https://liberty.ioCurated star highlights with fresh energy.en-usTue, 18 Aug 2026 22:23:05 -0400https://liberty.io/what-is-meant-by-passing-to-a-subsequence.htmlhttps://liberty.io/what-is-meant-by-passing-to-a-subsequence.htmlLet $\{f_k(x)\}_{k=1}^\infty$ be a sequence of differentiable functions defined on $[0,1]$. Suppose $f_k(0)=0$ and $|f_k'(x)|\le M$ for all $x\in[0,1]$ and for all $k\ge1$. Prove that after passing......Wed, 19 Aug 2026 02:22:33 -0400https://liberty.io/explanation-of-the-formula-for-combinations-without-repetition-probabi.htmlhttps://liberty.io/explanation-of-the-formula-for-combinations-without-repetition-probabi.htmlThe formula for combinations without the repetitions is as follows: $$ \frac{n!}{r!(n-r)!}$$ This is achieved by doing $$\frac{n!}{(n-r)!}*\frac{1}{r!}$$ What I don't understand where $\frac{1}{r......Wed, 19 Aug 2026 02:21:36 -0400https://liberty.io/why-do-you-need-three-perpendicular-bisectors-to-circumscribe-a-triang.htmlhttps://liberty.io/why-do-you-need-three-perpendicular-bisectors-to-circumscribe-a-triang.htmlAll the perpendicular bisectors of the sides of a triangle intersect at one point -- the circumcenter. So, why do we typically find all three perpendicular bisectors when trying to circumscribe a ...Wed, 19 Aug 2026 02:11:21 -0400https://liberty.io/homeomorphic-graphs.htmlhttps://liberty.io/homeomorphic-graphs.htmlAre these graphs homeomorphic? a) and the Peterson graph? b) I think both are not homeomorphic.Is it correct?Is there a way I can show that they are not homemorphic Also two graphs being isomo......Wed, 19 Aug 2026 02:09:55 -0400https://liberty.io/muddy-children-puzzle-when-more-than-2-kids-are-muddy.htmlhttps://liberty.io/muddy-children-puzzle-when-more-than-2-kids-are-muddy.htmlThis is about famous Muddy Children puzzle mentioned in section 1.1. I am trying to understand the significance of rounds when number of muddy children are 3 or more than 3. http://www.cs.cornell.......Wed, 19 Aug 2026 01:53:43 -0400https://liberty.io/how-to-find-an-angle-in-degrees-in-a-right-triangle-given-its-sides.htmlhttps://liberty.io/how-to-find-an-angle-in-degrees-in-a-right-triangle-given-its-sides.htmlI need to find out a degree of an angle. Pretty simple, or so I thought. I remember doing a crap-ton of these in high-school, sadly the details did not remain. Anyway, let's take a look at this tr......Wed, 19 Aug 2026 01:50:17 -0400https://liberty.io/finding-minimum-value-of-sqrt-x-2-y-2-duplicate.htmlhttps://liberty.io/finding-minimum-value-of-sqrt-x-2-y-2-duplicate.htmlFind the minimum value of $\sqrt{x^2+y^2}$, given $15x+8y=120$. My attempt: From $15x+8y=120$, I get $y=\frac{120-15x}{8}$. I substitute this value into $\sqrt{x^2+y^2}$, getting $\sqrt{\frac{28......Wed, 19 Aug 2026 01:49:26 -0400https://liberty.io/questions-tagged-definite-integrals.htmlhttps://liberty.io/questions-tagged-definite-integrals.htmlQ&A for people studying math at any level and professionals in related fields...Wed, 19 Aug 2026 01:36:31 -0400https://liberty.io/is-there-a-name-for-a-3d-shape-that-looks-like-a-circle-when-viewed-fr.htmlhttps://liberty.io/is-there-a-name-for-a-3d-shape-that-looks-like-a-circle-when-viewed-fr.htmlThe shape can be constructed by taking a cylinder with its height equal to its diameter and cutting a triangle out of it when viewing it from the side with the base of the triangle matching one end......Wed, 19 Aug 2026 01:28:36 -0400https://liberty.io/is-there-a-three-dimensional-version-of-curl-in-which-the-result-is-a-.htmlhttps://liberty.io/is-there-a-three-dimensional-version-of-curl-in-which-the-result-is-a-.htmlCurl is an operation from a vector field to a vector field in 3D. Is there an interpretation of Divergence that doesn't produce a scalar field but rather a vector field, such that at any point $|\t......Wed, 19 Aug 2026 01:23:40 -0400https://liberty.io/domain-of-exponential-function.htmlhttps://liberty.io/domain-of-exponential-function.htmlI have $$e^{\frac{x²+2}{x-1}}.$$ As the domain is the values for which the function is defined, I thought of making the denominator of the power equal to zero. So $x-1=0$ and the function is undefi......Wed, 19 Aug 2026 01:16:39 -0400https://liberty.io/bounded-vs-unbounded-linear-programs.htmlhttps://liberty.io/bounded-vs-unbounded-linear-programs.htmlConsider a (linear) optimization problem of the form "maximize $c^{\top}x$ subject to $\varphi(x)$". Consider the following definitions: The program is called unbounded iff it is feasible but its ...Wed, 19 Aug 2026 01:14:51 -0400https://liberty.io/real-analysis-continuity-of-a-composition-function.htmlhttps://liberty.io/real-analysis-continuity-of-a-composition-function.htmlSuppose $f$ and $g$ are functions such that $g$ is continuous at $a$, and $f$ is continuous at $g(a)$. Show the composition $f(g(x))$ is continuous at $a$. My idea: Can I go straight from definiti......Wed, 19 Aug 2026 01:12:20 -0400https://liberty.io/how-to-show-cubic-function-is-one-to-one-using-assumption.htmlhttps://liberty.io/how-to-show-cubic-function-is-one-to-one-using-assumption.htmlOkay I know the function $f(x)= x^3 +2x+3$ is one to one but how do I show that using the assumption that $f(a)=f(b)$ where $a=b$? So if I do the process I have $ a^3+2a+3 = b^3+2b+3$ my 3's ca......Wed, 19 Aug 2026 01:07:46 -0400https://liberty.io/integrate-over-an-area-of-a-parabola.htmlhttps://liberty.io/integrate-over-an-area-of-a-parabola.htmlHow do I figure out the integral bonds for r when integrating over the area M under a parabola y=2-x^2? Do I need to solve a quadratic equation? I think there must be an easier solution. Clarific......Wed, 19 Aug 2026 01:06:24 -0400https://liberty.io/centroid-root-locus.htmlhttps://liberty.io/centroid-root-locus.htmlI can't figure out how to find the root locus centroid for the poles of this simple equation in a positive feedback system. $$ H(s)=\frac{s}{s^2+3s+1} $$ I have read in many places that the centr......Wed, 19 Aug 2026 01:02:44 -0400https://liberty.io/why-is-establishing-absolute-consistency-of-zfc-impossible.htmlhttps://liberty.io/why-is-establishing-absolute-consistency-of-zfc-impossible.htmlWhy is establishing the absolute consistency of ZFC impossible? What are the fundamental limitations that prohibit us with coming up with a proof? EDIT: This post seems to make the most sense. In ......Wed, 19 Aug 2026 00:57:45 -0400https://liberty.io/what-is-the-conjugate-of-i-iota.htmlhttps://liberty.io/what-is-the-conjugate-of-i-iota.htmlAfter solving a bunch of stuff I got to $\ arg(1/-i)$. What does this mean geometrically, and moreover, will the conjugate of $i$ be $-i$ or just $i$ itself?...Wed, 19 Aug 2026 00:35:46 -0400https://liberty.io/prove-the-derivative-of-sin-1-x-exists.htmlhttps://liberty.io/prove-the-derivative-of-sin-1-x-exists.htmlHow do I prove the derivative of $$\sin(1/x)=-\frac{1}{x^2}\cos(1/x)$$? I understand that you use $$f'(x_0) = \lim_{x \to x_0} \frac{\sin(1/x) - \sin(1/x_0)}{x-x_0} = -\frac{1}{x_0^2}\cos(1/x_0)$$ ...Wed, 19 Aug 2026 00:26:12 -0400https://liberty.io/how-many-possible-3-character-combinations-using-letters-closed.htmlhttps://liberty.io/how-many-possible-3-character-combinations-using-letters-closed.htmlHow many combinations can be using 3 alphabet letters (from A to Z)? Note that string doesn't have to be unique, it can be eg. RPP or AZZ...Wed, 19 Aug 2026 00:08:29 -0400https://liberty.io/convert-quaternion-to-vector-closed.htmlhttps://liberty.io/convert-quaternion-to-vector-closed.htmlLets assume I have a rotation quaternion Q={1,1,1,0}, which represents a specific pointing direction, how can I extract this Vector(xyz) from it ? Or, how can I determine a matrixs determinante, be......Wed, 19 Aug 2026 00:03:11 -0400https://liberty.io/18-people-divided-into-3-groups-probability.htmlhttps://liberty.io/18-people-divided-into-3-groups-probability.html$18$ people are randomly divided into $3$ groups of size $5$, $6$ and $7$. Among the $18$ are John, Jack and James. What is the probability that: 1) What is the probability that both Jack and Jame......Wed, 19 Aug 2026 00:00:36 -0400https://liberty.io/into-function-vs-injective-function.htmlhttps://liberty.io/into-function-vs-injective-function.htmlIn many mathematical books that I have read and from lectures from professors, the words 'into' and 'injective' were used interchangeably, but in Patrick Suppes book Axiomatic Set Theory he gives a ...Tue, 18 Aug 2026 23:53:25 -0400https://liberty.io/transformations-from-n-sphere-coordinates-to-cartesian-coordinates.htmlhttps://liberty.io/transformations-from-n-sphere-coordinates-to-cartesian-coordinates.htmlI was wondering how one would proceed to convert between coordinate systems in $ \mathbb R^n $. For $ \mathbb R^2 $ the conversion is easy and just basic trigonometry. Given $(r, \theta)$ we can c......Tue, 18 Aug 2026 23:53:10 -0400https://liberty.io/solid-angle-integration.htmlhttps://liberty.io/solid-angle-integration.htmlCan somebody explain the equivalence between integrating over the surface of a unit sphere and integrating over solid angle? I have been trying to understand the following transformation using a Ja......Tue, 18 Aug 2026 23:40:21 -0400https://liberty.io/fundamental-theorem-of-calculus-of-a-definite-integral.htmlhttps://liberty.io/fundamental-theorem-of-calculus-of-a-definite-integral.html$$\frac{\mathrm{d}}{\mathrm{d}x} \int_{2x}^{3x+1}\! \sin\left(t^4\right)\, \mathrm{d}t$$ could you just use the Fundamental Theorem of Calculus to get $$\sin\left(t^4\right) \bigg|_{2x}^{3x+1}$$......Tue, 18 Aug 2026 23:39:27 -0400https://liberty.io/prime-numbers-6k-1-mod-rule-new-discovery.htmlhttps://liberty.io/prime-numbers-6k-1-mod-rule-new-discovery.htmlI've noticed that although all primes follow the pattern of $6k - 1$ and $6k + 1$ which seems to be a somewhat known fact. However, I also noticed that all the primes of the pattern of $6k - 1$ only ...Tue, 18 Aug 2026 23:38:22 -0400https://liberty.io/understanding-the-properties-and-use-of-the-laplacian-matrix-and-its-n.htmlhttps://liberty.io/understanding-the-properties-and-use-of-the-laplacian-matrix-and-its-n.htmlI am reading the wikipedia article on the Laplacian matrix: https://en.wikipedia.org/wiki/Laplacian_matrix I don't understand what is the particular use of it; having the diagonals as the degree an......Tue, 18 Aug 2026 23:27:55 -0400https://liberty.io/is-0-a-natural-number.htmlhttps://liberty.io/is-0-a-natural-number.htmlIs there a consensus in the mathematical community, or some accepted authority, to determine whether zero should be classified as a natural number? It seems as though formerly $0$ was considered i......Tue, 18 Aug 2026 23:21:00 -0400https://liberty.io/what-is-wrong-with-my-solution-int-cos-2-x-tan-3x-dx.htmlhttps://liberty.io/what-is-wrong-with-my-solution-int-cos-2-x-tan-3x-dx.htmlI am trying to do this problem completely on my own but I can not get a proper answer for some reason $$\begin{align} \int \cos^2 x \tan^3x dx &=\int \cos^2 x \frac{ \sin^3 x}{ \cos^3 x}dx\\ &......Tue, 18 Aug 2026 23:17:46 -0400https://liberty.io/what-allows-us-divide-multiply-dx-in-calculus.htmlhttps://liberty.io/what-allows-us-divide-multiply-dx-in-calculus.htmlI've read nearly all of the threads on this topic but none seem to answer my question or lead me in the best direction. When performing U-substituion or even in it's most basic form: $y = 2x$, $dy=......Tue, 18 Aug 2026 23:16:08 -0400https://liberty.io/what-is-general-term-of-the-sequence-1-2-4-4-8-8-8-8-16-16-16-16-16-16.htmlhttps://liberty.io/what-is-general-term-of-the-sequence-1-2-4-4-8-8-8-8-16-16-16-16-16-16.htmlI have a sequence as follows: $$1, 2, 4, 4, 8, 8, 8, 8, 16, 16, 16, 16, 16, 16, 16, 16, 32, 32 ...$$ What will be the closed form of the above sequence for the nth term? I clearly see the patter......Tue, 18 Aug 2026 23:16:02 -0400https://liberty.io/how-to-find-a-minimal-polynomial.htmlhttps://liberty.io/how-to-find-a-minimal-polynomial.htmlI need to find minimal polynomial of $\alpha = \sqrt 2 + \sqrt [3] 3 $ over $\mathbb Q$ and prove that my result is minimal polynomial. How do I do that?...Tue, 18 Aug 2026 23:11:00 -0400https://liberty.io/laplace-transform-of-sin-at-by-definition.htmlhttps://liberty.io/laplace-transform-of-sin-at-by-definition.htmlI want to find Laplace transform of $\sin(at)$ by definition $$\displaystyle \mathcal L \left\{{\sin at}\right\} = \int_0^{\to +\infty}e^{-st}\sin at \, \mathrm dt$$ After taking integration by pa......Tue, 18 Aug 2026 23:08:22 -0400https://liberty.io/what-is-temperature-distribution-equation.htmlhttps://liberty.io/what-is-temperature-distribution-equation.htmlI have $T(x,y)$ (a function of two variables $x$ and $y$) and it is the temperature distribution on a metal plate. What does temperature distribution mean? what does its equation say? if I have the ...Tue, 18 Aug 2026 23:04:52 -0400https://liberty.io/minimal-realization-of-a-state-space-model.htmlhttps://liberty.io/minimal-realization-of-a-state-space-model.htmlMinimal realization of a transfer function is about cancelling out poles against zeros. But the minimal realization of a state space model is about cancelling out non-controllable and non-observable ...Tue, 18 Aug 2026 23:03:41 -0400https://liberty.io/what-does-it-mean-to-solve-or-find-solutions-in-mathematics.htmlhttps://liberty.io/what-does-it-mean-to-solve-or-find-solutions-in-mathematics.htmlSomething that has been really confusing me lately is that this equation has four solutions $$3x(x+1)(x^2+x+2)=16x(x+1)(2x+1)$$ But what does that mean? Until now solutions to me has meant, what a......Tue, 18 Aug 2026 22:49:57 -0400https://liberty.io/find-a-vector-mathbf-x-whose-image-under-t-is-b.htmlhttps://liberty.io/find-a-vector-mathbf-x-whose-image-under-t-is-b.htmlI am having trouble with this question and how to get the answer. With $T$ defined by $T(\mathbf x)=A\mathbf x$, find a vector $x$ whose image under $T$ is $b$. $$ A = \begin{pmatrix} 1 & -3......Tue, 18 Aug 2026 22:24:16 -0400https://liberty.io/calculating-multivariable-limit.htmlhttps://liberty.io/calculating-multivariable-limit.html$$\lim_{(x,y) \to (0,0)} \frac{x^2y}{x^3 + y}$$ I tried the following, but im not sure if this is allowed: Let $$t = x^3+y$$ $$(x,y)\to (0,0)$$ $$t\to 0$$ $$y=t-x^3$$ Now we will calculate the ......Tue, 18 Aug 2026 22:20:48 -0400https://liberty.io/how-do-i-determine-if-a-matrix-is-in-a-span-of-two-other-matrices.htmlhttps://liberty.io/how-do-i-determine-if-a-matrix-is-in-a-span-of-two-other-matrices.htmlIf \begin{align} A =\begin{bmatrix} 1 & 1 \\ -1 & 1 \\ \end{bmatrix} \end{align} and \begin{align}B = \begin{bmatrix} 1 & -1 \\ 1 & 0 \\ ......Tue, 18 Aug 2026 22:13:32 -0400https://liberty.io/degeneracy-of-outerplanar-graphs.htmlhttps://liberty.io/degeneracy-of-outerplanar-graphs.htmlDoes anyone know an elegant proof to the fact that every outerplanar graph has a vertex of degree at most 2 (and hence is 2-degenerate, since every subgraph is also outerplanar). I have a proof by ...Tue, 18 Aug 2026 22:04:07 -0400https://liberty.io/period-vs-time-period-of-sine-wave.htmlhttps://liberty.io/period-vs-time-period-of-sine-wave.htmlIt's weird I'm still confused about this, but usually when we figure out the period of a sine wave from its graph, it's in radians. But the true period should be in time, like how fast we are revol......Tue, 18 Aug 2026 21:47:43 -0400https://liberty.io/determine-if-a-statement-is-true-or-false.htmlhttps://liberty.io/determine-if-a-statement-is-true-or-false.htmlI need to determine if this statement is true or false: For each non-zero integer x, there is a negative rational number y such that $x = \frac{3}{6y+4}$. If I say x is rational it can be written ......Tue, 18 Aug 2026 21:33:43 -0400https://liberty.io/manual-calculation-doesn-039-t-match-wolfram-alpha-why.htmlhttps://liberty.io/manual-calculation-doesn-039-t-match-wolfram-alpha-why.htmlSay I want to evaluate this sum: $$\sum_{x=2}^\infty \ln(x^3+1)-\ln(x^3-1)$$ We can rewrite the sum as $$\ln\left(\prod_{x=2}^\infty \frac{x^3+1}{x^3-1}\right)$$ We can split the product into two ...Tue, 18 Aug 2026 21:32:34 -0400https://liberty.io/simplifying-math-equation-with-factorials.htmlhttps://liberty.io/simplifying-math-equation-with-factorials.htmlI am trying to understand how simplified that equation: $$\frac{(x-2)!}{(x-4)!}-46=\frac{x!}{(x-1)!}$$ In my book it was solved like this: but I really don't understand how it happened? Where did ......Tue, 18 Aug 2026 21:28:43 -0400https://liberty.io/find-the-perimeter-of-the-trapezoid-from-the-3-side-lengths.htmlhttps://liberty.io/find-the-perimeter-of-the-trapezoid-from-the-3-side-lengths.htmlWhat we know about the trapezoid ABCD (AB and CD are respectively its large and small bases) is that AB=9 cm,AC=6 cm,AD=8 cm,CD=4 cm. Determine the perimeter of the trapezoid. It is quite tricky.......Tue, 18 Aug 2026 21:27:38 -0400https://liberty.io/there-is-any-relation-between-pi-sqrt-2-or-a-generic-polygon.htmlhttps://liberty.io/there-is-any-relation-between-pi-sqrt-2-or-a-generic-polygon.htmlI'm a programmer, I'm always looking for new formulas and new way of computing things, to satisfy my curiosity I would like to know if there are any formulas, or I should say equalities, that make ......Tue, 18 Aug 2026 21:24:00 -0400https://liberty.io/probability-of-3-people-in-a-room-of-30-having-the-same-birthday.htmlhttps://liberty.io/probability-of-3-people-in-a-room-of-30-having-the-same-birthday.htmlI have been looking at the birthday problem (http://en.wikipedia.org/wiki/Birthday_problem) and I am trying to figure out what the probability of 3 people sharing a birthday in a room of 30 people ......Tue, 18 Aug 2026 21:21:30 -0400https://liberty.io/solve-the-function-for-x-in-terms-of-y.htmlhttps://liberty.io/solve-the-function-for-x-in-terms-of-y.htmlEquation: $$y = \frac{x + 1}{x - 1}$$ Task: Solve for $x$ in terms of $y$. My attempt: I just dont know how to go about doing this because the variable $x$ is in both the numerator and denominator....Tue, 18 Aug 2026 21:19:39 -0400https://liberty.io/solve-equation-with-absolute-value-of-complex-numbers.htmlhttps://liberty.io/solve-equation-with-absolute-value-of-complex-numbers.htmlFind all complex numbers $z$ such that $|z| + \overline{z}=9-3i$. Thinking about using with the fact that $z=a+bi$ and $|z| = \sqrt{a^2+b^2}$ in some way. Not sure how....Tue, 18 Aug 2026 21:18:35 -0400https://liberty.io/how-to-perform-abstract-integration.htmlhttps://liberty.io/how-to-perform-abstract-integration.htmlSuppose $X$ is a Banach space. Here, page $124$, there is this sentence If $\mu$ is a finite support on $X$ we can define its barycenter $\beta(\mu)=\beta_X(\mu) \in X$ by $$\beta(\mu)=\int ......Tue, 18 Aug 2026 21:12:39 -0400https://liberty.io/zebra-groups-and-counting-stripes.htmlhttps://liberty.io/zebra-groups-and-counting-stripes.htmlHow many stripes can you paint on a $2$-group of fixed size? A group of order $2^ap^b$ is solvable, by Burnside's theorem, so its chief factors are either abelian $2$-groups or abelian $p$-groups.......Tue, 18 Aug 2026 20:53:10 -0400https://liberty.io/what-is-the-difference-between-shifting-functions-and-shifting-parabol.htmlhttps://liberty.io/what-is-the-difference-between-shifting-functions-and-shifting-parabol.htmlFor shifting a parabola to the right, do we write a "$+$" sign or "$-$" sign in the equation? Is this the same way for shifting a function, as well? Here is the equation of parabola just for the ...Tue, 18 Aug 2026 20:51:10 -0400https://liberty.io/limit-of-absolute-values.htmlhttps://liberty.io/limit-of-absolute-values.htmlHow can we solve $\lim_{h\rightarrow0}{\frac{|x+h|-|x|}{h}}$ with $x,h \in \mathbb{R}$, and similar limit problems with absolute values? Note that although this represents $\frac{d}{dx}|x|$, I w......Tue, 18 Aug 2026 20:50:10 -0400https://liberty.io/why-is-this-called-the-cocycle-condition.htmlhttps://liberty.io/why-is-this-called-the-cocycle-condition.htmlMy question is about why the condition below is called the cocycle condition. Surely it is named after some interpretation of it in cohomology. Let $C$ be a site, and let $F$ be a fibered category......Tue, 18 Aug 2026 20:38:56 -0400https://liberty.io/integration-involving-fractional-part-function.htmlhttps://liberty.io/integration-involving-fractional-part-function.htmlFind the value of $$ \int_{0}^{1}{(\{2x\}-1)(\{3x\}-1)}dx$$ where $\{x\}$ denotes the fractional part function. $\{x\}=x-[x]$ ,where $[x]$ denotes the greatest integer function. My try :: Since th......Tue, 18 Aug 2026 20:38:36 -0400https://liberty.io/is-a-linear-transformation-onto-or-one-to-one.htmlhttps://liberty.io/is-a-linear-transformation-onto-or-one-to-one.htmlThe definition of one-to-one was pretty straight forward. If the vectors are lin.indep the transformation would be one-to-one. But could it also be onto? The definition of onto was a little more ...Tue, 18 Aug 2026 20:34:37 -0400https://liberty.io/what-does-it-mean-for-media-to-play-at-1-5x-or-nx-speed.htmlhttps://liberty.io/what-does-it-mean-for-media-to-play-at-1-5x-or-nx-speed.htmlSuppose I have a video that plays for 60 minutes at normal "1x" speed. I know that if I set the video to play at "2x" speed, then it should play for 30 minutes. Now, what if the......Tue, 18 Aug 2026 20:31:20 -0400https://liberty.io/find-all-eigenvalues-of-a-matrix-that-holds-a-a-5.htmlhttps://liberty.io/find-all-eigenvalues-of-a-matrix-that-holds-a-a-5.htmlI need to find all eigenvalues of a matrix that holds $A = A^{5}$. I have no idea where to start. Any directions please?...Tue, 18 Aug 2026 20:20:48 -0400https://liberty.io/adjoint-of-a-differential-operator.htmlhttps://liberty.io/adjoint-of-a-differential-operator.htmlI am self-studying differential equations using MIT's publicly available materials. One of the recitation exercises runs as follows: Define an inner or dot product on $\mathcal{C}[a,b]$ by \......Tue, 18 Aug 2026 20:20:24 -0400https://liberty.io/why-is-congruence-sign-a-equiv-b-pmod-c-closed.htmlhttps://liberty.io/why-is-congruence-sign-a-equiv-b-pmod-c-closed.htmlWhy is congruence sign $A\equiv B\pmod{C}$? Wouldn't it be more natural to use $A \equiv_{C} B$ instead? The former notation suggests that you take B and make some operation on it (like in case of ...Tue, 18 Aug 2026 20:16:42 -0400https://liberty.io/vi-arnold-ode-need-help-coming-to-grips-with-early-notation-in-the-boo.htmlhttps://liberty.io/vi-arnold-ode-need-help-coming-to-grips-with-early-notation-in-the-boo.htmlThis seems to be a somewhat common theme among users on here trying to start reading Arnold's ODE book. I have found a few similar questions but none have sufficiently answered my problems. ...Tue, 18 Aug 2026 20:06:10 -0400https://liberty.io/fair-coin-and-weighted-coin.htmlhttps://liberty.io/fair-coin-and-weighted-coin.htmlI have a fair coin and a weighted coin which lands heads 75% of the time. I pick a coin at random and flip it 5 times and get heads 4 times and tails once. What is the probability that I picked the ...Tue, 18 Aug 2026 20:05:46 -0400https://liberty.io/how-can-2-100-2-98-2-99-2-97-be-written-in-terms-of-its-prime-factors.htmlhttps://liberty.io/how-can-2-100-2-98-2-99-2-97-be-written-in-terms-of-its-prime-factors.htmlHow can $(2^{100}-2^{98})(2^{99}-2^{97})$ be written in terms of its prime factors? I tried to expand it: $2^{199}-4^{197}+2^{195}$ What do I do next? The answer choices are: A. $2^{100}\cdot3 \......Tue, 18 Aug 2026 19:56:31 -0400https://liberty.io/domain-and-range-of-piecewise-function-help.htmlhttps://liberty.io/domain-and-range-of-piecewise-function-help.htmlI have a graph of a piecewise function below, and I am having trouble figuring out the domain of the function in interval notation. My answers are: Domain: $[-7, -1)\cup(-1, \infty)$ Range: $[-6, \...Tue, 18 Aug 2026 19:53:18 -0400https://liberty.io/how-to-find-sin-and-cos-of-0-pi-2-pi-without-calculator-closed.htmlhttps://liberty.io/how-to-find-sin-and-cos-of-0-pi-2-pi-without-calculator-closed.htmlIn my notes it shows how to calculate by using the unit circle. But I do not know why the value of sin is the y coordinate and the value of cos is the x coordinate....Tue, 18 Aug 2026 19:36:22 -0400https://liberty.io/degenerate-solution-in-linear-programming.htmlhttps://liberty.io/degenerate-solution-in-linear-programming.htmlHow can I determine if a solution in a linear programming problem is degenerate without I use any software or the graphical display of the solution; For example in the model: $$\max\{2x_1 + 4x_2\......Tue, 18 Aug 2026 19:21:47 -0400https://liberty.io/proof-of-bolzano-039-s-theorem.htmlhttps://liberty.io/proof-of-bolzano-039-s-theorem.htmlI know one proof of Bolzano's Theorem, which can be sketched as follows: Set $f$ a continuous function in $[a,b]$ such that ${f(a)<0<f(b)}$. ${A=\{x:a<x<b \text{ and } f <0\in[a,x]......Tue, 18 Aug 2026 19:19:37 -0400https://liberty.io/explanation-of-the-derivation-of-the-formula-for-the-sum-of-an-arithme.htmlhttps://liberty.io/explanation-of-the-derivation-of-the-formula-for-the-sum-of-an-arithme.htmlI am trying to understand the derivation of the formula for the sum of an arithmetic sequence of the first $n$ terms. I do not understand what rules or reasoning allow two sequences to be added in ...Tue, 18 Aug 2026 19:14:08 -0400https://liberty.io/polynomial-function-given-roots-and-a-point.htmlhttps://liberty.io/polynomial-function-given-roots-and-a-point.htmlA 3rd degree polynomial has roots at x=-2i and x=5. The y-intercept is (0,25). Write an equation for this function in factored form with real coefficients. My first hack is to say that in factored ...Tue, 18 Aug 2026 19:11:17 -0400https://liberty.io/how-to-find-the-transition-matrix-for-ordered-basis-of-2x2-diagonal-ma.htmlhttps://liberty.io/how-to-find-the-transition-matrix-for-ordered-basis-of-2x2-diagonal-ma.htmlThe problem: For the vector space of lower triangular matrices with zero trace, given ordered basis: $B=${$$ \begin{bmatrix} -5 & 0 \\ 4 & 5 \\ \end{bmatrix}......Tue, 18 Aug 2026 19:08:21 -0400https://liberty.io/what-does-inversion-mean.htmlhttps://liberty.io/what-does-inversion-mean.htmlI am in highschool taking some advanced math courses and I have some questions about terminology. There appears to be more definitions to the meaning of inversion in math than I can count. I'm hea......Tue, 18 Aug 2026 19:07:40 -0400https://liberty.io/why-does-a-flat-finite-type-morphism-of-irreducible-noetherian-schemes.htmlhttps://liberty.io/why-does-a-flat-finite-type-morphism-of-irreducible-noetherian-schemes.htmlBasically the title, I came across this statement reading some notes on dimension of fibers of flat maps. (I don't know if Noetherian actually matters for this question) I think that for maps of ...Tue, 18 Aug 2026 19:04:30 -0400https://liberty.io/let-f-k-be-two-fields-f-subset-k-and-suppose-f-x-g-x-in-f-x-are-relati.htmlhttps://liberty.io/let-f-k-be-two-fields-f-subset-k-and-suppose-f-x-g-x-in-f-x-are-relati.htmlLet $F,K$ be two fields $F \subset K$ and suppose $f(x),g(x) \in F[x]$ are relatively prime in $F[x].$ Prove they are relatively prime in $K[x].$ Suppose $f(x)$ and $g(x)$ are relatively prime in ......Tue, 18 Aug 2026 19:04:24 -0400https://liberty.io/four-men-hats-and-probability.htmlhttps://liberty.io/four-men-hats-and-probability.htmlI encountered the four men in hats puzzle for the first time today. My question is about a realisation I (think I) had while arriving at the solution, but I have no idea whether I've made a mistake ...Tue, 18 Aug 2026 19:04:10 -0400https://liberty.io/finding-polynomal-function-with-given-zeros-and-one-zero-is-a-square-r.htmlhttps://liberty.io/finding-polynomal-function-with-given-zeros-and-one-zero-is-a-square-r.htmlI've been having trouble with this problem: Find a polynomial function of minimum degree with $-1$ and $1-\sqrt{3}$ as zeros. Function must have integer coefficients. When I tried it, I got this: \...Tue, 18 Aug 2026 19:03:24 -0400https://liberty.io/what-is-the-domain-of-z-arcsin-dfrac-x-y.htmlhttps://liberty.io/what-is-the-domain-of-z-arcsin-dfrac-x-y.htmlI get that it should be $|y|>|x|$ and in the Wolfram it looks like this. But when I graph it by hand is that it should be only the "upper" part of intersection and not the "bottom" part as well,......Tue, 18 Aug 2026 19:02:19 -0400https://liberty.io/difference-between-variance-and-covariance.htmlhttps://liberty.io/difference-between-variance-and-covariance.htmlWhat exactly is the physical interpretation of variance (in terms of a data set) and the difference between variance and co-variance matrices?...Tue, 18 Aug 2026 18:54:23 -0400https://liberty.io/area-of-intersection-between-two-circles-duplicate.htmlhttps://liberty.io/area-of-intersection-between-two-circles-duplicate.htmlSuppose you have 2 circles that intersect each other in such a way that each circle passes through the other's center. What is the area between the circle(or common area) i.e. area between the cent......Tue, 18 Aug 2026 18:51:26 -0400https://liberty.io/how-does-one-prove-rodrigues-039-formula-for-legendre-polynomials.htmlhttps://liberty.io/how-does-one-prove-rodrigues-039-formula-for-legendre-polynomials.htmlI am trying to prove that $\frac{1}{n!\space2^n}\frac{d^n}{dx^n}\{(x^2-1)^n\}=P_n(x)$, where $P_n(x)$ is the Legendre Polynomial of order n. I've been told that the proof uses complex analysis, of......Tue, 18 Aug 2026 18:44:45 -0400https://liberty.io/what-algebraic-topology-book-to-read-after-hatcher-039-s.htmlhttps://liberty.io/what-algebraic-topology-book-to-read-after-hatcher-039-s.htmlI've currently finished chapter 2 of his book and done all the exercises of in chapter 0, 1 and 2. Was wondering when I finished reading this book what book do I read next in algebraic topology?...Tue, 18 Aug 2026 18:38:16 -0400https://liberty.io/finding-magnitude-of-a-function.htmlhttps://liberty.io/finding-magnitude-of-a-function.htmlI want to calculate the magnitude of the following function: $x(t)=\dfrac{i}{t}(e^{it}+1)$ I did the following to find the magnitude: $x(t)=\dfrac{i}{t}(1+\cos t+i \sin t)$ $|x(t)|=\sqrt{\dfrac{\si......Tue, 18 Aug 2026 18:29:02 -0400https://liberty.io/swing-twist-decomposition-for-quaternion-rotations-verification.htmlhttps://liberty.io/swing-twist-decomposition-for-quaternion-rotations-verification.htmlPRZEMYSLAW DOBROWOLSKI has written a paper that (I think) can be applied to swing-twist decompositions for quaternion rotations called "SWING-TWIST DECOMPOSITION IN CLIFFORD ALGEBRA" . I tried to ......Tue, 18 Aug 2026 18:15:28 -0400https://liberty.io/what-is-the-maximum-norm-induced-by-the-maximum-norm-for-mathbb-r-n.htmlhttps://liberty.io/what-is-the-maximum-norm-induced-by-the-maximum-norm-for-mathbb-r-n.htmlI'm reading a problem which uses the matrix norm induced by the maximum norm for $\mathbb{R}^n$. But I do know what it is exactly, and googling it gives different results....Tue, 18 Aug 2026 18:10:31 -0400https://liberty.io/how-is-the-joint-proportionality-joint-variation-equation-i-e-y-kxz-lo.htmlhttps://liberty.io/how-is-the-joint-proportionality-joint-variation-equation-i-e-y-kxz-lo.htmlIf $y$ is proportional to $x$, and $y$ is also proportional to $z$, then how are we able to arrive at the equation: $y = kxz$ ? My understanding so far of proportionality, which comes from this ...Tue, 18 Aug 2026 18:10:19 -0400https://liberty.io/does-the-series-sum-frac-1-n-ln-n-3-2-converge-or-diverge.htmlhttps://liberty.io/does-the-series-sum-frac-1-n-ln-n-3-2-converge-or-diverge.htmlConsider $$\sum \frac{1}{n\ \ln^{3/2}(n)}$$ The ratio test is inconclusive. The root test is inconclusive. And it seems right that $\frac{1}{n\ (\ln(n))^{3/2}}\leq\frac{1}{n}$ which diverges, but......Tue, 18 Aug 2026 18:03:03 -0400https://liberty.io/derivative-of-arcsin-x-in-1.htmlhttps://liberty.io/derivative-of-arcsin-x-in-1.htmlI need to find the derivative $\arcsin'(-1)$. I know that $(\arcsin)'(x)=\frac{1}{\sqrt{1-x^2}}$. I also know that $$f_{+}'(b)=\lim_{x \to b+}f'(x) \tag{1}$$ and that the right derivate is $+\infty......Tue, 18 Aug 2026 18:01:01 -0400https://liberty.io/rolling-a-fair-die.htmlhttps://liberty.io/rolling-a-fair-die.htmlYou and your friend play a game in which you and your friend take turns rolling a fair six-sided die and keep a running tally of the sum of the results of all rolls made. Play continues until either ...Tue, 18 Aug 2026 18:00:25 -0400https://liberty.io/find-the-area-common-to-the-inside-of-the-cardioid-r-1-sin-theta-and-t.htmlhttps://liberty.io/find-the-area-common-to-the-inside-of-the-cardioid-r-1-sin-theta-and-t.htmlFunction Plotter graph: I think the formula is $$A = \frac 1 2 \int_{\alpha}^{\beta} (\text{outer})^2 - (\text{inner})^2 d\theta$$ where $\alpha, \beta$ are where they intersect in $[0, 2\pi]$. ......Tue, 18 Aug 2026 18:00:14 -0400https://liberty.io/prove-sinh-x-x-for-all-x-0.htmlhttps://liberty.io/prove-sinh-x-x-for-all-x-0.htmlI did a proof for $\sinh x > x$ for all $x > 0$. But I am not sure if the proof is mathematically valid. I started by showing that $\frac{d}{dx} \sinh x = \cosh x$ and that the limit of $\co......Tue, 18 Aug 2026 17:57:25 -0400https://liberty.io/find-arcsin-frac-4-5-arccos-frac-1-sqrt-50.htmlhttps://liberty.io/find-arcsin-frac-4-5-arccos-frac-1-sqrt-50.htmlI can't solve this problem due to the irrational number. Used 2 methods: (1) with Pythagorean theorem I got: \begin{align} \arccos(\frac{1}{\sqrt{50}})&=\theta\\\frac{1}{\sqrt{50}}&=\cos(\t......Tue, 18 Aug 2026 17:55:25 -0400https://liberty.io/property-of-a-kite-inscribed-in-a-square.htmlhttps://liberty.io/property-of-a-kite-inscribed-in-a-square.htmlA friend of mine presented the following problem to me: Say you have a square $ABCD$ and a kite $BCEF$ (a kite is a quadrilateral object such that $\overline{BC}$ and $\overline{BF}$ have the same ......Tue, 18 Aug 2026 17:41:45 -0400https://liberty.io/what-distinguishes-039-family-039-vs-039-set-039-of-functions.htmlhttps://liberty.io/what-distinguishes-039-family-039-vs-039-set-039-of-functions.htmlSource: Stewart, James. Calculus: Early Transcendentals (6 edn 2007). [p. 50 Top:]   To understand how the expression for a function relates to its graph, it’s helpful to graph a family of ...Tue, 18 Aug 2026 17:41:18 -0400https://liberty.io/a-question-about-irreducible-representations-of-so-3-group.htmlhttps://liberty.io/a-question-about-irreducible-representations-of-so-3-group.htmlI have a perplexity about the representations of the group of 3D rotations $SO(3)$. I'm reading the following text which is describing the irreducible representations of $SO(3)$ that are associated......Tue, 18 Aug 2026 17:34:26 -0400https://liberty.io/evaluating-int-2x-2-1-e-x-2-dx.htmlhttps://liberty.io/evaluating-int-2x-2-1-e-x-2-dx.html$$\int(2x^2+1)e^{x^2}dx$$ The answer of course: $$\int(2x^2+1)e^{x^2}\,dx=xe^{x^2}+C$$ But what kind of techniques we should use with problem like this ?...Tue, 18 Aug 2026 17:18:29 -0400https://liberty.io/pentation-notation-how-does-it-work-duplicate.htmlhttps://liberty.io/pentation-notation-how-does-it-work-duplicate.htmlWhen going through with learning Grahams number, I got stuck at $$3↑↑↑3$$ Working it through, we have $$3↑3=3^3$$ $$3↑↑3=3^{3^3}=3↑(3↑3)$$ As such, it would appear to me that $$3↑↑↑3=3^{3^{3^......Tue, 18 Aug 2026 17:10:43 -0400https://liberty.io/is-there-a-way-to-solve-an-equation-with-both-sine-and-cosine-in-it.htmlhttps://liberty.io/is-there-a-way-to-solve-an-equation-with-both-sine-and-cosine-in-it.htmlI'm trying to solve a system of equations. I end up with sine and cosine in one equation and the parameter of both as the last unknown. Like this: num1 = num2 * sin(start) + num3 * cos(start) num1, ...Tue, 18 Aug 2026 17:06:23 -0400https://liberty.io/what-is-the-difference-between-a-unit-simplex-and-a-probability-simple.htmlhttps://liberty.io/what-is-the-difference-between-a-unit-simplex-and-a-probability-simple.htmlThe unit simplex is the $n$-dimensional simplex determined by the zero vector and the unit vectors, i.e., $0,e_1, \ldots,e_n\in\mathbf R^n$. It can be expressed as the set of vectors that satisfy ......Tue, 18 Aug 2026 16:57:16 -0400https://liberty.io/solving-ax-b-when-x-and-b-are-given.htmlhttps://liberty.io/solving-ax-b-when-x-and-b-are-given.htmlI am trying to study least square and linear regression and I understand the solution for $Ax = b$ when x is the unknown and the LS solution is given by $(A^TA)^{-1}A^TA$. Now, I was wondering if ...Tue, 18 Aug 2026 16:55:53 -0400https://liberty.io/calculate-average-wind-direction.htmlhttps://liberty.io/calculate-average-wind-direction.htmlWhat is the best way to average wind direction? I have found many conflicting suggestions elsewhere. Best one I saw is to average SINs of all angles in radians and take inverse SIN of the result. ......Tue, 18 Aug 2026 16:51:40 -0400