https://liberty.ioCurated star highlights with fresh energy.en-usMon, 10 Aug 2026 09:59:58 -0400https://liberty.io/confusion-between-the-definition-of-increasing-function-and-a-theorem-.htmlhttps://liberty.io/confusion-between-the-definition-of-increasing-function-and-a-theorem-.htmlThe definition of increasing function given in my school maths text book is Let $I$ be an open interval contained in the domain of a real valued function $f$. Then $f$ is said to be increasing on......Mon, 10 Aug 2026 13:59:46 -0400https://liberty.io/forward-euler-stability.htmlhttps://liberty.io/forward-euler-stability.htmlI am trying to understand the stability of the forward Euler method. I read there's a model problem to see the stability. $$y'(t) = \lambda y(t) \qquad t \in (0, \infty)$$ $$y(0) = 1$$ then the b......Mon, 10 Aug 2026 13:55:43 -0400https://liberty.io/definition-of-closed-subscheme.htmlhttps://liberty.io/definition-of-closed-subscheme.htmlLet $(X,\mathcal O_X)$ be a scheme. I'm using the following definition from Görtz-Wedhorn. A closed subscheme of $(X,\mathcal O_X)$ is a scheme $(Z,\mathcal O_Z)$, where $i\colon Z\hookrightarrow X......Mon, 10 Aug 2026 13:52:22 -0400https://liberty.io/how-to-take-the-derivative-of-a-matrix-with-respect-to-itself.htmlhttps://liberty.io/how-to-take-the-derivative-of-a-matrix-with-respect-to-itself.htmlCould someone please explain how to take the derivative of matrix with respect to itself? $$\frac{\partial \textbf{X}}{\partial \textbf{X}}$$ where $\textbf{X}$ is an M x N matrix...Mon, 10 Aug 2026 13:52:13 -0400https://liberty.io/finding-the-fourth-vertex-of-a-square.htmlhttps://liberty.io/finding-the-fourth-vertex-of-a-square.htmlI was given this problem: Without drawing a graph and given the following 3 vertices, find the coordinates of the last vertex of the square: $(3,2)$$(0,5)$$(-3,2)$ So first, I found the leng......Mon, 10 Aug 2026 13:34:16 -0400https://liberty.io/tangent-of-a-straight-line.htmlhttps://liberty.io/tangent-of-a-straight-line.htmlI just got back a math test in my EF Calc class, and I disagree with my teacher on this one problem. We are using derivatives to determine equations of lines tangent to a given equation. The equation ...Mon, 10 Aug 2026 13:33:13 -0400https://liberty.io/is-square-root-a-function.htmlhttps://liberty.io/is-square-root-a-function.htmlI just began the topic of functions in my Mathematics textbook and in the first paragraph itself, two conditions about a relation being a function were mentioned. They were : 1) For every a∈A, t......Mon, 10 Aug 2026 13:28:30 -0400https://liberty.io/the-speed-of-light-in-a-vacuum-is-2-998-times-10-8-m-sec-what-is-this-.htmlhttps://liberty.io/the-speed-of-light-in-a-vacuum-is-2-998-times-10-8-m-sec-what-is-this-.htmlPeace to all. Studying and trying to get a handle on how to calculate like-minded questions, I came across this question from the web. I don't understand how exactly it was solved. How were the u......Mon, 10 Aug 2026 13:19:10 -0400https://liberty.io/does-the-relation-in-follow-the-axiom-of-substitution-for-equality.htmlhttps://liberty.io/does-the-relation-in-follow-the-axiom-of-substitution-for-equality.htmlI am trying to self-learn set theory from Analysis 1 by Terence Tao. I am facing a problem understanding how he shows the relation $\in$ for sets follows the axiom of substitution. He begins by ...Mon, 10 Aug 2026 13:18:11 -0400https://liberty.io/show-that-pi-3-arccos-frac-5-sqrt-28-3-arctan-frac-sqrt-3-2.htmlhttps://liberty.io/show-that-pi-3-arccos-frac-5-sqrt-28-3-arctan-frac-sqrt-3-2.htmlQuestion: Show that, $$\pi =3\arccos(\frac{5}{\sqrt{28}}) + 3\arctan(\frac{\sqrt{3}}{2}) ~~~~~~ (*)$$ My proof method for this question has received mixed responses. Some people say it's fine, ot......Mon, 10 Aug 2026 13:17:35 -0400https://liberty.io/rationalize-the-denominator-with-4-terms-frac-1-1-sqrt-12-sqrt-3-3-sqr.htmlhttps://liberty.io/rationalize-the-denominator-with-4-terms-frac-1-1-sqrt-12-sqrt-3-3-sqr.htmlRationalize the denominator of this fraction $$\frac{1}{1+\sqrt{12}-\sqrt[3]{3}-\sqrt[3]{9}}$$...Mon, 10 Aug 2026 13:17:03 -0400https://liberty.io/find-all-the-relative-extrema-of-the-function-f-x-2x-24x-1-3.htmlhttps://liberty.io/find-all-the-relative-extrema-of-the-function-f-x-2x-24x-1-3.htmlFind all the relative extrema of the function $f(x)=2x-24x^{1/3}$ Solution: Step 1: Find the values of $x$ where $f'(x)=0$ and $f'(x)$ DNE. $f'(x)=2-8x^{\frac{-2}{3}}=2-\frac{8}{x^{\frac{2}{3}}}......Mon, 10 Aug 2026 13:12:40 -0400https://liberty.io/trapezoids-similarity.htmlhttps://liberty.io/trapezoids-similarity.htmlI found this geometry problem in an IGCSE text. Can't seem to find the missing lengths that they are asking for. I got an estimate for f (f=2.6666... cm) by drawing a line parallel to the left side......Mon, 10 Aug 2026 13:03:37 -0400https://liberty.io/how-to-implement-subscript-and-superscript-in-legend-matlab.htmlhttps://liberty.io/how-to-implement-subscript-and-superscript-in-legend-matlab.htmlI am wondering how to implement text in a legend with both a superscript and a subscript, and with an horizontal bar, something like this: Image...Mon, 10 Aug 2026 12:58:39 -0400https://liberty.io/proof-about-cardinality-of-sets-r-z-r.htmlhttps://liberty.io/proof-about-cardinality-of-sets-r-z-r.htmlProve that $|R-Z|=|R|$ where $R$ is reals and $Z$ is integers. New approach Maybe the problem is reduced to finding a bijection between (0,1) and [0,1] if we find a bijection between these interv......Mon, 10 Aug 2026 12:55:22 -0400https://liberty.io/why-are-self-adjoint-operators-important.htmlhttps://liberty.io/why-are-self-adjoint-operators-important.htmlI am learning about self-adjoint and normal operators. So far, they have come up in the Spectral theorem, which says self-adjoint operators have an eigenvalue basis and a corresponding diagonal ma......Mon, 10 Aug 2026 12:49:52 -0400https://liberty.io/cartesian-form-of-a-line.htmlhttps://liberty.io/cartesian-form-of-a-line.htmlI have a general question: when given the parametric form of a line such that: $$ x = a_1 + \lambda t_1 $$ $$ y = a_2 + \lambda t_2 $$ $$ z = a_3 + \lambda t_3 $$ The Cartesian form is: $$ \fr......Mon, 10 Aug 2026 12:49:31 -0400https://liberty.io/chinese-remainder-theorem-for-three-equations.htmlhttps://liberty.io/chinese-remainder-theorem-for-three-equations.htmlIs there a straightforward approach for solving the Chinese Remainder Theorem with three congruences? $$x \equiv a \bmod A$$ $$x \equiv b \bmod B$$ $$x \equiv c \bmod C$$ Assuming all values are ...Mon, 10 Aug 2026 12:45:46 -0400https://liberty.io/can-someone-explain-why-x-log-a-a-log-x-duplicate.htmlhttps://liberty.io/can-someone-explain-why-x-log-a-a-log-x-duplicate.htmlI'm trying to see why the below is true. $$ x^{\log(a)} = a^{\log(x)} $$ Anyone here know why this is? Thank you....Mon, 10 Aug 2026 12:40:14 -0400https://liberty.io/find-angle-between-two-curves-at-point-of-intersection-closed.htmlhttps://liberty.io/find-angle-between-two-curves-at-point-of-intersection-closed.htmlFind angle between these two curves at point of intersection : $$K_1: x^2y^2 + y^4 = 1$$ and $$K_2 : x^2 + y^2 = 4 $$ Thanks!...Mon, 10 Aug 2026 12:29:57 -0400https://liberty.io/fractal-fundamentals.htmlhttps://liberty.io/fractal-fundamentals.htmlI am a programmer by trade, and am very interested in fractals. To be very basic about the concept, one might say a 'circle of circles' is a fractal. Where each circle is made up of circles, and t......Mon, 10 Aug 2026 12:25:20 -0400https://liberty.io/equal-spaced-points-in-a-logarithmic-graph.htmlhttps://liberty.io/equal-spaced-points-in-a-logarithmic-graph.htmlI am plotting a graph with the x-axis as logarithmic. I want to select 10 point that are equally spaced in a logarithm scale. How can I determine the values if we have the range from 100 to 10000?...Mon, 10 Aug 2026 12:18:30 -0400https://liberty.io/when-a-fraction-is-raised-to-a-negative-exponent-do-you-normally-trans.htmlhttps://liberty.io/when-a-fraction-is-raised-to-a-negative-exponent-do-you-normally-trans.htmlMy text shows that $$\left(\frac{3a^2}{4b}\right)^{-3}=\frac{1}{\left(\frac{3a^2}{4b}\right)^{3}}.$$ It also shows that $$\frac{1}{\frac{144}{b}}=\frac{b}{144}.$$ In the first equation, it se......Mon, 10 Aug 2026 12:12:18 -0400https://liberty.io/questions-tagged-linear-programming.htmlhttps://liberty.io/questions-tagged-linear-programming.htmlQ&A for people studying math at any level and professionals in related fields...Mon, 10 Aug 2026 11:59:08 -0400https://liberty.io/finding-a-basis-for-the-columnspace-of-a-matrix.htmlhttps://liberty.io/finding-a-basis-for-the-columnspace-of-a-matrix.htmlFind a linearly independent set of vectors that spans the same subspace of $R^4$ as that spanned by the vectors - $$ \begin{bmatrix} 2 \\ -4 \\ -1 \\ -2 \\ \end{bmatrix} , \begin{bmatrix}......Mon, 10 Aug 2026 11:46:04 -0400https://liberty.io/questions-tagged-prime-numbers.htmlhttps://liberty.io/questions-tagged-prime-numbers.htmlQ&A for people studying math at any level and professionals in related fields...Mon, 10 Aug 2026 11:08:47 -0400https://liberty.io/ryll-nardzewski-theorem-and-types-with-parameters.htmlhttps://liberty.io/ryll-nardzewski-theorem-and-types-with-parameters.htmlBy Ryll-Nardzewski theorem we know that a theory $T$ is $\omega$-categorical iff for each $n$, every type in $S_n(T)$ is isolated. Question. What can we say about types with parameters. Does $\omega$-...Mon, 10 Aug 2026 11:07:03 -0400https://liberty.io/how-can-i-find-a-polynomial-that-fits-a-given-table-closed.htmlhttps://liberty.io/how-can-i-find-a-polynomial-that-fits-a-given-table-closed.htmlThe first difference is 1, 4, 7, 10, 13, 16 The second difference is 3, 3, 3, 3, 3, 3 Since the second difference is constant this would be quadratic and I would have $\frac{3}{2}n^{2}$ So now I ...Mon, 10 Aug 2026 10:58:54 -0400https://liberty.io/why-do-exponents-not-distribute-over-addition-closed.htmlhttps://liberty.io/why-do-exponents-not-distribute-over-addition-closed.htmlI understand that exponents don't distribute over addition and have seen plenty of examples i.e. $$ (x + y)^2\neq x^2 + y^2 $$ but I'm wondering why that is. Multiplication distributes over additio......Mon, 10 Aug 2026 10:54:12 -0400https://liberty.io/solving-trigonometric-equations-of-the-form-a-sin-x-b-cos-x-c.htmlhttps://liberty.io/solving-trigonometric-equations-of-the-form-a-sin-x-b-cos-x-c.htmlSuppose that there is a trigonometric equation of the form $a\sin x + b\cos x = c$, where $a,b,c$ are real and $0 < x < 2\pi$. An example equation would go the following: $\sqrt{3}\sin x + \c......Mon, 10 Aug 2026 10:43:10 -0400https://liberty.io/second-order-ode-in-matlab.htmlhttps://liberty.io/second-order-ode-in-matlab.htmlI am given an example file for solving a second order ODE using ODE 45 function ex_with_2eqs t0 = 0; tf = 20; y0 = [10;60]; a = .8; b = .01; c = .6; d = .1; [t,y] = ode45(@f,[t0,tf],y0,[],a,......Mon, 10 Aug 2026 10:38:22 -0400https://liberty.io/expected-value-finding-the-price-of-a-raffle-ticket.htmlhttps://liberty.io/expected-value-finding-the-price-of-a-raffle-ticket.htmlThis question has been bugging me for a while now and I want to know where I'm going wrong. There are $20$ tickets in a raffle with one prize. What should each ticket cost if the prize is \$80 ......Mon, 10 Aug 2026 10:37:47 -0400https://liberty.io/jacobian-matrix-in-039-polar-coordinates-039.htmlhttps://liberty.io/jacobian-matrix-in-039-polar-coordinates-039.htmlI'm pretty familiar with the del operator and I've just become familiar with the Jacobian Matrix. From my basic understanding of what the Jacobian Matrix is, the gradient is a subset of the Jacobian ...Mon, 10 Aug 2026 10:37:04 -0400https://liberty.io/why-can-039-t-a-rational-function-have-both-a-horizontal-and-an-obliqu.htmlhttps://liberty.io/why-can-039-t-a-rational-function-have-both-a-horizontal-and-an-obliqu.htmlWhy if a rational function has a horizontal asymptote, it excludes the possibility of this function to have an oblique azymptote, or viceversa?...Mon, 10 Aug 2026 10:26:49 -0400https://liberty.io/shannon-entropy-of-a-fair-dice.htmlhttps://liberty.io/shannon-entropy-of-a-fair-dice.htmlThe formula for Shannon entropy is as follows, $$\text{Entropy}(S) = - \sum_i p_i \log_2 p_i $$ Thus, a fair six sided dice should have the entropy, $$- \sum_{i=1}^6 \dfrac{1}{6} \log_2 \dfrac{1......Mon, 10 Aug 2026 10:17:16 -0400https://liberty.io/what-does-echelon-mean.htmlhttps://liberty.io/what-does-echelon-mean.htmlWhen you solve a system of linear equations, you write down the augmented matrix and reduce this to reduced row echelon form. What is the meaning of the word echelon?...Mon, 10 Aug 2026 10:06:31 -0400https://liberty.io/does-every-nonzero-turing-degree-adimit-a-nonzero-incompatible-degree.htmlhttps://liberty.io/does-every-nonzero-turing-degree-adimit-a-nonzero-incompatible-degree.htmlIt is well known that every nonzero Turing degree $\bf x$ adimits an incomparable degree $\bf y$, that is a degree such that $\bf x\not\le_T\bf y$ and $\bf y\not\le_T\bf x$. I was wondering whethe......Mon, 10 Aug 2026 10:01:14 -0400https://liberty.io/problems-with-simplifying-using-factoring-of-binomial-expressions.htmlhttps://liberty.io/problems-with-simplifying-using-factoring-of-binomial-expressions.htmlI am running into problems simplifying using factoring of binomial expressions. The problem at hand is this: $(x-1)^3*(2x-3)-(2x+12)*(x-1)^2$ I first expanded the left side of the minus sign, lik......Mon, 10 Aug 2026 09:57:49 -0400https://liberty.io/can-we-ever-get-an-irrational-number-by-dividing-two-rational-numbers.htmlhttps://liberty.io/can-we-ever-get-an-irrational-number-by-dividing-two-rational-numbers.htmlIf we try to divide any two random arbitrarily long rational numbers like 103850.2387209375029375092730958297836958623986868349693868398659825528365... and 127.123123123... Is it guaranteed th......Mon, 10 Aug 2026 09:53:18 -0400https://liberty.io/finding-measuring-points-in-three-point-perspective-drawing.htmlhttps://liberty.io/finding-measuring-points-in-three-point-perspective-drawing.htmlIn his Complete Guide to Perspective Drawing (page 39), Craig Attebery places Measuring Points of two-point perspective in a line perperdicular to that crossing the Station Point and the Center of ......Mon, 10 Aug 2026 09:19:03 -0400https://liberty.io/difference-in-the-definitions-of-glb-and-lub-in-real-analysis-and-abst.htmlhttps://liberty.io/difference-in-the-definitions-of-glb-and-lub-in-real-analysis-and-abst.htmlThe following text is from the book Abstract Algebra by T. W. Judson : Let $X = {\{1,2,3,4,6,8,12,24}\}$ be the set of divisors of $24$ with the partial order defined by $a\preceq b$ if $a | b$.......Mon, 10 Aug 2026 09:09:34 -0400https://liberty.io/in-a-parallelogram-does-the-diagonal-bisect-the-angles-that-they-meet.htmlhttps://liberty.io/in-a-parallelogram-does-the-diagonal-bisect-the-angles-that-they-meet.htmlI understand the following properties of the parallelogram: Opposite sides are parallel and equal in length. Opposite angles are equal. Adjacent angles add up to 180 degrees therefore adjacent ang......Mon, 10 Aug 2026 09:04:21 -0400https://liberty.io/two-circles-with-a-common-external-tangent-problem-closed.htmlhttps://liberty.io/two-circles-with-a-common-external-tangent-problem-closed.htmlTwo circles are tangent externally at A, and a common external tangent touches them at B and C. The line segment BA is extended, meeting the second circle at D. Prove that CD is a diameter....Mon, 10 Aug 2026 08:47:43 -0400https://liberty.io/find-normal-vector-of-a-3d-vector.htmlhttps://liberty.io/find-normal-vector-of-a-3d-vector.htmlI need to find the normal vector for the following 3d vector presented in the vectorial equation because I need to find a plane that is orthogonal to the following line: $(x,y,z)=(1,0,0)+k(1,2,3......Mon, 10 Aug 2026 08:37:46 -0400https://liberty.io/rank-theorem-on-manifolds.htmlhttps://liberty.io/rank-theorem-on-manifolds.htmlRank theorem on manifolds says that : Suppose $M$ and $N$ are two smooth manifolds of dimensions $m$ and $n$, respectively, and $F:M\rightarrow N$ be a smooth map with constant rank $r$. For eac......Mon, 10 Aug 2026 08:17:08 -0400https://liberty.io/prove-operatorname-rank-a-leq-1-iff-a-xy-t-for-some-x-y-in-mathbb-r-n-.htmlhttps://liberty.io/prove-operatorname-rank-a-leq-1-iff-a-xy-t-for-some-x-y-in-mathbb-r-n-.htmlLet $A \in \mathbb{R}^{n \times n}$. Then $\operatorname{rank} A \leq 1$ if and only if $A = xy^T$ for some $x, y \in \mathbb{R}^{n \times 1}$. Need help. I can't find sufficient facts nor theorem......Mon, 10 Aug 2026 08:15:48 -0400https://liberty.io/is-a-function-that-is-one-to-one-necessarily-onto.htmlhttps://liberty.io/is-a-function-that-is-one-to-one-necessarily-onto.htmlI'd would like to know that, because I don't want to prove a function is onto if I don't have to. If the answer is no, is there any particular case where it is true?...Mon, 10 Aug 2026 08:13:47 -0400https://liberty.io/orientation-geometry-closed.htmlhttps://liberty.io/orientation-geometry-closed.htmlA xy-plane is rotated about x and y-axis. But when it is rotated along z-axis, it doesn't change the orientation of xy-plane, why? I've tried to find the answer but those answer that i found didn't......Mon, 10 Aug 2026 08:10:21 -0400https://liberty.io/complex-variable-limit-at-infinity.htmlhttps://liberty.io/complex-variable-limit-at-infinity.htmlIs $\lim\limits_{z\to\infty} \frac{4z^2}{(z-1)^2}$, $z\in\mathbb{C}$, evaluated the same way as a real variable function limit? Or does one need to show separate cases for $x\to\infty$ and $y\to\in......Mon, 10 Aug 2026 08:09:03 -0400https://liberty.io/how-to-determine-what-these-vectors-are.htmlhttps://liberty.io/how-to-determine-what-these-vectors-are.htmlSo I'm not sure how to solves these problems. The Question asks to Determine whether the given vectors are orthogonal, parallel or neither. 1) a = <-5, 3, 7> and b = <6, -8, 2> 2) a = &lt......Mon, 10 Aug 2026 08:04:10 -0400https://liberty.io/solving-for-upper-limit-of-integration-given-a-lower-limit-and-integra.htmlhttps://liberty.io/solving-for-upper-limit-of-integration-given-a-lower-limit-and-integra.htmlConsider $f$ a real integrable function, usually we want to evaluate the integral $\int_a^bf(x)dx$ for some $a<b$ given. Now, suppose we know $a$ but we don't know $b$, further, we know the valu......Mon, 10 Aug 2026 08:00:53 -0400https://liberty.io/general-formula-for-sin-left-k-arcsin-x-right.htmlhttps://liberty.io/general-formula-for-sin-left-k-arcsin-x-right.htmlI'm wondering if there's a simple way to rewrite this in terms of $k$ and $x$, especially as a polynomial. It seems to me to crop up every so often, especially for $k=2$, when I integrate with trig ...Mon, 10 Aug 2026 07:57:29 -0400https://liberty.io/prove-that-a-parallelogram-is-1-rectangle-2-rhombus-3-square.htmlhttps://liberty.io/prove-that-a-parallelogram-is-1-rectangle-2-rhombus-3-square.htmlThe question is a follows. Midpoints of the sides of a quadrilateral are the vertices of a paralleogram. Determine under what conditions this parallelogram will be (1) a rectangle, (2) a rhombus,......Mon, 10 Aug 2026 07:39:21 -0400https://liberty.io/calculus-of-variations-lagrange-multipliers.htmlhttps://liberty.io/calculus-of-variations-lagrange-multipliers.htmlGiven a functional $$J(y)=\int_a^b F(x,y,y')dx, \tag{1}$$ where $y$ is a function of $x$, and a constraint $$\int_a^b K(x,y,y')dx=l, \tag{2}$$ if $y=y(x)$ is an extreme of (1) under the constr......Mon, 10 Aug 2026 07:34:46 -0400https://liberty.io/minimum-area-of-inscribed-square.htmlhttps://liberty.io/minimum-area-of-inscribed-square.htmlGRE study guide asks The perimeter of square S is 40. Square T is inscribed in square S. What is the least possible area of square T? Choices are 45 48 49 50 52 They say answer is 50. How d......Mon, 10 Aug 2026 07:34:36 -0400https://liberty.io/how-to-integrate-this-function-with-ln-and-u-substitution.htmlhttps://liberty.io/how-to-integrate-this-function-with-ln-and-u-substitution.htmlI can get started in the right direction, but cant seem to get all of the way there, and any examples I can find don't have the same complications. $$\int {2x\over (x-1)^2}\cdot dx$$ What I have......Mon, 10 Aug 2026 07:31:57 -0400https://liberty.io/predict-next-number-from-a-series.htmlhttps://liberty.io/predict-next-number-from-a-series.htmlWhich methods I can use to predict next number from a series of numbers ? I know the min & max possible number in advance....Mon, 10 Aug 2026 07:26:54 -0400https://liberty.io/finding-area-of-a-triangle-without-trignometric-ratios.htmlhttps://liberty.io/finding-area-of-a-triangle-without-trignometric-ratios.htmlHi I need to figure out the area of the following triangle, without using Trigonometric ratios. Any suggestions on the best approach. The answer is 12 square units Edit: I also think that the above ...Mon, 10 Aug 2026 07:20:33 -0400https://liberty.io/distance-between-centers-of-two-touching-circles.htmlhttps://liberty.io/distance-between-centers-of-two-touching-circles.htmlLets say we have two circles with radii 2 and 3 respectively. If we then put Circle radius 2 on a flat surface, then the other circle on the flat surface so they touch once, what is the distance b......Mon, 10 Aug 2026 07:18:30 -0400https://liberty.io/why-is-logistic-equation-called-logistic.htmlhttps://liberty.io/why-is-logistic-equation-called-logistic.htmlThe logistic function solves the logistic ODE which is the continuous version of the logistic map. However, I was not able to find why any of these things are called "logistic"....Mon, 10 Aug 2026 07:16:16 -0400https://liberty.io/how-to-understand-lindeberg-clt-compare-with-classical-clt.htmlhttps://liberty.io/how-to-understand-lindeberg-clt-compare-with-classical-clt.htmlFor Lindeberg CLT. Let$\{X_{nj},j=1,2,...k_n\}$ be independent r.v.s with $0<\sigma^2_n=Var(\sum_{j=1}^{k_n}X_{nj})<\infty$, $n=1,2,...$ and $k_n\rightarrow\infty$ as $n\rightarrow\infty$. If......Mon, 10 Aug 2026 07:09:45 -0400https://liberty.io/the-difference-between-87-and-71-of-a-number-is-448-what-is-59-of-that.htmlhttps://liberty.io/the-difference-between-87-and-71-of-a-number-is-448-what-is-59-of-that.htmlToday I came across an objective type question in that I have to choose from 4 optiona. I found myself struggling with this question since I have not been in touch with Math since I completed my ...Mon, 10 Aug 2026 07:07:24 -0400https://liberty.io/what-does-a-circle-in-4d-spacetime-look-like.htmlhttps://liberty.io/what-does-a-circle-in-4d-spacetime-look-like.htmlIn the normal metric for spacetime $ds^2=dx^2+dy^2+dz^2-cdt^2$ What does a "circle" of unit distance look like (the set of all points that satisfy $ds^2=1$?) That is does the negative give this any ...Mon, 10 Aug 2026 07:02:03 -0400https://liberty.io/what-is-the-motivation-behind-naming-identity-matrix-as-eye.htmlhttps://liberty.io/what-is-the-motivation-behind-naming-identity-matrix-as-eye.htmlMatlab, as well as some other PLs (e.g. Python's numpy package) use "eye" as a function name for creation of the identity matrix. Why is that so?...Mon, 10 Aug 2026 06:56:59 -0400https://liberty.io/find-four-groups-of-order-20-not-isomorphic-to-each-other.htmlhttps://liberty.io/find-four-groups-of-order-20-not-isomorphic-to-each-other.htmlFind four groups of order 20 not isomorphic to each other and prove why they aren't isomorphic. So far I thought of $\mathbb Z_{20}$, $\mathbb Z_2 \oplus\mathbb Z_{10}$, and $D_{10}$ (dihedral gro......Mon, 10 Aug 2026 06:51:50 -0400https://liberty.io/if-11-11-4-and-22-22-16-then-33-33-text-facebook-math-quiz-closed.htmlhttps://liberty.io/if-11-11-4-and-22-22-16-then-33-33-text-facebook-math-quiz-closed.htmlFrom a Facebook math quiz: $$\begin{align} 11 + 11 &= 4 \\ 22 + 22 &= 16 \\ 33 + 33 &= \text{???} \end{align}$$ Maybe I'm just stupid but the answer to this is $36$, however I think it ......Mon, 10 Aug 2026 06:45:48 -0400https://liberty.io/detailed-understandable-explanation-as-to-why-0-1-duplicate.htmlhttps://liberty.io/detailed-understandable-explanation-as-to-why-0-1-duplicate.htmlI have referred to many websites online regarding the proof, but I haven't understood it at all.. Please do help. As far as I know, $n! = 1 \times 2 \times 3 \times 4 \times \dots \times n$ Then ......Mon, 10 Aug 2026 06:32:18 -0400https://liberty.io/construction-of-the-lamplighter-group-as-wreath-product.htmlhttps://liberty.io/construction-of-the-lamplighter-group-as-wreath-product.htmlFor groups $(G,+)$ and $(H,\cdot)$ the wreath product $G \wr H$ is the set $W = G^H \times H$ (the set of all functions times $H$, so a typical element is $(f,h)$ where $f : H \to G$ and $h \in H$) ...Mon, 10 Aug 2026 06:25:40 -0400https://liberty.io/a-circle-of-radius-1-is-inscribed-inside-a-regular-octagon-a-polygon-w.htmlhttps://liberty.io/a-circle-of-radius-1-is-inscribed-inside-a-regular-octagon-a-polygon-w.htmlA circle of radius 1 is inscribed inside a regular octagon (a polygon with eight sides of length b). Calculate the octagon’s perimeter and its area. Hint: Split the octagon into eight isosceles tri......Mon, 10 Aug 2026 06:23:45 -0400https://liberty.io/what-is-ultrafinitism-and-why-do-people-believe-it.htmlhttps://liberty.io/what-is-ultrafinitism-and-why-do-people-believe-it.htmlI know there's something called "ultrafinitism" which is a very radical form of constructivism that I've heard said means people don't believe that really large integers actually exist. Could some......Mon, 10 Aug 2026 06:22:24 -0400https://liberty.io/algorithms-for-finding-the-prime-factorization-of-an-integer.htmlhttps://liberty.io/algorithms-for-finding-the-prime-factorization-of-an-integer.htmlAs practice, I am currently writing a program that takes a given integer $n$ as input, and then finds the (unique) prime factorization of $n$, provided $n$ is composite. My question is about algor......Mon, 10 Aug 2026 06:21:05 -0400https://liberty.io/definition-of-irreducible-element.htmlhttps://liberty.io/definition-of-irreducible-element.htmlIn my abstract algebra book, it says that an element $p \in R$ where $R$ is a commutative ring is irreducible if $\textbf{(i)}$: $p$ is not a unit $\textbf{(ii)}$: if $p=ab$ then either $a$ or $b$......Mon, 10 Aug 2026 06:19:23 -0400https://liberty.io/finding-the-optimal-mixed-strategy-of-a-3x3-matrix-game.htmlhttps://liberty.io/finding-the-optimal-mixed-strategy-of-a-3x3-matrix-game.htmlConsider a two player matrix game with payoff matrix : $$\begin{pmatrix}0 & 2 & -1\\ -2 & 0 & 1\ \\ 1 & -1 & 0\end{pmatrix}$$ I need to show that the game has no sad......Mon, 10 Aug 2026 06:14:59 -0400https://liberty.io/how-can-i-prove-sup-a-b-sup-a-sup-b-if-a-b-a-b-mid-a-in-a-b-in-b.htmlhttps://liberty.io/how-can-i-prove-sup-a-b-sup-a-sup-b-if-a-b-a-b-mid-a-in-a-b-in-b.htmlIf $A,B$ non empty, upper bounded sets and $A+B=\{a+b\mid a\in A, b\in B\}$, how can I prove that $\sup(A+B)=\sup A+\sup B$?...Mon, 10 Aug 2026 06:12:22 -0400https://liberty.io/99th-percentile-top-1-or-top-2-closed.htmlhttps://liberty.io/99th-percentile-top-1-or-top-2-closed.htmlIf one achieves a score in the 99th percentile on an exam, is that score considered in the top 1% or 2%? How is percentile defined in statistics? I read this somewhere: It’s top 2% - being in the x ...Mon, 10 Aug 2026 06:03:50 -0400https://liberty.io/funny-graph-of-x-by-a-graphing-program.htmlhttps://liberty.io/funny-graph-of-x-by-a-graphing-program.htmlI obtained a bizarre graph of $x!$, a function which I believe is only defined at positive integer domain. What causes such an error?. It would be interesting if someone can explain the method suc......Mon, 10 Aug 2026 05:55:28 -0400https://liberty.io/proof-by-mvt-no-calculator.htmlhttps://liberty.io/proof-by-mvt-no-calculator.htmlSo I need to prove that $$\frac{1}{16} <\sqrt{51} -7< \frac{1}{ 14}$$ using the MVT and no calculator but I can't find a function that would satisfy those outputs ![as for the question of w......Mon, 10 Aug 2026 05:52:49 -0400https://liberty.io/biased-coin-probability.htmlhttps://liberty.io/biased-coin-probability.html1) Biased coin flipped twice. There is 6/9 probability that the coin will land heads for at least one of the two flips. What is the smallest number of times that I can flip the coin for at least 99% ...Mon, 10 Aug 2026 05:51:00 -0400https://liberty.io/the-perimeter-of-an-isosceles-triangle-triangle-abc.htmlhttps://liberty.io/the-perimeter-of-an-isosceles-triangle-triangle-abc.htmlAn isosceles triangle $\triangle ABC$ is given with $\angle ACB=30^\circ$ and leg $BC=16$ $cm$. Find the perimeter of $\triangle ABC$. We have two cases, right? When 1) $AC=BC=16$ and 2) $AB=BC=16......Mon, 10 Aug 2026 05:46:54 -0400https://liberty.io/how-to-tell-whether-a-left-and-right-riemann-sum-are-overestiamtes-and.htmlhttps://liberty.io/how-to-tell-whether-a-left-and-right-riemann-sum-are-overestiamtes-and.htmlI know that in a positive and increasing function, the right riemann sum is an overestimate and the left is an underestimate, but what about if the function is negative and increasing like this? Wh......Mon, 10 Aug 2026 05:42:57 -0400https://liberty.io/difference-between-frac-df-dx-frac-delta-f-delta-x-and-frac-partial-f-.htmlhttps://liberty.io/difference-between-frac-df-dx-frac-delta-f-delta-x-and-frac-partial-f-.html[Beginning calculus question.] I've just been introduced to a number of ways of representing changes in a function value with respect to some variable in multivariable calculus. I don't get the ...Mon, 10 Aug 2026 05:42:06 -0400https://liberty.io/formula-to-find-an-angle-of-point-on-a-coordinate-plane.htmlhttps://liberty.io/formula-to-find-an-angle-of-point-on-a-coordinate-plane.htmlGiven a plane and an arbitrary (x,y) point, is there a succinct formula to find the angle of that point against the positive y-axis? For example, pictured below the green point is 0 degrees, blue 45 ...Mon, 10 Aug 2026 05:40:29 -0400https://liberty.io/how-to-prove-the-distributive-property-without-using-truth-tables.htmlhttps://liberty.io/how-to-prove-the-distributive-property-without-using-truth-tables.htmlI did it with using truth tables but I am inquisitive about how to proof the distributive property without using truth tables (i.e using the other rules of replacement or inference). $$ (P \land (......Mon, 10 Aug 2026 05:31:06 -0400https://liberty.io/how-do-i-find-the-equation-of-an-osculating-circle-when-i-039-m-given-.htmlhttps://liberty.io/how-do-i-find-the-equation-of-an-osculating-circle-when-i-039-m-given-.htmlThis is a question given out by my calculus professor, and I'm completely stumped as to how I need to go about solving it. Let the parabola $y=x^2$ be parameterized by $r(t)=ti+t^2j$. Find the equ......Mon, 10 Aug 2026 05:30:10 -0400https://liberty.io/theorem-18-1-of-munkres-topology.htmlhttps://liberty.io/theorem-18-1-of-munkres-topology.htmlLet $X$ and $Y$ be topological spaces; let $f:X\to Y$. Then the following are equivalent: (1) $f$ is continuous (2) for every subset $A$ of $X$, one has $f(\overline{A}) \subseteq \overline{f(A)}$.......Mon, 10 Aug 2026 05:07:31 -0400https://liberty.io/graph-the-polynomial-f-x-x-4-3x-3-4x-2-3x-2-using-a-graphing-calculato.htmlhttps://liberty.io/graph-the-polynomial-f-x-x-4-3x-3-4x-2-3x-2-using-a-graphing-calculato.htmlGraph the polynomial $f(x)=x^4-3x^3-4x^2+3x-2$ using a graphing calculator? a) The Range b) The Real Zeros c) The y-intercept d) relative minimum and Relative Maximum e) The interval where $f(x)\......Mon, 10 Aug 2026 05:03:12 -0400https://liberty.io/finding-angles-in-an-isosceles-trapezoid.htmlhttps://liberty.io/finding-angles-in-an-isosceles-trapezoid.htmlI was given the following question: PQRS is an isosceles trapezoid. If angle TRS= 100 degrees, angle T=35 degrees and PS=TS, find the measures of the numbered angles. Attached is a drawing of the ...Mon, 10 Aug 2026 05:00:23 -0400https://liberty.io/why-is-it-that-this-gives-a-good-approximation-of-pi.htmlhttps://liberty.io/why-is-it-that-this-gives-a-good-approximation-of-pi.htmlAt the end of a Physics examination, I decided to play around with my calculator, as I always do when I have time left over, and found that: $$\frac{1}{100} \cdot 11^{\ln(11)} \approx 3.1415978921......Mon, 10 Aug 2026 05:00:08 -0400https://liberty.io/midpoint-approximation-over-under-estimation.htmlhttps://liberty.io/midpoint-approximation-over-under-estimation.htmlSo left handed approximation underestimates the area under a increasing curve and over estimates for decreasing curves. And right handed approximation overestimates for increasing curves and ...Mon, 10 Aug 2026 04:55:04 -0400https://liberty.io/symmetry-group-of-equilateral-triangle.htmlhttps://liberty.io/symmetry-group-of-equilateral-triangle.htmlI have read at some places that the symmetry of equilateral triangle is C3v as well as some places mention it to be D3. The group tables for these two groups differ, hence they are not isomorphic......Mon, 10 Aug 2026 04:41:06 -0400https://liberty.io/linear-equation-to-matrix-form.htmlhttps://liberty.io/linear-equation-to-matrix-form.htmlHey everyone I'm practicing some linear equation matrix questions, so far they're easy to construct, but I am completely lost on this one since I cannot use the method of putting them in separate c......Mon, 10 Aug 2026 04:37:34 -0400https://liberty.io/is-there-a-rule-of-integration-that-corresponds-to-the-quotient-rule.htmlhttps://liberty.io/is-there-a-rule-of-integration-that-corresponds-to-the-quotient-rule.htmlWhen teaching the integration method of u-substitution, I like to emphasize its connection with the chain rule of integration. Likewise, the intimate connection between the product rule of derivati......Mon, 10 Aug 2026 04:17:27 -0400https://liberty.io/when-an-equation-has-no-solutions-denote-it-with-x-in-varnothing.htmlhttps://liberty.io/when-an-equation-has-no-solutions-denote-it-with-x-in-varnothing.htmlMy teacher claims that when an equation in variables $x_1,x_2,\ldots,x_n$ has no solutions, you should denote this fact with $(x_1,x_2,\ldots,x_n)\in\varnothing$. An empty set can't have an elemen......Mon, 10 Aug 2026 04:10:00 -0400https://liberty.io/probability-of-head-in-coin-flip-when-coin-is-flipped-two-times.htmlhttps://liberty.io/probability-of-head-in-coin-flip-when-coin-is-flipped-two-times.htmlProbability of getting a head in coin flip is $1/2$. If the coin is flipped two times what is the probability of getting a head in either of those attempts? I think both the coin flips are mutually ...Mon, 10 Aug 2026 04:08:07 -0400https://liberty.io/what-is-a-particular-and-a-homogenous-solution-of-a-differential-equat.htmlhttps://liberty.io/what-is-a-particular-and-a-homogenous-solution-of-a-differential-equat.htmlWhen solving linear nonhomogeneous equations, we deal with two types of solutions: particular homogeneous Why do we have these two types of solutions for differential equations? What does each of......Mon, 10 Aug 2026 04:03:19 -0400https://liberty.io/sum-of-fibonacci-numbers.htmlhttps://liberty.io/sum-of-fibonacci-numbers.htmlWhile trying to find find a formula to calculate the length of the golden spiral I came across the sum of the Fibonacci numbers. I noticed that $$\text{Fibonacci numbers: }1,1,2,3,5,8,13,21,34...$......Mon, 10 Aug 2026 03:57:25 -0400https://liberty.io/linear-combination-of-x.htmlhttps://liberty.io/linear-combination-of-x.htmlIs $3x + 4x^2 + y$ a valid linear combination of $x$? I'm not sure if "a linear combination of $x$" allows for other terms than $x$ which would just be treated as constants, or it should strictly c......Mon, 10 Aug 2026 03:55:01 -0400https://liberty.io/how-many-different-possible-wordle-tweets-can-exist.htmlhttps://liberty.io/how-many-different-possible-wordle-tweets-can-exist.htmlWordle is a viral web game where you have up to six attempts to guess a secret five-letter English word* using the letters A-Z. After each $5$-letter guess (which must itself be a real English word......Mon, 10 Aug 2026 03:40:38 -0400https://liberty.io/is-there-a-valid-shortcut-method-to-convert-a-quadratic-from-vertex-fo.htmlhttps://liberty.io/is-there-a-valid-shortcut-method-to-convert-a-quadratic-from-vertex-fo.htmlRecently, I had a couple of students give me an interesting way of converting from vertex form to factored form that I've never seen before. One told me he got it from his science teacher, and anot......Mon, 10 Aug 2026 03:29:38 -0400https://liberty.io/how-to-find-derivative-of-x-2-sin-x-using-only-the-limit-definition-of.htmlhttps://liberty.io/how-to-find-derivative-of-x-2-sin-x-using-only-the-limit-definition-of.htmlI'm trying to find the derivative of $x^2\sin x$ using only the limit definition of a derivative. I've tried two approaches, one using the difference quotient and another with the regular $x-a$ fo......Mon, 10 Aug 2026 03:16:24 -0400