https://track.liberty.ioFresh celebrity buzz with instant attention value.en-usSun, 06 Sep 2026 01:21:32 -0400https://track.liberty.io/lim-x-to0-sin-2x-2x-x-3.htmlhttps://track.liberty.io/lim-x-to0-sin-2x-2x-x-3.htmlEvaluate $$ \lim_{x\to0} \frac{\sin(2x)-2x}{x^3} $$ I thought that I could use L'Hopital's rule to get to an answer of $-1$ but according to the answer manual that isn't correct. What am I doing w......Sun, 06 Sep 2026 05:21:26 -0400https://track.liberty.io/is-there-any-difference-between-empty-graph-and-null-graph.htmlhttps://track.liberty.io/is-there-any-difference-between-empty-graph-and-null-graph.htmlI was going through one of the links Is the empty graph connected? here I found that it is mentioned that empty graph is a graph with no edges while null graph is a graph with no vertices but I ...Sun, 06 Sep 2026 05:18:27 -0400https://track.liberty.io/how-to-prove-operatorname-tr-ab-operatorname-tr-ba.htmlhttps://track.liberty.io/how-to-prove-operatorname-tr-ab-operatorname-tr-ba.htmlthere is a similar thread here Coordinate-free proof of $\operatorname{Tr}(AB)=\operatorname{Tr}(BA)$?, but I'm only looking for a simple linear algebra proof....Sun, 06 Sep 2026 05:08:57 -0400https://track.liberty.io/evaluating-limits-algebraically.htmlhttps://track.liberty.io/evaluating-limits-algebraically.htmlI bought the eighth edition of Stewart Calculus (metric version) and I'm up to the section about limits. It's been pretty easy so far, but I've come across a class of limit problems that don't seem ...Sun, 06 Sep 2026 04:52:52 -0400https://track.liberty.io/what-is-the-definition-of-tensor-contraction.htmlhttps://track.liberty.io/what-is-the-definition-of-tensor-contraction.htmlAccording to Wikipedia's page on tensor contraction: In general, a tensor of type $(m,n)$ (with $m \geq 1$ and $n \geq 1$) is an element of the vector space $V \otimes \ldots \otimes V \otimes V......Sun, 06 Sep 2026 04:41:54 -0400https://track.liberty.io/eigenfunction-fourier-series-relationship.htmlhttps://track.liberty.io/eigenfunction-fourier-series-relationship.htmlI'm having difficulty formulating a proof detailing that every eigenfunction of $D^2$ is either a constant or of the form $a$cos$(nx)$ + $b$sin($nx$) for some value $n$. I understand that the Fou......Sun, 06 Sep 2026 04:35:26 -0400https://track.liberty.io/property-of-dirac-delta-function-in-mathbb-r-n.htmlhttps://track.liberty.io/property-of-dirac-delta-function-in-mathbb-r-n.htmlHow does one prove the following identity? $$\int _Vf(\pmb{r})\delta (g(\pmb{r})) \, d\pmb{r}=\int_S \frac{f(\pmb{r})}{\left|\operatorname{grad} g(\pmb{r})\right|} \, d\sigma$$ where $S$ is the sur......Sun, 06 Sep 2026 04:34:30 -0400https://track.liberty.io/is-there-a-way-to-find-all-roots-of-a-polynomial-equation.htmlhttps://track.liberty.io/is-there-a-way-to-find-all-roots-of-a-polynomial-equation.htmlIs there a way to find all roots of a polynomial equation? Lets say $$x^5+ax^4+bx^3+cx^2+dx+e=0$$ how to find its all roots?...Sun, 06 Sep 2026 04:30:35 -0400https://track.liberty.io/what-is-the-mathematical-definition-of-index-set.htmlhttps://track.liberty.io/what-is-the-mathematical-definition-of-index-set.htmlI find some descriptions http://en.wikipedia.org/wiki/Index_set and http://mathworld.wolfram.com/IndexSet.html . But can't find any definition....Sun, 06 Sep 2026 04:27:20 -0400https://track.liberty.io/how-to-prove-that-derivatives-have-the-intermediate-value-property.htmlhttps://track.liberty.io/how-to-prove-that-derivatives-have-the-intermediate-value-property.htmlI'm reading a book which gives this theorem without proof: If a and b are any two points in an interval on which ƒ is differentiable, then ƒ' takes on every value between ƒ'(a) and ƒ'(b). As ......Sun, 06 Sep 2026 04:25:37 -0400https://track.liberty.io/what-is-a-wavenumber.htmlhttps://track.liberty.io/what-is-a-wavenumber.htmlI have been struggling to understand the concept of wavenumber for a number of years. Is a wavenumber we talk about in the context of wave propagation in physics (say ocean waves, or acoustics) sa......Sun, 06 Sep 2026 04:16:05 -0400https://track.liberty.io/testing-for-associativity-using-the-multiplication-table-cayley-table-.htmlhttps://track.liberty.io/testing-for-associativity-using-the-multiplication-table-cayley-table-.htmlI seem to recall that there is a relatively easy method for determining the associativity of an operation by using its Cayley table. What is it?...Sun, 06 Sep 2026 04:12:40 -0400https://track.liberty.io/feynman-integration-problem.htmlhttps://track.liberty.io/feynman-integration-problem.html$$ I = \frac{\pi^2}{8} - \int_0^1 \frac{\arctan(x)}{\sqrt{1-x^2}} \,dx $$ Evaluate $I$ $$ I = \frac{\pi^2}{8} - \int_0^1 \frac{\arctan(x)}{\sqrt{1-x^2}} \,dx$$ $$f(a) = \int_0^1 \frac{\arctan(ax......Sun, 06 Sep 2026 03:57:44 -0400https://track.liberty.io/9x-multiplication-table-trick-how-does-it-work.htmlhttps://track.liberty.io/9x-multiplication-table-trick-how-does-it-work.htmlThere is a trick that is taught to children to help them remember their 9 times tables. As illustrated below, the digits of the multiplication problem in question can be determined by: Splitting ......Sun, 06 Sep 2026 03:56:46 -0400https://track.liberty.io/user-quinn-greicius-mathematics-stack-exchange.htmlhttps://track.liberty.io/user-quinn-greicius-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Sun, 06 Sep 2026 03:54:17 -0400https://track.liberty.io/less-than-or-equal-sign.htmlhttps://track.liberty.io/less-than-or-equal-sign.htmlIf I know for two numbers a and b that $${a < b }$$ Then is it correct to say that $$ a \leq b $$ I know that the second statement is true as long as the first one is. It seems OK as it is ...Sun, 06 Sep 2026 03:51:05 -0400https://track.liberty.io/would-going-first-in-egg-russian-roulette-increase-or-decrease-your-ch.htmlhttps://track.liberty.io/would-going-first-in-egg-russian-roulette-increase-or-decrease-your-ch.htmlEgg Russian Roulette: 6 eggs, 5 of which are hard boiled & 1 of which is raw. Players (1v1) each choose eggs in a turn based system whereby the egg is smashed against the forehead. The f......Sun, 06 Sep 2026 03:49:44 -0400https://track.liberty.io/why-average-value-of-a-function-is-not-calculated-by-using-the-formula.htmlhttps://track.liberty.io/why-average-value-of-a-function-is-not-calculated-by-using-the-formula.htmlI know the formula for calculating the average value of a function as $$ \frac{1}{b-a}\int_a^b f(x)\,dx$$ But in elementry level maths and physics problems we generally use a very simple approach to ...Sun, 06 Sep 2026 03:38:12 -0400https://track.liberty.io/what-is-the-meaning-of-the-symbol-pitchfork.htmlhttps://track.liberty.io/what-is-the-meaning-of-the-symbol-pitchfork.htmlI have seen this symbol in the formulation of this question. There, it is said: Let $f: X \to Y$ be a smooth map with $f \pitchfork Z$. I was googling, but I haven't found any answers....Sun, 06 Sep 2026 03:37:33 -0400https://track.liberty.io/what-does-the-definition-of-curvature-mean.htmlhttps://track.liberty.io/what-does-the-definition-of-curvature-mean.htmlFirst question, am I right in saying that curvature measure how quickly the direction of a cruve changes? Also, we have been given the "definition": $$T'(t) = \kappa(t) | \gamma'(t) | U(t)$$ wh......Sun, 06 Sep 2026 03:30:08 -0400https://track.liberty.io/is-a-non-repeating-and-non-terminating-decimal-always-an-irrational.htmlhttps://track.liberty.io/is-a-non-repeating-and-non-terminating-decimal-always-an-irrational.htmlWe can build $\frac{1}{33}$ like this, $.030303$ $\cdots$ ($03$ repeats). $.0303$ $\cdots$ tends to $\frac{1}{33}$. So,I was wondering this: In the decimal representation, if we start writing the ......Sun, 06 Sep 2026 03:21:55 -0400https://track.liberty.io/proof-that-eigenvalues-are-the-diagonal-entries-of-the-upper-triangula.htmlhttps://track.liberty.io/proof-that-eigenvalues-are-the-diagonal-entries-of-the-upper-triangula.htmlThis is 5.18 from Axler's Linear Algebra Done Right: Theorem: Suppose $T \in L(V)$ has an upper-triangular matrix with respect to some basis of $V$. Then the eigenvalues of $T$ consist precisely o......Sun, 06 Sep 2026 03:21:32 -0400https://track.liberty.io/block-diagonal-matrix-diagonalizable.htmlhttps://track.liberty.io/block-diagonal-matrix-diagonalizable.htmlI am trying to prove that: The matrix $C = \left(\begin{smallmatrix}A& 0\\0 & B\end{smallmatrix}\right)$ is diagonalizable, if only if $A$ and $B$ are diagonalizable. If $A\in GL(\mathbb{......Sun, 06 Sep 2026 03:16:48 -0400https://track.liberty.io/an-infinite-union-of-closed-sets-is-a-closed-set.htmlhttps://track.liberty.io/an-infinite-union-of-closed-sets-is-a-closed-set.htmlQuestion: {$B_n$}$ \in \Bbb R$ is a family of closed sets. Prove that $\cup _{n=1}^\infty B_n$ is not necessarily a closed set. What I thought: Using a counterexample: If I say that each $B_i$ is ......Sun, 06 Sep 2026 03:14:04 -0400https://track.liberty.io/learning-projective-geometry.htmlhttps://track.liberty.io/learning-projective-geometry.htmlMy ultimate goal is to learn some algebraic geometry with the more concrete immediate goal of understanding things like how $\mathbb R\mathrm P^2$ is embedded in $\mathbb C\mathrm P^2$ or how $\mat......Sun, 06 Sep 2026 03:07:48 -0400https://track.liberty.io/absolute-maximum-and-minimum-values-of-a-function-over-a-region.htmlhttps://track.liberty.io/absolute-maximum-and-minimum-values-of-a-function-over-a-region.htmlThe prompt is to find the absolute maximum and minimum values of the function $f(x, y) = x^2 + 2x - y^2$ on the upper half of a unit disc, $D =\{(x, y): x^2 + y^2 \leq 1, y \geq 0 \}$ I began by f......Sun, 06 Sep 2026 03:00:26 -0400https://track.liberty.io/help-with-proving-that-the-transpose-of-the-product-of-any-number-of-m.htmlhttps://track.liberty.io/help-with-proving-that-the-transpose-of-the-product-of-any-number-of-m.htmlSpecifically I am trying to show that (An)T = (AT)n where A is an mxm square matrix and n is a positive integer. This is where I'm stuck: To prove the theorem I would like to show that ((An)T)ij =......Sun, 06 Sep 2026 02:55:15 -0400https://track.liberty.io/calculate-matrix-a-from-null-space-basis-of-a-4i.htmlhttps://track.liberty.io/calculate-matrix-a-from-null-space-basis-of-a-4i.htmlHow to find a matrix $A$ when you are given some parameters and the basis for the null space? The problem I've been scratching my head over is this. The basis for the null space of $A-4I$ is $$\......Sun, 06 Sep 2026 02:54:24 -0400https://track.liberty.io/methods-to-prove-frac-21n-4-14n-3-is-irreducible-for-every-natural-num.htmlhttps://track.liberty.io/methods-to-prove-frac-21n-4-14n-3-is-irreducible-for-every-natural-num.htmlProve $\frac{21n + 4}{14n + 3}$ is irreducible for every natural number $n$. I was thinking of taking a number-theory based approach. Can you suggest the following method Calculus/Number theory ......Sun, 06 Sep 2026 02:31:52 -0400https://track.liberty.io/why-is-the-function-f-x-x-3-pmod-10-periodic-with-this-strange-propert.htmlhttps://track.liberty.io/why-is-the-function-f-x-x-3-pmod-10-periodic-with-this-strange-propert.htmlI've noticed that the function $f(x)=x^3 \pmod{10}$ is periodic. For example, listing mod(x^3,10) we get: 0,1,8,7,4,5,6,3,2,9,0,1,8,7,4,5,6,3,2,9,0,1,8,7,4,5,6,3,2,9,0,1,8,7,4,5,6,3,2,9,0,1,8,7,4......Sun, 06 Sep 2026 02:29:45 -0400https://track.liberty.io/summation-symbol-with-just-one-letter-below-it-how-to-verbally-state.htmlhttps://track.liberty.io/summation-symbol-with-just-one-letter-below-it-how-to-verbally-state.htmlI'm used to seeing summation capital sigmas with a "start" at bottom and a "stop" at top and would say the sum from i to 100 or whatever. But in this case, there is just a y at bottom. How would I ...Sun, 06 Sep 2026 02:29:30 -0400https://track.liberty.io/what-trig-identity-would-help-solve-2-cos-2x-3-cos-x.htmlhttps://track.liberty.io/what-trig-identity-would-help-solve-2-cos-2x-3-cos-x.htmlI need help with a homework question that has me puzzled. I need to solve the following equation: $$2 + \cos(2x) = 3\cos(x)$$ I don't see a good trig identity to apply. I tried $\cos(2x) = 2\cos^......Sun, 06 Sep 2026 02:25:00 -0400https://track.liberty.io/cos-4x-sin-2x-in-terms-of-first-power-of-cosine.htmlhttps://track.liberty.io/cos-4x-sin-2x-in-terms-of-first-power-of-cosine.htmlI believe that I have his correct but if someone could check it and see that'd be great. Here's a pic!...Sun, 06 Sep 2026 02:21:14 -0400https://track.liberty.io/simplifying-a-square-root-fraction.htmlhttps://track.liberty.io/simplifying-a-square-root-fraction.htmlSimplify the following $$\frac{\sqrt{3}}{\sqrt{2}(\sqrt{6} - \sqrt{3})}$$ Apparently the answer is $\frac{1}{2} (2 + \sqrt{2})$ but can't for the life of me see how to get it. Any help is massiv......Sun, 06 Sep 2026 02:13:15 -0400https://track.liberty.io/related-rates-how-fast-is-the-water-leaking-from-a-conical-shaped-tank.htmlhttps://track.liberty.io/related-rates-how-fast-is-the-water-leaking-from-a-conical-shaped-tank.htmlWater is poured at the rate of 8 cubic feet per minute into a conical-shaped tank, 20 ft deep and 10 ft in diameter at the top. If the tank has a leak in the bottom and the water level is rising at......Sun, 06 Sep 2026 02:12:14 -0400https://track.liberty.io/why-do-we-care-about-dual-spaces.htmlhttps://track.liberty.io/why-do-we-care-about-dual-spaces.htmlWhen I first took linear algebra, we never learned about dual spaces. Today in lecture we discussed them and I understand what they are, but I don't really understand why we want to study them within ...Sun, 06 Sep 2026 02:11:19 -0400https://track.liberty.io/definition-of-concave-set.htmlhttps://track.liberty.io/definition-of-concave-set.htmlWe know that a set is convex if the straight line joining any two points of the set lies completely in the set. Or, mathematically, a set $X$ is convex if $$x_1, x_2 \in X, 0 \leq \lambda \leq 1 \...Sun, 06 Sep 2026 02:08:11 -0400https://track.liberty.io/multivariable-polynomial-long-division-calculator.htmlhttps://track.liberty.io/multivariable-polynomial-long-division-calculator.htmlIs there code/online calculator that divides polynomials in multiple variables? The answers are always other polynomials, but they have many variables....Sun, 06 Sep 2026 02:05:48 -0400https://track.liberty.io/what-are-these-bracketing-symbols-and-what-do-they-mean.htmlhttps://track.liberty.io/what-are-these-bracketing-symbols-and-what-do-they-mean.htmlWhat do the matching "L" shapes (near .5 and 20) mean in this forumla? The document where I found this formula can be found here: http://www.cs.cmu.edu/~hcwong/Pdfs/icip02.ps...Sun, 06 Sep 2026 01:54:20 -0400https://track.liberty.io/x-1-x-2-iid-standard-normal-with-polar-coordinates-r-and-p-are-r-and-p.htmlhttps://track.liberty.io/x-1-x-2-iid-standard-normal-with-polar-coordinates-r-and-p-are-r-and-p.htmlI have two scalar random variables $X_1$ and $X_2$ and they are both independent and both have standard normal distribution $N(0,1)$. I am letting $r$ and $p$ be the polar coordinates of the point ......Sun, 06 Sep 2026 01:49:48 -0400https://track.liberty.io/why-don-039-t-parabolas-have-asymptotes.htmlhttps://track.liberty.io/why-don-039-t-parabolas-have-asymptotes.htmlAs given on the Wikipedia page, an asymptote is a line which becomes the tangent of the curve as the $x$ or $y$ cordinates of the curve tends to infinity. Hyperbola has asymptotes but parabolas ......Sun, 06 Sep 2026 01:48:43 -0400https://track.liberty.io/finding-the-diagonal-of-a-rhombus.htmlhttps://track.liberty.io/finding-the-diagonal-of-a-rhombus.htmlI am trying to show that the diagonals of a RHOMBUS intersect each other at 90 degrees However I need to find the length of the diagonal without using Trig. ratios. Any ideas how i could find t......Sun, 06 Sep 2026 01:43:32 -0400https://track.liberty.io/what-is-the-most-general-definition-of-a-graph.htmlhttps://track.liberty.io/what-is-the-most-general-definition-of-a-graph.htmlI would like to give the most general definition of a graph. Based on what I read about graphs, I came up with the following definition: Let us introduce some symbols. They are (a) $V$:: a non-em......Sun, 06 Sep 2026 01:42:42 -0400https://track.liberty.io/area-under-the-graph-y-ln-x.htmlhttps://track.liberty.io/area-under-the-graph-y-ln-x.htmlFind the Area of the shaded region $y= \ln x \rightarrow x = e^y $ I found that the curve cuts the $x$ axis at $1$ through substituting $0$ to $y$ Why I can't find the area under the graph thr......Sun, 06 Sep 2026 01:41:44 -0400https://track.liberty.io/what-is-the-topology-of-a-simplicial-complex.htmlhttps://track.liberty.io/what-is-the-topology-of-a-simplicial-complex.htmlI know what a simplicial complex is, but when reading about triangulations on surfaces I found that there must exist a homeomorphism betwen the space underlying the surface and some simplicial comp......Sun, 06 Sep 2026 01:40:52 -0400https://track.liberty.io/does-sum-n-1-infty-frac-n-1-n-2-converge-or-diverge.htmlhttps://track.liberty.io/does-sum-n-1-infty-frac-n-1-n-2-converge-or-diverge.htmlIs my logic OK? $a_{n}=\frac{n-1}{n^2}$ $\frac{1}{n} \leq b_{n}=\frac{n-\frac{n}{2}}{n^2}=\frac{n}{2n^2}=\frac{1}{2n} \leq a_{n}=\frac{n-1}{n^2}$ and there for the initial series diverges....Sun, 06 Sep 2026 01:37:24 -0400https://track.liberty.io/what-is-the-result-of-3-digit-chopping-for-0-000234.htmlhttps://track.liberty.io/what-is-the-result-of-3-digit-chopping-for-0-000234.htmlI am trying to understand if the 0."000" part counted as digit or not. If 0."000" is not "digit", then the result should 0.000234. If yes, then the result should be 0.00 Which one is correct? He......Sun, 06 Sep 2026 01:36:00 -0400https://track.liberty.io/factorize-a-third-degree-polynomial.htmlhttps://track.liberty.io/factorize-a-third-degree-polynomial.htmlI'm currently trying to solve a problem which asks if a 3x3 matrix is diagonalizable, I know the method but when it comes to finding the roots, I have a third degree polynomial and I don't know how......Sun, 06 Sep 2026 01:32:02 -0400https://track.liberty.io/finding-extreme-values-in-constraints-using-lagrange-multipliers.htmlhttps://track.liberty.io/finding-extreme-values-in-constraints-using-lagrange-multipliers.htmlThis question may sound repetitive, but please give it a read. I read about Lagrange multipliers, and I know the formula. However, I was trying to get the intuition behind this. I read the posts Ho......Sun, 06 Sep 2026 01:31:34 -0400https://track.liberty.io/prove-that-if-ab-is-invertible-then-b-is-invertible.htmlhttps://track.liberty.io/prove-that-if-ab-is-invertible-then-b-is-invertible.htmlI know this proof is short but a bit tricky. So I suppose that $AB$ is invertible then $(AB)^{-1}$ exists. We also know $(AB)^{-1}=B^{-1}A^{-1}$. If we let $C=(B^{-1}A^{-1}A)$ then by the invertible ...Sun, 06 Sep 2026 01:22:46 -0400https://track.liberty.io/difference-between-sq-km-and-km-sq.htmlhttps://track.liberty.io/difference-between-sq-km-and-km-sq.htmlIf I have a square with side $2\ \text{km}$, what is its area: $2\ \text{sq km}$ or $4\ \text{km}^2$?...Sun, 06 Sep 2026 01:18:48 -0400https://track.liberty.io/how-to-find-a-unit-normal-vector-to-the-given-curve.htmlhttps://track.liberty.io/how-to-find-a-unit-normal-vector-to-the-given-curve.htmlFind the unit normal vector to the curve $r(t) = (3\sin t)i + (3\cos t)j + 4t k$ at point $( \pi /2, 0,1)$. The tangent vector of this curve is $(3 \cos t)i -(3 \sin t)j + 4k$ and unit normal vector ...Sun, 06 Sep 2026 01:11:31 -0400https://track.liberty.io/question-about-the-irwin-hall-distribution-uniform-sum-distribution.htmlhttps://track.liberty.io/question-about-the-irwin-hall-distribution-uniform-sum-distribution.htmlSo I have been reading about the Irwin-Hall distribution online, it is a sum of uniform distributions on $[0,1]$, and it seems very interesting: http://en.wikipedia.org/wiki/Irwin%E2%80%...Sun, 06 Sep 2026 01:11:15 -0400https://track.liberty.io/difference-between-closed-and-open-intervals-for-continuous-functions.htmlhttps://track.liberty.io/difference-between-closed-and-open-intervals-for-continuous-functions.htmlSay I have a continuous function $f(x)$ on the open interval $(0, 1)$. There is no limit on how large $|f(x)|$ can be apart from the fact that it can't be infinity, it will always be a real number.......Sun, 06 Sep 2026 00:57:37 -0400https://track.liberty.io/is-there-an-analogue-to-the-delta-symbol-for-ratios.htmlhttps://track.liberty.io/is-there-an-analogue-to-the-delta-symbol-for-ratios.htmlA capital delta ($\Delta$) is commonly used to indicate a difference (especially an incremental difference). For example, $\Delta x = x_1 - x_0$ My question is: is there an analogue of this notati......Sun, 06 Sep 2026 00:55:23 -0400https://track.liberty.io/show-that-101-is-a-prime-number-given-the-fact-that-10-squared-is-100-.htmlhttps://track.liberty.io/show-that-101-is-a-prime-number-given-the-fact-that-10-squared-is-100-.htmlQuestion: Use the fact that $10^2 = 100$ and $11^2 = 121$ to show that the number 101 is a prime number. Could someone please given me a hint on how to solve this problem. I can't seem to relate ......Sun, 06 Sep 2026 00:42:22 -0400https://track.liberty.io/example-for-adjacency-matrix-of-a-bipartite-graph.htmlhttps://track.liberty.io/example-for-adjacency-matrix-of-a-bipartite-graph.htmlCan someone explain to me with an example how to create the adjacency matrix of a bipartite graph? And why the diagonal elements of it are not zero? Thanks....Sun, 06 Sep 2026 00:40:59 -0400https://track.liberty.io/what-039-s-the-use-of-composition-series-in-group-theory.htmlhttps://track.liberty.io/what-039-s-the-use-of-composition-series-in-group-theory.htmlI'm going through Dummit & Foote, and I stumbled upon a definition of composition series (for groups) and Jordan-Holder theorem. I get the ''it's something like a factoring of a group'' intuit......Sun, 06 Sep 2026 00:40:48 -0400https://track.liberty.io/why-is-mathematical-induction-called-mathematical.htmlhttps://track.liberty.io/why-is-mathematical-induction-called-mathematical.htmlOne of my whims is that I never write "mathematical induction" but just "induction". We are doing maths, so what is the point about precising? We don't say "Let $f$ be a mathematical function from ......Sun, 06 Sep 2026 00:22:46 -0400https://track.liberty.io/when-can-we-interchange-integration-and-differentiation.htmlhttps://track.liberty.io/when-can-we-interchange-integration-and-differentiation.htmlLet $f$ be a Riemann Integrable function over $\mathbb{R}^2$. When can we do this? $$\frac{\partial}{\partial\theta}\int_{a}^{b}f(x,\theta)dx=\int_{a}^{b}\frac{\partial}{\partial\theta}f(x,\theta)......Sun, 06 Sep 2026 00:18:23 -0400https://track.liberty.io/definition-of-conditional-expectation-of-y-given-x.htmlhttps://track.liberty.io/definition-of-conditional-expectation-of-y-given-x.htmlLet $(S,F,P)$ be probability space. Let $X,Y$ be continuous random variables from $S$ to $\mathbb{R}.$ Formal Definition of Conditional Expectation $E(Y|X)$ of $Y$ to X is $\sigma{(X)} - Borel$ ...Sun, 06 Sep 2026 00:14:38 -0400https://track.liberty.io/are-there-any-good-discrete-mathematics-video-online.htmlhttps://track.liberty.io/are-there-any-good-discrete-mathematics-video-online.htmlI want to learn discrete mathematics by reading book by myself but I find sometime it's very hard to understand what author trying to say. I want to know, are there any good online video that teach ...Sun, 06 Sep 2026 00:10:19 -0400https://track.liberty.io/the-number-of-integer-solutions-of-equations.htmlhttps://track.liberty.io/the-number-of-integer-solutions-of-equations.htmlThe Number Of Integer Solutions Of Equations An approach is to find the number of distinct non-negative integer-valued vectors $(x_1,x_2,...,x_r)$ such that $$x_1 + x_2 + ... + x_r = n$$ Firstly, ...Sun, 06 Sep 2026 00:07:44 -0400https://track.liberty.io/equation-of-the-line-passing-through-3-2-5-and-3-2-6.htmlhttps://track.liberty.io/equation-of-the-line-passing-through-3-2-5-and-3-2-6.htmlFind the Cartesian equation of the line passing through $(3,-2,-5)$ and $(3,-2,6)$ in $3$D. The equation of the line through the points $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ is given by $$ \vec{r}=\......Sat, 05 Sep 2026 23:48:35 -0400https://track.liberty.io/rotation-matrix-between-right-and-left-handed-systems-with-an-addition.htmlhttps://track.liberty.io/rotation-matrix-between-right-and-left-handed-systems-with-an-addition.htmlI have two coordinate frames, A and B. I want to create the rotation matrix RAB which takes you from A to B. A is a right-handed system, and B is a left-handed system. Furthermore, after moving fro......Sat, 05 Sep 2026 23:48:24 -0400https://track.liberty.io/lu-decomposition-vs-cholesky-decomposition.htmlhttps://track.liberty.io/lu-decomposition-vs-cholesky-decomposition.htmlWhat is the difference between LU Decomposition and Cholesky Decomposition about using these methods to solving linear equation systems? Could you explain the difference with a simple example? Also ...Sat, 05 Sep 2026 23:47:53 -0400https://track.liberty.io/complete-residue-system-proof.htmlhttps://track.liberty.io/complete-residue-system-proof.htmlI was given a proof of this in class, but it does not make sense to me, can someone provide a better proof, or explain the one I have provided, thanks. Suppose $\gcd(a,n) = 1$, where $a$ and $n$ are ...Sat, 05 Sep 2026 23:45:36 -0400https://track.liberty.io/finding-the-regression-line-given-the-mean-correlation-and-standard-de.htmlhttps://track.liberty.io/finding-the-regression-line-given-the-mean-correlation-and-standard-de.htmlSo we have $100$ observations for $(x, y)$. The mean of $x$ is $1.06$, and for $y$ it is $3$. The standard deviation is $0.52$ for $x$ and for $y$ it is $1.13$. the correlation between $x$ and $y......Sat, 05 Sep 2026 23:25:43 -0400https://track.liberty.io/area-of-an-ellipse.htmlhttps://track.liberty.io/area-of-an-ellipse.htmlI need to find the area of the image of a circle centred at the origin with radius 3 under the transformation: $ \begin{pmatrix} 3 & 0\\ 0 & \frac{1}{3} \end{pmatrix} $ The image is the ...Sat, 05 Sep 2026 23:12:40 -0400https://track.liberty.io/trouble-integrating-1-x-from-riemann-sum.htmlhttps://track.liberty.io/trouble-integrating-1-x-from-riemann-sum.htmlPreface: I'm a A-Level student, so much of the maths I'm speaking about here is quite new to me, in particular Riemann Sums. I apologise if this already has an answer, I couldn't find it. I'm tryi......Sat, 05 Sep 2026 23:02:31 -0400https://track.liberty.io/prove-that-if-events-a-and-b-are-independent-then-the-complement-event.htmlhttps://track.liberty.io/prove-that-if-events-a-and-b-are-independent-then-the-complement-event.htmlI know that: $$\begin{gathered}P\left(A\cap B\right)=P\left(A\right)P\left(B\right)\\ P\left(A^{C}\right)=1-P\left(A\right)\\ P\left(B^{C}\right)=1-P\left(B\right) \end{gathered} $$ My proof so fa......Sat, 05 Sep 2026 22:59:06 -0400https://track.liberty.io/damped-wave-equation-with-fourier-method.htmlhttps://track.liberty.io/damped-wave-equation-with-fourier-method.htmlThe problem I was trying to solve is the following PDE problem $$\begin{cases} \partial_{tt}^2 u = \partial_{xx}^2 u -\gamma\partial_t u \\[5 pt] u(0,t)= u(\pi, t) = 0 \\[5 pt] u(x,0) = (\sin2x)......Sat, 05 Sep 2026 22:58:40 -0400https://track.liberty.io/bipartite-graph-and-non-connected-node.htmlhttps://track.liberty.io/bipartite-graph-and-non-connected-node.htmlIs this one bipartite graph or not? It is a simple question for you but i can't find the answer....Sat, 05 Sep 2026 22:37:34 -0400https://track.liberty.io/deriving-formula-for-surface-area-of-an-ellipsoid.htmlhttps://track.liberty.io/deriving-formula-for-surface-area-of-an-ellipsoid.htmlI am doing some research on ellipsoids. I am not sure where the formula for the surface area of a prolate ellipsoid comes from. Can anyone please help me with how to derive the formula. I have the ...Sat, 05 Sep 2026 22:37:23 -0400https://track.liberty.io/finding-volume-using-integration-for-area-bound-by-two-curves.htmlhttps://track.liberty.io/finding-volume-using-integration-for-area-bound-by-two-curves.htmlI am trying to find the volume of the solid formed by rotating the region of the region bounded by $$ y=x $$ and $$ y=\sqrt x $$ about the line $$ x=2 $$ I don't understand how to set up this int......Sat, 05 Sep 2026 22:30:24 -0400https://track.liberty.io/solve-ln-x-ln-x-1-0-for-x.htmlhttps://track.liberty.io/solve-ln-x-ln-x-1-0-for-x.htmlSolve the following equation for x; $$\ln(x)+\ln(x-1)=0$$ What I did is the following but I'm pretty sure its wrong.. $$\ln(x)+\ln(x-1)=0$$ $$\ln(x)=-\ln(x-1)$$ $$e^{\ln(x)}=e^{-\ln(x-1))}$$ $$x......Sat, 05 Sep 2026 22:28:20 -0400https://track.liberty.io/is-it-coincidental-that-the-definition-of-similar-matrices-seems-to-be.htmlhttps://track.liberty.io/is-it-coincidental-that-the-definition-of-similar-matrices-seems-to-be.htmlTwo matrices $A$ and $B$ are similar if there exists an invertible matrix $P$ for which $B = PAP^{-1}$. This definition seems presented without any motivation behind it. However, I did notice that......Sat, 05 Sep 2026 22:27:11 -0400https://track.liberty.io/properties-of-zero-diagonal-symmetric-matrices.htmlhttps://track.liberty.io/properties-of-zero-diagonal-symmetric-matrices.htmlI'm interested in the properties of zero-diagonal symmetric (or hermitian) matrices, also known as hollow symmetric (or hermitian) matrices. The only thing I can come up with is that it cannot be ...Sat, 05 Sep 2026 22:14:44 -0400https://track.liberty.io/how-to-calculate-an-unknown-vector-from-a-known-vector-and-an-angle.htmlhttps://track.liberty.io/how-to-calculate-an-unknown-vector-from-a-known-vector-and-an-angle.htmlGiven three vectors in 3D space $\vec{a}$, $\vec{b}$, & $\vec{c}$ lay in the same plane. Let $\vec{c}$ be unknown. Knowing the vectors $\vec{a}$ & $\vec{b}$ and the angle between angle bet......Sat, 05 Sep 2026 22:04:28 -0400https://track.liberty.io/what-is-the-quotient-mathbb-z-sqrt-3-1-2-sqrt-3.htmlhttps://track.liberty.io/what-is-the-quotient-mathbb-z-sqrt-3-1-2-sqrt-3.htmlI am currently doing a past paper and it asks the following: Prove that for $I=(1+2\sqrt{3})$ we have $\mathbb Z[\sqrt{3}]/I$ a field with $11$ elements. If I assume standard algebraic number t......Sat, 05 Sep 2026 22:01:33 -0400https://track.liberty.io/prove-that-open-half-planes-are-open-sets.htmlhttps://track.liberty.io/prove-that-open-half-planes-are-open-sets.htmlAn open half plane is a subset of $\Bbb{R}^2$ in the form $\{(x, y)\in \Bbb{R}^2\vert \space Ax + By<C\}$ for some $A,C,B\in \Bbb{R}$ with either $A$ or $B$ nonzero. I need to prove that open ......Sat, 05 Sep 2026 21:52:29 -0400https://track.liberty.io/determine-the-number-of-real-solutions-of-an-equation.htmlhttps://track.liberty.io/determine-the-number-of-real-solutions-of-an-equation.htmlI need to calculate the number of real solutions in a polynomial equation. From the Fundamental Theorem of Algebra, I know that an equation of degree n has exactly n roots (but, here the complex on......Sat, 05 Sep 2026 21:49:40 -0400https://track.liberty.io/how-to-use-finite-differences-to-determine-an-equation-of-a-polynomial.htmlhttps://track.liberty.io/how-to-use-finite-differences-to-determine-an-equation-of-a-polynomial.htmlThis chart is given: for $x=-3$, $y=-9$ for $x=-2$, $y=3$ for $x=-1$, $y=3$ for $x=0$, $y=-3$ for $x=1$, $y=-9$ for $x=2$, $y=-9$ for $x=3$, $y=3$ I found the finite differences to be 6 and ...Sat, 05 Sep 2026 21:21:15 -0400https://track.liberty.io/proof-about-hessian-matrix-and-vectors.htmlhttps://track.liberty.io/proof-about-hessian-matrix-and-vectors.htmlI would like some help with the following question: Given a $ C^{2} $ function $ f: \mathbb{R}^n \rightarrow \mathbb{R} $ and $a \in \mathbb{R}^n $, a local minima point of $f$, and $v\in \math......Sat, 05 Sep 2026 21:21:14 -0400https://track.liberty.io/1992-ahsme-problem-20.htmlhttps://track.liberty.io/1992-ahsme-problem-20.html$1992$ Ahsme Problem $20$: Part of an "n-pointed regular star" is shown. It is a simple closed polygon in which all $2n$ edges are congruent, angles $A_1,A_2,\cdots,A_n$ are congruent, and angles $......Sat, 05 Sep 2026 21:20:29 -0400https://track.liberty.io/is-the-following-sentence-contrapositive-of-the-sentence-if-it-039-s-s.htmlhttps://track.liberty.io/is-the-following-sentence-contrapositive-of-the-sentence-if-it-039-s-s.htmlI need to find the contrapositive, which is "If the birds do not sing, it is not spring" But if it's written like, "If it's not spring, the birds will not sing", can this be ...Sat, 05 Sep 2026 21:18:28 -0400https://track.liberty.io/how-many-triangles-can-be-formed.htmlhttps://track.liberty.io/how-many-triangles-can-be-formed.htmlI have an excersie that says: ¿How many triangles can be formed, with the vertex of a hexagon? I think this: $1$) Ok, it is requesting triangles, therefore, it is not permutation since it take......Sat, 05 Sep 2026 21:09:12 -0400https://track.liberty.io/shortest-distance-between-two-general-curves-using-matlab.htmlhttps://track.liberty.io/shortest-distance-between-two-general-curves-using-matlab.htmlGiven two functions: $F(x)$ and $G(x)$ is there is a way to find out the shortest distance between $F(x)$ and $G(x)$ provided we know that they do not intersect. I tried to consider parametric poin......Sat, 05 Sep 2026 21:04:19 -0400https://track.liberty.io/basic-arithmetic-trick-question.htmlhttps://track.liberty.io/basic-arithmetic-trick-question.htmlI have the following question: A baker filled a measuring cup with $3/4$ cup of water. He poured $1/2$ of the water into the batter, and then spilled $1/8$ of the water on the floor. How much wate......Sat, 05 Sep 2026 21:01:41 -0400https://track.liberty.io/how-to-find-the-vertices-angle-after-rotation.htmlhttps://track.liberty.io/how-to-find-the-vertices-angle-after-rotation.htmlI drew this picture to interpret my question. I have the x and y axis for all the vertices before rotating the object. And I have the angle of rotation, how can I find the x and y axis for the ver......Sat, 05 Sep 2026 20:55:26 -0400https://track.liberty.io/compute-number-of-comparisons-in-quicksort-pivoting-on-median-or-third.htmlhttps://track.liberty.io/compute-number-of-comparisons-in-quicksort-pivoting-on-median-or-third.htmlWith a few friends we read the Algorithm Design Manual from Skiena. One of his (chapter 4) exercises asks for the number of comparisons that the quicksort algorithm does (comparing an element to the ...Sat, 05 Sep 2026 20:42:37 -0400https://track.liberty.io/finding-the-geodesic-equations-on-s-2.htmlhttps://track.liberty.io/finding-the-geodesic-equations-on-s-2.htmlThe metric on $S^2$ is given as $\bar{g} = ds^2 = d\theta^2 + \sin^2\theta d\phi^2$ where $x^1 = \theta$ and $x^2 = \phi$. The only non-zero components of the Christoffel symbols are $\Gamma^{\ 1}_......Sat, 05 Sep 2026 20:42:04 -0400https://track.liberty.io/convergence-divergence-of-the-sum-sum-n-2-infty-1-ln-n.htmlhttps://track.liberty.io/convergence-divergence-of-the-sum-sum-n-2-infty-1-ln-n.htmlIs the sum $$ \sum_{n =2}^{\infty} \frac{1}{\ln n!} $$ convergent or divergent? I have tried different methods and it doesn't work. Perhaps comparing with a divergent series will work? I'm think......Sat, 05 Sep 2026 20:37:19 -0400https://track.liberty.io/what-is-the-point-of-quadratic-residues.htmlhttps://track.liberty.io/what-is-the-point-of-quadratic-residues.htmlWhat is the most motivating way to introduce quadratic residues? Are there any real life examples of quadratic residues? Why is the Law of Quadratic Reciprocity considered as one of the most impor......Sat, 05 Sep 2026 20:35:53 -0400https://track.liberty.io/population-vs-sampling-frame-vs-sample.htmlhttps://track.liberty.io/population-vs-sampling-frame-vs-sample.htmlCould someone please explain how the sampling frame is different from population and sample? I understand that the population is all the sampling unit that match our criteria for the study. And the ...Sat, 05 Sep 2026 20:34:53 -0400https://track.liberty.io/how-to-calculate-limits-of-sequences.htmlhttps://track.liberty.io/how-to-calculate-limits-of-sequences.htmlCalculate the limits of the following sequences $$ a_n = \frac{n+4n-n^5}{n^3+3n^5-2n}$$ $$ b_n = \sqrt{n^2+4} - \sqrt{n^2+2} $$ I know the formulas to transform sequences to find out its limit bu......Sat, 05 Sep 2026 20:23:24 -0400https://track.liberty.io/understanding-zorn-039-s-lemma.htmlhttps://track.liberty.io/understanding-zorn-039-s-lemma.htmlA lot of authors assume Zorn's lemma. I am told it is not an obvious mathematical fact, but I am having problems understanding why that is. Zorn's lemma states that if every chain in a partially ...Sat, 05 Sep 2026 20:10:54 -0400https://track.liberty.io/minimum-distance-between-two-boats-travelling-in-different-velocities.htmlhttps://track.liberty.io/minimum-distance-between-two-boats-travelling-in-different-velocities.htmlQuestion: Boat $A$ was $100$ km south of boat $B$ at $9:00$ AM. If boat $A$ travels towards the north with the speed of $20$ km/h, and boat $B$ travels towards the east with the speed of $15$ km/h,......Sat, 05 Sep 2026 20:07:43 -0400https://track.liberty.io/would-it-be-fine-to-use-serge-lang-039-s-two-calculus-books-as-textboo.htmlhttps://track.liberty.io/would-it-be-fine-to-use-serge-lang-039-s-two-calculus-books-as-textboo.htmlI'm a freshman in Maths major, but the recommanded textbook(Calculus:A Complete Course by Robert A. Adams) by Prof. of Calculus course is too much expensive, well, I found there're Serge Lang's two ...Sat, 05 Sep 2026 20:00:17 -0400https://track.liberty.io/calculating-the-length-norm-of-a-vector-duplicate.htmlhttps://track.liberty.io/calculating-the-length-norm-of-a-vector-duplicate.htmlHow do you interpret this equation? I've seen far simpler equations for solving the magnitude, like the square root of all of the elements of a vector squared added to one another. What exactly is ......Sat, 05 Sep 2026 20:00:13 -0400