} = \left[\begin{array}{cc} 0 & 1 \\ 1 & -3 \\ 0 & 0 \end{array}\right]. $$ My task at hand is to all left-inverse matrices of $\mathbf{B}$, without using the Moore-Penr......Sun, 23 Aug 2026 21:01:23 -0400https://liberty.io/what-needs-to-be-done-to-prove-that-a-vector-bundle-is-trivial.htmlhttps://liberty.io/what-needs-to-be-done-to-prove-that-a-vector-bundle-is-trivial.htmlWhat needs to be done to prove that a vector bundle is trivial? equivalently, This can be thought of as proving that the vector bundle satisfies the criteria of being trivial, then what is this cri......Sun, 23 Aug 2026 20:58:56 -0400https://liberty.io/what-is-the-formal-definition-of-a-mathematical-expression.htmlhttps://liberty.io/what-is-the-formal-definition-of-a-mathematical-expression.htmlI am interested in what the definition of a mathematical expression is. For example $x_1 + x_2$ is a different expression than $x_2 + x_1$, even though they both evaluate to the same function on the ...Sun, 23 Aug 2026 20:52:14 -0400https://liberty.io/positively-homogeneous-function.htmlhttps://liberty.io/positively-homogeneous-function.htmlA function $f:X→\mathbb{R}$ is said to be positive homogeneous of degree $k\in\mathbb{R}$ if $f(tx)=t^kf(x)$ for every $x\in X$ and every $t\in\mathbb{R}_{++}$. For $X=\mathbb{R}_+$, the sample ex......Sun, 23 Aug 2026 20:44:22 -0400https://liberty.io/can-i-say-that-if-a-kernel-is-translation-invariant-or-shift-invariant.htmlhttps://liberty.io/can-i-say-that-if-a-kernel-is-translation-invariant-or-shift-invariant.htmlLet $x,y \in \mathbb{R}$ where $y=x-t$. Translation-invariant (or shift-invariant) kernel $\kappa(\cdot,\cdot)$ is defined as $\kappa(x,y) = \kappa(x,x-t) = \kappa(t)$. Can I say that the functio......Sun, 23 Aug 2026 20:35:22 -0400https://liberty.io/how-to-find-perimeter-of-right-angle-triangle-given-area-and-three-ang.htmlhttps://liberty.io/how-to-find-perimeter-of-right-angle-triangle-given-area-and-three-ang.htmlHello everyone, I’ve come across a question where I need to find the perimeter of a right angle triangle given its area and three sides (the only angle written in the picture is 40 degrees, but the......Sun, 23 Aug 2026 20:23:30 -0400https://liberty.io/what-039-s-another-name-for-the-integral-function.htmlhttps://liberty.io/what-039-s-another-name-for-the-integral-function.htmlI know this is certainty a basic question, but I'm wondering what you could use as an alternate name for the integral of a function. That is to say; $$\text{In } \int f(x)dx=F(x) \text{, } f(x) \te......Sun, 23 Aug 2026 20:15:05 -0400https://liberty.io/what-is-the-inverse-of-the-mbox-vec-operator.htmlhttps://liberty.io/what-is-the-inverse-of-the-mbox-vec-operator.htmlThere is a well known vectorization operator $\mbox{vec}$ in matrix analysis. I've vectorized my matrix equations, did some transformation of vectorized equations and now I want to get back to the ...Sun, 23 Aug 2026 20:09:48 -0400https://liberty.io/proof-of-squeeze-theorem-for-functions.htmlhttps://liberty.io/proof-of-squeeze-theorem-for-functions.htmlSuppose for all $x$ we know $g(x)\le f(x)\le h(x)$ and $\lim_{x\to c} g(x)=L=\lim_{x\to c} h(x)$. Does the following argument work to conclude that $\lim_{x\to c} f(x)=L$? Let $\epsilon\gt 0$ be ......Sun, 23 Aug 2026 20:08:25 -0400https://liberty.io/how-to-solve-the-differential-equation-y-039-y-1-y.htmlhttps://liberty.io/how-to-solve-the-differential-equation-y-039-y-1-y.htmlUp until now, we simply rearranged and integrated both sides, so $$y'=y(1-y)$$ $$\frac{dy}{dx}=y(1-y)$$ $$\frac{dy}{y(1-y)}=dx$$ $$\int\frac{dy}{y(1-y)}=\int dx$$ With partial fraction decompositio......Sun, 23 Aug 2026 19:45:55 -0400https://liberty.io/prove-e-ln-x-x.htmlhttps://liberty.io/prove-e-ln-x-x.htmlIs it possible to prove $e^{\ln{x}} = x$ for a student or can you only say that exponentiation is defined to be the inverse of natural logarithm and stop there?...Sun, 23 Aug 2026 19:43:36 -0400https://liberty.io/constructing-a-graph-from-a-degree-sequence.htmlhttps://liberty.io/constructing-a-graph-from-a-degree-sequence.htmlLet's say I'm given several degree sequences like $$\begin{array}{l} \{4,3,3,2,2\} \\ \{3,3,3,3\} \\ \{5,3,3,2,2,1\} \end{array}$$ I can find the number of edges using the handshaking lemma $$ \sum......Sun, 23 Aug 2026 19:25:54 -0400https://liberty.io/calculate-overlapping-area-of-two-squares.htmlhttps://liberty.io/calculate-overlapping-area-of-two-squares.htmlGiven task: Two $10 \times 10$ cm square napkins were thrown on the table, as shown in the figure. They covered an area of ​​the table equal to $172$ cm$^2$. What is their overlap area? Answer is......Sun, 23 Aug 2026 19:24:56 -0400https://liberty.io/definition-of-weak-topology.htmlhttps://liberty.io/definition-of-weak-topology.htmlThough I learned some basic theory about topological vector spaces, I always confused about the definition of weak-$*$ topology. Given $x\in X$, let $\phi_x: X^*\to \Bbb R$ denote the evaluation m......Sun, 23 Aug 2026 19:24:55 -0400https://liberty.io/how-do-i-solve-algebraic-fractions-with-two-fractions-on-one-side.htmlhttps://liberty.io/how-do-i-solve-algebraic-fractions-with-two-fractions-on-one-side.htmlHow do I solve algebraic fractions with two fractions on one side? $$\frac{a}{4}-\frac{a+2}{3}=9$$ Just like that one, what would I do? Please explain why you did what you did. I can't show worki......Sun, 23 Aug 2026 19:24:15 -0400https://liberty.io/best-way-to-find-orthonormal-vector.htmlhttps://liberty.io/best-way-to-find-orthonormal-vector.htmlWhat is the easiest method to find two vectors that make an orthonormal basis with a given vector? For example, I have a vector \begin{pmatrix} 2^{-1/2} \\ -2^{-1/2} \\ 0 \end{pmatrix} I understand ...Sun, 23 Aug 2026 19:19:42 -0400https://liberty.io/in-which-order-do-i-graph-transformations-of-functions.htmlhttps://liberty.io/in-which-order-do-i-graph-transformations-of-functions.htmlIn which order do I graph transformations of functions? The 6 function transformations are: Vertical Shifts Horizontal Shifts Reflection about the x-axis Reflection about the y-axis Vertical sh......Sun, 23 Aug 2026 19:15:12 -0400https://liberty.io/missing-statement-in-a-proof-jordan-normal-form.htmlhttps://liberty.io/missing-statement-in-a-proof-jordan-normal-form.htmlI am reading the proof for the existence of the Jordan normal form. I know the Fitting lemma. I am missing one thing though, I mean the proof in my book seems to have this particular thing missing. I ...Sun, 23 Aug 2026 19:14:44 -0400https://liberty.io/integrate-int-1-0-frac-r-3-sqrt-4-r-2-dr.htmlhttps://liberty.io/integrate-int-1-0-frac-r-3-sqrt-4-r-2-dr.html$$\int_0^1\frac{r^3}{\sqrt{4+r^2}}\ \mathrm dr$$ I have attached my work. I am stuck....Sun, 23 Aug 2026 19:12:48 -0400https://liberty.io/determine-the-values-of-x-y-and-z-using-one-equation-with-three-variab.htmlhttps://liberty.io/determine-the-values-of-x-y-and-z-using-one-equation-with-three-variab.htmlSolve for x, y, and z using one equation with three variables: $xyz = x! + y! + z!$ where factorials are defined in the usual way, namely $n! = n(n-1)...2*1$ I'm not sure where to start with this ...Sun, 23 Aug 2026 19:11:14 -0400https://liberty.io/polyhedrons-exclusively-made-out-of-even-sided-polygons.htmlhttps://liberty.io/polyhedrons-exclusively-made-out-of-even-sided-polygons.htmlI know that the cube is the only 3 d shape which falls in polyhedrons but still is composed of squares, exclusively although its a even sided shape. I have noticed that after square there is no sin......Sun, 23 Aug 2026 19:06:00 -0400https://liberty.io/if-sin-x-sin-pi-4-x-then-why-isn-039-t-x-x-pi-4.htmlhttps://liberty.io/if-sin-x-sin-pi-4-x-then-why-isn-039-t-x-x-pi-4.htmlI've been solving a question, If $\cos(x) + \sin(x)=\sqrt{2} \cos(\pi/2 - x)$ then find the value of $x$. We know that $\cos(x) + \sin(x)= \sqrt{2} \sin(\pi/4 + x)$. So, $$\sin(\pi/4 + x) = \......Sun, 23 Aug 2026 19:05:51 -0400https://liberty.io/motivation-for-spectral-graph-theory.htmlhttps://liberty.io/motivation-for-spectral-graph-theory.htmlWhy do we care about eigenvalues of graphs? Of course, any novel question in mathematics is interesting, but there is an entire discipline of mathematics devoted to studying these eigenvalues, so ......Sun, 23 Aug 2026 19:03:24 -0400https://liberty.io/for-which-values-of-x-does-this-series-converge.htmlhttps://liberty.io/for-which-values-of-x-does-this-series-converge.htmlFor which values of $x$ does the series presented below converge? $$\sum_{n=1}^{+\infty}\frac{x^n(1-x^n)}{n}$$ Neither the root test nor the ratio test is of much help - I've tried for awhile now ......Sun, 23 Aug 2026 18:58:41 -0400https://liberty.io/meaning-of-the-symbol-oplus-closed.htmlhttps://liberty.io/meaning-of-the-symbol-oplus-closed.htmlAnybody who knows/can explain what this character means?...Sun, 23 Aug 2026 18:58:25 -0400https://liberty.io/how-to-change-the-curvature-of-a-function.htmlhttps://liberty.io/how-to-change-the-curvature-of-a-function.htmlWe have learnt that the curvature of a function like $f(x)=x^2$ can be computed by its second derivative. However, a question came to my mind on the possible transformations which saves the shape o......Sun, 23 Aug 2026 18:56:14 -0400https://liberty.io/how-many-solutions-ax-0-has.htmlhttps://liberty.io/how-many-solutions-ax-0-has.htmlProblem How many solutions equation $Ax=0$ has when $A$ is defined as: $$ A = \begin{bmatrix} 2 & 3 & 5 \\ 1 & 4 & 0 \\ -4 & 2 & 1 \end{bmatrix} $$ Attemtp to solve ......Sun, 23 Aug 2026 18:54:06 -0400https://liberty.io/about-trichotomy-law-for-real-numbers.htmlhttps://liberty.io/about-trichotomy-law-for-real-numbers.htmlI must proof the "Trichotomy Law for Real Numbers": Prop. 1: let be $\Bbb{R}$ a complete ordered field, then $$\forall x,y \in \Bbb{R}(x=y \vee x < y \vee x > y)$$ Proof 1: by definition of......Sun, 23 Aug 2026 18:53:13 -0400https://liberty.io/find-the-curvature-of-a-curve-at-the-given-point.htmlhttps://liberty.io/find-the-curvature-of-a-curve-at-the-given-point.htmlEdit: I've revised this to clear up any confusion. The $T(0), N(0)$, etc were just to help me. Suppose we have a space curve $r(t)=<3t,\cos t,\sin t>$. I'm asked to find the curvature at $P......Sun, 23 Aug 2026 18:50:48 -0400https://liberty.io/if-x-y-and-z-are-sets-prove-set-related-questions.htmlhttps://liberty.io/if-x-y-and-z-are-sets-prove-set-related-questions.htmlnot Z ⊆ not Y implies Y ⊆ Z by contradiction not Z ⊆ not Y implies X ∩ Y ⊆ X ∩ Z (using part 1) So far for 1 I have if not Z is subset of not Y then all elements of not Z are in not Y which mean......Sun, 23 Aug 2026 18:49:37 -0400https://liberty.io/explanation-of-linearity-of-expectation.htmlhttps://liberty.io/explanation-of-linearity-of-expectation.htmlLinearity of expectation is the property that the expected value of the sum of random variables is equal to the sum of their individual expected values, regardless of whether they are independe......Sun, 23 Aug 2026 18:46:08 -0400https://liberty.io/subsets-in-linear-algebra.htmlhttps://liberty.io/subsets-in-linear-algebra.htmlLooking to get some help with some problems. Not quite understanding the concepts of vectors subspaces. So I need to determine which of the following are subsets in $ \Bbb R^3$ are actually subspac......Sun, 23 Aug 2026 18:35:01 -0400