https://liberty.ioCurated star highlights with fresh energy.en-usWed, 19 Aug 2026 22:23:12 -0400https://liberty.io/creating-a-rotating-partner-round-robin-schedule.htmlhttps://liberty.io/creating-a-rotating-partner-round-robin-schedule.htmlFirst, I'm going to apologize for a lack of proper terminology on this post. It's been well over a decade since my days in the classroom, so I've regressed back to a simple math layman here. (As a ...Thu, 20 Aug 2026 02:22:44 -0400https://liberty.io/automorphism-and-inner-automorphism-of-d-4.htmlhttps://liberty.io/automorphism-and-inner-automorphism-of-d-4.htmlI want to find $Aut(D_4)$ and $Inn(D_4)$. The group $D_4 = \{1,r,r^2,r^3, s,rs,r^2s,r^3s\}$. An automorphism of a group $G$ is an isomorphism from the group to itself. An inner automorphism of a g......Thu, 20 Aug 2026 01:54:55 -0400https://liberty.io/notation-what-does-this-summation-mean.htmlhttps://liberty.io/notation-what-does-this-summation-mean.htmlI need to understand what the summation below mean. I specifically do not understand the $i<j$ below the summation. $$\sum_{\substack{ i<j}}^{N} d_{ij}$$ Insight about the equation There......Thu, 20 Aug 2026 01:53:28 -0400https://liberty.io/what-is-a-level-set.htmlhttps://liberty.io/what-is-a-level-set.htmlI have the ellipsoid $x^2 + 2y^2 + z^2 = 4$. It is mentioned that I can rewrite this equation as a level set $f(x,y) = x^2 + 2y^2 + z^2$. Of course I can look at Wikipedia where a level set is ...Thu, 20 Aug 2026 01:32:43 -0400https://liberty.io/online-video-course-on-topology.htmlhttps://liberty.io/online-video-course-on-topology.htmlI am at the beginning of learning topology, I found PDFs like topology without tears, but I wonder if there is a full university-like course online, on MIT open courseware I did not found anything ......Thu, 20 Aug 2026 01:19:52 -0400https://liberty.io/d-dx-notation-explanation-please.htmlhttps://liberty.io/d-dx-notation-explanation-please.htmlI know how to derive, I know how to integrate. I know what to do when I see $\frac{d}{dx}$ and such but what does it really mean? I know it means something like derive in terms of $x$, but whats the ...Thu, 20 Aug 2026 01:18:29 -0400https://liberty.io/a-slightly-problematic-integral-int-1-x-4-1-1-4-mathrm-d-x.htmlhttps://liberty.io/a-slightly-problematic-integral-int-1-x-4-1-1-4-mathrm-d-x.htmlQuestion. To find the integral of- $$\int {\frac{1}{(x^4+1)^\frac{1}{4}} \, \mathrm{d}x}$$ I have tried substituting $x^4+1$ as $t$, and as $t^4$, but it gives me an even more complex integral. An......Thu, 20 Aug 2026 01:14:11 -0400https://liberty.io/heron-039-s-formula.htmlhttps://liberty.io/heron-039-s-formula.htmlHeron's Formula gives the area of a triangle when the length of all three sides are known. There is no need to calculate angles or other distances in the triangle first to determine its area. The f......Thu, 20 Aug 2026 01:14:00 -0400https://liberty.io/how-do-i-solve-y-039-039-4y-0.htmlhttps://liberty.io/how-do-i-solve-y-039-039-4y-0.htmlThis problem is in Penney's Elementary Differential Equations, listed as a reducible, 2nd-order DE. The chapter has taught two techniques to be used, which are for when either $x$ or $y(x)$ is mis......Thu, 20 Aug 2026 01:11:51 -0400https://liberty.io/probability-of-a-given-neg-b.htmlhttps://liberty.io/probability-of-a-given-neg-b.htmlIf we have the probabilities of $P(A)$, $P(B)$ and $P(A\mid B)$, how can we calculate the probability of $P(A\mid\neg B)$ ? Does $A$ depends on $\neg B$ if it Actually depends on $B$ ?...Thu, 20 Aug 2026 01:06:29 -0400https://liberty.io/n-th-derivative-of-e-1-x.htmlhttps://liberty.io/n-th-derivative-of-e-1-x.htmlI am trying to find the $n$'th derivative of $f(x)=e^{1/x}$. When looking at the first few derivatives I noticed a pattern and eventually found the following formula $$\frac{\mathrm d^n}{\mathrm d......Thu, 20 Aug 2026 01:05:25 -0400https://liberty.io/determinant-of-matrix-exponential.htmlhttps://liberty.io/determinant-of-matrix-exponential.htmlSuppose $A$ is a $n \times n$ constant matrix. How can I prove $\det(e^A) = e^{\displaystyle \sum_{\lambda_i\in\sigma(A)} \lambda_i}$, where $\sigma(A)$ is the multiset of eigenvalues of $A$? The ...Thu, 20 Aug 2026 01:02:53 -0400https://liberty.io/is-littlewood-039-s-law-true.htmlhttps://liberty.io/is-littlewood-039-s-law-true.htmlLittlewood's law states that a person can expect to experience an event with odds of one in a million (defined by the law as a "miracle") at the rate of about one per month. This is attributed to the ...Thu, 20 Aug 2026 01:00:29 -0400https://liberty.io/finding-the-area-of-a-15-75-90-triangle-with-the-length-of-the-hypoten.htmlhttps://liberty.io/finding-the-area-of-a-15-75-90-triangle-with-the-length-of-the-hypoten.htmlSo there is a right triangle $ABC$ with $m∠C=90°$, $m∠B=75°$, and $BC\ (the \ hypotenuse)=12 cm$. I want to find the area of this triangle It would look something like this: Note: I have already ...Thu, 20 Aug 2026 00:58:39 -0400https://liberty.io/how-do-i-calculate-the-indefinite-integral-with-a-square-root.htmlhttps://liberty.io/how-do-i-calculate-the-indefinite-integral-with-a-square-root.htmlFor example, how do I calculate the indefinite integral of the square root of x? I understand how the power rule works, but im confused by the square root. I would appreciate any help given!...Thu, 20 Aug 2026 00:53:09 -0400https://liberty.io/calculating-inverse-tangent.htmlhttps://liberty.io/calculating-inverse-tangent.htmlWhen calculating the inverse tangent of a degree, the calculator will always give an angle between 90 degrees and –90 degrees. But I want to find the positive value of the negative angle. Do I add ......Thu, 20 Aug 2026 00:40:57 -0400https://liberty.io/what-is-the-significance-of-convergence-divergence-of-series.htmlhttps://liberty.io/what-is-the-significance-of-convergence-divergence-of-series.htmlThere are many ways and mathematical methods to test the convergence or Divergence of a series. What does Convergence or divergence indicate practically in the real world? Why is the convergence of ...Thu, 20 Aug 2026 00:37:40 -0400https://liberty.io/choosing-4-men-from-6-and-4-women-from-7-with-a-restriction.htmlhttps://liberty.io/choosing-4-men-from-6-and-4-women-from-7-with-a-restriction.htmlA team of 4 men and 4 women has to be chosen from 6 men and 7 women, so total combinations: $^6C_{4} \times ^7C_{4} = 525$ If a restriction is placed that some two women will always be chosen tog......Thu, 20 Aug 2026 00:37:25 -0400https://liberty.io/logic-symbol-forall-before-or-after-statement.htmlhttps://liberty.io/logic-symbol-forall-before-or-after-statement.htmlI have question about what is difference about two statements below. $$\forall a,b,c, \ \ a+(b+c)=(a+b)+c $$ or $$a+(b+c)=(a+b)+c , \ \forall a,b,c$$ I usaually see second in textbooks but my ...Thu, 20 Aug 2026 00:16:52 -0400https://liberty.io/epsilon-delta-proof-of-a-quadratic-limit.htmlhttps://liberty.io/epsilon-delta-proof-of-a-quadratic-limit.htmlI just started learning about limits and their definition. I'm practicing with some proofs that involve the epsilon-delta definition of a limit.While I absolutely have no problem with linear functi......Wed, 19 Aug 2026 23:58:27 -0400https://liberty.io/what-is-the-actual-curvature-of-earth-either-per-kilometer-km-of-sea-l.htmlhttps://liberty.io/what-is-the-actual-curvature-of-earth-either-per-kilometer-km-of-sea-l.htmlFirst of all, the following question with it's answers are the main reason I put this question, because as I will show the question wording is bad. How to calculate curvature of Earth per surface ...Wed, 19 Aug 2026 23:57:26 -0400https://liberty.io/theorem-of-existence-and-uniqueness-for-first-order-linear-differentia.htmlhttps://liberty.io/theorem-of-existence-and-uniqueness-for-first-order-linear-differentia.htmlThe theorem of existence and uniqueness is: Let $ y'+p(x)y=g(x) $, $ y(x_{0})=y_{0} $ be a first order linear differential equation such that $ p(x) $ and $ g(x) $ are both continuous for $ a<x......Wed, 19 Aug 2026 23:45:46 -0400https://liberty.io/conditions-for-taking-a-limit-into-an-infinite-sum.htmlhttps://liberty.io/conditions-for-taking-a-limit-into-an-infinite-sum.htmlSuppose $f\left(x\right)={\displaystyle \sum_{n=0}^{\infty}g_{n}\left(x\right)}$ under what conditions is it true that: $$\lim_{x\to c}f\left(x\right)={\displaystyle \sum_{n=0}^{\infty}\lim_{x\to c......Wed, 19 Aug 2026 23:30:51 -0400https://liberty.io/full-and-reduced-svd-of-a-3x3-matrix.htmlhttps://liberty.io/full-and-reduced-svd-of-a-3x3-matrix.htmlI currently studying for an exam, and I'm currently working my way through some old exam problems and I'm currently at the following. First, we have a matrix $A= \begin{bmatrix} 2&0&0\\2&......Wed, 19 Aug 2026 23:29:52 -0400https://liberty.io/what-would-base-1-be.htmlhttps://liberty.io/what-would-base-1-be.htmlBase $10$ uses these digits: $\{0,1,2,3,4,5,6,7,8,9\};\;$ base $2$ uses: $\{0,1\};\;$ but what would base $1$ be? Let's say we define Base $1$ to use: $\{0\}$. Because $10_2$ is equal to $010_2$, ......Wed, 19 Aug 2026 23:29:18 -0400https://liberty.io/dot-product-of-a-vector-and-del-operator.htmlhttps://liberty.io/dot-product-of-a-vector-and-del-operator.htmlHow did this expansion come about. What is the physical significance of this expansion. And what is the significance of $$\vec{A} . \nabla {\vec{A}}$$ And what does this mathematically represent....Wed, 19 Aug 2026 23:25:29 -0400https://liberty.io/find-the-interpolating-polynomial-of-degree-3-that-interpolates-f-x-x-.htmlhttps://liberty.io/find-the-interpolating-polynomial-of-degree-3-that-interpolates-f-x-x-.htmlFind the interpolating polynomial of degree $3$ that interpolates $f(x) = x^3$ at the nodes $x_0=0, x_1 = 1, x_2=2, x_3 = 3$. Here are my workings below The basic Lagrange polynomials are: $$L_......Wed, 19 Aug 2026 23:12:55 -0400https://liberty.io/intuition-of-euclidean-norm.htmlhttps://liberty.io/intuition-of-euclidean-norm.htmlI know the following : A norm is Euclidean iff it respects the parallelogram law. The problem is that the parallelogram law is not very intuitive geometrically. So I am wondering if there is a ...Wed, 19 Aug 2026 23:03:23 -0400https://liberty.io/can-you-give-me-a-specific-representation-of-q-weyl-algebra.htmlhttps://liberty.io/can-you-give-me-a-specific-representation-of-q-weyl-algebra.htmlCan you give me a specific representation of q-Weyl algebra over field k,i.e $K<x,y>/<yx=qxy>$ for proving that {$x^{i}y^{j}$, i,j$\in$$Z$} is indeed a basis in $k<x,y>/<yx=qxy......Wed, 19 Aug 2026 22:56:48 -0400https://liberty.io/setting-up-an-english-pool-table.htmlhttps://liberty.io/setting-up-an-english-pool-table.htmlA friend and I were playing some English pool yesterday. When you rack the balls it has the specific set up in the picture where the 8 ball must be in that central position and the balls are laid out ...Wed, 19 Aug 2026 22:48:44 -0400https://liberty.io/what-is-the-inverse-function-of-cot-x.htmlhttps://liberty.io/what-is-the-inverse-function-of-cot-x.htmlMy Math skills got real rustic over time so bear with a (hopefully) simple question: I'm looking for a way to compute $n$ from this formula, where $\cot(x)$ is the cotangent. $$\cot(n) = \frac{\pi}......Wed, 19 Aug 2026 22:39:02 -0400https://liberty.io/why-does-a-times-its-inverse-equal-to-the-identity-matrix-duplicate.htmlhttps://liberty.io/why-does-a-times-its-inverse-equal-to-the-identity-matrix-duplicate.htmlI was trying to come up with a proof of why: $AA^{-1} = I$. If we know that: $A^{-1}A = I$, then $A(A^{-1}A) = A \implies (AA^{-1})A = A$. However I don't like setting $AA^{-1} = I$ for fear that......Wed, 19 Aug 2026 22:36:28 -0400https://liberty.io/even-numbers-have-more-factors-than-odd-numbers.htmlhttps://liberty.io/even-numbers-have-more-factors-than-odd-numbers.htmlThis was an exercise to show that, in a sense, the even numbers have more prime factors than the odds, but--if it's right-- I still have a question. As an heuristic calculation, we could take a la......Wed, 19 Aug 2026 22:32:12 -0400https://liberty.io/finding-the-real-potential-and-streamfunction-from-a-complex-potential.htmlhttps://liberty.io/finding-the-real-potential-and-streamfunction-from-a-complex-potential.htmlI'm trying to solve the following problem given on a previous exam in my fluid mechanics course: A stationary flow is given as the following complex velocity potential: $$ \beta(z) = \ln(z - a) + \......Wed, 19 Aug 2026 22:31:07 -0400https://liberty.io/what-are-the-classic-or-great-books-for-algebra-geometry-and-trigonome.htmlhttps://liberty.io/what-are-the-classic-or-great-books-for-algebra-geometry-and-trigonome.htmlThere are classic textbooks for Calculus like Spivak, Courant, Apostol, etc that do a fantastic job at explaining the fundamental concepts and theory along with great problem sets. My question is the ...Wed, 19 Aug 2026 22:29:44 -0400https://liberty.io/why-the-derivative-is-the-rate-of-change-of-the-function.htmlhttps://liberty.io/why-the-derivative-is-the-rate-of-change-of-the-function.htmlThe price of an action follow the function $f(t)=e^{-t}$. The question is, what is the rate of change of the price ? For me, the rate of change is the rate of change by unit of time $h$, i.e. $$\fr......Wed, 19 Aug 2026 22:24:56 -0400https://liberty.io/about-the-definition-of-local-fields.htmlhttps://liberty.io/about-the-definition-of-local-fields.htmlI read in a book that the definition of local fields is: A field $K$ for which with respect to a discrete valuation $v$, the residue field is finite and $K$ is complete with respect to $v$. Howeve......Wed, 19 Aug 2026 22:23:24 -0400https://liberty.io/trying-to-understand-discontinuities-and-domains.htmlhttps://liberty.io/trying-to-understand-discontinuities-and-domains.htmlI am really struggling with this concept and I feel like no math book clarifies it sufficiently. First off: I assume that if a one-sided limit exists then it is a real value and not $\pm \infty$.......Wed, 19 Aug 2026 22:22:15 -0400https://liberty.io/stars-and-bars-everyone-gets-at-least-one.htmlhttps://liberty.io/stars-and-bars-everyone-gets-at-least-one.htmlIn how many ways can we distribute $11$ pieces of candy among $4$ kids, provided that every kid gets at least $1$ piece of candy? I know how to solve this problem one way, but why does this way no......Wed, 19 Aug 2026 22:10:38 -0400https://liberty.io/what-symbol-expresses-less-than-approximately.htmlhttps://liberty.io/what-symbol-expresses-less-than-approximately.htmlSuppose, I want to state that $a$ is less than $b$. However, I do not know $b$ exactly, but only that it is approximately $c$. With other words I want to state that $a$ is less than some value that......Wed, 19 Aug 2026 21:58:16 -0400https://liberty.io/rounding-up-to-two-decimal-points.htmlhttps://liberty.io/rounding-up-to-two-decimal-points.htmlI just have a question regarding my answers in my exam paper, my teacher said to give the answer to two decimal points, I got the correct answer (2.89) but I rounded it up to 2.90. Now my teacher d......Wed, 19 Aug 2026 21:53:47 -0400https://liberty.io/solving-a-neumann-problem-in-a-disk.htmlhttps://liberty.io/solving-a-neumann-problem-in-a-disk.html$\Delta u(x,y) = x^2 $ $x^2 + y^2 <9$ On the boundary $x^2 + y^2 = 9$, $\frac {\delta u}{\delta n} = y$. I've seen similar problems solved in texts (Strauss, Bleecker, Stavroulakis), but whe......Wed, 19 Aug 2026 21:49:38 -0400https://liberty.io/let-x-n-and-y-n-be-given-sequences-and-define-z-n-to-be-the-shuffled-s.htmlhttps://liberty.io/let-x-n-and-y-n-be-given-sequences-and-define-z-n-to-be-the-shuffled-s.htmlLet $(x_n)$ and $(y_n)$ be given sequences and define $(z_n)$ to be the "shuffled" sequence $(x_1,y_1,x_2,y_2,x_3,y_3,..., x_n,y_n,...)$. Prove that $(z_n)$ is convergent if and only if $(x_n)$ and......Wed, 19 Aug 2026 21:42:53 -0400https://liberty.io/2-log-2-3-cdot-log-3-2-without-using-a-calculator.htmlhttps://liberty.io/2-log-2-3-cdot-log-3-2-without-using-a-calculator.htmlI want to express this into a single logarithm without using the calculator. $$2\log_2 3 \cdot \log_3 2$$ My calculator's log function has only log base 10. It's easy to change the base to 10 an......Wed, 19 Aug 2026 21:33:41 -0400https://liberty.io/derivation-of-formula-for-finding-median-for-grouped-data.htmlhttps://liberty.io/derivation-of-formula-for-finding-median-for-grouped-data.htmlI know the formula of formula for finding median for grouped data that is $$\mathrm{Median} = L_m + \left [ \frac { \frac{n}{2} - F_{m-1} }{f_m} \right ] \times c$$ and I know what all the letters ......Wed, 19 Aug 2026 21:31:48 -0400https://liberty.io/volumes-of-n-balls-what-is-so-special-about-n-5.htmlhttps://liberty.io/volumes-of-n-balls-what-is-so-special-about-n-5.htmlThe volume of an $n$-dimensional ball of radius $1$ is given by the classical formula $$V_n=\frac{\pi^{n/2}}{\Gamma(n/2+1)}.$$ For small values of $n$, we have $$V_1=2\qquad$$ $$V_2\approx 3.14$$ ......Wed, 19 Aug 2026 21:15:00 -0400https://liberty.io/period-of-sin-x-cos-x.htmlhttps://liberty.io/period-of-sin-x-cos-x.htmlThe period of $\sin(x)$ is $2\pi$ and $\cos(x)$ is $2\pi$. And the period of $\sin(x)+\cos(x)$ is also $2\pi$. Why it is so?...Wed, 19 Aug 2026 21:14:44 -0400https://liberty.io/difference-between-rolle-039-s-theorem-and-mean-value-theorem.htmlhttps://liberty.io/difference-between-rolle-039-s-theorem-and-mean-value-theorem.htmlThe Rolle's theorem is defined as: Suppose f is continous on $[\alpha,\beta]$ such that $f(\alpha) = f(\beta)$ $f′$ exists on $(\alpha,\beta)$ Then f has a local maximun or minimum at some $c \in (\...Wed, 19 Aug 2026 21:13:19 -0400https://liberty.io/ball-thrown-straight-up-modern-elementary-differential-equations.htmlhttps://liberty.io/ball-thrown-straight-up-modern-elementary-differential-equations.htmlI'm reading this book: Modern Elementary Differential Equations I understand that the second derivative of x (position) with respect to t is the downward effect gravity (acceleration) has on the b......Wed, 19 Aug 2026 21:10:30 -0400https://liberty.io/solve-a-pde-with-feynman-kac-formula.htmlhttps://liberty.io/solve-a-pde-with-feynman-kac-formula.htmlSo there is the following PDE given: $\frac{\partial}{\partial t}f(t,x) + rx\frac{\partial}{\partial x}f(t,x)+\frac{\sigma^2 x^2}{2}\frac{{\partial}^2}{\partial x^2}f(t,x) = rf(t,x)$ With boundary ...Wed, 19 Aug 2026 21:01:54 -0400https://liberty.io/how-many-ways-to-make-scalene-triangles-in-a-nonagon.htmlhttps://liberty.io/how-many-ways-to-make-scalene-triangles-in-a-nonagon.htmlThe title pretty much says it all, but: How many distinct scalene triangles can be formed by connecting three vertices of a regular nonagon? Any answers would be greatly appreciated, thank you!......Wed, 19 Aug 2026 20:58:31 -0400https://liberty.io/proof-question-let-v-be-a-finite-dimensional-vector-space-and-let-t-v-.htmlhttps://liberty.io/proof-question-let-v-be-a-finite-dimensional-vector-space-and-let-t-v-.htmlLet V be a finite-dimensional vector space, and let $T:V \rightarrow V$ be linear. a). if $\mathrm{rank} (T)= \mathrm{rank} (T^2)$, prove $R(T) \bigcap N(T)$={0}. $N(T)$ is defined by $T(0_V)=0_......Wed, 19 Aug 2026 20:55:57 -0400https://liberty.io/solve-z-5-32-include-complex-solutions.htmlhttps://liberty.io/solve-z-5-32-include-complex-solutions.htmlQuestion: Solve $z^5 =32$ (including complex solutions) My Working : $$ z^5=32 $$ $$ z^5=32cis(0+2k\pi) $$ $$ z=2cis(0+\frac{2k\pi}5) $$ I then went through all the solutions for $k=0\......Wed, 19 Aug 2026 20:51:53 -0400https://liberty.io/how-do-i-show-a-mapping-is-a-homomorphism.htmlhttps://liberty.io/how-do-i-show-a-mapping-is-a-homomorphism.htmlI don't want to make this question too broad, or non-specific. I'll will discuss a simple situation so we can all share a common context, but my question is less about this particular group, and m......Wed, 19 Aug 2026 20:42:38 -0400https://liberty.io/find-all-solutions-for-dy-dx-y-2.htmlhttps://liberty.io/find-all-solutions-for-dy-dx-y-2.htmlStarted to learn ODE, and the second example was of the equation $y^\prime=y^2$ We showed that the group of solutions is $\{y(x)=0 \}\cup \{ y(x)=\frac{1}{c-x} | c \in \mathbb{R} \}$ and mentioned ......Wed, 19 Aug 2026 20:35:51 -0400https://liberty.io/percentage-change-over-multiple-years.htmlhttps://liberty.io/percentage-change-over-multiple-years.htmlI have data for the number of mice in an area of 13 years. $\begin{bmatrix} \text{Year} & \text{#mice} \\ 1995 & 30 \\ 1996 & 113 \\ 1997 & 106 \\ 1998 & 50 \\ 1999 & 19 \\......Wed, 19 Aug 2026 20:30:57 -0400https://liberty.io/finding-the-mean-and-median-of-a-probability-density-function.htmlhttps://liberty.io/finding-the-mean-and-median-of-a-probability-density-function.htmlI suspect this is super-easy, but I haven't done any math in about ten years and I'm working with concepts that have been woefully explained... I need to find the mean and median of a continuous r......Wed, 19 Aug 2026 20:27:12 -0400https://liberty.io/express-if-true-then-1-else-0-in-a-formula-suitable-for-desmos-calcula.htmlhttps://liberty.io/express-if-true-then-1-else-0-in-a-formula-suitable-for-desmos-calcula.htmlIn programming, often the value of True is also 1, and False is 0. This means that: (x>5)*4 will return 4 if x is greater than 5 (because (x>5)==1), else 0. I need to accomplish a similar......Wed, 19 Aug 2026 20:26:04 -0400https://liberty.io/l-039-hospital-rule-in-multivariable-calculus.htmlhttps://liberty.io/l-039-hospital-rule-in-multivariable-calculus.htmlI was just wondering. Given functions $f(x,y), g(x,y) $, and the corresponding limit $\lim_{(x,y) \to (0,0) } \frac{f(x,y)}{g(x,y)} $ , where $\lim_{(x,y) \to (0,0) } f = \lim_{(x,y) \to (0,0) }g =......Wed, 19 Aug 2026 20:19:18 -0400https://liberty.io/what-039-s-the-intuition-behind-a-split-exact-sequence.htmlhttps://liberty.io/what-039-s-the-intuition-behind-a-split-exact-sequence.htmlI've learned the following definition for a split exact sequence: A short exact sequence of $R$-module homomorphisms $0\to A \stackrel{f}{\to} B \stackrel{g}{\to} C \to 0$ is split if there exists a ...Wed, 19 Aug 2026 20:01:58 -0400https://liberty.io/torque-and-angular-frequency-of-a-merry-go-round.htmlhttps://liberty.io/torque-and-angular-frequency-of-a-merry-go-round.htmlI have been trying to approach this by finding the gravitational torque of the person on the merry-go-round, however I can't find a way to relate that to angular velocity in the given circumstances......Wed, 19 Aug 2026 19:57:35 -0400https://liberty.io/finding-the-vertex-of-a-quadratic-function.htmlhttps://liberty.io/finding-the-vertex-of-a-quadratic-function.htmlI'm in the process of relearning math and came across a problem when calculating the vertex for $5x^2-20x+15$ function. I took the following steps to find the vertex: I divided every term by 5 to ......Wed, 19 Aug 2026 19:46:43 -0400https://liberty.io/is-there-such-a-thing-as-implicit-integration.htmlhttps://liberty.io/is-there-such-a-thing-as-implicit-integration.htmlA problem I'm working on asks me at a certain point to integrate the following: $$x=2\left(\sin\left(y\right)\right)^2.$$ I've never integrated anything in this form. I can't isolate for $y$ eith......Wed, 19 Aug 2026 19:44:56 -0400https://liberty.io/what-are-equations-with-a-degree-more-than-3-called.htmlhttps://liberty.io/what-are-equations-with-a-degree-more-than-3-called.htmlSorry if I am using the incorrect terminology. I am beginning to learn Mathematics. I had it in school and college but have forgotten most of it, so please be gentle with me if my questions are stu......Wed, 19 Aug 2026 19:43:19 -0400https://liberty.io/line-integral-around-a-triangle.htmlhttps://liberty.io/line-integral-around-a-triangle.htmlLet $R$ be the interior of the triangle with vertices $(0,0), (4,2),$ and $(0,2)$. Let $C$ be the boundary of $R$, oriented counterclockwise. Now evaluate the integral below. $$\int_C(y+e^\sqrt{x......Wed, 19 Aug 2026 19:39:05 -0400https://liberty.io/039-normally-distributed-random-numbers-039-vs-039-uniformly-distribut.htmlhttps://liberty.io/039-normally-distributed-random-numbers-039-vs-039-uniformly-distribut.htmlwhat is the difference between 'normally distributed random numbers' and 'uniformly distributed random number'? A answer in a layman language is appreciated :)...Wed, 19 Aug 2026 19:38:41 -0400https://liberty.io/biased-coin-probability-uneven-probability.htmlhttps://liberty.io/biased-coin-probability-uneven-probability.htmlGetting at least $2$ heads when flipping a coin $3$ times but the coin is biased so that heads are $3$ times more likely than tails. Can anyone explain how uneven probability works? Thank you...Wed, 19 Aug 2026 19:38:12 -0400https://liberty.io/graph-of-y-log-1-x.htmlhttps://liberty.io/graph-of-y-log-1-x.htmlTo draw the graph of $|y|=|\log(1-|x|)|$, I see that $x=0, y=0$ satisfies it. Also, the asymptotes would be $x=1, x=-1$. I have made a graph which has sort of eight curves in it but I don't know ho......Wed, 19 Aug 2026 19:31:23 -0400https://liberty.io/why-can-039-t-there-be-a-quintic-formula-duplicate.htmlhttps://liberty.io/why-can-039-t-there-be-a-quintic-formula-duplicate.htmlPossible Duplicate: Why is it so hard to find the roots of polynomial equations? For polynomials (with real coefficients), in degrees 2, 3, 4, there are the quadratic, cubic, and quartic formu......Wed, 19 Aug 2026 19:30:35 -0400https://liberty.io/solve-tan2x-1.htmlhttps://liberty.io/solve-tan2x-1.htmlAre there any errors in my work? Thanks in advance! (Sorry for the bad format. I'm still new to this) $\tan2x=1$ $\frac{2\tan x}{1-\tan^2x}=1$ $2\tan x=1-\tan^2x$ $0=1-\tan^2x-2\tan x$ $0 =-\t......Wed, 19 Aug 2026 19:17:09 -0400https://liberty.io/what-local-system-really-is.htmlhttps://liberty.io/what-local-system-really-is.htmlI know a local system is a locally constant sheaf. But why does a local system on the topological space $X$ correspond to $\tilde{X}\times_G V$, where $G$ is the fundamental group of $X$, $\tilde{X......Wed, 19 Aug 2026 19:10:34 -0400https://liberty.io/why-h-p-coprod-i-in-i-x-i-g-cong-prod-i-in-i-h-p-x-i-g.htmlhttps://liberty.io/why-h-p-coprod-i-in-i-x-i-g-cong-prod-i-in-i-h-p-x-i-g.htmlMy question is the following; why $\displaystyle H^p\big(\coprod_{i\in I}X_i;G\big)$ is isomorphic to $\displaystyle\prod_{i\in I}H^p(X_i;G)$? I have tried the following: one can prove the followin......Wed, 19 Aug 2026 19:08:01 -0400https://liberty.io/what-does-exterior-algebra-actually-mean.htmlhttps://liberty.io/what-does-exterior-algebra-actually-mean.htmlThis question may be too basic and even silly, but I am new to exterior algebra and reading Wikipedia. Given $e_1, e_2,\cdots, e_n$ is a standard basis for a vector space $V$, what does $e_1\wedge......Wed, 19 Aug 2026 19:01:38 -0400https://liberty.io/definite-integral-in-matlab.htmlhttps://liberty.io/definite-integral-in-matlab.htmlI have just started learning how to solve definite integrals in matlab, and from the documentation, I picked up this example: $\int^{\infty}_{0}(e^{-x^2} )dx$ but when I run it in my matlab, I ge......Wed, 19 Aug 2026 18:58:06 -0400https://liberty.io/fast-algorithm-for-computing-integer-square-roots-on-machines-that-doe.htmlhttps://liberty.io/fast-algorithm-for-computing-integer-square-roots-on-machines-that-doe.htmlFast algorithm for computing integer square roots on machines that doesn't support floating-point arithmetic? I'm looking for a fast algorithm for computing the integer square root of an integer $......Wed, 19 Aug 2026 18:54:45 -0400https://liberty.io/mathematical-structures-and-signature.htmlhttps://liberty.io/mathematical-structures-and-signature.htmlFrom Wikipedia: In mathematics, a structure on a set, or more generally a type, consists of additional mathematical objects that in some manner attach (or relate) to the set, making it easier to ...Wed, 19 Aug 2026 18:46:24 -0400https://liberty.io/square-roots-modulo-powers-of-2.htmlhttps://liberty.io/square-roots-modulo-powers-of-2.htmlExperimentally, it seems like every $a\equiv1 \pmod 8$ has $4$ square roots mod $2^n$ for all $n \ge 3$ (i.e. solutions to $x^2\equiv a \pmod {2^n}$) Is this true? If so, how can I prove it? If n......Wed, 19 Aug 2026 18:44:04 -0400https://liberty.io/constructing-a-confidence-interval-for-population-variance.htmlhttps://liberty.io/constructing-a-confidence-interval-for-population-variance.htmlGiven the following data set, how does one construct a 99% confidence interval for the population variance? I should mention now I don't have a very good understanding of when to divide by n and wh......Wed, 19 Aug 2026 18:43:39 -0400https://liberty.io/what-does-the-symbol-nabla-indicate.htmlhttps://liberty.io/what-does-the-symbol-nabla-indicate.htmlFirst up, this question differs from the other ones on this site as I would like to know the isolated meaning of nabla if that makes sense. Meanwhile, other questions might ask what it means in rel......Wed, 19 Aug 2026 18:40:18 -0400https://liberty.io/calculate-the-confidence-interval-of-parameter-of-exponential-distribu.htmlhttps://liberty.io/calculate-the-confidence-interval-of-parameter-of-exponential-distribu.htmlHow can I calculate the confidence interval for parameter $\alpha$ of exponential distribution ? I think I can use test-t. Knowing that: $$mean = {1\over\alpha}$$ I found that : $${1\over {\bar......Wed, 19 Aug 2026 18:34:59 -0400https://liberty.io/list-of-functions-not-integrable-in-elementary-terms.htmlhttps://liberty.io/list-of-functions-not-integrable-in-elementary-terms.htmlWhen teaching integration to beginning calculus students I always tell them that some integrals are "impossible" (with a bit of expansion on what that actually means). However I must admit that the ...Wed, 19 Aug 2026 18:32:52 -0400https://liberty.io/find-5th-degree-polynomial-equation-from-points.htmlhttps://liberty.io/find-5th-degree-polynomial-equation-from-points.htmlI have a set of estimated points, listed below, and I would like to know how to calculate the polynomial. I've looked at Wikipedia, and I don't quite understand how at works. Example 1 on http://en....Wed, 19 Aug 2026 18:28:54 -0400https://liberty.io/rational-parametrization-of-circle-in-wikipedia.htmlhttps://liberty.io/rational-parametrization-of-circle-in-wikipedia.htmlIn http://en.wikipedia.org/wiki/Circle but also in the corresponding article in the German Wikipedia I find this formulation ( sorry, I exchange x and y as I am accustomed to it in this way ) : "An ...Wed, 19 Aug 2026 18:19:57 -0400https://liberty.io/find-the-domain-of-root-of-a-logarithmic-function.htmlhttps://liberty.io/find-the-domain-of-root-of-a-logarithmic-function.htmlI'm a little confused about this question since output of a logarithmic function varies from $ -\infty $ to $\infty$ .I should find the domain of this function: $ y=\sqrt{\log_x(10-x^2)} $ . How c......Wed, 19 Aug 2026 17:57:03 -0400https://liberty.io/how-many-32-digit-binary-number-combinations-are-possible.htmlhttps://liberty.io/how-many-32-digit-binary-number-combinations-are-possible.htmlHow many 32 digit binary number combinations are possible? For example: $$00000000-00000000-00000000-00000000$$ $$00000000-00000000-00000000-00000001$$ $$00000000-00000000-00000000-00000010$$ $$.$......Wed, 19 Aug 2026 17:54:25 -0400https://liberty.io/propositional-equivalence.htmlhttps://liberty.io/propositional-equivalence.htmlAre the following two propositions equivalent? p IMPLIES (q IMPLIES r) p IMPLIES (q AND r) From what I can tell, using the logical equivalences, this should be false, correct? p IMPLIES (q IMPLI......Wed, 19 Aug 2026 17:51:14 -0400https://liberty.io/how-lim-h-0-sin-h-h-is-equal-to-1-duplicate.htmlhttps://liberty.io/how-lim-h-0-sin-h-h-is-equal-to-1-duplicate.htmlHow $\lim_{h\to 0} \sin h$ when divided by $h$ gives the value $1$ ? Does it also follows with other ratios that for example. $\lim_{h\to 0} \cos x/x = 1$...Wed, 19 Aug 2026 17:39:27 -0400https://liberty.io/intuitive-way-to-calculate-volume-of-the-solid-bounded-by-a-plane.htmlhttps://liberty.io/intuitive-way-to-calculate-volume-of-the-solid-bounded-by-a-plane.htmlFor a National Board Exam Review: What is the volume of the solid bounded by the plane $3x+4y+6z=12$ and the coordinate axes? Answer is $4$. I am looking for a quick and intuitive way to so......Wed, 19 Aug 2026 17:35:42 -0400https://liberty.io/elliptic-coordinates-inverting-the-transformation.htmlhttps://liberty.io/elliptic-coordinates-inverting-the-transformation.htmlThe standard way to transform elliptic coordinates $(\mu, \nu)$ $\ to$ Cartesian coordinates $(x,y)$: $x = a \cosh(\mu) \cos(\nu)$ $y = a \sinh(\mu) \sin(\nu)$ Is there any way to get the ...Wed, 19 Aug 2026 17:31:07 -0400https://liberty.io/system-of-linear-equations-how-many-solutions.htmlhttps://liberty.io/system-of-linear-equations-how-many-solutions.htmlFor which real values of t does the following system of linear equations: $$ \left\{ \begin{array}{c} tx_1 + x_2 + x_3 = 1 \\ x_1 + tx_2 + x_3 = 1 \\ x_1 + x_2 + tx_3 = 1 \end{array} \right. $$ ...Wed, 19 Aug 2026 17:22:49 -0400https://liberty.io/geometric-interpretation-of-element-wise-vector-multiplication.htmlhttps://liberty.io/geometric-interpretation-of-element-wise-vector-multiplication.htmlGeometrically speaking, vector dot product is interpreted as a projection of one vector on the other, while vector cross product gives another vector that is orthogonal to the multiplied vectors(ri......Wed, 19 Aug 2026 17:02:41 -0400https://liberty.io/how-do-we-use-derivatives-in-our-daily-lives-closed.htmlhttps://liberty.io/how-do-we-use-derivatives-in-our-daily-lives-closed.htmlHow can we use derivatives in our real lives, I know that we have a lot of formulas to find the derivatives of a function,but why do we need them ?...Wed, 19 Aug 2026 16:43:27 -0400https://liberty.io/how-to-graph-sec-x-2-by-manipulating-the-cosine-function.htmlhttps://liberty.io/how-to-graph-sec-x-2-by-manipulating-the-cosine-function.html$\sec(x) = 1/\cos(x)$. Usually, I graph $\sec(x)$ by graphing $\cos(x)$ first. $\sec x/2= 1/2\cdot (1/\cos x) = (1/2)\cos x$. Following the same logic above, I graphed $2\cos(x)$. However, what I ...Wed, 19 Aug 2026 16:42:00 -0400https://liberty.io/is-it-true-that-a-3rd-order-polynomial-must-have-at-least-one-real-roo.htmlhttps://liberty.io/is-it-true-that-a-3rd-order-polynomial-must-have-at-least-one-real-roo.htmlWhile solving a problem a friend said - this polynomial is $3^{rd}$ order ($ax^3+bx^2+cx+d$), with $\{a,b,c,d\}$ real coefficients, so it must have a real root. I didn't want to sound stupid and I ......Wed, 19 Aug 2026 16:39:23 -0400https://liberty.io/is-the-product-of-uniquely-decipherable-codes-uniquely-decipherable.htmlhttps://liberty.io/is-the-product-of-uniquely-decipherable-codes-uniquely-decipherable.htmlA code $C$ is prefix-free if no codeword in $C$ is a prefix of another codeword in $C$. For example, $C=\{0,10,110\}$ is prefix-free but $D=\{0,10,100\}$ is not. A code $C$ is uniquely decipherabl......Wed, 19 Aug 2026 16:38:10 -0400https://liberty.io/is-the-given-ode-system-a-gradient-system.htmlhttps://liberty.io/is-the-given-ode-system-a-gradient-system.htmlConsider the planar system $$\dot{x} = 2x-y-\frac{2x^3+2xy^2-xy}{\sqrt{x^2+y^2}} \\ \dot{y} = 2y+x-\frac{2x^2y+2y^3+x^2}{\sqrt{x^2+y^2}}$$ which in polar coordinates is given by $$\ \dot{r}=2r(1......Wed, 19 Aug 2026 16:13:33 -0400https://liberty.io/coherent-sheaves-vs-sheaves-locally-of-finite-presentation.htmlhttps://liberty.io/coherent-sheaves-vs-sheaves-locally-of-finite-presentation.htmlI think it is clear for most people that finitely generated and finitely presented modules are important and, in a certain sense, the "right" notions. However, the corresponding notions for ...Wed, 19 Aug 2026 16:11:38 -0400https://liberty.io/how-do-you-obtain-the-2-3-7-triangle-group-as-a-quotient-of-the-modula.htmlhttps://liberty.io/how-do-you-obtain-the-2-3-7-triangle-group-as-a-quotient-of-the-modula.htmlSo, at various places on Wikipedia (like here), its claimed that the $(2,3,7)$ triangle group can be obtained as quotient of the modular group. How is this done? Treating the integers as mod $7$ ......Wed, 19 Aug 2026 16:01:42 -0400https://liberty.io/is-this-the-notation-you-use.htmlhttps://liberty.io/is-this-the-notation-you-use.htmlI've noticed that my terminology is a bit haggard. I do math on my own so I'm not entirely sure how everyone else refers to things and so I need a check. so is this correct: $\lim\limits_{\delta ......Wed, 19 Aug 2026 15:54:52 -0400https://liberty.io/is-1-3-and-or-17-in-the-set-of-integers-modulo-7.htmlhttps://liberty.io/is-1-3-and-or-17-in-the-set-of-integers-modulo-7.htmlIs $1/3$ and/or $-17$ in the set of integers modulo $7$? So I say that $1/3$ is in the set of integers modulo $7$ because $3 \cdot 1/3 = 1$, so $3x \bmod 7 = 1$. For $x = 5$, this works, so $1/3$ is ...Wed, 19 Aug 2026 15:49:35 -0400