https://liberty.ioCurated star highlights with fresh energy.en-usSun, 09 Aug 2026 09:21:44 -0400https://liberty.io/find-the-solution-of-the-recurrence-relation-fibonacci.htmlhttps://liberty.io/find-the-solution-of-the-recurrence-relation-fibonacci.htmlFind the solution of the recurrence relation $f_n = f_{n-1} + f_{n-2}$ with $f_0 = f_1 = 1$ I had someone show me how to solve/prove this using the linear difference equation, but I had never been ...Sun, 09 Aug 2026 13:14:34 -0400https://liberty.io/why-do-dummit-foote-require-an-ideal-also-be-a-subring.htmlhttps://liberty.io/why-do-dummit-foote-require-an-ideal-also-be-a-subring.htmlOn page 242 of Abstract Algebra, 3rd Ed., by Dummit & Foote, they write: However, in my lecture notes, there is a line that says Definition. (Ideal.) Let $R$ be a ring. A subgroup $I$ of the ...Sun, 09 Aug 2026 12:51:29 -0400https://liberty.io/find-derivative-of-taylor-series.htmlhttps://liberty.io/find-derivative-of-taylor-series.htmlHow would one differentiate a function in this form? $$f(x) = f(a)+{\frac {f'(a)}{1!}}(x-a)+{\frac {f''(a)}{2!}}(x-a)^{2}+{\frac {f'''(a)}{3!}}(x-a)^{3}$$ I'm sorry if this is something obvious, ......Sun, 09 Aug 2026 12:47:11 -0400https://liberty.io/is-the-percentage-symbol-a-constant.htmlhttps://liberty.io/is-the-percentage-symbol-a-constant.htmlIsn't the percentage symbol actually just a constant with the value $0.01$? As in $$ 15\% = 15 \times \% = 15 \times 0.01 = 0.15. $$ Isn't every unit actually just a constant? But why do we treat......Sun, 09 Aug 2026 12:45:20 -0400https://liberty.io/gradient-descent-in-n-dimensional-space-in-the-context-of-an-rbf-netwo.htmlhttps://liberty.io/gradient-descent-in-n-dimensional-space-in-the-context-of-an-rbf-netwo.htmlI am trying to implement an algorithm to perform gradient descent in a $n$-dimensional space in the context of an RBF network. My network has 5 inputs and 1 output. It has the following Gaussian ra......Sun, 09 Aug 2026 12:44:06 -0400https://liberty.io/proof-for-why-the-conceptual-and-computation-formulas-for-sum-of-squar.htmlhttps://liberty.io/proof-for-why-the-conceptual-and-computation-formulas-for-sum-of-squar.htmlThe Sum of Squared Deviates has a conceptual formula of: $SS = ∑(Xi — Mx)^2$ (where $Mx$ is the sample mean) The corresponding computational formula is: $$SS = ∑Xi^2 —\frac{(∑Xi)^2}{N}$$ How are ......Sun, 09 Aug 2026 12:35:41 -0400https://liberty.io/understanding-what-a-gradient-vector-is.htmlhttps://liberty.io/understanding-what-a-gradient-vector-is.htmlDefinition: let $A$ be an open set in a Euclidean space $X$. Let $f : A \to \mathbb R$ be a real valued differentiable function. Then,for each $a \in A$, there is a unique vector $\nabla f(a) x \in......Sun, 09 Aug 2026 12:22:16 -0400https://liberty.io/why-is-delta-used-to-describe-difference-between-two-entities.htmlhttps://liberty.io/why-is-delta-used-to-describe-difference-between-two-entities.htmlI hear a lot about extracting "delta" between two properties in my current job. I come from a User Interface programming background and I do not really have much math background. I looked up delta ......Sun, 09 Aug 2026 12:14:05 -0400https://liberty.io/what-is-the-fisher-information-for-a-uniform-distribution.htmlhttps://liberty.io/what-is-the-fisher-information-for-a-uniform-distribution.htmlIf X is U[$0$,$\theta$], then the likelihood is given by $f(X,\theta) = \dfrac{1}{\theta}\mathbb{1}\{0\leq x \leq \theta\}$. The definition of Fisher information is $I(\theta) = \mathbb{E} \left[ \......Sun, 09 Aug 2026 12:08:50 -0400https://liberty.io/proof-for-theorem-of-upper-and-lower-bounds-on-zeroes-of-polynomials.htmlhttps://liberty.io/proof-for-theorem-of-upper-and-lower-bounds-on-zeroes-of-polynomials.htmlI'm currently a high school Pre-Calculus student and my textbook presents the following theorem without proof: Let $f(x)$ be a polynomial with real coefficients and a positive leading coefficient.......Sun, 09 Aug 2026 11:56:23 -0400https://liberty.io/can-an-arc-be-greater-than-360-degrees-or-less-than-0-degrees.htmlhttps://liberty.io/can-an-arc-be-greater-than-360-degrees-or-less-than-0-degrees.htmlCan an arc be greater than 360 degrees or less than 0 degrees? Is the (angular) arc length a signed scalar? I'm creating a Graphics and basic geometry software library. I believe an arc can be defi......Sun, 09 Aug 2026 11:54:52 -0400https://liberty.io/matlab-plot-vertical-line.htmlhttps://liberty.io/matlab-plot-vertical-line.htmlI have a system of equations, about 3 equations, that I need to plot that are dependent on 2 variables, R and z. But one of the equations is R={constant value}. Here's the code: R = 0:.1:500; z = ......Sun, 09 Aug 2026 11:52:40 -0400https://liberty.io/why-would-we-use-the-radius-of-a-circle-instead-of-the-diameter-when-c.htmlhttps://liberty.io/why-would-we-use-the-radius-of-a-circle-instead-of-the-diameter-when-c.htmlForgive me if this question is a little too strange or maybe even off. Mathematics has never been my strong point, but I definitely think it's the coolest... Anyway, I was looking into tau, pi's ......Sun, 09 Aug 2026 11:52:17 -0400https://liberty.io/any-way-to-easily-rip-blu-ray-disks.htmlhttps://liberty.io/any-way-to-easily-rip-blu-ray-disks.htmlI've seen the documentation on Blu-Ray Restricted formats here but I'm looking to see if anyone has come up with an easy solution yet? Also it would be awesome if someone knows of any type of prog......Sun, 09 Aug 2026 11:46:11 -0400https://liberty.io/strategy-for-7-up-7-down-game.htmlhttps://liberty.io/strategy-for-7-up-7-down-game.htmlThis is a dice game where the host throws the dice and the players bet on 3 numbers - numbers below 7, 7 and numbers above 7. On numbers below and above 7 the players get double their stake and on ......Sun, 09 Aug 2026 11:21:16 -0400https://liberty.io/ubuntu-sdk-error-039-could-not-find-or-load-the-qt-platform-plugin-xcb.htmlhttps://liberty.io/ubuntu-sdk-error-039-could-not-find-or-load-the-qt-platform-plugin-xcb.htmlI am trying to write my first ubuntu app using the Ubuntu SDK but keep running into an error. This error turns up both with QML and HTML5 apps. I am trying this just on the template for both project ...Sun, 09 Aug 2026 11:16:35 -0400https://liberty.io/how-to-state-a-recurrence-equation-for-running-time-of-a-divide-and-co.htmlhttps://liberty.io/how-to-state-a-recurrence-equation-for-running-time-of-a-divide-and-co.htmlI'm working on my homework and I saw this problem: A divide-and-conquer algorithm $ X $ divides any given problem instance into exactly two smaller problem instances and solve them recursively. ...Sun, 09 Aug 2026 11:14:56 -0400https://liberty.io/find-an-equation-of-the-plane-through-a-point-and-parallel-to-another-.htmlhttps://liberty.io/find-an-equation-of-the-plane-through-a-point-and-parallel-to-another-.htmlFind an equation of the plane through the point (1, -1, -1) and parallel to the plane $5{x} - {y} - {z} = 6$ I was shown an example of this problem but it was in a problem where it had to be ...Sun, 09 Aug 2026 11:14:16 -0400https://liberty.io/cheese-has-holes-more-cheese-more-holes-more-holes-less-cheese-more-ch.htmlhttps://liberty.io/cheese-has-holes-more-cheese-more-holes-more-holes-less-cheese-more-ch.htmlI found this statement on the internet and I would like to know why this is a fallacy. Cheese has holes. More cheese = more holes More holes = less cheese More cheese = less cheese Why is this fa......Sun, 09 Aug 2026 11:01:33 -0400https://liberty.io/calculating-the-nth-root-in-simple-calculator.htmlhttps://liberty.io/calculating-the-nth-root-in-simple-calculator.htmlRecently during my physics class that how to take a cube root in a simple calculator. Follow the steps given below Step1. Press the square root button 13 times Step2. subtract 1 from it Step3. ...Sun, 09 Aug 2026 10:54:37 -0400https://liberty.io/what-does-the-steady-state-represent-to-a-markov-chain.htmlhttps://liberty.io/what-does-the-steady-state-represent-to-a-markov-chain.htmlI'm a little confused as to the interpretation of the steady state in the context of a Markov chain. I know Markov chains are memoryless, in that each state only depends on its immediate predecesso......Sun, 09 Aug 2026 10:50:42 -0400https://liberty.io/find-the-remainder-using-fermat-039-s-little-theorem-when-5-119-is-div.htmlhttps://liberty.io/find-the-remainder-using-fermat-039-s-little-theorem-when-5-119-is-div.htmlFind the remainder using Fermat's little theorem when $5^{119}$ is divided by $59$. Fermat's little theorem states that if $p$ is prime and $\operatorname{gcd}(a,p)=1$,then $a^{p-1} -1$ is a multi......Sun, 09 Aug 2026 10:42:19 -0400https://liberty.io/max-min-inequality.htmlhttps://liberty.io/max-min-inequality.htmlIt is known that $\underset{x}{\max} \underset{y}{\min} f(x,y) \leq \underset{y}{\min} \underset{x}{\max} f(x,y)$ . When does equality hold in this expression?...Sun, 09 Aug 2026 10:41:10 -0400https://liberty.io/can-systems-of-3-linear-equations-with-3-unknowns-have-more-than-one-s.htmlhttps://liberty.io/can-systems-of-3-linear-equations-with-3-unknowns-have-more-than-one-s.htmlIn each part,determine whether the given vector is a solution of the linear system \begin{align} 2x-4y-z&=1\\ x-3y+z&=1\\ 3x-5y-3z&=1 \end{align} (a) $(3,1,1)$ (b) $(3,-1,1)$ (c) ......Sun, 09 Aug 2026 10:39:30 -0400https://liberty.io/integration-by-substitution-called-weierstrass-substitution.htmlhttps://liberty.io/integration-by-substitution-called-weierstrass-substitution.htmlI have seen this recent question What's wrong in my calculation of $\int_0^{3 \pi/4} \frac{\cos x}{1 + \cos x}dx$? and I have read that when I operate for integration by substitution this type ......Sun, 09 Aug 2026 10:33:52 -0400https://liberty.io/invertible-and-non-invertible-linear-transformation.htmlhttps://liberty.io/invertible-and-non-invertible-linear-transformation.htmlIf a linear transformation is represented by an non-invertible matrix $P$, then it might happen that two different vectors (points in $\mathbb{R^n}$) will be mapped to the same point. However, if the ...Sun, 09 Aug 2026 10:27:11 -0400https://liberty.io/noetherian-ring-not-descending-chain-condition.htmlhttps://liberty.io/noetherian-ring-not-descending-chain-condition.htmlAnalogous to the ascending chain condition we can define a descending chain condition: if $I_1 \supseteq I_2 \supseteq I_3 \supseteq \cdots$ is a descending chain of ideals then there exists a pos......Sun, 09 Aug 2026 10:24:52 -0400https://liberty.io/angles-in-a-nonagon.htmlhttps://liberty.io/angles-in-a-nonagon.htmlThe following figure was draw by taking midpoints on three sides of a regular nonagon. The angles are 20 degrees, 50 degrees, and 50 degrees. It's not hard to find the angles by using trigonometry......Sun, 09 Aug 2026 10:21:29 -0400https://liberty.io/power-regression-y-ax-b-c.htmlhttps://liberty.io/power-regression-y-ax-b-c.htmlI have to do reproduce a power regression but I don't have any experience in procedures like that. I read a little bit about power fit/power regression and that a formula like $y = ax^b$ is used for ...Sun, 09 Aug 2026 10:07:48 -0400https://liberty.io/differentiate-frac-ln-x-cos-x.htmlhttps://liberty.io/differentiate-frac-ln-x-cos-x.htmlPlease help me with this question. $$y= \frac{\ln(x)}{\cos(x)}$$ Just starting with calculus. Thank you....Sun, 09 Aug 2026 10:01:31 -0400https://liberty.io/what-does-calculating-the-inverse-of-a-matrix-mean.htmlhttps://liberty.io/what-does-calculating-the-inverse-of-a-matrix-mean.htmlAssume I have 3 equations, $x+2y+z=2, 3x+8y+z=12, 4y+z=2$ which could be represented in matrix form ($Ax = b$) like this: $\begin{pmatrix} 1 & 2 & 1\\ 3 & 8 & 1\\ 0 & 4 & ......Sun, 09 Aug 2026 09:58:09 -0400https://liberty.io/difference-between-probability-and-probability-density.htmlhttps://liberty.io/difference-between-probability-and-probability-density.htmlThis question is from DeGroot's "Probability and Statistics" : Unbounded p.d.f.’s. Since a value of a p.d.f.(probability density function) is a probability density, rather than a probability, ......Sun, 09 Aug 2026 09:35:36 -0400https://liberty.io/limit-of-x-ln-x-as-x-approaches-0.htmlhttps://liberty.io/limit-of-x-ln-x-as-x-approaches-0.htmlHow could I solve the limit in this form $ \frac{x}{x+1}$ using l'Hospital's rule? I know how to solve it in this way: $\frac{\ln x}{\frac{1}{x}}$ Thanks...Sun, 09 Aug 2026 09:31:36 -0400https://liberty.io/how-to-swap-rows-in-square-a-matrix-algebraically.htmlhttps://liberty.io/how-to-swap-rows-in-square-a-matrix-algebraically.htmlIs there some way to achieve swapping of rows of a 3x3 square matrix(for example exchanging rows 0 and 2) by using matrix algebra? Or is it something that cannot be done with algebra? What about row ...Sun, 09 Aug 2026 09:30:58 -0400https://liberty.io/find-an-equation-of-the-tangent-line-to-the-curve.htmlhttps://liberty.io/find-an-equation-of-the-tangent-line-to-the-curve.htmlFind an equation of the tangent line to the curve $2·(x^2+y^2)^2=25·(x^2−y^2)$ at the point $(3,1)$. Any hints?...Sun, 09 Aug 2026 09:29:54 -0400https://liberty.io/in-a-calculator-which-button-should-be-bigger-sinh-or-rm-asin.htmlhttps://liberty.io/in-a-calculator-which-button-should-be-bigger-sinh-or-rm-asin.htmlI am making a calculator app and would like to know: Which of the following functions would be used more often: $\sin()$ $\sinh()$ $\text{arsin}()$ ($\sin^{-1}$) $\text{arsinh}()$ ($\sinh^{-1}$)...Sun, 09 Aug 2026 09:29:42 -0400https://liberty.io/why-is-a-circle-in-a-plane-surrounded-by-6-other-circles.htmlhttps://liberty.io/why-is-a-circle-in-a-plane-surrounded-by-6-other-circles.htmlWhen you draw a circle in a plane you can perfectly surround it with 6 other circles of the same radius. This works for any radius. What's the significance of 6? Why not some other numbers? I'm lo......Sun, 09 Aug 2026 09:28:05 -0400https://liberty.io/ordinal-tetration-the-issue-of-epsilon-0-omega.htmlhttps://liberty.io/ordinal-tetration-the-issue-of-epsilon-0-omega.htmlSo in the past few months when trying to learn about the properties of the fixed points in ordinals as I move from $0$ to $\epsilon_{\epsilon_0}$ I noticed when moving from $\epsilon_n$ to the next......Sun, 09 Aug 2026 09:11:55 -0400https://liberty.io/what-is-the-exact-and-precise-definition-of-an-angle.htmlhttps://liberty.io/what-is-the-exact-and-precise-definition-of-an-angle.htmlOn wikipedea I found a definition of an Angle as such: "In order to measure an angle θ, a circular arc centered at the vertex of the angle is drawn, e.g. with a pair of compasses. The length o......Sun, 09 Aug 2026 09:00:44 -0400https://liberty.io/how-to-find-equation-of-a-graph-using-vertical-asymptote.htmlhttps://liberty.io/how-to-find-equation-of-a-graph-using-vertical-asymptote.htmlGraph Based on the graph, I need to find the equation. What I know: vertical asymptote x = 4, and opening at x = -4. I am struggling to find the rational function of the graph. y = 1/-x+4 is what ......Sun, 09 Aug 2026 08:58:51 -0400https://liberty.io/functions-linear-function-f-x-3x-5.htmlhttps://liberty.io/functions-linear-function-f-x-3x-5.htmlHey if i wanted to get a linear function how would i do it? Im being shown an example and the results are supposedly: x 3x+5 -2 -1 -1 2 0 5 1 8 2 11 But i cant figure out how to get it.......Sun, 09 Aug 2026 08:56:15 -0400https://liberty.io/calculate-the-image-and-a-basis-of-the-image-matrix.htmlhttps://liberty.io/calculate-the-image-and-a-basis-of-the-image-matrix.htmlWhat's the image of the matrix? What's the basis of the image? $M=\begin{pmatrix} -1 & 1 & 1\\ -2 & -3 & 6\\ 0 & -1 & 1 \end{pmatrix}$ First transposed the matrix:......Sun, 09 Aug 2026 08:55:14 -0400https://liberty.io/why-can-039-t-pi-be-expressed-as-a-fraction-duplicate.htmlhttps://liberty.io/why-can-039-t-pi-be-expressed-as-a-fraction-duplicate.htmlWhy can't $\pi$ be expressed as a fraction? If pi is the ratio of a circle's circumference to its diameter, why can't we simply take a circle, measure its circumference and diameter, and derive the ...Sun, 09 Aug 2026 08:43:52 -0400https://liberty.io/what-is-a-powerset-with-base-larger-than-2.htmlhttps://liberty.io/what-is-a-powerset-with-base-larger-than-2.htmlA powerset $P(S) $of some set $S$ can be treated as all different possible ways of partition $S$ into 2 ordered pair of disjoint subsets. And I'm curious what is the equivalence of partition $S$ into ...Sun, 09 Aug 2026 08:35:49 -0400https://liberty.io/inverse-of-a-vector.htmlhttps://liberty.io/inverse-of-a-vector.htmlHow would we calculate the inverse of a $3$-vector $v$ like we would do with the matrix? Basically the inverse of $1\times3$ (?) matrix. Would we get the negated value of the vector $-v$ or $1/v$? ......Sun, 09 Aug 2026 08:29:52 -0400https://liberty.io/why-is-delta-x-tending-to-be-zero-in-dy-dx-and-not-delta-y-tending-to-.htmlhttps://liberty.io/why-is-delta-x-tending-to-be-zero-in-dy-dx-and-not-delta-y-tending-to-.htmlWhy is $\Delta x $ and not $\Delta y $ written to be tending towards 0 in case when we give limits to form dy/dx. , $$\lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x} = \frac{dy}{dx}$$ Can we solve......Sun, 09 Aug 2026 08:24:55 -0400https://liberty.io/how-to-find-the-directional-derivative-of-the-following-function.htmlhttps://liberty.io/how-to-find-the-directional-derivative-of-the-following-function.htmlI am stuck with the following problem: Consider the function $f(x,y)=\frac{x^2}{y^2}$; where $(x,y)\in [1/2,3/2]\times[1/2,3/2]$.Then what is the derivative of $f$ at $(1,1)$along the direction ......Sun, 09 Aug 2026 08:17:51 -0400https://liberty.io/what-do-curly-brackets-mean-in-this-number-theory-equation.htmlhttps://liberty.io/what-do-curly-brackets-mean-in-this-number-theory-equation.html$$(\lfloor y\rfloor-1)(\lfloor x\rfloor-1)=\{x\}+\{y\}+1$$ From looking at the LaTeX, I can see the left-hand side symbols mean the floor of the variable, but the right-hand side doesn't give much......Sun, 09 Aug 2026 08:13:37 -0400https://liberty.io/what-is-the-equal-sign-with-3-lines-mean-in-wilson-039-s-theorem.htmlhttps://liberty.io/what-is-the-equal-sign-with-3-lines-mean-in-wilson-039-s-theorem.htmlI'm reading up on Wilson's Theorem, and see a symbol I don't know... what does an equal sign with three lines mean? I'm looking at the example table and I still can't infer what they are trying to......Sun, 09 Aug 2026 08:11:18 -0400https://liberty.io/what-is-the-purpose-of-port-1234-tcp-running-by-039-systemd-039-with-p.htmlhttps://liberty.io/what-is-the-purpose-of-port-1234-tcp-running-by-039-systemd-039-with-p.htmlWhen I used $ nmap 192.168.1.X it shows that 1234/tcp OPEN hotline is open. I am running behind firewall ufw, and there is no rule for port 1234. $ lsof -i showed that it is running by systemd with ...Sun, 09 Aug 2026 08:09:35 -0400https://liberty.io/a-question-about-incompressible-vector-field.htmlhttps://liberty.io/a-question-about-incompressible-vector-field.htmlLet $ X $ be a unit vector field on $\mathbb{R}^2$,with canonical metric $g$ and connection $\nabla$ .Show that if $$divX=0$$ then $$\nabla X=0$$ I tried: Let $X=(f,g)$,$Y=(-g,f,)$,then $X,Y$ ......Sun, 09 Aug 2026 08:07:19 -0400https://liberty.io/proof-about-complete-residue-system.htmlhttps://liberty.io/proof-about-complete-residue-system.htmlTheorem 3-5. If $a_1,a_2,\ldots,a_m$ is a complete residue system $\pmod m$ and $\gcd(k,m)=1$, then $ka_1,ka_2,\ldots,ka_m$ also is a complete residue system $\pmod m$ Proof: We show directly that ...Sun, 09 Aug 2026 08:04:54 -0400https://liberty.io/what-is-mathematical-structure.htmlhttps://liberty.io/what-is-mathematical-structure.htmlWhen we have an isomorphism, between 2 groups or vector spaces let us say, then it is said to be structure preserving. An isomorphism exists when there is at least one mutually invertible morphism ...Sun, 09 Aug 2026 07:51:02 -0400https://liberty.io/wine-visual-c-redistributable-for-visual-studio-2015.htmlhttps://liberty.io/wine-visual-c-redistributable-for-visual-studio-2015.htmlWhen I want to run Chessbase 14 I get this error output: gratis@Aurora:~$ env WINEPREFIX="/home/gratis/.wine" wine C:\\Program\ Files\\ChessBase\\CBase14\\CBase14.exe err:module:import_dll Librar......Sun, 09 Aug 2026 07:47:00 -0400https://liberty.io/confusion-with-showing-that-hn-n-cong-h-h-cap-n.htmlhttps://liberty.io/confusion-with-showing-that-hn-n-cong-h-h-cap-n.htmlSuppose $G$ is a group, $H \le G, $ and $N \triangleleft G$. Show that $$HN/N \cong H/(H \cap N)$$ Well, I know there was a similar question on math.stackexchange and I know there is a solution by ......Sun, 09 Aug 2026 07:42:17 -0400https://liberty.io/how-to-convert-a-right-angled-triangle-into-a-equilateral-triangle.htmlhttps://liberty.io/how-to-convert-a-right-angled-triangle-into-a-equilateral-triangle.htmlI want to use the Apophysis program to make a right angled sierpinski triangle into an equilateral triangle. But how can i do so? i have tried the second picture one but that is not correct....Sun, 09 Aug 2026 07:40:49 -0400https://liberty.io/figuring-out-price-demand-equation.htmlhttps://liberty.io/figuring-out-price-demand-equation.htmlWhen selling burgers at $2.50$ dollar per unit, average sales are $875$ units per night. When the price was raised to $3.20$ dollar per unit, average sales dropped to $700$ units per night. Find......Sun, 09 Aug 2026 07:35:19 -0400https://liberty.io/is-n-sin-n-dense-on-the-real-line.htmlhttps://liberty.io/is-n-sin-n-dense-on-the-real-line.htmlIs $\{n \sin n | n \in \mathbb{N}\}$ dense on the real line? If so, is $\{n^p \sin n | n \in \mathbb{N}\}$ dense for all $p>0$? This seems much harder than showing that $\sin n$ is dense on [-......Sun, 09 Aug 2026 07:33:00 -0400https://liberty.io/finding-poles-indicating-their-order-and-then-computing-their-residues.htmlhttps://liberty.io/finding-poles-indicating-their-order-and-then-computing-their-residues.htmlI really don't understand the concept behind finding poles in Complex Analysis and I can't find anything on the internet or in books that helps me grasp the concept... The following are past exam ...Sun, 09 Aug 2026 07:28:16 -0400https://liberty.io/prove-that-if-in-a-tetrahedron-if-two-pairs-of-opposite-edges-are-perp.htmlhttps://liberty.io/prove-that-if-in-a-tetrahedron-if-two-pairs-of-opposite-edges-are-perp.htmlProve that if in a tetrahedron if two pairs of opposite edges are perpendicular then the third pair is also perpendicular. Method: $\vec a+ \vec b + \vec c + \vec d +\vec e + \vec f = 0 \tag0$ ...Sun, 09 Aug 2026 07:22:37 -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 -0400