https://liberty.ioCurated star highlights with fresh energy.en-usSun, 09 Aug 2026 03:21:57 -0400https://liberty.io/if-i-roll-two-fair-dice-the-probability-that-i-would-get-at-least-one-.htmlhttps://liberty.io/if-i-roll-two-fair-dice-the-probability-that-i-would-get-at-least-one-.html$11$ out of $36$? I got this by writing down the number of possible outcomes ($36$) and then counting how many of the pairs had a $6$ in them: $(1,6)$, $(2,6)$, $(3,6)$, $(4,6)$, $(5,6)$, $(6,6)$, ......Sun, 09 Aug 2026 07:21:17 -0400https://liberty.io/matrix-multiplication-notation.htmlhttps://liberty.io/matrix-multiplication-notation.htmlThis is from my textbook: If $A=(a_{ij})\in M_{mn}(\Bbb F), B=(b_{ij})\in M_{np}(\Bbb F)$ then $C=A\times B=(c_{ij})\in M_{mp}(\Bbb F)$. $c_{ij}=\sum_{k=1}^{n} a_{ik}b_{kj}$ where $i=1,...m,......Sun, 09 Aug 2026 07:20:37 -0400https://liberty.io/meaning-of-closure-of-a-set.htmlhttps://liberty.io/meaning-of-closure-of-a-set.htmlDoes closure of a set mean, only adding boundary values if the set is open and leave it as it is if the set is closed?...Sun, 09 Aug 2026 07:19:14 -0400https://liberty.io/qn-is-bipartite.htmlhttps://liberty.io/qn-is-bipartite.htmlHi i'd like to know if this proves that $Q_n$ is bipartite or I should write it different, THANKS!! Let $G$ be a $Q_n$ graph. In $Q_n$ vertices are represented by vectors ($v_1$,$v_2$,....$v_n$) ......Sun, 09 Aug 2026 07:18:35 -0400https://liberty.io/factoring-polynomials-with-four-terms-and-two-variables.htmlhttps://liberty.io/factoring-polynomials-with-four-terms-and-two-variables.htmlI've been working on this for hours and cannot figure it out. When I search, I find factorization techniques that I already know but don't seem to be able to apply here, or that are for polynomial......Sun, 09 Aug 2026 07:06:38 -0400https://liberty.io/when-and-how-many-times-a-day-do-a-clock-s-hands-overlap.htmlhttps://liberty.io/when-and-how-many-times-a-day-do-a-clock-s-hands-overlap.htmlThe time is 0:00.00. Hands are moving continuously. What is the next angle when they are meet again? Answer for the 2nd question is 22 I think. And for the 1st question I think like this, but is cant ...Sun, 09 Aug 2026 07:00:35 -0400https://liberty.io/how-do-even-and-odd-functions-relate-to-even-and-odd-numbers.htmlhttps://liberty.io/how-do-even-and-odd-functions-relate-to-even-and-odd-numbers.htmlHow do the notions of oddness and evenness apply to both functions and numbers? If Even and Odd functions share nothing in common with Even and Odd numbers, then why were Even and Odd adopted for ...Sun, 09 Aug 2026 06:58:43 -0400https://liberty.io/matlab-coding-finding-zeros-without-using-fzero-or-roots-function.htmlhttps://liberty.io/matlab-coding-finding-zeros-without-using-fzero-or-roots-function.htmlSo i am a completely new at Matlab. I'm basically suppose to develop a function in Matlab that finds the zeros of a cubic polynomial. real and complex. I'm pasting below what I have so far. I star......Sun, 09 Aug 2026 06:52:25 -0400https://liberty.io/finding-the-equation-of-the-tangent-line-at-a-point.htmlhttps://liberty.io/finding-the-equation-of-the-tangent-line-at-a-point.htmlWith the equation $$f(x)=\sin^2(x)-\cos^2(x),$$ I need to find the equation of the tangent line at $(\frac{\pi}{4},0).$ I am stuck. I took the derivative of the function, but I don't know where ......Sun, 09 Aug 2026 06:51:48 -0400https://liberty.io/using-cardano-039-s-formula-to-find-roots.htmlhttps://liberty.io/using-cardano-039-s-formula-to-find-roots.htmlThe equation I am considering is $$x^{3}-63x=162$$ Here is how far I got: $$3st=-63$$ $$s=\frac{-63}{3t}=-21t$$ $$s^{3}t^{3}=-162$$ $$(-21t)^{3}-t^{3}=-162$$ $$-9261t^{3}-t^{3}=-162$$ Where ...Sun, 09 Aug 2026 06:51:16 -0400https://liberty.io/variation-on-the-imo-2011-windmill-problem.htmlhttps://liberty.io/variation-on-the-imo-2011-windmill-problem.htmlThis question references the "windmill" question C3 of the 2011 Int'l Math Olympiad, documented here, p. 32: https://www.imo-official.org/problems/IMO2011SL.pdf In the first solution case......Sun, 09 Aug 2026 06:45:09 -0400https://liberty.io/can-someone-identify-this-math-font.htmlhttps://liberty.io/can-someone-identify-this-math-font.htmlI'm doing my master's thesis and trying to find a good font for math symbols and formulas. I would like to use the following font style: Can anyone recognize this? How to get X the way it is in t......Sun, 09 Aug 2026 06:44:15 -0400https://liberty.io/eigenvectors-of-a-circulant-matrix.htmlhttps://liberty.io/eigenvectors-of-a-circulant-matrix.htmlShow that for the circulant matrix $$C = \begin{bmatrix} c_0 & c_{n-1} & \dots & c_{2} & c_{1} \\ c_{1} & c_0 & c_{n-1} & & c_{2} \\ \vdots ......Sun, 09 Aug 2026 06:43:33 -0400https://liberty.io/questions-about-the-addtion-of-injective-and-surjective-functions.htmlhttps://liberty.io/questions-about-the-addtion-of-injective-and-surjective-functions.htmlI have a question which is as follows, Consider $f:\mathbb{R} \to \mathbb{R}$ and $g:\mathbb{R} \to \mathbb{R}$. Define the sum of $f$ and $g$ as the funtion $h:\mathbb{R} \to \mathbb{R}$ such t......Sun, 09 Aug 2026 06:42:57 -0400https://liberty.io/sum-of-two-invertible-matrices-duplicate.htmlhttps://liberty.io/sum-of-two-invertible-matrices-duplicate.htmlIf A and B are two n x n invertible matrices, would the matrix result from A+B be invertible? I think it would because for a matrix to be invertible its determinant would have to be greater than 0......Sun, 09 Aug 2026 06:42:57 -0400https://liberty.io/implicit-equation-for-double-torus-genus-2-orientable-surface.htmlhttps://liberty.io/implicit-equation-for-double-torus-genus-2-orientable-surface.htmlThe embedded torus in $\mathbb R^3$ can be described by the set of points in $(x,y,z)\in \mathbb R^3$ satisfying $T(x,y,z)=0$, where $T$ is the polynomial $T(x,y,z)=(x^2+y^2+z^2+R^2-r^2)^2-4R^2(x^2......Sun, 09 Aug 2026 06:15:11 -0400https://liberty.io/proof-that-1729-is-the-smallest-taxicab-number.htmlhttps://liberty.io/proof-that-1729-is-the-smallest-taxicab-number.htmlFor homework I have to produce the proof (algebraic or otherwise) to show that $1729$ HAS to be the smallest taxi cab number. A taxicab number means that it is the sum of two different cubes and ca......Sun, 09 Aug 2026 06:12:24 -0400https://liberty.io/full-subcategory-of-abelian-category-is-abelian.htmlhttps://liberty.io/full-subcategory-of-abelian-category-is-abelian.htmlI'm trying to understand a proof in Rotman's 'Introduction to Homological Algebra', Proposition 5.92, p.310. Proposition: Let $\mathcal S$ be a full subcategory of an abelian category $\mathcal A$......Sun, 09 Aug 2026 05:59:08 -0400https://liberty.io/how-to-evaluate-log-1-x-in-terms-of-log-x.htmlhttps://liberty.io/how-to-evaluate-log-1-x-in-terms-of-log-x.htmlI can do this using the following relation: $$\log(1 - x) = \log(1 - \exp(y))$$ Here $y = \log(x)$ is always a negative number. However, I was wondering whether it's possible to compute $\log(1 -......Sun, 09 Aug 2026 05:39:53 -0400https://liberty.io/differentiate-sin-x-x-using-definition-of-a-derivative-first-principle.htmlhttps://liberty.io/differentiate-sin-x-x-using-definition-of-a-derivative-first-principle.htmlHello I have a question I have had soem difficulty with for a long time with no answer I could find. How do you differentiate the sine of x divided by x using first principles? I know to use sin(x+......Sun, 09 Aug 2026 05:39:24 -0400https://liberty.io/how-to-calculate-10-0-4-without-using-calculator.htmlhttps://liberty.io/how-to-calculate-10-0-4-without-using-calculator.htmlHow to calculate $10^{0.4}$ without using calculator or if not what is the closest answer you can get just using pen and paper within say $2$ min?...Sun, 09 Aug 2026 05:38:37 -0400https://liberty.io/why-graphs-are-so-important.htmlhttps://liberty.io/why-graphs-are-so-important.htmlThe notion of graph seems to be of a huge importance in Computer Science. A very rich Graph Theory was developed, and theoretical problems are being solved all the time. What I don't understand, i......Sun, 09 Aug 2026 05:33:21 -0400https://liberty.io/extensions-of-the-turing-jump.htmlhttps://liberty.io/extensions-of-the-turing-jump.htmlThe Turing jump $0^{(\alpha)}$ is defined for ordinals $\alpha<\omega_1^{\mathit{CK}}$ with $0^{(0)} = \varnothing$, $0^{(\alpha+1)}$ is the diagonal halting problem using $0^{(\alpha)}$ as an ......Sun, 09 Aug 2026 05:32:49 -0400https://liberty.io/can-i-get-some-example-coordinates-of-an-equilateral-triangle-but-only.htmlhttps://liberty.io/can-i-get-some-example-coordinates-of-an-equilateral-triangle-but-only.htmlI'm trying to implement Bresenham's Circle drawing algorithm and I intend to draw a circumscribed and inscribed circle (circumcircle and incircle) in a triangle. I need some simple ways to find ...Sun, 09 Aug 2026 05:07:40 -0400https://liberty.io/xor-gate-with-4-inputs-truth-table-mdnf.htmlhttps://liberty.io/xor-gate-with-4-inputs-truth-table-mdnf.htmlI have difficulties with one equation. It's an XOR gate with 4 inputs. F(A,B,C,D)=(A⊕B)C+(B⊕C)(A⊕C) 'B' from the second bracket has a line above it , so does the 'A' from the third. I did the ......Sun, 09 Aug 2026 05:03:56 -0400https://liberty.io/show-me-some-pigeonhole-problems-closed.htmlhttps://liberty.io/show-me-some-pigeonhole-problems-closed.htmlI'm preparing myself to a combinatorics test. A part of it will concentrate on the pigeonhole principle. Thus, I need some hard to very hard problems in the subject to solve. I would be thankful if......Sun, 09 Aug 2026 05:00:05 -0400https://liberty.io/how-to-find-a-linearly-independent-vector.htmlhttps://liberty.io/how-to-find-a-linearly-independent-vector.htmlGiven two vectors $(1,2,8),(0,1,9)$ find a 3rd vector that is linearly independent from these two vectors. I sort of have an idea how to go about solving the problem but I'm not 100% sure. I'm kno......Sun, 09 Aug 2026 04:54:14 -0400https://liberty.io/how-to-change-the-trigonometric-identity-to-sec-2-x.htmlhttps://liberty.io/how-to-change-the-trigonometric-identity-to-sec-2-x.htmlEvaluate the definite integral: $$\int_{\frac{\pi}{8}}^\frac{\pi}{4}(\csc(2\theta)-\cot(2\theta)\ d\theta$$ Finding the derivative gives me this, which is confirmed by the steps in Wolfram Alpha.......Sun, 09 Aug 2026 04:54:00 -0400https://liberty.io/what-039-s-the-specific-definition-of-a-direct-sum-of-vector-spaces.htmlhttps://liberty.io/what-039-s-the-specific-definition-of-a-direct-sum-of-vector-spaces.htmlI understand a couple people have posted about this before, but different sources are giving me different answers on what exactly a direct sum is with regards to vector spaces. I've seen both of th......Sun, 09 Aug 2026 04:52:08 -0400https://liberty.io/visual-proof-of-isosceles-base-angle-congruency.htmlhttps://liberty.io/visual-proof-of-isosceles-base-angle-congruency.htmlA geometric proof (without algebra or trigonometry), and ideally presented visually (a proof without words). EDITED (I'm especially curious if it's possible without without using triangle congruen......Sun, 09 Aug 2026 04:51:57 -0400https://liberty.io/please-explain-definition-of-determinant-using-permutations.htmlhttps://liberty.io/please-explain-definition-of-determinant-using-permutations.htmlMany people (in different texts) use the following famous definition of the determinant of a matrix $A$: \begin{align*} \det(A) = \sum_{\tau \in S_n}\operatorname{sgn}(\tau)\,a_{1,\tau(1)}a_{2,\tau......Sun, 09 Aug 2026 04:51:36 -0400https://liberty.io/multiplication-of-adjacency-matrix-of-two-graphs-duplicate.htmlhttps://liberty.io/multiplication-of-adjacency-matrix-of-two-graphs-duplicate.htmlConsider two $N \times N$ adjacency matrices $A$ and $B$ belonging to different graphs(unweighted and undirected). What does their product $AB$ represent? I know that the powers of the adjacency m......Sun, 09 Aug 2026 04:50:58 -0400https://liberty.io/can-a-sphere-be-considered-the-limit-of-a-regular-polyhedron-with-infi.htmlhttps://liberty.io/can-a-sphere-be-considered-the-limit-of-a-regular-polyhedron-with-infi.htmlI was trying to approximate the surface area of the sphere. I remember that Archimede approximated the circumference of a circle by calcualting the perimeter of its inscribed and circumscribed regu......Sun, 09 Aug 2026 04:50:33 -0400https://liberty.io/how-to-calculate-the-gradient-of-matrix-equation.htmlhttps://liberty.io/how-to-calculate-the-gradient-of-matrix-equation.htmlShort question: How do I calculate the gradient of the $MSE(a, b)$ equation below? Longer explanation: This problem arises, while I'm following a derivation of a term for an optimal beamvector $a$......Sun, 09 Aug 2026 04:46:42 -0400https://liberty.io/calculate-the-inverse-laplace-transform-by-convolution-5-6-26.htmlhttps://liberty.io/calculate-the-inverse-laplace-transform-by-convolution-5-6-26.htmlSynopsis: I cannot duplicate the answer in my text although I do get somewhat close. This tells me that my method is correct but I am making another kind of error -- perhaps in my integration? The ...Sun, 09 Aug 2026 04:44:49 -0400https://liberty.io/find-the-exact-value-of-cos-11-pi-12.htmlhttps://liberty.io/find-the-exact-value-of-cos-11-pi-12.htmlThis question I look at as being similar to $\sin(7\pi/12)$. You can break it up using the special triangles into $3\pi/12 + 4\pi/12$. However with this one, I can't find one of the angles in whi......Sun, 09 Aug 2026 04:44:18 -0400https://liberty.io/how-to-find-a-polynomial-from-a-given-root.htmlhttps://liberty.io/how-to-find-a-polynomial-from-a-given-root.htmlI was asked to find a polynomial with integer coefficients from a given root/solution. Lets say for example that the root is: $\sqrt{5} + \sqrt{7}$. How do I go about finding a polynomial that has ...Sun, 09 Aug 2026 04:30:52 -0400https://liberty.io/proving-additivity-of-bernoulli-binomial-distribution.htmlhttps://liberty.io/proving-additivity-of-bernoulli-binomial-distribution.htmlThis is about proving the additivity of Bernoulli's binomial distribution. On another words, $$X\sim\mathrm{Bin}(n,p),\quad Y\sim\mathrm{Bin}(m,p)$$ implies that $$ X+Y\sim\mathrm{Bin}(n+m,p).$$ ...Sun, 09 Aug 2026 04:23:38 -0400https://liberty.io/find-all-points-for-intersection-between-2-polar-equations.htmlhttps://liberty.io/find-all-points-for-intersection-between-2-polar-equations.htmlI stumped at one of the exercise in my multivariable calculus textbook. I try to search online but I can't seem to search on how answer no 3 and 4 below is derived. I also plot both of polar coord......Sun, 09 Aug 2026 04:21:55 -0400https://liberty.io/integrate-sqrt-x-2-2x.htmlhttps://liberty.io/integrate-sqrt-x-2-2x.html$$\int{\sqrt{x^2 - 2x}}$$ I think I should be doing trig substitution, but which? I completed the square giving $$\int{\sqrt{(x-1)^2 -1}}$$ But the closest I found is for $$\frac{1}{\sqrt{a^2......Sun, 09 Aug 2026 04:19:19 -0400https://liberty.io/if-f-is-continous-on-j-1-times-j-2-and-g-is-riemann-integrable-on-j-1-.htmlhttps://liberty.io/if-f-is-continous-on-j-1-times-j-2-and-g-is-riemann-integrable-on-j-1-.htmlIf $J_1=[a,b], J_2=[c,d]$, and if $f$ is continous on $J_1\times J_2$ to $\mathbb{R}$ and $g$ is Riemann integrable on $J_1$, then the function $F$, defined on $J_2$ by $$F(t)=\int_a^bf(x,t)g(x)dx$......Sun, 09 Aug 2026 04:18:31 -0400https://liberty.io/calculate-sides-of-the-right-triangle-if-i-know-one-side-and-all-the-a.htmlhttps://liberty.io/calculate-sides-of-the-right-triangle-if-i-know-one-side-and-all-the-a.htmlHow to calculate the sides and hypotenuse length of the right triangle if I know $ \text{bigger side} = 60$, $\text{one angle} = 60^o$ & $\text{second angle} = 30^o$ ($\text{third angle} = 90^o$)...Sun, 09 Aug 2026 04:18:24 -0400https://liberty.io/notation-for-sequences.htmlhttps://liberty.io/notation-for-sequences.htmlI am trying to write a small article, and I just want to know how would be a good way to present the maths I have written so that it looks professional. I am trying to define a sequence $x_n$ of r......Sun, 09 Aug 2026 04:14:50 -0400https://liberty.io/power-series-expansion-e-2x.htmlhttps://liberty.io/power-series-expansion-e-2x.htmlI'm currently finding the power series expansion centered at 0 for a bunch of functions. My answers are starting not to add up with the 'correct' ones and I'm looking for some assistance as to what......Sun, 09 Aug 2026 04:13:52 -0400https://liberty.io/scale-rotate-skew-a-2d-shape-to-look-like-3d.htmlhttps://liberty.io/scale-rotate-skew-a-2d-shape-to-look-like-3d.htmlWhat would be the math to scale/rotate/skew a simple 2D shape to give the appearance of being a 2D shape in a 3D world? For example, say I have a circle, I can scale (+/-100%) the X-axis to look l......Sun, 09 Aug 2026 03:49:43 -0400https://liberty.io/if-we-know-the-congruence-of-the-apex-and-the-base-of-two-isosceles-ar.htmlhttps://liberty.io/if-we-know-the-congruence-of-the-apex-and-the-base-of-two-isosceles-ar.htmlWell, there are general criteria for congruence of triangles: SAS (Side-Angle-Side) ASA (Angle-Side-Angle) SSS (Side-Side-Side) SAA (Side-Angle-Angle) And there are some criteria for specific tri......Sun, 09 Aug 2026 03:47:13 -0400https://liberty.io/predicates-quantifiers.htmlhttps://liberty.io/predicates-quantifiers.htmlHow to express these in terms of predicates & quantifiers : Some properties are tautologies The negation of a contradiction is a tautology The dis junction of two contingencies can be a tautol......Sun, 09 Aug 2026 03:41:33 -0400https://liberty.io/which-sets-are-present-in-every-model-of-zf.htmlhttps://liberty.io/which-sets-are-present-in-every-model-of-zf.htmlAs in the title: the existence of which sets is implied by the axioms of $\mathsf{ZF}$? For example one such set would be the empty set whose existence is demanded by the Axiom of the Empty Set. ......Sun, 09 Aug 2026 03:35:13 -0400https://liberty.io/what-is-the-fourier-transform-of-a-functional.htmlhttps://liberty.io/what-is-the-fourier-transform-of-a-functional.htmlGenerally Fourier transforms are defined as follows, $$ F(s)=\int dxf(x)e^{-isx}.$$ Similarly, is it possible to define Fourier Space for a functional $$\psi[\phi(x)]~?$$...Sun, 09 Aug 2026 03:27:25 -0400https://liberty.io/finding-the-derivative-of-x-using-the-limit-definition.htmlhttps://liberty.io/finding-the-derivative-of-x-using-the-limit-definition.htmlPlease Help me derive the derivative of the absolute value of x using the following limit definition. $$\lim_{\Delta x\rightarrow 0}\frac{f(x+\Delta x)-f(x)}{\Delta x} $$ I have no idea as to how ......Sun, 09 Aug 2026 03:26:57 -0400https://liberty.io/propositional-logic-entailment.htmlhttps://liberty.io/propositional-logic-entailment.htmlI'm trying to understand propositional logics and the concepts of entailment, but I'm struggling. The concepts don't seem to be difficult in theory, but are very strange-looking when examined. For ...Sun, 09 Aug 2026 03:23:17 -0400https://liberty.io/simplifying-using-boolean-algebra.htmlhttps://liberty.io/simplifying-using-boolean-algebra.htmlI was doing the following question. Using the following rules of boolean algebra: _ law 1: X+X=1 law 2: X.1=X law 3:X.Y+X.Z = X.(Y+Z) simplify: __ _ __ ABC+ABC+ABC+ABC I have ...Sun, 09 Aug 2026 03:21:24 -0400https://liberty.io/user-kemish-jimenez-mathematics-stack-exchange.htmlhttps://liberty.io/user-kemish-jimenez-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Sun, 09 Aug 2026 03:17:53 -0400https://liberty.io/are-the-actuarial-exams-hard-closed.htmlhttps://liberty.io/are-the-actuarial-exams-hard-closed.htmlI heard that they are difficult. Is this true? Are they like the qualifying exams in grad school? For example, is the probability exam and the financial math exam comparable to qualifying exams (e.g. ...Sun, 09 Aug 2026 03:16:50 -0400https://liberty.io/integration-by-parts-of-x-5-ln-x.htmlhttps://liberty.io/integration-by-parts-of-x-5-ln-x.htmlI have to integrate the function: $x^5\ln(x)$ My Attempt $$\int(x^5\ln(x))dx$$ $u=\ln(x)$ and $du=\frac{1}{x}dx$ $dv=x^5dx$ and $v=\frac{x^6}{6}$ Using this I can then integrate the function using ......Sun, 09 Aug 2026 02:47:38 -0400https://liberty.io/mutually-orthogonal-set-of-vectors.htmlhttps://liberty.io/mutually-orthogonal-set-of-vectors.htmlShow that the standard basis: $$..$$ $\mathscr{B} = \left\{ \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 0\\ 1 \\ 0 \end{bmat......Sun, 09 Aug 2026 02:43:34 -0400https://liberty.io/non-integral-powers-of-a-matrix.htmlhttps://liberty.io/non-integral-powers-of-a-matrix.htmlQuestion Given a square complex matrix $A$, what ways are there to define and compute $A^p$ for non-integral scalar exponents $p\in\mathbb R$, and for what matrices do they work? My thoughts Int......Sun, 09 Aug 2026 02:30:07 -0400https://liberty.io/geometric-interpretation-of-linear-transformation.htmlhttps://liberty.io/geometric-interpretation-of-linear-transformation.htmlI have a linear transformation, given by the following matrix $$ \begin{pmatrix} x_1\\ x_2 \end{pmatrix} \mapsto \begin{pmatrix} 2 & 2\\ -1 & -1\\ \end{pmatrix} \begin{pmatrix} x_1\\ x_2 \end{...Sun, 09 Aug 2026 02:24:54 -0400https://liberty.io/finding-the-radius-of-a-third-tangent-circle.htmlhttps://liberty.io/finding-the-radius-of-a-third-tangent-circle.htmlSorry if this is a foolish question, but I'm having difficulty understanding how to solve for $r_3$ in the following diagram... According to WolframAlpha's page on tangent circles, the radius of $......Sun, 09 Aug 2026 02:23:43 -0400https://liberty.io/proving-the-double-differential-of-z-z-implies-z-sinx.htmlhttps://liberty.io/proving-the-double-differential-of-z-z-implies-z-sinx.htmlIn common usage, we know that $$\frac{d^2z}{dx^2}=-z\,$$ implies $z$ is of the form $a\sin x + b\cos x$. Is there a proof for the same. I was trying to arrive at the desired function but couldn't ...Sun, 09 Aug 2026 02:17:17 -0400https://liberty.io/find-a-formula-for-sin-5x-in-terms-of-sin-x-and-cos-x.htmlhttps://liberty.io/find-a-formula-for-sin-5x-in-terms-of-sin-x-and-cos-x.htmlI was asked to find a formula for $\sin(5x)$ in terms of $\sin(x)$ and $\cos(x)$. I thought about using euler formula which gives: $$\sin(5x) = e^{i\sin(5x)} = \cos(\sin(5x))+i\cdot sin(\sin(5x))$$......Sun, 09 Aug 2026 02:12:55 -0400https://liberty.io/what-does-a-polynomial-function-039-s-increasing-and-decreasing-interv.htmlhttps://liberty.io/what-does-a-polynomial-function-039-s-increasing-and-decreasing-interv.htmlNow I know that smarter and more interesting question were asked over here, but bear with me. When someone says,"the increasing interval of this polynomial function is (-infinity, 4)," when does it ...Sun, 09 Aug 2026 02:10:43 -0400https://liberty.io/integration-by-parts-done-fast.htmlhttps://liberty.io/integration-by-parts-done-fast.htmlEven though I've been through countless instances where I needed to use integration by parts, to this day, I still derive it from the chain rule, identify my 'parts' and apply the formula. At some ...Sun, 09 Aug 2026 02:09:21 -0400https://liberty.io/what-does-cap-product-actually-mean.htmlhttps://liberty.io/what-does-cap-product-actually-mean.htmlI have read the Wikipedia definition of cap product. Here is the Wikipedia definition that I know. https://en.wikipedia.org/wiki/Cap_product What I understand: For a good space, for example, an ...Sun, 09 Aug 2026 02:07:41 -0400https://liberty.io/proposition-2-9-of-hatcher-039-s-algebraic-k-theory-and-vector-bundles.htmlhttps://liberty.io/proposition-2-9-of-hatcher-039-s-algebraic-k-theory-and-vector-bundles.htmlhttps://www.math.cornell.edu/~hatcher/VBKT/VB.pdf Reading through Hatcher's proof of the the induced exact sequence of $\widetilde{K}$ groups, I've run into a few issues. I'm unsure of how there......Sun, 09 Aug 2026 01:59:47 -0400https://liberty.io/origin-of-vectors.htmlhttps://liberty.io/origin-of-vectors.htmlBackground I was reviewing notes of physics, and i realized that something about the mathematics of vectors was wrong in my head. Example-problem Suppose a vector is $A=5\textbf{i} + 3\textbf{......Sun, 09 Aug 2026 01:49:50 -0400https://liberty.io/how-to-find-the-derivative-of-a-fraction.htmlhttps://liberty.io/how-to-find-the-derivative-of-a-fraction.htmlSo I have this fancy problem I've been working on for two days: I need to find two things: 1) $f'(t)$ 2) $f'(2)$ I have tried plugging it into the definition of a derivative, but do not know how......Sun, 09 Aug 2026 01:48:43 -0400https://liberty.io/understanding-variable-of-integration.htmlhttps://liberty.io/understanding-variable-of-integration.htmlI am currently trying to get a deeper understanding of multivariable calculus and I am finding my one stumbling block to be the variable of integration. In single variable calculus, I have always ...Sun, 09 Aug 2026 01:44:56 -0400https://liberty.io/is-it-possible-to-apply-cofactor-expansion-to-a-2-x-2-matrix-why-is-th.htmlhttps://liberty.io/is-it-possible-to-apply-cofactor-expansion-to-a-2-x-2-matrix-why-is-th.htmlSolved: the determinant of a constant has the same value as that constant. It is possible to apply cofactor expansion to a two-by-two matrix. For example, this matrix: $$ \left[ \begin{array}{cc}......Sun, 09 Aug 2026 01:37:01 -0400https://liberty.io/method-of-characteristics-lagrange-charpit-equations.htmlhttps://liberty.io/method-of-characteristics-lagrange-charpit-equations.htmlI need to solve the following PDE with initial condition $U(x,0)=U_0(x)$. Once this is one of my first times, I'd like to get second opinions. Many Thanks. \begin{equation} \partial_t U + (\te......Sun, 09 Aug 2026 01:35:39 -0400https://liberty.io/number-of-elementary-multiplications-for-multiplying-4x4-matrices.htmlhttps://liberty.io/number-of-elementary-multiplications-for-multiplying-4x4-matrices.htmlApplying recursively Strassen's algorithm on 4x4 matrices results in 49 elementary multiplications. Are there methods tailored for 4x4 matrices which can do better? Links to articles are highly ...Sun, 09 Aug 2026 01:34:36 -0400https://liberty.io/understanding-the-order-of-approximation-big-o-notation.htmlhttps://liberty.io/understanding-the-order-of-approximation-big-o-notation.htmlAssume that $n+1$ different grid points $x_0,...,x_n$ are given. Exercise: determine the coefficients $a_i$ such that de order $(k)$ of the error: $$\left|\,f'(x) - Q(h)\right| = O(h^k)$$ is maxi......Sun, 09 Aug 2026 01:31:10 -0400https://liberty.io/why-is-there-a-constant-of-integration.htmlhttps://liberty.io/why-is-there-a-constant-of-integration.html$\int (2x^2 +1)dx = \dfrac{2}{3}x^3 + x +C$. This is a simple integral, and as you all know, the answer to an integral will always have a '$+C$' at the end, the constant of integration. Why is it......Sun, 09 Aug 2026 01:31:02 -0400https://liberty.io/rate-of-change-area-and-perimeter.htmlhttps://liberty.io/rate-of-change-area-and-perimeter.htmlThe side of rectangle $x = 20m$ increases at the rate of $5m/s$, the other side $y=30m$ decreases at $4m/sec$. What is the rate of change o the perimeter and area of the retangle? If we put this ...Sun, 09 Aug 2026 00:59:38 -0400https://liberty.io/minimize-total-cost-of-box.htmlhttps://liberty.io/minimize-total-cost-of-box.htmlSo there is a rectangular box that has a volume of $8 m^3$. The top and bottom of the box is made with some material that has a cost of $8$ dollars per square meter. The sides are made with another ...Sun, 09 Aug 2026 00:56:43 -0400https://liberty.io/what-will-be-installed-windows-or-ubuntu-on-factory-data-reset-on-pre-.htmlhttps://liberty.io/what-will-be-installed-windows-or-ubuntu-on-factory-data-reset-on-pre-.htmlI have Dell Inspiron 15 5000 Series which had Windows 10 Preinstalled but I love Ubuntu so I have installed Ubuntu 20 LTS. My question is what if I do factory data rest to this machine, what will be ...Sun, 09 Aug 2026 00:56:27 -0400https://liberty.io/simplify-square-root-expression-sqrt-125-sqrt-5.htmlhttps://liberty.io/simplify-square-root-expression-sqrt-125-sqrt-5.html$\sqrt{125}-\sqrt5$ simplify it. I thought it would be $\sqrt {5\cdot5\cdot5}-\sqrt 5$ which would be the square root of 25 which is 5 but it is not. Can you show how to simplify this?...Sun, 09 Aug 2026 00:56:23 -0400https://liberty.io/the-complement-of-a-graph-g.htmlhttps://liberty.io/the-complement-of-a-graph-g.htmlI have a homework problem where I have a graph $G$ and I am tasked with proving that at least one of $G$ and $G$ complement is connected. However, I am unclear on the exact meaning of $G$ complemen......Sun, 09 Aug 2026 00:29:52 -0400https://liberty.io/alternative-proof-that-base-angles-of-an-isosceles-triangle-are-equal.htmlhttps://liberty.io/alternative-proof-that-base-angles-of-an-isosceles-triangle-are-equal.htmlThe "classic textbook proof" of equality of base angles of an isosceles triangles which I studied in my school days is as follows: Let $\Delta ABC$ be a triangle with $AB = AC$ and let $D$ be the ......Sun, 09 Aug 2026 00:23:25 -0400https://liberty.io/find-the-equation-of-the-sine-function-graphed-below.htmlhttps://liberty.io/find-the-equation-of-the-sine-function-graphed-below.htmlFind the equation of the sine function graphed below. Write a cosine function for the graph below. Assume the least possible phase shift....Sun, 09 Aug 2026 00:22:49 -0400https://liberty.io/solving-a-matrix-with-one-variable.htmlhttps://liberty.io/solving-a-matrix-with-one-variable.htmlI have a system of equations as follows: $$1x + hy = -5$$ $$2x - 8y = 6$$ My question is how do I work with such a variable in a matrix? Thank you....Sun, 09 Aug 2026 00:12:48 -0400https://liberty.io/can-i-use-only-the-super-key-as-a-shortcut.htmlhttps://liberty.io/can-i-use-only-the-super-key-as-a-shortcut.htmlUsually in Gnome and KDE it is only possible to define shortcuts with the Super key in combination with another key. But is it possible to use only the Super key as a shortcut, and how could I achi......Sat, 08 Aug 2026 23:51:23 -0400https://liberty.io/fan-noise-reduction-asrock-motherboard-amd-ryzen.htmlhttps://liberty.io/fan-noise-reduction-asrock-motherboard-amd-ryzen.htmlI am asking this question because I noticed there wasn't an obvious answer for beginners. The problem is very simple: I have a pretty noisy Wraith Spire CPU fan. Normally, it could run at a silent ......Sat, 08 Aug 2026 23:44:13 -0400https://liberty.io/monoprice-drawing-tablet-mp1060-ha60.htmlhttps://liberty.io/monoprice-drawing-tablet-mp1060-ha60.htmlI've purchased this model 6814 tablet from Monoprice and am trying to use it in Gimp and Inkscape in 14.04. It seems that previously, there was a driver "wizardpen" for non-Wacom tablets which is no ...Sat, 08 Aug 2026 23:38:24 -0400https://liberty.io/osu-installer-restarts-all-the-time-wine.htmlhttps://liberty.io/osu-installer-restarts-all-the-time-wine.htmlWhen I try to install osu! with wine the installer goes too 100% and then restarts in an endless loop. What can I do to fix this?...Sat, 08 Aug 2026 23:34:42 -0400https://liberty.io/unable-to-print-with-brother-mfc-290c.htmlhttps://liberty.io/unable-to-print-with-brother-mfc-290c.htmlI have an Ubuntu 12.04 laptop from which I want to print in a Brother MFC-290C printer wired to a Win7 PC. Printer is already being shared by PC. I've tried to install the printer via Alt+F2 system-...Sat, 08 Aug 2026 23:31:26 -0400https://liberty.io/gnu-radio-radar-toolbox.htmlhttps://liberty.io/gnu-radio-radar-toolbox.htmlI am trying to download the GNU Radio Radar Toolbox from this link(https://github.com/kit-cel/gr-radar). I followed the steps listed in that link, but when I reach the {cmake ../} it gives that: -- ...Sat, 08 Aug 2026 23:30:48 -0400https://liberty.io/how-to-disable-screen-saver-in-totem-when-playing-movie.htmlhttps://liberty.io/how-to-disable-screen-saver-in-totem-when-playing-movie.htmlI'm using Ubuntu 11.10. There's a option in Totem that says to disable screen saver while playing movie, but it doesn't work. Is there anything else I can do to make it work ?...Sat, 08 Aug 2026 23:26:49 -0400https://liberty.io/syntax-error-039-unexpected-end-of-file.htmlhttps://liberty.io/syntax-error-039-unexpected-end-of-file.htmlWhen I open a terminal, I get: bash: /root/.bashrc: line 115: syntax error: unexpected end of file bashrc has # ~/.bashrc: executed by bash(1) for non-login shells. # see /usr/share/doc/bash/exa......Sat, 08 Aug 2026 23:13:22 -0400https://liberty.io/how-to-merge-multiple-pdf-files-into-a-single-file-duplicate.htmlhttps://liberty.io/how-to-merge-multiple-pdf-files-into-a-single-file-duplicate.htmlIs there any software or methods available to merge multiple PDF files into a single file?...Sat, 08 Aug 2026 23:11:33 -0400https://liberty.io/questions-tagged-xrdp.htmlhttps://liberty.io/questions-tagged-xrdp.htmlQ&A for Ubuntu users and developers...Sat, 08 Aug 2026 23:09:40 -0400https://liberty.io/how-do-i-disable-headphone-controls.htmlhttps://liberty.io/how-do-i-disable-headphone-controls.htmlI'm running Ubuntu 18.04 on a Lenovo Thinkpad X1 Carbon 7th Generation. I have Bose QC20 headphones which have volume and start/stop controls on them. The headphones controls have started continu......Sat, 08 Aug 2026 23:00:24 -0400https://liberty.io/when-trying-to-install-on-mac-no-mountable-file-system.htmlhttps://liberty.io/when-trying-to-install-on-mac-no-mountable-file-system.htmlI downloaded Ubuntu 14.04.2 - desktop. When I try to open the file, I have the following error message: NO MOUNTABLE FILE SYSTEM. What should I do to solve this problem?...Sat, 08 Aug 2026 22:53:19 -0400https://liberty.io/move-a-file-to-a-directory-requiring-root-permission.htmlhttps://liberty.io/move-a-file-to-a-directory-requiring-root-permission.htmlI'm a complete noob to Linux and Ubuntu. I'm running Ubuntu Studio 20.04. I downloaded a plugin for Audacity that needs to go from home/greg/downloads into usr/share/audacity/plug-ins. I tried the ...Sat, 08 Aug 2026 22:47:47 -0400https://liberty.io/can-intel-i7-1065g7-run-ubuntu-20-04-duplicate.htmlhttps://liberty.io/can-intel-i7-1065g7-run-ubuntu-20-04-duplicate.htmlI am new on Linux based operation systems.It will be my first time so I need your help. I just bought HP Pavilion 14-CE3005NT and it has great hardwares but the processor. The hardware list of this ...Sat, 08 Aug 2026 22:47:35 -0400https://liberty.io/mpif-h-not-found-error-install-openmpi.htmlhttps://liberty.io/mpif-h-not-found-error-install-openmpi.htmlI was trying to install openpi on ubuntu. I have an Ubuntu 18.04.4 LTS (64 bit) desktop. I installed openmpi using sudo apt-get install openmpi-bin openmpi-common openmpi-doc libopenmpi2 libopenmp......Sat, 08 Aug 2026 22:45:04 -0400https://liberty.io/how-can-i-enable-custom-svg-fonts-for-image-viewer-eye-of-gnome.htmlhttps://liberty.io/how-can-i-enable-custom-svg-fonts-for-image-viewer-eye-of-gnome.htmlI have some SVG files with custom fonts. I can open them and fonts look nice in Firefox, but the font is not properly displayed in Eye of Gnome (Image Viewer) application. Why is this so, and how c......Sat, 08 Aug 2026 22:44:01 -0400https://liberty.io/how-to-assign-one-node-among-specific-nodes-in-slurm.htmlhttps://liberty.io/how-to-assign-one-node-among-specific-nodes-in-slurm.htmlI want to submit a job using one node among a specific nodelist. Suppose, nodes 151-154 are my specific node. So, Here is my script: #!/bin/sh #SBATCH --job-name=k89a1 #SBATCH --nodes=1 #SBATCH --t......Sat, 08 Aug 2026 22:36:18 -0400https://liberty.io/fan-not-recognized-on-an-hp-nc6400.htmlhttps://liberty.io/fan-not-recognized-on-an-hp-nc6400.htmlI'm using Ubuntu 11.10 and an HP nc6400 laptop, and it doesn't recognize my fans, which means that my laptop keeps overheating, is there anyway to make it recognize them? Thanks in advance :)...Sat, 08 Aug 2026 22:18:52 -0400https://liberty.io/what-to-do-after-failed-to-start-load-kernel-modules.htmlhttps://liberty.io/what-to-do-after-failed-to-start-load-kernel-modules.htmlI tried to upgrade my system from Ubuntu 14.04. My laptop froze for a few hours. Not even the pointer did anything. So I switched off by pressing the power button for a few seconds. This uncompleted ...Sat, 08 Aug 2026 22:17:25 -0400