https://liberty.ioCurated star highlights with fresh energy.en-usSun, 09 Aug 2026 06:15:21 -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 -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 -0400