https://liberty.ioCurated star highlights with fresh energy.en-usThu, 03 Sep 2026 21:29:56 -0400https://liberty.io/compute-the-double-sum-sum-n-m-0-n-neq-m-frac-1-n-left-m-2-n-2-right-f.htmlhttps://liberty.io/compute-the-double-sum-sum-n-m-0-n-neq-m-frac-1-n-left-m-2-n-2-right-f.htmlI am trying to compute the following double sum $$\boxed{\sum_{n, m>0, n \neq m} \frac{1}{n\left(m^{2}-n^{2}\right)}=\frac{3}{4} \zeta(3)}$$ I proceeded as following $$\sum_{n=1}^{\infty}\sum_{m......Fri, 04 Sep 2026 01:25:06 -0400https://liberty.io/definition-of-good-reduction.htmlhttps://liberty.io/definition-of-good-reduction.htmlLet $C/\mathbb{Q}$ be a curve. I am trying to understand the difference between (1) There is a model $\mathcal{C}/\mathbb{Z}$ of $C$ such that the fiber $\mathcal{C}_p$ is regular and (2) ......Fri, 04 Sep 2026 01:24:26 -0400https://liberty.io/symbol-for-the-set-of-integers-from-1-to-n-duplicate.htmlhttps://liberty.io/symbol-for-the-set-of-integers-from-1-to-n-duplicate.htmlIs there a special symbol for the set: $$ \{1, 2, 3, \dots, n\}$$, or $$\{x| 1\leq x\leq n, n \in \mathbb{Z} \} $$?...Fri, 04 Sep 2026 01:14:06 -0400https://liberty.io/an-open-ball-is-an-open-set.htmlhttps://liberty.io/an-open-ball-is-an-open-set.htmlProve that for any $x_0 \in X$ and any $r>0$, the open ball $B_r(x_o)$ is open. My attempt: Let $y\in B_r(x_0)$. By definition, $d(y,x_0)<r$. I want to show there exists an $r_1\in\mathbb{R^......Fri, 04 Sep 2026 01:12:47 -0400https://liberty.io/finding-plane-equation-given-two-lines.htmlhttps://liberty.io/finding-plane-equation-given-two-lines.htmlDetermine an equation of the plane containing the lines $\frac{x-1}{2}=\frac{y+1}{-1}=\frac{z-5}{6}$; $r=\lt1,-1,5\gt+t\lt1,1,-3\gt$. I calculated the cross product between the directional vector......Fri, 04 Sep 2026 01:09:26 -0400https://liberty.io/could-someone-explain-me-what-is-a-clopen-set.htmlhttps://liberty.io/could-someone-explain-me-what-is-a-clopen-set.htmlI came across this term while studying for a test and got curious as to what this actually is. could someone please the concept in a relatively easy language?...Fri, 04 Sep 2026 01:00:20 -0400https://liberty.io/what-is-the-point-of-non-commutative-probability.htmlhttps://liberty.io/what-is-the-point-of-non-commutative-probability.htmlI read this article by Terry Tao about non-commutative probability. I honestly don't really get it. Apparently, we can generalize Kolmogorov's system of probability theory, in such a way that it i......Fri, 04 Sep 2026 01:00:13 -0400https://liberty.io/solving-cos4-theta-cos3-theta-to-find-the-roots-of-8x-3-4x-2-4x-1-0.htmlhttps://liberty.io/solving-cos4-theta-cos3-theta-to-find-the-roots-of-8x-3-4x-2-4x-1-0.htmlFind all the solution to the equation $\cos 4\theta=\cos 3\theta$. Hence or otherwise, show that the roots of the equation $$8x^3+4x^2-4x-1=0$$ are $$\cos \frac{2\pi}{7},\cos \frac{4\pi}{7},\cos \f......Fri, 04 Sep 2026 00:59:49 -0400https://liberty.io/what-are-the-common-abbreviation-for-minimum-in-equations.htmlhttps://liberty.io/what-are-the-common-abbreviation-for-minimum-in-equations.htmlI'm searching for some symbol representing minimum that is commonly used in math equations....Fri, 04 Sep 2026 00:59:48 -0400https://liberty.io/is-the-laplacian-a-vector-or-a-scalar.htmlhttps://liberty.io/is-the-laplacian-a-vector-or-a-scalar.htmlNeed to prove $\operatorname{div}(\nabla u)=\nabla ^2 u$ where $u=g(x,y,z)$ The RHS is the Lapacian which we were told is a vector. But $\nabla u=(g_x,g_y,g_z)$ and the divergence of that is $g_{x......Fri, 04 Sep 2026 00:56:13 -0400https://liberty.io/how-to-check-if-5-is-a-square-mod-701-without-a-calculator.htmlhttps://liberty.io/how-to-check-if-5-is-a-square-mod-701-without-a-calculator.htmlThe Legendre symbol tells us to calculate $5^{350} \mod 701$, but this question was on an exam where no calculators are allowed, so I wasn't able to do this question. How can you find if $5$ is a s......Fri, 04 Sep 2026 00:55:34 -0400https://liberty.io/difference-between-polynomial-functions-and-polynomials-and-why-these-.htmlhttps://liberty.io/difference-between-polynomial-functions-and-polynomials-and-why-these-.htmlHere's an excerpt from abstract algebra book that I'm reading and my question is given later: The difference between a polynomial and a polynomial function is mainly a difference of viewpoint. Giv......Fri, 04 Sep 2026 00:50:39 -0400https://liberty.io/what-is-the-function-fsolve-in-python-doing-mathematically.htmlhttps://liberty.io/what-is-the-function-fsolve-in-python-doing-mathematically.htmlIn the Python documentation for fsolve it says "Return the roots of the (non-linear) equations defined by func(x) = 0 given a starting estimate" f(x, *args). I wondered if anyone knew the mathemati......Fri, 04 Sep 2026 00:45:29 -0400https://liberty.io/constructibility-of-the-17-gon.htmlhttps://liberty.io/constructibility-of-the-17-gon.htmlComment: I greatly shortened and simplified the question. As a drawback, some comments/answers might not make any sense anymore. Assume we are using this set of axioms $A$ for plane euclidean geo......Fri, 04 Sep 2026 00:43:17 -0400https://liberty.io/cosine-rule-to-find-a-side-of-a-triangle.htmlhttps://liberty.io/cosine-rule-to-find-a-side-of-a-triangle.htmlI would like to ask you for help for the following problem: The sides of a $\triangle ABC$ are $BC=2,AB=5,AC=\sqrt{22}$ and point $D$ lies on $AC$, such that $BD=3$. Find $CD$. I have just studied......Fri, 04 Sep 2026 00:29:28 -0400https://liberty.io/drawing-a-phase-portrait-given-eigenvectors.htmlhttps://liberty.io/drawing-a-phase-portrait-given-eigenvectors.htmlI am a bit confused on how the author here drew the phase portraits in the following picture. The second eigenvalue is larger than the first. For large and positive t’s this means that the solu......Fri, 04 Sep 2026 00:25:50 -0400https://liberty.io/what-is-the-definition-of-2-5-2-5-factorial.htmlhttps://liberty.io/what-is-the-definition-of-2-5-2-5-factorial.htmlI was messing around with my TI-84 Plus Silver Edition calculator and discovered that it will actually give me values when taking the factorial of any number $n/2$ where $n$ is any integer greater ......Fri, 04 Sep 2026 00:22:06 -0400https://liberty.io/grandi-039-s-series-tends-to-1-2-but-why-is-this-considered-a-valid-su.htmlhttps://liberty.io/grandi-039-s-series-tends-to-1-2-but-why-is-this-considered-a-valid-su.htmlGrandi's series, $$1+1-1+1-1+1-1+...$$ can be expressed as the below: $$\sum_{n=0}^\infty(-1)^n$$ Two valid sums that make sense to me are $1$, and $0$, depending on how you approach the series......Fri, 04 Sep 2026 00:14:39 -0400https://liberty.io/how-can-i-show-that-a-function-is-non-decreasing.htmlhttps://liberty.io/how-can-i-show-that-a-function-is-non-decreasing.htmlI have the following function: $$F(x)=\begin{cases} \dfrac{x}{x+1},& \text{if }x\ge 0,\\ 0,& \text{otherwise}\end{cases}$$ I want to prove that it is right-continuous and non-decreasing. ......Fri, 04 Sep 2026 00:14:12 -0400https://liberty.io/questions-tagged-factoring.htmlhttps://liberty.io/questions-tagged-factoring.htmlQ&A for people studying math at any level and professionals in related fields...Fri, 04 Sep 2026 00:12:16 -0400https://liberty.io/how-can-i-simply-prove-that-the-pearson-correlation-coefficient-is-bet.htmlhttps://liberty.io/how-can-i-simply-prove-that-the-pearson-correlation-coefficient-is-bet.htmlFor building a recommendation system, I also use the Pearson correlation coefficient. This is the definition: $r(x, y)=\frac{\sum_{i=1}^n (x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\sum_{i=1}^n (x_i-\bar{x......Fri, 04 Sep 2026 00:07:51 -0400https://liberty.io/optimisation-in-two-variables-second-order-conditions.htmlhttps://liberty.io/optimisation-in-two-variables-second-order-conditions.htmlI have a problem understanding second-order conditions at a critical point in finding critical points of two-variable functions. Let's consider $f(x,y)$. The first-order conditions are $\frac{\pa......Fri, 04 Sep 2026 00:01:53 -0400https://liberty.io/why-is-it-that-0-5-0-5-equals-sin-pi-4.htmlhttps://liberty.io/why-is-it-that-0-5-0-5-equals-sin-pi-4.htmlIf you put $0.5^{0.5}$ into a calculator you will see that: $$0.5^{0.5}\approx0.707106781187$$ And, if you also put $\sin(\frac{\pi}{4})$ into that same calculator you will get: $$\sin(\frac{\pi}......Thu, 03 Sep 2026 23:56:40 -0400https://liberty.io/counting-the-number-of-real-roots-of-a-polynomial.htmlhttps://liberty.io/counting-the-number-of-real-roots-of-a-polynomial.htmlI am interested in solving problems which involve finding the number of real roots of any polynomial. Suppose I take a function $$f(x)=x^6+x^5+x^4+x^3+x^2+x+1$$ This does not have any real roots b......Thu, 03 Sep 2026 23:49:41 -0400https://liberty.io/what-is-the-modulus-of-a-matrix.htmlhttps://liberty.io/what-is-the-modulus-of-a-matrix.htmlI came across the following: "Let f be continuously differentiable with Lipschitz gradient, i.e., $ ||\nabla f(\mathbf{x}) - \nabla f(\mathbf{y})|| \leq L||(\mathbf{x} - \mathbf{y})|| $ where L ......Thu, 03 Sep 2026 23:48:58 -0400https://liberty.io/is-upper-hemi-continuity-and-upper-semi-continuity-the-same-thing-for-.htmlhttps://liberty.io/is-upper-hemi-continuity-and-upper-semi-continuity-the-same-thing-for-.htmlDespite of difference in their name, they seem to be the same definition. Some texts use the name "hemi-continuity", others use the other name; none of those notes clarify the difference between hemi-...Thu, 03 Sep 2026 23:47:46 -0400https://liberty.io/confused-with-natural-logarithms.htmlhttps://liberty.io/confused-with-natural-logarithms.htmlHow can we solve the following natural logarithms? I'm confused with this stuff: $\ln(x+1) - \ln x = \ln 3$ $\ln(x+1) + \ln x = \ln 2$...Thu, 03 Sep 2026 23:38:17 -0400https://liberty.io/spectrum-of-adjacency-matrix-of-complete-graph.htmlhttps://liberty.io/spectrum-of-adjacency-matrix-of-complete-graph.htmlFooling around in matlab, I did an eigenvalue decomposition of the adjacency matrix of $K_5$. A = [0 1 1 1 1; 1 0 1 1 1; 1 1 0 1 1; ......Thu, 03 Sep 2026 23:28:20 -0400https://liberty.io/what-is-the-taylor-expansion-of-x-2-closed.htmlhttps://liberty.io/what-is-the-taylor-expansion-of-x-2-closed.htmlWhat is the Taylor Expansion of the expression $x^2$ ? Is it possible to express this as a general expression evaluated at any point? Edit I have been asked to explain why this question is differ......Thu, 03 Sep 2026 23:22:07 -0400https://liberty.io/weird-integral-symbol-mathrel-int.htmlhttps://liberty.io/weird-integral-symbol-mathrel-int.htmlWhat does this integral sign mean ($\int$ with line going through the middle)? $$ \mathrel{\int\!\!\!\!\!\!-} $$ (It had something to do with the Beckenbach-Radó Theorem)...Thu, 03 Sep 2026 23:18:49 -0400https://liberty.io/formal-symbol-for-the-integer-division-operation.htmlhttps://liberty.io/formal-symbol-for-the-integer-division-operation.htmlThe integer division is a common and useful operation in Computer Science. It comes up in many domains, as in the manipulation of matrices and grids. Is there any formal symbol for this operation?......Thu, 03 Sep 2026 23:13:11 -0400https://liberty.io/if-an-ideal-contains-the-multiplicative-identity-then-it-is-the-whole-.htmlhttps://liberty.io/if-an-ideal-contains-the-multiplicative-identity-then-it-is-the-whole-.htmlI have to prove that if $I\subseteq R$ is an ideal, and $1\in I$, then $I=R\,$. So I know $I\subseteq R$ is an ideal if $a,b\in I$ implies $a+b\in I$, and if $a\in I$, $r\in R$, then $ra\in I$. I'm ...Thu, 03 Sep 2026 23:12:13 -0400https://liberty.io/taylor-series-of-frac-1-1-x-2.htmlhttps://liberty.io/taylor-series-of-frac-1-1-x-2.htmlI have to construct the Taylor series of $$\frac 1 {1+x^2}$$ around $0$ and $1$ and analyze the convergence in both cases. Also (but this is a consequence of the previous series) I have to constru......Thu, 03 Sep 2026 23:07:18 -0400https://liberty.io/does-the-order-of-operations-matter-with-just-addition-and-subtraction.htmlhttps://liberty.io/does-the-order-of-operations-matter-with-just-addition-and-subtraction.htmlHad a debate on whether you could do addition/subtraction in any order you want. Specifically, for the following: $9 - 4 + 3$ We both agree that the answer is 8. I argue that, by giving addition a ...Thu, 03 Sep 2026 22:59:55 -0400https://liberty.io/help-with-finding-the-directional-derivative-of-a-f-x-y-z-xy-xz-yz-at-.htmlhttps://liberty.io/help-with-finding-the-directional-derivative-of-a-f-x-y-z-xy-xz-yz-at-.htmlI need to find the directional derivative of $f(x,y,z)=xy+xz+yz$ at $P(1,2,3)$ in the direction of $\overrightarrow{v}=\langle 2,1,-1 \rangle$ I think I started this incorrectly and would greatly ...Thu, 03 Sep 2026 22:57:32 -0400https://liberty.io/team-balancing-score-normalisation-for-gamification.htmlhttps://liberty.io/team-balancing-score-normalisation-for-gamification.htmlThis is the first time I post in Mathematics as I'm normally in programing. But for this question, I think I can get better help here. I'm trying to create a gamification system where different tea......Thu, 03 Sep 2026 22:48:18 -0400https://liberty.io/convective-derivative-vs-divergence-of-velocity.htmlhttps://liberty.io/convective-derivative-vs-divergence-of-velocity.htmlWhat is the physical significance or difference between Convective derivative : $\vec{v} \cdot \nabla $ and the Divergence of velocity $\nabla \cdot \vec{v}$? I have understood the convective deriv......Thu, 03 Sep 2026 22:47:48 -0400https://liberty.io/a-ladder-leans-against-a-wall-but-the-problem-asks-how-long-is-the-lad.htmlhttps://liberty.io/a-ladder-leans-against-a-wall-but-the-problem-asks-how-long-is-the-lad.htmlhere's the problem: A ladder leans against a wall with its base 10 ft from it. When the ladder is pulled 3 ft further from the wall, the upper end slides down 7 ft. How long is the ladder? Here's w......Thu, 03 Sep 2026 22:31:13 -0400https://liberty.io/does-the-series-sum-frac-1-ln-3-k-converge-or-diverge.htmlhttps://liberty.io/does-the-series-sum-frac-1-ln-3-k-converge-or-diverge.htmlMy question is if $\displaystyle\sum\limits_{k=2}^\infty \frac{1}{\ln^3(k)}$ converges or diverges. I could so far use the Integral Test, and solved $\displaystyle\int_2^\infty\dfrac{1}{\ln^3(x)}dx$, ...Thu, 03 Sep 2026 22:29:36 -0400https://liberty.io/how-to-calculate-the-coordinates-of-a-triangle-039-s-orthocentre.htmlhttps://liberty.io/how-to-calculate-the-coordinates-of-a-triangle-039-s-orthocentre.htmlHow does one calculate the coordinates of a triangle's orthocentre? I was surfing through the net and got this formula: $$x-\rm{coordinate}= \frac{x_1\tan A+x_2\tan B+x_3\tan C}{\tan A+\tan B+\tan......Thu, 03 Sep 2026 22:22:41 -0400https://liberty.io/find-intersection-of-two-3d-lines.htmlhttps://liberty.io/find-intersection-of-two-3d-lines.htmlI have two lines $(5,5,4) (10,10,6)$ and $(5,5,5) (10,10,3)$ with same $x$, $y$ and difference in $z$ values. Please some body tell me how can I find the intersection of these lines. EDIT: By usi......Thu, 03 Sep 2026 22:21:30 -0400https://liberty.io/determine-if-sum-n-1-infty-frac-1-nn-2-n-n-3-1-converges-or-diverges.htmlhttps://liberty.io/determine-if-sum-n-1-infty-frac-1-nn-2-n-n-3-1-converges-or-diverges.htmlAnother series I found I'm struggling with. Determine if the following series converges or diverges.$$\sum_{n=1}^{\infty}\frac{(-1)^nn^2+n}{n^3+1}$$ Ratio test and n-th root test are both ...Thu, 03 Sep 2026 22:15:55 -0400https://liberty.io/what-does-039-card-039-stands-for.htmlhttps://liberty.io/what-does-039-card-039-stands-for.htmlI read some book where 'card' symbol is used to signify number of elements in the set. But where is it from? What the full word could be for 'card'?...Thu, 03 Sep 2026 22:07:40 -0400https://liberty.io/geometry-how-to-determine-if-two-lines-are-parallel-in-3d-based-on-coo.htmlhttps://liberty.io/geometry-how-to-determine-if-two-lines-are-parallel-in-3d-based-on-coo.htmlI am a Belgian engineer working on software in C# to provide smart bending solutions to a manufacturer of press brakes. In this context I am searching for the best way to determine if two lines are ...Thu, 03 Sep 2026 22:02:23 -0400https://liberty.io/show-that-measure-is-trivial.htmlhttps://liberty.io/show-that-measure-is-trivial.htmlLet $\mu(dx,dy)$ be a Borel measure with compact support on $\left\{(x,y) \mid x>0, y>0 \right\}$ and let $\mu(dx,dy)$ satisfy $$ \mu(dx,dy) = \lambda^{\alpha} \mu \left( d \frac{x}{\lambd......Thu, 03 Sep 2026 21:44:49 -0400https://liberty.io/how-do-you-distinguish-the-difference-between-a-negative-sign-and-a-mi.htmlhttps://liberty.io/how-do-you-distinguish-the-difference-between-a-negative-sign-and-a-mi.htmlBoth the negative sign symbol and subtraction's minus symbol look the same, so how does anyone tell them apart?...Thu, 03 Sep 2026 21:44:02 -0400https://liberty.io/proving-a-tautology-by-applying-a-chain-of-logical-identities.htmlhttps://liberty.io/proving-a-tautology-by-applying-a-chain-of-logical-identities.htmlI need help showing that $[ (p \land q) \Rightarrow (p \Rightarrow q) ]$ is a tautology by applying a chain of logical identities. The question also asks to identify each identity I use. I have no ......Thu, 03 Sep 2026 21:42:39 -0400https://liberty.io/can-someone-explain-geometric-multiplicity.htmlhttps://liberty.io/can-someone-explain-geometric-multiplicity.htmlI'm reading my textbook and I'm really confused about geometric multiplicity. I've read the definition and they have given an example but I'm still lost. I've tried looking it up on other websites.......Thu, 03 Sep 2026 21:41:37 -0400https://liberty.io/why-is-sqrt-2-sqrt-3-neq-sqrt-6.htmlhttps://liberty.io/why-is-sqrt-2-sqrt-3-neq-sqrt-6.htmlWhy is $\sqrt{-2} \cdot \sqrt{-3} \neq \sqrt{6}$? Are there other examples where regular arithmetic goes wrong for complex numbers?...Thu, 03 Sep 2026 21:40:50 -0400https://liberty.io/show-that-tan-pi-over-8-sqrt-2-1.htmlhttps://liberty.io/show-that-tan-pi-over-8-sqrt-2-1.htmlShow that $\tan {\pi \over 8} = \sqrt 2 - 1$, using the identity $\tan 2\theta = {{2\tan \theta } \over {1 - {{\tan }^2}\theta }}$ Using $\tan 2\theta = {{2\tan \theta } \over {1 - {{\tan }^2}\...Thu, 03 Sep 2026 21:29:32 -0400https://liberty.io/in-general-what-does-the-location-of-the-rightmost-1-in-a-binary-numbe.htmlhttps://liberty.io/in-general-what-does-the-location-of-the-rightmost-1-in-a-binary-numbe.htmlIn general, what does the location of the rightmost “1” in a binary number tell you? Is it different for positive and for negative numbers?...Thu, 03 Sep 2026 21:27:43 -0400https://liberty.io/linear-bound-on-angles-in-an-euclidean-triangle.htmlhttps://liberty.io/linear-bound-on-angles-in-an-euclidean-triangle.htmlI am trying to understand a proof in the book of Burago "A Course in metric geometry" (Lemma 10.8.13 page 383). I have difficulties with a certain inequality for the angle of euclidean triangles: ...Thu, 03 Sep 2026 21:27:23 -0400https://liberty.io/about-algebraic-integer-and-algebraic-number.htmlhttps://liberty.io/about-algebraic-integer-and-algebraic-number.htmlLet $u$ a algebraic number. Prove that exists a natural number $n\in \mathbb{Z}$ such that $nu$ is a algebraic integer If $u$ is algebraic integer and $n\in \mathbb{Z}$ then $u+n$ and $nu$ are ...Thu, 03 Sep 2026 21:25:46 -0400https://liberty.io/pre-multiplying-and-post-multiplying-matrices-give-the-same-diagonal-e.htmlhttps://liberty.io/pre-multiplying-and-post-multiplying-matrices-give-the-same-diagonal-e.htmlIf $$X = \left[ \begin{array}{ccc} 3 & 4 & 1\\ 4 & 1 & 3\\ 1 & 3 & 4\end{array} \right]$$ find the possible matrix $Y$ such that: $$XY - YX = I$$ The method my ...Thu, 03 Sep 2026 21:13:44 -0400https://liberty.io/question-about-heart-shaped-graph.htmlhttps://liberty.io/question-about-heart-shaped-graph.htmlFirstly, I am sorry for my poor english since it is only my second language. During my spare time, I tried to form a hearshaping function. Here's what I did. If I drew a ellipse that looks like......Thu, 03 Sep 2026 21:09:48 -0400https://liberty.io/lagrangian-mechanics-derivatives.htmlhttps://liberty.io/lagrangian-mechanics-derivatives.htmlI don't really know whether to put this in Physics forums since it is relating to Mechanics, or Math since the question is actually about the math being done. Don't criticize me over it. So for the ...Thu, 03 Sep 2026 21:09:44 -0400https://liberty.io/how-to-write-reciprocal-squared-function-shifted-right-by-3-and-down-b.htmlhttps://liberty.io/how-to-write-reciprocal-squared-function-shifted-right-by-3-and-down-b.htmlGiven a reciprocal squared function that is shifted right by $3$ and down by $4$, write this as a rational function. I am uncertain how to denote this. My attempt: $$\frac{1}{x^2-3}-4$$ But I need......Thu, 03 Sep 2026 21:05:57 -0400https://liberty.io/tail-sum-for-expectation.htmlhttps://liberty.io/tail-sum-for-expectation.htmlIn Pitman's Probability, the tail sum formula for expectation is introduced for a nonnegative (0,1,...) discrete random variable $X$: $$E(X) = \sum_{i=0}^\infty P(X > i).$$ I wonder if there i......Thu, 03 Sep 2026 20:57:59 -0400https://liberty.io/particular-solution-of-second-order-differential-equation.htmlhttps://liberty.io/particular-solution-of-second-order-differential-equation.htmlGiven the ode: $$ y''-2y'+y=e^t, $$ how can I find the form of the particular solution? At first, I tried the form $y=Ae^t$ but $$ \begin{split} &\frac{d^2y}{dt^2}Ae^t=\frac{dy}{dt}Ae^t=......Thu, 03 Sep 2026 20:52:33 -0400https://liberty.io/quadratic-equations-and-factorized-form.htmlhttps://liberty.io/quadratic-equations-and-factorized-form.htmlI have trouble understanding why: Given the equation $x^2+4x-21=0$, the solution given is $(x+7)(x-3)=0$ in its factorization form. Using quadratic equations, $$ x = \frac{-b\pm \sqrt{b^2-4ac}}{......Thu, 03 Sep 2026 20:46:57 -0400https://liberty.io/an-infinite-dictionary-countably-infinite-or-uncountably-infinite.htmlhttps://liberty.io/an-infinite-dictionary-countably-infinite-or-uncountably-infinite.htmlThis question concerns Ian Stewart's "Hyperwebster", an uncountable dictionary. Say a publishing company wants to publish every possible permutation (of any length) of the characters A-Z. The ...Thu, 03 Sep 2026 20:43:46 -0400https://liberty.io/linear-combinations-of-sine-and-cosine.htmlhttps://liberty.io/linear-combinations-of-sine-and-cosine.htmlIf you take a linear combination of the cosine and sine function, then the result is again a sinusoid, but with a new amplitude and phase shift. $$a \cos(\theta) + b \sin(\theta) = A \cos(\theta + \...Thu, 03 Sep 2026 20:42:55 -0400https://liberty.io/why-does-squaring-of-the-radius-of-a-circle-times-pi-give-us-the-area.htmlhttps://liberty.io/why-does-squaring-of-the-radius-of-a-circle-times-pi-give-us-the-area.htmlWhy does squaring the radius of a circle times pi equal the area? (I.e., why does the area of circle representable as $\pi r^2$ ?)What is the relationship between the radius of a circle and the area?...Thu, 03 Sep 2026 20:38:31 -0400https://liberty.io/how-does-one-find-the-common-area-shared-between-two-polar-curves.htmlhttps://liberty.io/how-does-one-find-the-common-area-shared-between-two-polar-curves.htmlThe problem: Consider the circle $r = (\sqrt{2} - 1)\cos(\theta)$ and the cardioid $r = 1 - \cos(\theta)$. Find the area $A$ of the region $R$ common to the inside of the circle and the inside of the ...Thu, 03 Sep 2026 20:24:22 -0400https://liberty.io/is-there-a-math-symbol-meaning-if-or-when.htmlhttps://liberty.io/is-there-a-math-symbol-meaning-if-or-when.htmlTo introduce a logic premise normally people use the word If, or Iff (if and only if). Is there a math symbol with the same meaning?...Thu, 03 Sep 2026 20:19:35 -0400https://liberty.io/what-does-the-dot-over-x-or-y-mean.htmlhttps://liberty.io/what-does-the-dot-over-x-or-y-mean.htmlI am just starting to read up on differential equations. The problem is (in my materials) nowhere is explained what do these dots mean. Can anyone shed some light? $$\begin{align}&\begin{cases......Thu, 03 Sep 2026 20:04:04 -0400https://liberty.io/prove-3-n-ge-n-3-by-induction.htmlhttps://liberty.io/prove-3-n-ge-n-3-by-induction.htmlYep, prove $3^n \ge n^3$, $n \in \mathbb{N}$. I can do this myself, but can't figure out any kind of "beautiful" way to do it. The way I do it is: Assume $3^n \ge n^3$ Now, $(n+1)^3 = n^3 + ......Thu, 03 Sep 2026 19:57:56 -0400https://liberty.io/given-the-equation-x-sqrt-y-does-the-equation-represent-y-as-a-functio.htmlhttps://liberty.io/given-the-equation-x-sqrt-y-does-the-equation-represent-y-as-a-functio.htmlI can justify why the answer might be no, but also why it might be yes, so which line of reasoning is correct? I. $x=\sqrt{y}$ does not represent y as a function of x because if $x$ is a negative ......Thu, 03 Sep 2026 19:46:15 -0400https://liberty.io/how-to-understand-force-of-interest-in-simple-interest.htmlhttps://liberty.io/how-to-understand-force-of-interest-in-simple-interest.htmlI am struggling to understand "force of interest" in simple interest. I understand that "force of interest" can be interpreted as compound interest with infinitely small time interval. That is, the......Thu, 03 Sep 2026 19:45:26 -0400https://liberty.io/help-solving-a-differential-equation.htmlhttps://liberty.io/help-solving-a-differential-equation.htmlmy Calculus II class is nearing the end of the quarter and we've just started differential equations to get ready for Calculus III. In my homework, I came upon these problems. One of the problems ......Thu, 03 Sep 2026 19:37:00 -0400https://liberty.io/equation-of-a-spherocylinder-capsule.htmlhttps://liberty.io/equation-of-a-spherocylinder-capsule.htmlI want to know if a 3D spherocylinder/capsule, that looks like this: Can be approximated by a known equation. I want a shape with volume. Edit: In my program I already have a cylinder with two ...Thu, 03 Sep 2026 19:36:44 -0400https://liberty.io/not-getting-what-it-means-v-ai-bj-ck-for-some-vector-v.htmlhttps://liberty.io/not-getting-what-it-means-v-ai-bj-ck-for-some-vector-v.htmlI am watching Khan Academy Linear Algebra series and I am really getting confused when Sal writes some sort of equation for the vector, such as $v = ai + bj + ck$. I understand that vector $v$ shou......Thu, 03 Sep 2026 19:34:29 -0400https://liberty.io/how-may-one-show-that-sin-x-sin-y-le-x-y-duplicate.htmlhttps://liberty.io/how-may-one-show-that-sin-x-sin-y-le-x-y-duplicate.htmlI think I saw the inequality $$|\sin x-\sin y|\le |x-y|$$ somewhere recently. I've been trying to see why it could be true but so far have been unable to come up with any success. Could someone s......Thu, 03 Sep 2026 19:33:36 -0400https://liberty.io/fractional-linear-transformations-and-their-matrix-form.htmlhttps://liberty.io/fractional-linear-transformations-and-their-matrix-form.htmlFor $z \in \mathbb{C}$, a fractional linear transformation of $z$, with an associated matrix $M$ is: $$M = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \\ T_{M}(z) = \frac{az + b}{cz+d}.$$ ...Thu, 03 Sep 2026 19:29:03 -0400https://liberty.io/proving-the-cauchy-schwarz-inequality-by-induction.htmlhttps://liberty.io/proving-the-cauchy-schwarz-inequality-by-induction.htmlI ran across this problem in some old notes, and I frustratingly can't figure out how to do it Let $a_i$ and $b_i$ be sequences of natural numbers, use induction to show $\sum_{i=1}^n (a_ib_i)^{......Thu, 03 Sep 2026 19:25:40 -0400https://liberty.io/how-can-i-find-the-equation-of-a-parabola-only-given-it-039-s-x-interc.htmlhttps://liberty.io/how-can-i-find-the-equation-of-a-parabola-only-given-it-039-s-x-interc.htmlI received a problem in my math class the other day that left me stumped. The problem went something like this. Mr. Lots-O-Cash would like to order a parabola that passes through the points $(-4......Thu, 03 Sep 2026 19:17:47 -0400https://liberty.io/a-jacobi-elliptic-integral-with-cosine.htmlhttps://liberty.io/a-jacobi-elliptic-integral-with-cosine.htmlAfter reduction of a problem, I find myself in front of these integrals, $$ \int_0^{2\pi} cn \left( \frac{2K(1/2)}{\pi} \theta, \frac{1}{2} \right) \cos(m \theta) \mathrm{d}\theta, $$ with $m$ posi......Thu, 03 Sep 2026 19:11:00 -0400https://liberty.io/using-shell-method-find-the-volume-of-the-solid-generated-when-y-3-x-2.htmlhttps://liberty.io/using-shell-method-find-the-volume-of-the-solid-generated-when-y-3-x-2.htmlQuestion Via the Shell Method, what is the volume of the solid of revolution formed by revolving a portion of $y=3-x^2$ about the x-axis bounded by $y=0$ and $y=3$. Context The answer from Wolfram......Thu, 03 Sep 2026 19:10:37 -0400https://liberty.io/intuitive-explanation-of-duhamel-039-s-principle.htmlhttps://liberty.io/intuitive-explanation-of-duhamel-039-s-principle.htmlThis is with regards to the first section of Wikipedia article Duhamel's principle (revision from July 2012). I want to see if I am understanding this. Basically the inhomogeneous equation says tha......Thu, 03 Sep 2026 18:58:39 -0400https://liberty.io/translate-to-first-order-logic.htmlhttps://liberty.io/translate-to-first-order-logic.htmlI have this two sentences: All men love the woman to whom they are married. No woman loves a man who does not respect all women. and in the exercise they are translated to First Order Logic as fo......Thu, 03 Sep 2026 18:53:30 -0400https://liberty.io/variance-of-x-y.htmlhttps://liberty.io/variance-of-x-y.htmlIf X and Y are random variables with correlation coefficient 0.7, each of which has variance 6, what is the variance of X−Y? Enter your answer as a decimal. Using the information given, I was abl......Thu, 03 Sep 2026 18:50:06 -0400https://liberty.io/ordinary-differential-equations-v-dx-dt.htmlhttps://liberty.io/ordinary-differential-equations-v-dx-dt.htmlThe question is: What ODE does v=dx/dt solve? And what is the limit as t approaches infinity of v(t). It sounds almost too easy, but I am confused about what to do exactly. Intuitively, as the ques......Thu, 03 Sep 2026 18:49:05 -0400https://liberty.io/how-the-product-of-two-integrals-is-iterated-integral-int-cdot-int-iin.htmlhttps://liberty.io/how-the-product-of-two-integrals-is-iterated-integral-int-cdot-int-iin.htmlLet $\large{I} = \Large \int\limits_{-\infty}^{\infty} \normalsize e^{\small -\frac{y^2}{2}}\ dy$. Then, my textbook says, $$\tag{1} I^2 = \left( \int\limits_{-\infty}^{\infty} \normalsize e^{\......Thu, 03 Sep 2026 18:47:24 -0400https://liberty.io/simplify-sqrt-sqrt-3-5-sqrt-3-4.htmlhttps://liberty.io/simplify-sqrt-sqrt-3-5-sqrt-3-4.htmlDenest $\sqrt {\sqrt[3]{5}-\sqrt[3]{4}}$. I have tried completing square by several method but all failed. Can anyone help me please? Thank you. p.s. I'm a poor question-tagger....Thu, 03 Sep 2026 18:46:11 -0400https://liberty.io/cauchy-schwarz-inequality-for-nonsymmetric-matrices.htmlhttps://liberty.io/cauchy-schwarz-inequality-for-nonsymmetric-matrices.htmlI am interested in Cauchy-Schwarz inequalities for inner products of the form $\langle Ax,y\rangle$ where $A$ is some matrix. It is rather easy to see that the Cauchy-Schwarz inequality continues t......Thu, 03 Sep 2026 18:45:59 -0400https://liberty.io/consistency-of-a-system-of-linear-equations-using-unknowns-in-the-matr.htmlhttps://liberty.io/consistency-of-a-system-of-linear-equations-using-unknowns-in-the-matr.htmlCan anyone tell me how would I answer these type of questions? I already know how to answer the normal way which doesn't include any variables such as Alpha in the question....Thu, 03 Sep 2026 18:40:20 -0400https://liberty.io/how-find-this-sum-sin-2-x-sin-2-2x-sin-2-3x-cdots-sin-2-nx.htmlhttps://liberty.io/how-find-this-sum-sin-2-x-sin-2-2x-sin-2-3x-cdots-sin-2-nx.htmlQuestion: Find the value $$f^{(2)}_{n}(x)=\sin^2{x}+\sin^2{(2x)}+\sin^2{(3x)}+\cdots+\sin^2{(nx)}$$ My solution: since $$\sin^2{x}=\dfrac{1}{2}(1-\cos{(2x)})$$ so $$f_{n}(x)=\dfrac{n}{2}-\dfra......Thu, 03 Sep 2026 18:37:29 -0400https://liberty.io/change-of-polynomial-basis-exercise.htmlhttps://liberty.io/change-of-polynomial-basis-exercise.htmlCan someone please check if my solution is correct? Thanks. Considering the following basis: $$ B= \left\{ x^2,1+x^2,x-1 \right\}\\ C= \left\{ x,1-x,x^2 \right\} $$ a) Find the change of basis mat......Thu, 03 Sep 2026 18:26:53 -0400https://liberty.io/derivative-of-matrix-exponential-w-r-t-to-each-element-of-the-matrix.htmlhttps://liberty.io/derivative-of-matrix-exponential-w-r-t-to-each-element-of-the-matrix.htmlI have $x= \exp(At)$ where $A$ is a matrix. I would like to find derivative of $x$ with respect to each element of $A$. Could anyone help with this problem?...Thu, 03 Sep 2026 18:20:43 -0400https://liberty.io/i-wonder-if-the-set-of-all-characteristic-function-on-a-set-a-is-simil.htmlhttps://liberty.io/i-wonder-if-the-set-of-all-characteristic-function-on-a-set-a-is-simil.htmlRecently I studied something known as characteristic function (Indicator function), after going through it deeply I want know if the set of all characteristic functions on A will be similar to its ......Thu, 03 Sep 2026 18:14:27 -0400https://liberty.io/riemann-sheets-for-combined-roots.htmlhttps://liberty.io/riemann-sheets-for-combined-roots.htmlSo, I am aware of the fact that if I do have a function like $$f(z) = z^{1/2} + z^{1/3}$$ being $2, 3$ relatively primes, I have $$\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z}$$ hence ......Thu, 03 Sep 2026 18:14:11 -0400https://liberty.io/proof-for-rational-roots.htmlhttps://liberty.io/proof-for-rational-roots.htmlLet $f(x) = a_0+a_1x+\cdots+a_nx^n$ be a polynomial of degree $n$ over $\Bbb Z$. A: If a rational number $\frac{p}{q}$ is a root of $f(X)$, show that $p\mid a_0$ and $q\mid a_n$. Assume $\gcd(p, q......Thu, 03 Sep 2026 18:03:50 -0400https://liberty.io/derivative-of-operatorname-arcsec-3x.htmlhttps://liberty.io/derivative-of-operatorname-arcsec-3x.htmlThis seems straightforward to me, but it was marked wrong on my test with no clarification and no ability to ask questions. $y=\operatorname{arcsec}(3x)$. Calculate the derivative. I take the se......Thu, 03 Sep 2026 17:57:58 -0400https://liberty.io/prove-that-in-any-triangle-abc-cos-2a-cos-2b-cos-2c-geq-frac-3-4.htmlhttps://liberty.io/prove-that-in-any-triangle-abc-cos-2a-cos-2b-cos-2c-geq-frac-3-4.htmlI have two similar looking questions. (1) Prove that in triangle $ABC$, $$\cos^2A+\cos^2B+\cos^2C\geq\frac{3}{4}.$$ (2) If $\Delta ABC$ is an acute angled, then prove that $$\cos^2A+\cos^2B+\cos^2......Thu, 03 Sep 2026 17:56:16 -0400https://liberty.io/can-i-subtract-infinity-from-infinity.htmlhttps://liberty.io/can-i-subtract-infinity-from-infinity.htmlI was stuck when solving a problem on limits. It was like----> $\lim_{x\to\infty} (x-x)$. What should I do now?...Thu, 03 Sep 2026 17:55:32 -0400https://liberty.io/aaa-similarity-theorem.htmlhttps://liberty.io/aaa-similarity-theorem.htmlDef: Two triangles are called similar if and only if one can be scaled into the other. AAA Theorem: If two triangles have the same angles if and only if they are similar. How does one prove the T......Thu, 03 Sep 2026 17:36:53 -0400https://liberty.io/how-to-find-the-nth-term-in-the-following-sequence-1-1-2-2-4-4-8-8-16-.htmlhttps://liberty.io/how-to-find-the-nth-term-in-the-following-sequence-1-1-2-2-4-4-8-8-16-.htmlI'm having difficulty in finding the formula for the sequence above, when I put this in WolframAlpha it gave me a rather complex formula which I'm not convinced even works properly but I'm sure the......Thu, 03 Sep 2026 17:34:17 -0400https://liberty.io/is-x-1-a-linear-function.htmlhttps://liberty.io/is-x-1-a-linear-function.htmlI want to know if $\frac1x$ is a linear function or not. I read that a linear function can also be defined as a function of the first degree. Since $\frac1x=x^{-1}$, the function is of the first d......Thu, 03 Sep 2026 17:28:59 -0400https://liberty.io/orthogonal-set-vs-orthogonal-basis.htmlhttps://liberty.io/orthogonal-set-vs-orthogonal-basis.htmlFor some reason my book distinguishes the two names. If a set is an orthogonal set, doesn't that make it immediately a basis for some subspace $W$ since all the vectors in the orthogonal set are ...Thu, 03 Sep 2026 17:26:11 -0400https://liberty.io/why-are-all-non-prime-numbers-divisible-by-a-prime-number.htmlhttps://liberty.io/why-are-all-non-prime-numbers-divisible-by-a-prime-number.htmlIn Euclid's infinite prime numbers proof, the logic is as follows: Assume a set $S$ of all prime numbers in existence is finite (there are a finite amount of primes) Then there must be a greatest......Thu, 03 Sep 2026 17:14:21 -0400