https://liberty.ioCurated star highlights with fresh energy.en-usMon, 10 Aug 2026 18:51:07 -0400https://liberty.io/integral-test-error-approximation.htmlhttps://liberty.io/integral-test-error-approximation.htmlFind an N so that: $\sum_{n=1}^\infty \frac{1}{n^4}$ is between $\sum_{n=1}^N \frac{1}{n^4}$ and $\sum_{n=1}^N \frac{1}{n^4} + 0.005$ I am getting N = 200. Is this correct? I don't know if I am d......Mon, 10 Aug 2026 22:48:37 -0400https://liberty.io/curious-integrals-for-jacobi-theta-functions-int-0-1-vartheta-n-0-q-dq.htmlhttps://liberty.io/curious-integrals-for-jacobi-theta-functions-int-0-1-vartheta-n-0-q-dq.htmlThere are various identities for the Jacobi Theta Functions $\vartheta_n(z,q)$ on the MathWorld page and on the Wikipedia page. But I found no integral identities for these functions. Meanwhile, t......Mon, 10 Aug 2026 22:48:36 -0400https://liberty.io/drawing-marbles-out-of-a-bag.htmlhttps://liberty.io/drawing-marbles-out-of-a-bag.htmlThe infamous drawing marbles from a bag type question that I am used to seeing has became more complicated. Given a bag with 2 green marbles, 5 red marbles, and 3 yellow marbles. I then draw 2 ma......Mon, 10 Aug 2026 22:48:16 -0400https://liberty.io/relationship-between-complex-number-and-vectors.htmlhttps://liberty.io/relationship-between-complex-number-and-vectors.htmlWhat is the relation between complex numbers and vectors? I have read in some places "a complex number a 2-dimensional vector". One can easily observe that $i\cdot i=-1$ in complex multiplicatio......Mon, 10 Aug 2026 22:42:49 -0400https://liberty.io/calculate-the-wronskian-of-f-t-t-t-and-g-t-t-2-on-the-following-interv.htmlhttps://liberty.io/calculate-the-wronskian-of-f-t-t-t-and-g-t-t-2-on-the-following-interv.htmlHow to Calculate the Wronskian of $f(t)=t|t|$ and $g(t)=t^2$ on the following intervals: $(0,+\infty)$, $(-\infty, 0)$ and $0$. And then how would I show that the Wronskian of the two functions $f......Mon, 10 Aug 2026 22:41:34 -0400https://liberty.io/probability-of-one-pair-hand-in-poker-5-cards.htmlhttps://liberty.io/probability-of-one-pair-hand-in-poker-5-cards.htmlI have been working on the problem of probability of poker hands, I have been able to calculate the probability of each hand except one pair and high card hand. Here is what I have P1 P2 X1 X2 X......Mon, 10 Aug 2026 22:36:28 -0400https://liberty.io/how-to-integrate-displaystyle-1-e-1-x-2.htmlhttps://liberty.io/how-to-integrate-displaystyle-1-e-1-x-2.htmlHow to integrate $\displaystyle 1-e^{-1/x^2}$ ? as hint is given: $\displaystyle\int_{\mathbb R}e^{-x^2/2}=\sqrt{2\pi}$ If i substitute $u=\dfrac{1}{x}$, it doesn't bring anything: $\,\displaystyl......Mon, 10 Aug 2026 22:34:42 -0400https://liberty.io/parallel-postulate-from-playfair-039-s-axiom.htmlhttps://liberty.io/parallel-postulate-from-playfair-039-s-axiom.htmlParallel postulate: If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely......Mon, 10 Aug 2026 22:33:43 -0400https://liberty.io/what-is-the-difference-between-operatorname-arcsin-operatorname-arcsin.htmlhttps://liberty.io/what-is-the-difference-between-operatorname-arcsin-operatorname-arcsin.htmlThis is a homework problem I have. I don't remember learning the difference, and searching hasn't helped explain the difference between the capitalization....Mon, 10 Aug 2026 22:08:35 -0400https://liberty.io/definition-of-a-subfield.htmlhttps://liberty.io/definition-of-a-subfield.htmlGiven $\mathbb{K}$ a field, what is the definition subfield of $\mathbb{K}$? Intuitively, I think a subfield as a subset of a field which itself is a field, in other words, satisfy the axioms that a ...Mon, 10 Aug 2026 22:07:38 -0400https://liberty.io/questions-tagged-uniform-convergence.htmlhttps://liberty.io/questions-tagged-uniform-convergence.htmlQ&A for people studying math at any level and professionals in related fields...Mon, 10 Aug 2026 21:57:34 -0400https://liberty.io/how-to-solve-int-1-0-1-7x-1-3-dx.htmlhttps://liberty.io/how-to-solve-int-1-0-1-7x-1-3-dx.htmlI worked through $\int^1_0 (1+7x)^{1/3}dx$ and I got $\frac{3}{4}+C$ for the answer. However, I forgot about the exponent when I found the difference of the sides of the integral. I am retrying it......Mon, 10 Aug 2026 21:54:42 -0400https://liberty.io/gravitational-potential-energy-near-the-surface-of-the-earth.htmlhttps://liberty.io/gravitational-potential-energy-near-the-surface-of-the-earth.htmlOkay so for an object of mass m at a distance r from the Earth's centre, the GPE is $$U(r) = {{ - GMm} \over r}$$ For an object at height z above the surface (which obviously means at radius $r = {r_{...Mon, 10 Aug 2026 21:49:33 -0400https://liberty.io/roots-of-sum-of-two-polynomials-with-known-roots.htmlhttps://liberty.io/roots-of-sum-of-two-polynomials-with-known-roots.htmlI am writing a piece of software and I'm trying to avoid root finding polynomials for efficiency purposes. I have two polynomials with complex coefficients, where the roots of both polynomials are ......Mon, 10 Aug 2026 21:48:39 -0400https://liberty.io/finding-the-area-of-a-quadrilateral.htmlhttps://liberty.io/finding-the-area-of-a-quadrilateral.htmlI have a quadrilateral whose four sides are given $2341276$, $34374833$, $18278172$, $17267343$ units. How can I find out its area? What would be the area?...Mon, 10 Aug 2026 21:48:17 -0400https://liberty.io/derivative-of-e-x-with-respect-to-y.htmlhttps://liberty.io/derivative-of-e-x-with-respect-to-y.htmlI recently came across a question that asked for the derivative of $e^x$ with respect to $y$. I answered $\frac{d}{dy}e^x$ but the answer was $e^x\frac{dx}{dy}$. How is that the answer? I am confused....Mon, 10 Aug 2026 21:34:43 -0400https://liberty.io/how-is-3-equiv-3-bmod-5.htmlhttps://liberty.io/how-is-3-equiv-3-bmod-5.htmlJust tried googling but couldn't find any example, but how $3\equiv 3\bmod{5}$ Googled it...Mon, 10 Aug 2026 21:34:13 -0400https://liberty.io/the-row-reducing-matrix-with-complex-entries.htmlhttps://liberty.io/the-row-reducing-matrix-with-complex-entries.html$$\begin{pmatrix} -2i&-3&7\\ 0&-1-2i&5\\ 0&-1&1-2i \end{pmatrix} \mathbf{v} =0 \implies \begin{pmatrix} -2i&-3&7\\ 0&-1-2i&5\\ 0&0&0 \end{pmatrix} \m......Mon, 10 Aug 2026 21:31:40 -0400https://liberty.io/notation-of-the-second-derivative-where-does-the-d-go.htmlhttps://liberty.io/notation-of-the-second-derivative-where-does-the-d-go.htmlIn school I was taught that we use $\frac{du}{dx}$ as a notation for the first derivative of a function $u(x)$. I was also told that we could use the $d$ just like any variable. After some time we ......Mon, 10 Aug 2026 21:31:16 -0400https://liberty.io/getting-value-of-sin-x-cos-x-with-complex-exponential.htmlhttps://liberty.io/getting-value-of-sin-x-cos-x-with-complex-exponential.htmlI have a bit of difficulty with this. I am trying to express $\sin x + \cos x$ with complex exponential. I started by using Euler's equations. Then, I used the trigonometric substitution $\sin x =......Mon, 10 Aug 2026 21:24:09 -0400https://liberty.io/roots-of-equation-x-3-x-1-0.htmlhttps://liberty.io/roots-of-equation-x-3-x-1-0.htmlIf α , β , γ are the roots of equation $x^3 -x -1 =0$ then $$ \frac{1+\alpha}{1-\alpha} + \frac{1+\beta}{1-\beta} + \frac{1+\gamma}{1-\gamma} $$ My ...Mon, 10 Aug 2026 21:23:57 -0400https://liberty.io/most-general-antiderivative-involving-sec-x.htmlhttps://liberty.io/most-general-antiderivative-involving-sec-x.htmlI'm stumped on how to get the most general antiderivative, $F(x)$, of $f(x)=e^x+3secx(tan x + sec x)$. First, I split the equation on addition, since $\int[f(x)+g(x)]dx=\int f(x)dx+\int g(x)dx$ ......Mon, 10 Aug 2026 21:17:18 -0400https://liberty.io/analytical-solution-to-2d-diffusion-equation-with-a-drift-term.htmlhttps://liberty.io/analytical-solution-to-2d-diffusion-equation-with-a-drift-term.htmlI have been trying to solve the 2D PDE $$\frac{\partial p(x,y,t)}{\partial t}=-\frac{\partial }{\partial x}p(x,y,t)+\frac{\partial ^2}{\partial x^2}p(x,y,t)+\frac{\partial ^2}{\partial y^2}p(x,y,t)$$ ...Mon, 10 Aug 2026 21:10:37 -0400https://liberty.io/should-we-define-something-after-we-have-prove-that-exists-based-on-th.htmlhttps://liberty.io/should-we-define-something-after-we-have-prove-that-exists-based-on-th.htmlIn last days I have a hard time to understand what a definition is in mathematics. Until today I thought definition had a dual role in mathematics. Dictionary role The first role is that it mereley ...Mon, 10 Aug 2026 21:09:33 -0400https://liberty.io/sides-of-a-triangle-square-roots.htmlhttps://liberty.io/sides-of-a-triangle-square-roots.htmlThis is the exercise: Let $a,b,c\in \mathbb{R}^+$. Prove that the following propositions are equivalents: $a,b,c$ are sides of a triangle. $\sqrt{a},\sqrt{b},\sqrt{c}$ are sides of an acute triang......Mon, 10 Aug 2026 21:04:12 -0400https://liberty.io/how-do-i-find-all-points-of-discontinuity-of-the-function-f-x-x-2-x-3-.htmlhttps://liberty.io/how-do-i-find-all-points-of-discontinuity-of-the-function-f-x-x-2-x-3-.htmlBasically it is a modulus $\operatorname{fun}^c$. To begin with we'll redefine the function, i.e. $f(x)=|x+2|+|x-3|$. Taking $$ u = |x+2| = \begin{cases} -(x+2), & \text{if $x<0$,} \\ x+2, &......Mon, 10 Aug 2026 20:59:39 -0400https://liberty.io/why-is-triangle-in-p-p-np.htmlhttps://liberty.io/why-is-triangle-in-p-p-np.htmlI'm learning about $P/NP$ and my friend used an example in which he said that if you have a triangle in an undirected graph which is basically a set of three nodes in which all pairs of nodes are ...Mon, 10 Aug 2026 20:51:16 -0400https://liberty.io/euler-product-formula-for-riemann-zeta-function-proof.htmlhttps://liberty.io/euler-product-formula-for-riemann-zeta-function-proof.htmlIn class we introduced Reimann Zeta function $$ \zeta (x)=\sum_{n=1}^{+\infty} \frac{1}{n^x} $$ And we proved its domain was $D=(1,+\infty)$ Now Euler proved that $$ \zeta(x)=\prod_{p\text{ pr......Mon, 10 Aug 2026 20:51:05 -0400https://liberty.io/computing-the-derivative-of-a-system-of-equations-in-the-neighborhood-.htmlhttps://liberty.io/computing-the-derivative-of-a-system-of-equations-in-the-neighborhood-.htmlI'm solving the following problem on an old exam in real analysis. Thus, only such methods may be used. The system \begin{align*} \begin{cases} \sin(x+y)+\sin(y+z)+z=0 \\ \cos(x+y)......Mon, 10 Aug 2026 20:36:07 -0400https://liberty.io/can-i-assume-that-a-biologist-will-know-what-lhs-and-rhs-mean-or-what-.htmlhttps://liberty.io/can-i-assume-that-a-biologist-will-know-what-lhs-and-rhs-mean-or-what-.htmlI am writing a scientific article with a few mathematical equations. Can I assume that my audience will know what lhs and rhs mean?...Mon, 10 Aug 2026 20:33:50 -0400https://liberty.io/utility-function-plotting-indifference-curves.htmlhttps://liberty.io/utility-function-plotting-indifference-curves.htmlI know how to plot indifference curves; simply take the utility function and plot some level curves in $2D$. But how to plot a specific indifference curve, so all bundles on it are indifferent to a ...Mon, 10 Aug 2026 20:16:33 -0400https://liberty.io/laplace-equation-fourier-transform.htmlhttps://liberty.io/laplace-equation-fourier-transform.htmlSuppose $$ \begin{cases} u_{xx} + u_{yy} = 0, & (x,y) \in \mathbb R \times (0, \infty) \\ u_{y}(x,0) = f(x). \end{cases} $$ Show that $$u(x,y)= \frac{1}{2 \pi} \int_{-\infty}^{\infty} \ln((x-z)......Mon, 10 Aug 2026 20:13:16 -0400https://liberty.io/how-can-i-generate-the-binary-representation-of-any-real-number.htmlhttps://liberty.io/how-can-i-generate-the-binary-representation-of-any-real-number.htmlIn p. 30 of Baby Rudin, I find a reference to the fact that the binary representation of a real number implies the uncountablity of the set of real numbers. But I have two questions: Does every real ...Mon, 10 Aug 2026 20:00:35 -0400https://liberty.io/where-to-start-with-the-inverse-galois-problem.htmlhttps://liberty.io/where-to-start-with-the-inverse-galois-problem.htmlI have always been interested in the Inverse Galois Problem since I read the story of Évariste Galois. I have recently finished up courses in rings, fields, groups, and finally Galois theory. I rea......Mon, 10 Aug 2026 19:56:14 -0400https://liberty.io/smooth-sawtooth-wave-y-x-cos-x-cos-x-cos-x-dots.htmlhttps://liberty.io/smooth-sawtooth-wave-y-x-cos-x-cos-x-cos-x-dots.htmlConsider an infinite recursive function $$y(x)=\cos(x-\cos(x-\cos(x-\dots)))$$ $$y=\cos(x-y)$$ Plotting the function $y(x)$ implicitly we get a smooth sawtooth-like wave: Was this function s......Mon, 10 Aug 2026 19:56:06 -0400https://liberty.io/moment-generating-function-of-poisson.htmlhttps://liberty.io/moment-generating-function-of-poisson.htmlI'm unable to understand the proof behind determining the Moment Generating Function of a Poisson which is given below: $N \sim \mathrm{Poiss}(\lambda)$ $$ E[e^{\theta N}] = \sum\limits_{k=0}^\inf......Mon, 10 Aug 2026 19:53:06 -0400https://liberty.io/double-integral-using-riemann-sums.htmlhttps://liberty.io/double-integral-using-riemann-sums.htmlSo there is $R=\left[-1,3\right]\times\left[0,2\right]$. I have to use a Riemann sum with m=4,n=2 to estimate the value of double integral $\int \int(y^2-2x^2)\ \mathrm{d}A$, taking the sample poin......Mon, 10 Aug 2026 19:28:56 -0400https://liberty.io/understanding-reflexive-relations.htmlhttps://liberty.io/understanding-reflexive-relations.htmlI'm reviewing some problems to try and get a better understanding of relations. I get how a reflexive relation works on a defined relation with numbers, but not so much when its done with a set bui......Mon, 10 Aug 2026 19:23:45 -0400https://liberty.io/039-algebraic-039-way-to-prove-the-boolean-identity-a-overline-a-b-a-b.htmlhttps://liberty.io/039-algebraic-039-way-to-prove-the-boolean-identity-a-overline-a-b-a-b.htmlFor me, it is pretty clear that $a + \overline{a}*b = a + b$, because the first $a$ in the or will make sure that if the second term must be 'evaluated', $a$ will always be false, and therefore won't ...Mon, 10 Aug 2026 19:21:53 -0400https://liberty.io/prove-that-the-each-angle-of-regular-hexagon-is-120.htmlhttps://liberty.io/prove-that-the-each-angle-of-regular-hexagon-is-120.htmlI need a theoretical proof that each angle of a regular hexagon is $120^\circ$....Mon, 10 Aug 2026 19:19:16 -0400https://liberty.io/how-to-show-that-mathfrak-sl-n-mathbb-r-and-mathfrak-sl-n-mathbb-c-are.htmlhttps://liberty.io/how-to-show-that-mathfrak-sl-n-mathbb-r-and-mathfrak-sl-n-mathbb-c-are.htmlWe defined a Lie algebra to be simple, if it has no proper Lie ideals and is not $k$ (the ground field). We have the proposition that $\mathfrak{sl}_n(\mathbb{R})$ and $\mathfrak{sl}_n(\mathbb{C})$......Mon, 10 Aug 2026 19:13:34 -0400https://liberty.io/how-does-the-determinant-link-to-the-cross-product.htmlhttps://liberty.io/how-does-the-determinant-link-to-the-cross-product.htmlFor a $2\times 2$ matrix $$\begin{pmatrix}a&b\\c&d \end{pmatrix} $$ The determinant is given by $ad-bc$. And the cross product of $$\begin{pmatrix} a\\b\\0\end{pmatrix}\times \begin{pmatrix......Mon, 10 Aug 2026 19:00:57 -0400https://liberty.io/how-to-compare-periodically-and-continuously-compounding-interest-when.htmlhttps://liberty.io/how-to-compare-periodically-and-continuously-compounding-interest-when.htmlI am working on a textbook problem, and I think I disagree with the solution. The problem is (10.6) Bruce deposits 100 into a bank account. His account is credited interest at a nominal rate of ...Mon, 10 Aug 2026 18:55:29 -0400https://liberty.io/proof-verification-every-sequence-in-bbb-r-contains-a-monotone-sub-seq.htmlhttps://liberty.io/proof-verification-every-sequence-in-bbb-r-contains-a-monotone-sub-seq.htmlCame across the following exercise in Bartle's Elements of Real Analysis. This is the solution I came up with. Would be grateful if someone could verify it for me and maybe suggest better/alternate ...Mon, 10 Aug 2026 18:43:14 -0400https://liberty.io/divisibility-of-9-trick-why-does-it-work-duplicate.htmlhttps://liberty.io/divisibility-of-9-trick-why-does-it-work-duplicate.htmlI've been wondering ever since I learned about the divisibility of nine trick. Add up the sum of the digits. If the sum is a multiple of nine, the entire number is divisible by nine. If the sum is ......Mon, 10 Aug 2026 18:27:57 -0400https://liberty.io/difference-of-convexity-and-strict-convexity.htmlhttps://liberty.io/difference-of-convexity-and-strict-convexity.htmlI know what is a function convex and what is a strict convex function. But I was wondering what is concretely the difference, haw can we differentiate both on a graph ? For example, I know that a c......Mon, 10 Aug 2026 18:24:14 -0400https://liberty.io/evaluate-cos-1-2-arcsin-3-5-i-039-m-not-sure-what-i-039-m-doing-wrong.htmlhttps://liberty.io/evaluate-cos-1-2-arcsin-3-5-i-039-m-not-sure-what-i-039-m-doing-wrong.html$x=\arcsin(-3/5), \; \sin x = -3/5$ **Drew a triangle to find $\cos x$ $\cos x = 4/5$ Now, I don't know what to do from here. I know I have to use a double angle formula, but when I evaluate the......Mon, 10 Aug 2026 18:14:28 -0400https://liberty.io/examples-of-set-functions.htmlhttps://liberty.io/examples-of-set-functions.htmlI have recently got acquainted with a special kind of function known as a set function . I've a series of questions in my mind with respect to this . Firstly it is hard on my part, at this level to ...Mon, 10 Aug 2026 18:14:18 -0400https://liberty.io/variance-of-n-bernoulli-trials.htmlhttps://liberty.io/variance-of-n-bernoulli-trials.htmlCount the variance of n Bernoulli trials with each probability of success is p. Let random variable $X_i$ be $1$ if trial is success, or $0$ if trial fails. Then expected value $E(X_i) = 1 \......Mon, 10 Aug 2026 18:13:12 -0400https://liberty.io/why-does-a-square-pyramid-have-a-third-of-the-volume-of-a-square-prism.htmlhttps://liberty.io/why-does-a-square-pyramid-have-a-third-of-the-volume-of-a-square-prism.htmlWhy is it that a square pyramid with the exact same dimensions (length, width and height) as a square prism have a third of its volume? I know this is because the square pyramid formula is the exac......Mon, 10 Aug 2026 18:12:01 -0400https://liberty.io/what-is-the-meaning-of-a-discriminant-graphically.htmlhttps://liberty.io/what-is-the-meaning-of-a-discriminant-graphically.htmlI know that if $b^2-4ac>0$, then there are real solutions and so on. But for any quadratic function, where is the discriminant present in the plot? For example, $x^2-8x+7=f(x)$. Now how is $......Mon, 10 Aug 2026 17:57:28 -0400https://liberty.io/proving-that-the-second-derivative-of-a-convex-function-is-nonnegative.htmlhttps://liberty.io/proving-that-the-second-derivative-of-a-convex-function-is-nonnegative.htmlMy task is as follows: Let $f:\mathbb{R}\to\mathbb{R}$ be a twice-differentiable function, and let $f$'s second derivative be continuous. Let $f$ be convex with the following definition of ...Mon, 10 Aug 2026 17:57:23 -0400https://liberty.io/roll-a-4-sided-die-whatever-value-appears-flip-exactly-that-many-fair-.htmlhttps://liberty.io/roll-a-4-sided-die-whatever-value-appears-flip-exactly-that-many-fair-.htmlRoll a $4$-sided die. Whatever value appears, flip exactly that many fair coins. [For instance, if the die shows $3$, then flip $3 $fair coins.] Let $X$ denote the number of heads that appear on the ...Mon, 10 Aug 2026 17:46:12 -0400https://liberty.io/what-is-the-probability-of-multiple-independent-events-occurring-in-a-.htmlhttps://liberty.io/what-is-the-probability-of-multiple-independent-events-occurring-in-a-.htmlI thought about this problem the other day and my limited knowledge in basic probability theory was not enough to help me figure this one out. Say you have 3 probabilities: You have a 50% chance of ...Mon, 10 Aug 2026 17:43:31 -0400https://liberty.io/the-prolate-cycloid.htmlhttps://liberty.io/the-prolate-cycloid.htmlA cycloid is given by the parametric equations: $ x = 2 - \pi \cos(t)$ and $ y = 2t - \pi \sin(t)$. The problem asks for the slope of the tangents on the cycloid at a point where the cycloid inte......Mon, 10 Aug 2026 17:43:12 -0400https://liberty.io/proving-that-the-inverse-function-exists-and-finding-it.htmlhttps://liberty.io/proving-that-the-inverse-function-exists-and-finding-it.htmlLet $f$ be a function defined as $$f:=\sqrt[3]{x^{3}+bx^{2}}, b\in \mathbb{R}$$ for all$$x\leq 0$$ In order for a function ta have an inverse, it must be both injective and surjective(hope I spe......Mon, 10 Aug 2026 17:42:33 -0400https://liberty.io/how-to-find-the-stabilizer-of-a-subgroup-of-s-4.htmlhttps://liberty.io/how-to-find-the-stabilizer-of-a-subgroup-of-s-4.htmlGiven $H = \{ id , (12) , (34) , (12)(34)\}$ find $Stab(H)$ and the bijection from the set of left cosets of $Stab(H) $ in $ S_4 $ to $ \mathcal{O} (H) $ ; $S_4$ acts on $H$ by conjugation. ie: $$\...Mon, 10 Aug 2026 17:25:13 -0400https://liberty.io/for-a-matrix-what-is-a-solution-vector.htmlhttps://liberty.io/for-a-matrix-what-is-a-solution-vector.htmlIf given a value $N$ and told to generate the following: A random matrix $A[N][N]$ A solution vector $S[N]$ What would be a "solution vector" of a randomly generated matrix? I am not sure I have ......Mon, 10 Aug 2026 17:20:31 -0400https://liberty.io/given-a-latitude-how-many-miles-is-the-corresponding-longitude.htmlhttps://liberty.io/given-a-latitude-how-many-miles-is-the-corresponding-longitude.htmlOK so lines of longitude (the distance/circumference around the earth horizontally) differ based on what latitude you are at (0 at north and south poles up to ~25k at the equator.) So given a lati......Mon, 10 Aug 2026 17:16:20 -0400https://liberty.io/why-does-the-geometric-series-formula-intuitively-work.htmlhttps://liberty.io/why-does-the-geometric-series-formula-intuitively-work.htmlI understand the proof for the geometric series formula, but I don't understand how the formula, $S_n = a_1\frac{(r^n -1)}{(r-1)}$ actually relates to the sum of all the terms. What operations are ...Mon, 10 Aug 2026 17:14:58 -0400https://liberty.io/find-the-derivative-of-f-gmm-over-r-2.htmlhttps://liberty.io/find-the-derivative-of-f-gmm-over-r-2.htmlNewton's Law of Gravitation says that the magnitude F of the force exerted by a body of mass M on a body of mass m is $F =$ $(GmM)\over r^2$ Where G is the gravitational constant and r is the dis......Mon, 10 Aug 2026 16:57:45 -0400https://liberty.io/is-it-possible-to-calculate-the-standard-deviation-with-a-given-mean-a.htmlhttps://liberty.io/is-it-possible-to-calculate-the-standard-deviation-with-a-given-mean-a.htmlI have been going in rounds with this problem... I may be thinking "complicated", any advice? I have the mean and total sample size (=number of data points) and I need to know what is the standard ...Mon, 10 Aug 2026 16:47:13 -0400https://liberty.io/how-to-convert-from-cartesian-to-polar-equation.htmlhttps://liberty.io/how-to-convert-from-cartesian-to-polar-equation.htmlI am trying to convert the equation $y=4/x$ into a polar equation. I have done this work but I am not sure if it is right. I just subsituted $r\sin(\theta)$ for $y$ and $r\cos(\theta)$ for $x$ ......Mon, 10 Aug 2026 16:45:14 -0400https://liberty.io/scaling-rigid-or-non-rigid-transformation.htmlhttps://liberty.io/scaling-rigid-or-non-rigid-transformation.htmlI am trying to look for a precise definition of what rigid and non-rigid transformation is, and to which categories does 'scaling' belong. This is connected to a Point-Set registration problem that......Mon, 10 Aug 2026 16:43:34 -0400https://liberty.io/how-to-find-the-vector-equation-of-a-plane-given-the-scalar-equation-c.htmlhttps://liberty.io/how-to-find-the-vector-equation-of-a-plane-given-the-scalar-equation-c.htmlHow would I find the vector equation of the plane: $x + 2y + 7z - 3 = 0$ So far, I found the normal vector: it's $(1, 2, 7)$....Mon, 10 Aug 2026 16:43:06 -0400https://liberty.io/the-coercivity-of-uniformly-positive-definite-matrix-of-sobolev-functi.htmlhttps://liberty.io/the-coercivity-of-uniformly-positive-definite-matrix-of-sobolev-functi.htmlFor $u=(u^1,\ldots, u^N)\in W^{1,2}(\Omega,R^N)$ where $\Omega$ is bounded. We define $$ E[u]=\int_\Omega g_{ij}(u)\nabla u^i\nabla u^jdx$$ where $G=(g_{ij})_{1\leq i,j\leq N}$ is an given uniformly ...Mon, 10 Aug 2026 16:39:28 -0400https://liberty.io/in-english-is-there-a-difference-between-square-and-squared.htmlhttps://liberty.io/in-english-is-there-a-difference-between-square-and-squared.htmlFor example, if I say "one over $x$ square/squared" Do I mean: $1/\sqrt{x}$ or $1/x^2$? Is there some other way to distinguish between the English pronounciation between $x^2$ and $\sqrt{......Mon, 10 Aug 2026 16:37:06 -0400https://liberty.io/real-analysis-folland-problem-1-5-29-lebesgue-measurable-set.htmlhttps://liberty.io/real-analysis-folland-problem-1-5-29-lebesgue-measurable-set.html1.5.29 - Let $E$ be a Lebesgue measurable set. a.) If $E\subset N$ where $N$ is the nonmeasurable set described in section 1.1, then $m(E) = 0$. b.) If $m(E) > 0$, then $E$ contains a nonmeasura......Mon, 10 Aug 2026 16:32:06 -0400https://liberty.io/evaluating-surface-integral-using-stokes-039-theorem.htmlhttps://liberty.io/evaluating-surface-integral-using-stokes-039-theorem.htmlLet $\vec{F}$ be the vector field $\vec{F}\left(x,y,z\right)=\left(z,x,y\right)$. Let $\rm S$ be portion of the surface $x^{2}+y^{2}+z=1$ lying above the $\rm XY$-plane, oriented upward. Evaluate the ...Mon, 10 Aug 2026 16:30:34 -0400https://liberty.io/prove-that-the-planar-curve-obtained-by-projecting-alpha-into-its-oscu.htmlhttps://liberty.io/prove-that-the-planar-curve-obtained-by-projecting-alpha-into-its-oscu.htmlLet $\alpha(s)$ be a regular curve with $\kappa\neq0$ at P, where $\kappa$ is the curvature. Prove that the planar curve obtained by projecting $\alpha(s)$ into its osculating plane at $P$ has the ......Mon, 10 Aug 2026 16:29:25 -0400https://liberty.io/write-2-log-3-x-log-9-x-4-log-27-x-5-into-one-single-logarithm.htmlhttps://liberty.io/write-2-log-3-x-log-9-x-4-log-27-x-5-into-one-single-logarithm.htmlI'm trying to answer this problem: Write $$2+\log_3 x+\log_9 x^4-\log_{27} x^5$$ into one single logarithm. This is what I'm stuck with: $$\frac{\log x}{\log 3}+ \frac{\log x^4}{\log 3^2}+\frac......Mon, 10 Aug 2026 16:25:04 -0400https://liberty.io/how-to-find-the-potential-function-of-a-vector-field.htmlhttps://liberty.io/how-to-find-the-potential-function-of-a-vector-field.htmlI calculated that $\frac{dP}{dy} = \cos(y) = \frac{dQ}{dx}$ This tells me that the potential function exists, however I can't figure out what it is. So far I have found that $\frac{df}{dx} = \si......Mon, 10 Aug 2026 16:23:31 -0400https://liberty.io/is-the-term-monotone-used-fairly-consistently-to-mean-non-decreasing-o.htmlhttps://liberty.io/is-the-term-monotone-used-fairly-consistently-to-mean-non-decreasing-o.htmlIn BBFSK, (~1960, Germany) at least in the section I am currently reading, the authors use the term monotone increasing (decreasing) to mean what I often see called strictly increasing (decreasing)......Mon, 10 Aug 2026 16:18:54 -0400https://liberty.io/orthogonal-projection-on-an-eigenspace.htmlhttps://liberty.io/orthogonal-projection-on-an-eigenspace.htmlFor different symmetric matrix $A$ ( eigenspaces are orthogonals ), I've noticed that when I build the matrix of the orthogonal projection of an eigenspace I get identity. Are my calculation wrong ......Mon, 10 Aug 2026 16:18:31 -0400https://liberty.io/how-to-solve-sqrt-3-x-cos-2x-0-for-sketching-the-graph-of-y-sqrt-3-x-c.htmlhttps://liberty.io/how-to-solve-sqrt-3-x-cos-2x-0-for-sketching-the-graph-of-y-sqrt-3-x-c.htmlSolve the following question. $$\begin{align} \\\sqrt{3}x-\cos 2x=0\\ \end{align}$$ I have no idea how to solve the above question with x inside and outside the cos. Since I have to sketch the......Mon, 10 Aug 2026 16:07:36 -0400https://liberty.io/software-for-testing-relational-algebra.htmlhttps://liberty.io/software-for-testing-relational-algebra.htmlDoes anyone know of any software to let you test relational algebra queries? By this I don't mean a database such as MySQL, something where the query can be input in some for of mathematical notation ...Mon, 10 Aug 2026 16:07:17 -0400https://liberty.io/using-power-series-to-evaluate-a-limit.htmlhttps://liberty.io/using-power-series-to-evaluate-a-limit.htmlIf we want to evaluate the limit of $$\lim_{x\to 0} \frac{\ln(1-8x)+8x+32x^2}{3x^2}$$ I would imagine we need to use the power series expansion of the function $${\ln(1-8x)}$$ and then determine ......Mon, 10 Aug 2026 15:57:48 -0400https://liberty.io/finding-a-derivative-with-imaginary-numbers.htmlhttps://liberty.io/finding-a-derivative-with-imaginary-numbers.htmlIf I have some function $f = (1 + 7i - x)(7 + 5i - x)(3 + 1i - x)$ How do I find its derivative? I know it is $(13/3 + 11i/3 - x)(3 + 5i - x)$ but I don't know how to find it....Mon, 10 Aug 2026 15:45:19 -0400https://liberty.io/what-is-the-meaning-of-a-s.htmlhttps://liberty.io/what-is-the-meaning-of-a-s.htmlWhat is the meaning of a.s. behind a limit formula (I found this in a paper about stochastic processes) , or sometimes P-a.s.?...Mon, 10 Aug 2026 15:42:34 -0400https://liberty.io/why-are-there-12-pentagons-and-20-hexagons-on-a-soccer-ball.htmlhttps://liberty.io/why-are-there-12-pentagons-and-20-hexagons-on-a-soccer-ball.htmlEdge-attaching many hexagons results in a plane. Edge-attaching pentagons yields a dodecahedron. Is there some insight into why the alternation of pentagons and hexagons yields an approximated sph......Mon, 10 Aug 2026 15:40:55 -0400https://liberty.io/explanation-of-method-for-finding-the-intersection-of-two-parabolas.htmlhttps://liberty.io/explanation-of-method-for-finding-the-intersection-of-two-parabolas.htmlI am trying to understand the steps in a modified version of the Fortune's algorithm for voronoi generation (http://blog.ivank.net/fortunes-algorithm-and-implementation.html). I've stumbled on the ...Mon, 10 Aug 2026 15:40:01 -0400https://liberty.io/how-to-find-the-inverse-of-y-x-3-5x-2-3x-c.htmlhttps://liberty.io/how-to-find-the-inverse-of-y-x-3-5x-2-3x-c.htmlI know how to find the inverse of $y = x^3$, but once you add/subtract another term, that is where I become lost. I have used wolframalpha to get an answer. I am lost upon how this was obtained....Mon, 10 Aug 2026 15:37:25 -0400https://liberty.io/easy-optimization-problem-checking-to-see-my-reasoning-maximum-volume-.htmlhttps://liberty.io/easy-optimization-problem-checking-to-see-my-reasoning-maximum-volume-.htmlA cylindrical package to be sent by a postal service can have a maximum combined length and girth (perimeter of a cross section) of 108 inches. Find the dimensions of the package of maximum volume ......Mon, 10 Aug 2026 15:30:27 -0400https://liberty.io/probability-of-getting-at-least-2-heads.htmlhttps://liberty.io/probability-of-getting-at-least-2-heads.htmlLet assume, I through a fair coin three times, and I want to get at least two heads. I want to find the the probability. To find that, At first I evaluated the probability of getting 1 head whic......Mon, 10 Aug 2026 15:19:55 -0400https://liberty.io/gauss-sum-in-regular-7-gon.htmlhttps://liberty.io/gauss-sum-in-regular-7-gon.htmlThe heptagon on the picture is a regular heptagon with side 1. What is the length of the dashed interval? This is a (kind of) 'geometric version of a quadratic Gauss sum for p=7' (this observation......Mon, 10 Aug 2026 15:12:52 -0400https://liberty.io/symbol-for-such-that-not-in-set.htmlhttps://liberty.io/symbol-for-such-that-not-in-set.htmlIf $A$ is a set, we can use the set notation $$A= \{ b \mid\text{property $p_1$ of $b$}\}$$ But say $A$ is an element like $b$, $$A = b \mid \text{property $p_1$ of $b$}$$ is this a usual nota......Mon, 10 Aug 2026 15:01:32 -0400https://liberty.io/what-is-the-difference-between-sigma-sigma-bar-x-s-s-and-s-bar-x.htmlhttps://liberty.io/what-is-the-difference-between-sigma-sigma-bar-x-s-s-and-s-bar-x.htmlWhat is the difference between $\sigma, \sigma_{\bar{x}}, S, s,$ and $s_{\bar{x}}$? My textbook uses lots of different symbols, and it's not clear to me what the difference between all of them are......Mon, 10 Aug 2026 14:57:31 -0400https://liberty.io/what-is-cos-pi-12-i-sin-pi-12-16-cos-pi-12-i-sin-pi-12-16.htmlhttps://liberty.io/what-is-cos-pi-12-i-sin-pi-12-16-cos-pi-12-i-sin-pi-12-16.htmlWhat is $[\cos(\pi/12)+i\sin(\pi/12)]^{16}+[\cos(\pi/12)-i\sin(\pi/12)]^{16}$? I can use De Moivre's formula for the left part: $[\cos(\pi/12)+i\sin(\pi/12)]^{16} = \cos(4\pi/3) + i\sin(4\pi/3) = -\...Mon, 10 Aug 2026 14:54:58 -0400https://liberty.io/absolute-value-theorem.htmlhttps://liberty.io/absolute-value-theorem.htmlWhen trying to prove the inequality $$ |a +b| \leq |a| + |b| \text{, for any real numbers a and b} $$ I manage to use the absolute value definition to get to following inequality: $$ -\big(|a|+|......Mon, 10 Aug 2026 14:54:16 -0400https://liberty.io/decimal-form-of-irrational-numbers.htmlhttps://liberty.io/decimal-form-of-irrational-numbers.htmlIn the decimal form of an irrational number like: $$\pi=3.141592653589\ldots$$ Do we have all the numbers from $0$ to $9$. I verified $\pi$ and all the numbers are there. Is this true in general for ...Mon, 10 Aug 2026 14:51:46 -0400https://liberty.io/tessellate-the-plane-with-arbitrary-surfaces.htmlhttps://liberty.io/tessellate-the-plane-with-arbitrary-surfaces.htmlIt is known that every surface can be represented by a polygon and glueing the sides appropriately (A source). It is also known that the only tessellations of the plane are by squares, triangles and ...Mon, 10 Aug 2026 14:30:45 -0400https://liberty.io/markov-039-s-inequality-tight-in-general.htmlhttps://liberty.io/markov-039-s-inequality-tight-in-general.htmlFrom here: Even though Markov’s and Chebyshev’s Inequality only use information about the expectation and thevariance of the random variable under consideration, they are essentially tight for a ...Mon, 10 Aug 2026 14:29:18 -0400https://liberty.io/what-did-whitehead-and-russell-039-s-principia-mathematica-achieve.htmlhttps://liberty.io/what-did-whitehead-and-russell-039-s-principia-mathematica-achieve.htmlIn philosophical contexts, the Principia Mathematica is sometimes held in high regard as a demonstration of a logical system. But what did Whitehead and Russell's Principia Mathematica achieve for ...Mon, 10 Aug 2026 14:19:01 -0400https://liberty.io/how-does-the-difference-quotient-with-a-square-root-in-the-numerator-e.htmlhttps://liberty.io/how-does-the-difference-quotient-with-a-square-root-in-the-numerator-e.htmlI don't understand when I apply the difference quotient to: $f(x) = \sqrt{x} $ , to get: $$\frac{\sqrt{x+h} - \sqrt{x}}{h}$$ To simplify it.. How does it end up like this?: $$\frac{x + h - x}......Mon, 10 Aug 2026 14:16:08 -0400https://liberty.io/square-brackets-in-indices.htmlhttps://liberty.io/square-brackets-in-indices.htmlWhat do these brackets within the indices mean in an equation like $$ \delta ^\mu _{[\alpha} \eta _{\beta ]\nu} ?$$ I can't find a text, which uses this notation, that explains it....Mon, 10 Aug 2026 14:11:04 -0400https://liberty.io/one-sided-limits-of-f-039-x-at-a-point-vs-one-sided-limits-of-the-diff.htmlhttps://liberty.io/one-sided-limits-of-f-039-x-at-a-point-vs-one-sided-limits-of-the-diff.htmlI know that if a function is piecewise, then when differentiating it one needs to deal separately with the points of intersection of the pieces, because the derivative may not be defined there. Al......Mon, 10 Aug 2026 14:04:29 -0400https://liberty.io/can-the-delta-function-be-even-or-odd.htmlhttps://liberty.io/can-the-delta-function-be-even-or-odd.htmlSay I have the function, $$ \delta\left(\tau-\frac{T}{2}\right)\mathrm d\tau $$ where T = $$\frac{2π}{ω}$$ Is this even or odd? Or neither? Reason I ask is to find out whether I can cancel some ...Mon, 10 Aug 2026 14:03:15 -0400https://liberty.io/confusion-between-the-definition-of-increasing-function-and-a-theorem-.htmlhttps://liberty.io/confusion-between-the-definition-of-increasing-function-and-a-theorem-.htmlThe definition of increasing function given in my school maths text book is Let $I$ be an open interval contained in the domain of a real valued function $f$. Then $f$ is said to be increasing on......Mon, 10 Aug 2026 13:59:46 -0400https://liberty.io/forward-euler-stability.htmlhttps://liberty.io/forward-euler-stability.htmlI am trying to understand the stability of the forward Euler method. I read there's a model problem to see the stability. $$y'(t) = \lambda y(t) \qquad t \in (0, \infty)$$ $$y(0) = 1$$ then the b......Mon, 10 Aug 2026 13:55:43 -0400https://liberty.io/definition-of-closed-subscheme.htmlhttps://liberty.io/definition-of-closed-subscheme.htmlLet $(X,\mathcal O_X)$ be a scheme. I'm using the following definition from Görtz-Wedhorn. A closed subscheme of $(X,\mathcal O_X)$ is a scheme $(Z,\mathcal O_Z)$, where $i\colon Z\hookrightarrow X......Mon, 10 Aug 2026 13:52:22 -0400