Buzz Spill Daily

Buzz Spill Dailyhttps://liberty.ioCurated star highlights with fresh energy.en-usFri, 14 Aug 2026 03:24:40 -0400Difference between postulates, axioms, and theorems?https://liberty.io/difference-between-postulates-axioms-and-theorems.htmlhttps://liberty.io/difference-between-postulates-axioms-and-theorems.htmlI'm trying to get an overarching understanding of the components of mathematical systems so that in my self study of each category of math I can break them down by their unique aspects, i.e. the ...Fri, 14 Aug 2026 07:10:23 -0400Convolution of a function with itselfhttps://liberty.io/convolution-of-a-function-with-itself.htmlhttps://liberty.io/convolution-of-a-function-with-itself.htmlFunction $\phi (x)$ is defined as: $$\phi(x) = \begin{cases} 1 & \text{ if } 0 \leq x \leq 1\\0 & \text{otherwise} \end{cases} $$ How do I find the convolution of $\phi(x)$ with itself? I t......Fri, 14 Aug 2026 07:01:28 -0400Graphing the surface $z = xy$https://liberty.io/graphing-the-surface-z-xy.htmlhttps://liberty.io/graphing-the-surface-z-xy.htmlLet the surface $S \subset \mathbb{R}^3$ be the graph of the function $f:\mathbb{R}^2 \to \mathbb{R}, f (x, y) = xy$. Let $U$ be the portion of $S$ for which $x^2 + y^2 ≤ 2$ and let $C$ be the boun......Fri, 14 Aug 2026 06:57:36 -0400Probability of 2-digit appearing in random 3 digit number?https://liberty.io/probability-of-2-digit-appearing-in-random-3-digit-number.htmlhttps://liberty.io/probability-of-2-digit-appearing-in-random-3-digit-number.htmlLet's say I have {1, 2} what is the probability of them both appearing in three digit number? [1x2, 2x1, 12x, 21x, x12, x21] I tried the following but is not getting the correct answer 1 - P(Bo......Fri, 14 Aug 2026 06:45:57 -0400Difference between a Reduced Residue Class and a Reduced Residue Systemhttps://liberty.io/difference-between-a-reduced-residue-class-and-a-reduced-residue-syste.htmlhttps://liberty.io/difference-between-a-reduced-residue-class-and-a-reduced-residue-syste.htmlCurrently studying Dirichlets proof of Primes in arithmetic progressions. Is there a difference between a Reduced Residue Class and a Reduced Residue System? Any subset R of the integers is called a ...Fri, 14 Aug 2026 06:43:50 -0400Finding the Moment Generating Function of Standard Normal Random Variable from Normal Random Variablehttps://liberty.io/finding-the-moment-generating-function-of-standard-normal-random-varia.htmlhttps://liberty.io/finding-the-moment-generating-function-of-standard-normal-random-varia.htmlGiven $Y\sim N(\mu, \sigma^2)$. I'm trying to find the moment generating function of $Z=\frac{Y-\mu}{\sigma}$ using the MGF transform method. Here's what I've tried: $$M_Y(t)=e^{\mu t+\frac{\sigma^......Fri, 14 Aug 2026 06:40:19 -0400What technique should I apply to find the derivative of a ceiling or floor function e.g d/dx(x*⌈x⌉) and d/dx(x*⌊x⌋)?https://liberty.io/what-technique-should-i-apply-to-find-the-derivative-of-a-ceiling-or-f.htmlhttps://liberty.io/what-technique-should-i-apply-to-find-the-derivative-of-a-ceiling-or-f.htmlMay I know the technique to apply to find the derivative, whenever I see a ceiling function of floor function. Thank You! e.g $$ \frac{d}{dx}(x*\lceil x \rceil )$$ and $$ \frac{d}{dx}(x*\lfloor......Fri, 14 Aug 2026 06:39:55 -0400Subtracting two points, the math.https://liberty.io/subtracting-two-points-the-math.htmlhttps://liberty.io/subtracting-two-points-the-math.htmlI would like to confirm the math of subtracting two points. I've looked online but all of the sources I have seen want to end up with a vector, I need to end with a point. I am looking at this fu......Fri, 14 Aug 2026 06:38:55 -0400I have 100 hundred different items. How to I calculate all possible combinations?https://liberty.io/i-have-100-hundred-different-items-how-to-i-calculate-all-possible-com.htmlhttps://liberty.io/i-have-100-hundred-different-items-how-to-i-calculate-all-possible-com.htmlELI5 the formula to figure out all possible combinations without repeating any items but no upper or lower limit of the number to include. Order does not matter....Fri, 14 Aug 2026 06:38:54 -0400How would I set up the Taylor's Inequality to prove that the function is equal to its Taylor Series expansion?https://liberty.io/how-would-i-set-up-the-taylor-039-s-inequality-to-prove-that-the-funct.htmlhttps://liberty.io/how-would-i-set-up-the-taylor-039-s-inequality-to-prove-that-the-funct.htmlHow would I set up the Taylor's Inequality to prove that the function $f(x) = \frac{1}{x}$ is equal to its Taylor Series expansion centered at $x=1$? I've done the Taylor series expansion, but I'm......Fri, 14 Aug 2026 06:36:36 -0400Find area of the shaded portion.https://liberty.io/find-area-of-the-shaded-portion.htmlhttps://liberty.io/find-area-of-the-shaded-portion.htmlThe distances between the dots in the figure are equal to 1, both horizontally and vertically. What is the measure of the common area of the triangle and the square which you can see shaded in the ......Fri, 14 Aug 2026 06:35:55 -0400Infinite row of squares, some of them white with probability $p$. What is the average width of a white rectangle?https://liberty.io/infinite-row-of-squares-some-of-them-white-with-probability-p-what-is-.htmlhttps://liberty.io/infinite-row-of-squares-some-of-them-white-with-probability-p-what-is-.htmlSuppose that you have an infinite horizontal row of squares, each square of size $1\times1$ (like a row of squares in a grid of infinite size). The probability of square being white is $p$. Some wh......Fri, 14 Aug 2026 06:25:51 -0400Difference between $\mathrm{rot}$ and $\mathrm{Rot}$ (Curl)https://liberty.io/difference-between-mathrm-rot-and-mathrm-rot-curl.htmlhttps://liberty.io/difference-between-mathrm-rot-and-mathrm-rot-curl.htmlI'm not sure whether you've heard of that difference at all and if it's of importance in English notations. In German at least you define: $$\mathrm{rot\,\textbf{V}} = \nabla \times \textbf{V}, \quad\...Fri, 14 Aug 2026 06:25:36 -0400Does $\sum\frac{4^n(n!)^2}{(2n)!}$ converge?https://liberty.io/does-sum-frac-4-n-n-2-2n-converge.htmlhttps://liberty.io/does-sum-frac-4-n-n-2-2n-converge.htmlDoes $\sum\frac{4^n(n!)^2}{(2n)!}$ converge? The ratio test is inconclusive. I know that $\sum\frac{2^n(n!)^2}{(2n)!}$ converges, if that's of any help....Fri, 14 Aug 2026 06:20:50 -0400Number of directed graph with $n$ vertices [closed]https://liberty.io/number-of-directed-graph-with-n-vertices-closed.htmlhttps://liberty.io/number-of-directed-graph-with-n-vertices-closed.htmlIn terms of $n$, how many distinct digraphs are there whose vertex set is $X$, where $X$ is a nonempty finite set with $|X| = n$? How does the answer change if we allow loops, but not parallel ar......Fri, 14 Aug 2026 06:13:52 -0400If $x\in\mathbb R$, solve $4x^2-40\lfloor x\rfloor+51=0$.https://liberty.io/if-x-in-mathbb-r-solve-4x-2-40-lfloor-x-rfloor-51-0.htmlhttps://liberty.io/if-x-in-mathbb-r-solve-4x-2-40-lfloor-x-rfloor-51-0.htmlIf $x\in\mathbb R$, solve $$4x^2-40\lfloor x\rfloor+51=0$$ where $\lfloor x\rfloor$ denotes the integer part of the number. $\lfloor x\rfloor\le x$ and $\lfloor x\rfloor=x-\{x\}$, where $\{x\}$ m......Fri, 14 Aug 2026 06:07:34 -0400Using the Inverse matrix to solve a linear system.https://liberty.io/using-the-inverse-matrix-to-solve-a-linear-system.htmlhttps://liberty.io/using-the-inverse-matrix-to-solve-a-linear-system.htmlI have the following question: $$x - 2y = 1$$ $$2x + 3y = 4$$ So let's say: $$A = \left[ \begin{matrix} 1 & -2 \\ 2 & 3 \end{matrix} \right]$$ So according to some formula to $2 \ti......Fri, 14 Aug 2026 06:01:23 -0400Calculate the distance $d$ from $\vec{y}$ to the straight line through $\vec{u}$ and the originhttps://liberty.io/calculate-the-distance-d-from-vec-y-to-the-straight-line-through-vec-u.htmlhttps://liberty.io/calculate-the-distance-d-from-vec-y-to-the-straight-line-through-vec-u.htmlBackground Let $$\vec{y}= \left[ {\begin{array}{cc} 5 \\ 1 \\ 1......Fri, 14 Aug 2026 05:59:57 -0400Evaluate $\int \sqrt{\sec x} \tan x \ \mathrm dx$https://liberty.io/evaluate-int-sqrt-sec-x-tan-x-mathrm-dx.htmlhttps://liberty.io/evaluate-int-sqrt-sec-x-tan-x-mathrm-dx.htmlI tried to use integration by parts but it led to results which I could not integrate further. Also substituting $\sec x$ in terms of $\tan x$ or vice-versa led me to nothing proper. Update: Takin......Fri, 14 Aug 2026 05:54:20 -0400If $B$ is nilpotent and $AB=BA$ then $\det(A+B) = \det(A)$ [duplicate]https://liberty.io/if-b-is-nilpotent-and-ab-ba-then-det-a-b-det-a-duplicate.htmlhttps://liberty.io/if-b-is-nilpotent-and-ab-ba-then-det-a-b-det-a-duplicate.htmlThe following stumps me: Let $\mathbb K$ be a field. Let $A, B \in \mathbb K^{n \times n}$ where $B$ is nilpotent and commutes with $A$, i.e., $A B = B A$. Show that $$ \det(A+B)=\det(A) $$ I h......Fri, 14 Aug 2026 05:53:04 -0400What is the difference between a random variable and an event?https://liberty.io/what-is-the-difference-between-a-random-variable-and-an-event.htmlhttps://liberty.io/what-is-the-difference-between-a-random-variable-and-an-event.htmlIn layman terms, what is the difference between a random variable and an event? To my understanding a random variable is a function outputting a real number. And an event is an outcome or a set of ...Fri, 14 Aug 2026 05:33:29 -0400Explanation for the Wilson Score Interval?https://liberty.io/explanation-for-the-wilson-score-interval.htmlhttps://liberty.io/explanation-for-the-wilson-score-interval.htmlI'm looking at this blog to try to understand the Wilson Score interval. I understand it somewhat, but I'm confused by the part under the title "Excerpt". In particular, I don't understand what he's ...Fri, 14 Aug 2026 05:32:55 -0400Dot product for 3 vectorshttps://liberty.io/dot-product-for-3-vectors.htmlhttps://liberty.io/dot-product-for-3-vectors.htmlThe dot product can be used to write the sum: $$\sum_{i=1}^n a_i b_i$$ as $$a^T b$$ Is there an equivalent notation for the following sum: $$\sum_{i=1}^n a_i b_i c_i$$...Fri, 14 Aug 2026 05:31:32 -0400Difference between algebra and geometryhttps://liberty.io/difference-between-algebra-and-geometry.htmlhttps://liberty.io/difference-between-algebra-and-geometry.htmlLooking up dictionary definitions of algebra geometry is pretty unsatisfying as they are usually along the lines of "the branch of mathematics in which letters and other general symbols are us......Fri, 14 Aug 2026 05:31:08 -0400Is there a sense in which $4^{2} = -16$? [closed]https://liberty.io/is-there-a-sense-in-which-4-2-16-closed.htmlhttps://liberty.io/is-there-a-sense-in-which-4-2-16-closed.htmlHere's my train of thought: I. $4^2=4^{(4/2)}$ II. $4^{(4/2)}=(4^4)^{(1/2)}$ III. $(4^4)^{(1/2)}=256^{(1/2)}$ IV. $256^{(1/2)}=\sqrt{256}$ square root of 256 has 2 solutions: 16 and -16 Can you exp......Fri, 14 Aug 2026 05:26:34 -0400Questions tagged [pochhammer-symbol] https://liberty.io/questions-tagged-pochhammer-symbol.htmlhttps://liberty.io/questions-tagged-pochhammer-symbol.htmlQ&A for people studying math at any level and professionals in related fields...Fri, 14 Aug 2026 05:25:25 -0400Dual quaternion inversehttps://liberty.io/dual-quaternion-inverse.htmlhttps://liberty.io/dual-quaternion-inverse.htmlIs it true that for every dual quaternion $Q$ I can find it's inverse such that $QQ^{-1} = 1?$ Using the usual definition $Q^{-1}=\frac{Q^{*}}{||Q||^2}$ doesn't work for me, since the dual part doe......Fri, 14 Aug 2026 05:20:49 -0400Optimization problem solution (water-filling)https://liberty.io/optimization-problem-solution-water-filling.htmlhttps://liberty.io/optimization-problem-solution-water-filling.htmlLet $x\in\mathbb{R}^n$ be an optimization variable and $\alpha\in\mathbb{R}^n$ be an n-dimensional vector. The standard water-filling problem is formulated as \begin{equation} \begin{array}{ll} \...Fri, 14 Aug 2026 05:16:11 -0400How do you explain to a 5th grader why division by zero is meaningless?https://liberty.io/how-do-you-explain-to-a-5th-grader-why-division-by-zero-is-meaningless.htmlhttps://liberty.io/how-do-you-explain-to-a-5th-grader-why-division-by-zero-is-meaningless.htmlI wish to explain my younger brother: he is interested and curious, but he cannot grasp the concepts of limits and integration just yet. What is the best mathematical way to justify not allowing di......Fri, 14 Aug 2026 05:15:24 -0400Proving If and Only If When X and Y are independenthttps://liberty.io/proving-if-and-only-if-when-x-and-y-are-independent.htmlhttps://liberty.io/proving-if-and-only-if-when-x-and-y-are-independent.htmlLet $X, Y$ be two random variables, defined on some probability space $(\Omega , A , P)$., each only has two district values: $X \to \{x_1,x_2\}$ , $Y \to \{y_1,y_2\}$. Recall that, in this ca......Fri, 14 Aug 2026 05:13:13 -0400Two doors, 2 guards logic problem ("is XOR a hypothetical statement?")https://liberty.io/two-doors-2-guards-logic-problem-is-xor-a-hypothetical-statement.htmlhttps://liberty.io/two-doors-2-guards-logic-problem-is-xor-a-hypothetical-statement.htmlYou're trapped in a room with 2 doors, one on your left and one on your right. One door leads to heaven, one door leads to hell. There are two birds in the room. One is a bad bird, which always lie......Fri, 14 Aug 2026 05:13:03 -0400How can I generate a random number on my ti-84 without the rand functionhttps://liberty.io/how-can-i-generate-a-random-number-on-my-ti-84-without-the-rand-functi.htmlhttps://liberty.io/how-can-i-generate-a-random-number-on-my-ti-84-without-the-rand-functi.htmlIs there an equation that is possible where I enter a seed, and it will give me the number from the seed. I want it to give me the same results as the rand function would give with the seed as 0. I......Fri, 14 Aug 2026 04:57:35 -0400How to multiply radicals?https://liberty.io/how-to-multiply-radicals.htmlhttps://liberty.io/how-to-multiply-radicals.htmlThe problem is $$0 = -(\sqrt{7 -2 b} - 2) * -\sqrt{7-2 b} - 2) $$ the original question was this $$-2 = \sqrt{7 -2 b} - \sqrt{2 b + 3}$$ and than i found out the you had to isolate one radical v......Fri, 14 Aug 2026 04:56:19 -0400What is a Fourier Symbolhttps://liberty.io/what-is-a-fourier-symbol.htmlhttps://liberty.io/what-is-a-fourier-symbol.htmlI was wondering what exactly the term Fourier symbol, or sometimes said the symbol of the Fourier transform means. Also, are any books that cover this topic in detail? Edit: After the comments, I h......Fri, 14 Aug 2026 04:46:32 -0400What does ^ sign mean on a vector?https://liberty.io/what-does-sign-mean-on-a-vector.htmlhttps://liberty.io/what-does-sign-mean-on-a-vector.htmlI have a maths homework question Given the vector b find bˆ The symbol is actually above the b, but I am not sure what it means. Guessing it means to find the unit vector of b?...Fri, 14 Aug 2026 04:44:24 -0400Circle revolutions rolling around another circlehttps://liberty.io/circle-revolutions-rolling-around-another-circle.htmlhttps://liberty.io/circle-revolutions-rolling-around-another-circle.htmlI just watched this video, and I'm a bit perplexed. Problem: The radius of Circle A is 1/3 the radius of Circle B. Circle A rolls around Circle B one trip back to its starting point. How many t......Fri, 14 Aug 2026 04:36:47 -0400How do I integrate $\sec(x)$?https://liberty.io/how-do-i-integrate-sec-x.htmlhttps://liberty.io/how-do-i-integrate-sec-x.htmlMy HW asks me to integrate $\sin(x)$, $\cos(x)$, $\tan(x)$, but when I get to $\sec(x)$, I'm stuck....Fri, 14 Aug 2026 04:26:49 -0400Let B be the solid tetrahedron bounded by x = 0, y = 0, z = 0, and x+y+z = 1.https://liberty.io/let-b-be-the-solid-tetrahedron-bounded-by-x-0-y-0-z-0-and-x-y-z-1.htmlhttps://liberty.io/let-b-be-the-solid-tetrahedron-bounded-by-x-0-y-0-z-0-and-x-y-z-1.htmlLet $B$ be the solid tetrahedron bounded by $x = 0, y = 0, z = 0$, and $x+y+z = 1$. Assume that the density of $B$ is given by the function $y$. Compute the mass of $B$. I am having trouble solvin......Fri, 14 Aug 2026 04:22:12 -0400Proof by induction: Matriceshttps://liberty.io/proof-by-induction-matrices.htmlhttps://liberty.io/proof-by-induction-matrices.htmlGiven the matrix $A=\begin{pmatrix} 1 & 2 \\ 0 & 1\end{pmatrix}$, I want to prove that $A^k = \begin{pmatrix}1 & 2k \\ 0 & 1 \end{pmatrix}$ (=induction hypothesis). Since I struggle......Fri, 14 Aug 2026 04:21:27 -0400Math Error evaluating ${ \int \cot^3(ax) csc^2(ax)dx }$ [closed]https://liberty.io/math-error-evaluating-int-cot-3-ax-csc-2-ax-dx-closed.htmlhttps://liberty.io/math-error-evaluating-int-cot-3-ax-csc-2-ax-dx-closed.htmlFor a National Board Exam Review: Evaluate: $${ \int \cot^3(ax) csc^2(ax)dx }$$ Answer is $${ -\frac{1}{4a}cot^4ax +C }$$ Ok. I certainly know how to integrate it from scratch. What I'm ......Fri, 14 Aug 2026 04:12:24 -0400Multivariable calculus - find derivative using implicit differentiationhttps://liberty.io/multivariable-calculus-find-derivative-using-implicit-differentiation.htmlhttps://liberty.io/multivariable-calculus-find-derivative-using-implicit-differentiation.htmlShort simple question which i managed to solve partially. we are given the equation $x^2+y^2-z^2+xz-yz-1=0$. Show using the implicit function theorem that this equation sets in the neighborhood o......Fri, 14 Aug 2026 04:06:38 -0400Weighted Jacobi Methodhttps://liberty.io/weighted-jacobi-method.htmlhttps://liberty.io/weighted-jacobi-method.htmlConsider solving a linear system $A \boldsymbol{x}=\boldsymbol{b}$. Let $D$ be the diagonal matrix whose diagonal entries are those of $A,-L$ be the strictly lower triangular part of $A$, and $-U$ ......Fri, 14 Aug 2026 03:53:53 -0400What is the general equation of the ellipse that is not in the origin and rotated by an angle?https://liberty.io/what-is-the-general-equation-of-the-ellipse-that-is-not-in-the-origin-.htmlhttps://liberty.io/what-is-the-general-equation-of-the-ellipse-that-is-not-in-the-origin-.htmlI have the equation not in the center, i.e. $$\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1.$$ But what will be the equation once it is rotated?...Fri, 14 Aug 2026 03:43:59 -0400Convert a n by n matrix to upper triangularhttps://liberty.io/convert-a-n-by-n-matrix-to-upper-triangular.htmlhttps://liberty.io/convert-a-n-by-n-matrix-to-upper-triangular.htmlHow can we mathematically and algorithmic-ally convert a $n\times n$ matrix to a upper triangular matrix...Fri, 14 Aug 2026 03:38:07 -0400Solve the inequality $x^2 - 3 > 0$https://liberty.io/solve-the-inequality-x-2-3-0.htmlhttps://liberty.io/solve-the-inequality-x-2-3-0.htmlFor the inequality $x^2 - 3 > 0$, we have \begin{align} x^2 - 3 & = (x+\sqrt 3)(x- \sqrt 3) > 0 \end{align} Therefore, \begin{align} x > -\sqrt 3 \end{align} and \begin{align} x ......Fri, 14 Aug 2026 03:35:29 -0400Can decimal numbers be considered "even" or "odd"?https://liberty.io/can-decimal-numbers-be-considered-even-or-odd.htmlhttps://liberty.io/can-decimal-numbers-be-considered-even-or-odd.htmlIs the concept of even/odd numbers applicable to decimal numbers? For e.g. - 4.222 is a even number?...Fri, 14 Aug 2026 03:33:45 -0400The number of integers 'a' ($1\leq a\leq 200$) such that a^a is a perfect square are:$ A). 105$ $ B) 103$ $C)107$ $D)109?$https://liberty.io/the-number-of-integers-039-a-039-1-leq-a-leq-200-such-that-a-a-is-a-pe.htmlhttps://liberty.io/the-number-of-integers-039-a-039-1-leq-a-leq-200-such-that-a-a-is-a-pe.htmlThe number of integers $a$ ($1\leq a\leq 200$) such that $a^a$ is a perfect square are is: $A)\;105$ $B)\;103$ $C)\;107$ $D)\;109$ My work $a^a$ is a perfect square when $a$ is even. $2^2 = 4. 4^4......Fri, 14 Aug 2026 03:32:05 -0400Simplifying/Finding the natural log of two terms without logarithm laws.https://liberty.io/simplifying-finding-the-natural-log-of-two-terms-without-logarithm-law.htmlhttps://liberty.io/simplifying-finding-the-natural-log-of-two-terms-without-logarithm-law.htmlIf there's a natural log of two terms, which I cannot simplify with the laws of logarithms, how should I simplify it? e.g. $\ln(e^{6x} + 17)$ The full equation could be something like this: $$\......Fri, 14 Aug 2026 03:22:44 -0400Cayley table for $D_4/\langle r^2\rangle$.https://liberty.io/cayley-table-for-d-4-langle-r-2-rangle.htmlhttps://liberty.io/cayley-table-for-d-4-langle-r-2-rangle.htmlWrite out the Cayley table for $D_4/\langle r^2\rangle$. I know that $D_4$ is the dihedral group with 4 sides (square) and $\langle r^2\rangle$ is the algebra generated by $r^2$ (a.k.a. taking the ...Fri, 14 Aug 2026 03:11:08 -0400Interior Extremum Theoremhttps://liberty.io/interior-extremum-theorem.htmlhttps://liberty.io/interior-extremum-theorem.htmlInterior Extremum Theorem. Let $f$ be differentiable on an open interval $(a,b)$. If $f$ attains a maximum value at some point $c \in (a,b)$ ($f(c) \ge f(x)$ for all $x \in(a,b)$), then $f'(c) = 0$......Fri, 14 Aug 2026 02:59:29 -0400In simple English, what does it mean to be transcendental?https://liberty.io/in-simple-english-what-does-it-mean-to-be-transcendental.htmlhttps://liberty.io/in-simple-english-what-does-it-mean-to-be-transcendental.htmlFrom Wikipedia A transcendental number is a real or complex number that is not algebraic A transcendental function is an analytic function that does not satisfy a polynomial equation However these ...Fri, 14 Aug 2026 02:58:42 -0400Partial Orders and Hasse Diagrams: Totally confusedhttps://liberty.io/partial-orders-and-hasse-diagrams-totally-confused.htmlhttps://liberty.io/partial-orders-and-hasse-diagrams-totally-confused.htmlSeems I've stumbled upon one of the most confusing topics in discrete mathematics...My book does a terrible job of explaining how you actually create Hasse diagrams and only shows premade diagrams ...Fri, 14 Aug 2026 02:38:20 -0400How can I find the value of $\ln( |x|)$ without using the calculator?https://liberty.io/how-can-i-find-the-value-of-ln-x-without-using-the-calculator.htmlhttps://liberty.io/how-can-i-find-the-value-of-ln-x-without-using-the-calculator.htmlI want to know if there is a way to find for example $\ln(2)$, without using the calculator ? Thanks...Fri, 14 Aug 2026 02:34:52 -0400Upside-down triangle symbol on functionhttps://liberty.io/upside-down-triangle-symbol-on-function.htmlhttps://liberty.io/upside-down-triangle-symbol-on-function.htmlI came across the symbol that looks like an upside-down triangle, and coming in front of a function $f(x,y)$. What does that mean?...Fri, 14 Aug 2026 02:32:17 -0400Principal branch of logarithmhttps://liberty.io/principal-branch-of-logarithm.htmlhttps://liberty.io/principal-branch-of-logarithm.htmlI need to compute $\log(1+i)$ and $\text{Log}(1+i)$, $\text{Log}$ meaning the principal branch. What I do is to express $(1+i)$ in polar coordinates, then equate it with $$\sqrt {2}(\cos \pi/4+ i\......Fri, 14 Aug 2026 02:29:11 -0400Negation of if and only if?https://liberty.io/negation-of-if-and-only-if.htmlhttps://liberty.io/negation-of-if-and-only-if.htmlLet a statement P is "X is true if and only if Y is true". What is the negation of P? I am little confused. It seems that digital equivalent of this statement is P = X and Y. Hence negation of P is......Fri, 14 Aug 2026 02:25:38 -0400Equidistant points on a circlehttps://liberty.io/equidistant-points-on-a-circle.htmlhttps://liberty.io/equidistant-points-on-a-circle.htmlI would like to obtain/generate points on a circle in Cartesian coordinates such that the distance between two consecutive points will be always equal. For example, plotting a circle with radius 10......Fri, 14 Aug 2026 02:25:19 -0400Relative version of Kunneth Theorem for $CW$ pairs.https://liberty.io/relative-version-of-kunneth-theorem-for-cw-pairs.htmlhttps://liberty.io/relative-version-of-kunneth-theorem-for-cw-pairs.htmlI am working on problem 3.2.12 in Hatcher's algebraic topology which asks to show that the spaces $(S^1 \times \mathbb{C}P^{\infty})/(S^1 \times \{x_0\})$ and $S^3 \times \mathbb{C}P^{\infty}$ have ...Fri, 14 Aug 2026 02:25:05 -0400What is the relationship between cubic B-splines and cubic splines?https://liberty.io/what-is-the-relationship-between-cubic-b-splines-and-cubic-splines.htmlhttps://liberty.io/what-is-the-relationship-between-cubic-b-splines-and-cubic-splines.htmlWhat is the relationship between cubic B-splines and cubic splines?...Fri, 14 Aug 2026 02:21:34 -0400Find a point on a line segment which is the closest to other point not on the line segmenthttps://liberty.io/find-a-point-on-a-line-segment-which-is-the-closest-to-other-point-not.htmlhttps://liberty.io/find-a-point-on-a-line-segment-which-is-the-closest-to-other-point-not.htmlMy question is in the context of a line segment and not in the context of a line which has infinite length. Lets say I have two points in 2d which represent for me a line segment and I got another......Fri, 14 Aug 2026 02:05:33 -0400How to study abstract algebrahttps://liberty.io/how-to-study-abstract-algebra.htmlhttps://liberty.io/how-to-study-abstract-algebra.htmlI am taking Abstract Algebra course at the university. We are doing chapters 1-20 from Gallian's abstract algebra text book. I am just doing assigned homework everyweek ( About 5 questions from each ...Fri, 14 Aug 2026 02:05:11 -0400Find number of vertices when given number of edgeshttps://liberty.io/find-number-of-vertices-when-given-number-of-edges.htmlhttps://liberty.io/find-number-of-vertices-when-given-number-of-edges.htmlA Simple Connected Graph G has $M$ vertices and 4 edges, find $M$ Now lets say we didn't have any more info than what's mentioned above. By drawing out a couple of graphs I know that $M$ could pos......Fri, 14 Aug 2026 02:03:30 -0400The Definition of the Absolute Valuehttps://liberty.io/the-definition-of-the-absolute-value.htmlhttps://liberty.io/the-definition-of-the-absolute-value.htmlThe Absolute Value can be defined in many ways, but these are the two most common : 1. As a Piecewise Function $$ |x|= \begin{cases} -x&\text{if } x < 0\\ x&\text{if } x\geq 0 \end{cas......Fri, 14 Aug 2026 01:56:05 -0400Skew lines and what's between themhttps://liberty.io/skew-lines-and-what-039-s-between-them.htmlhttps://liberty.io/skew-lines-and-what-039-s-between-them.htmlIs it always possible to find a line perpendicular to two skew lines in space? And how can we visualise the proof geometrically? And if anyone could present the proof that it is always possible to ......Fri, 14 Aug 2026 01:53:23 -0400Determine Direction of an Arc (CW/CCW)https://liberty.io/determine-direction-of-an-arc-cw-ccw.htmlhttps://liberty.io/determine-direction-of-an-arc-cw-ccw.htmlI'm trying to write a function that will determine whether a circular arc travels clockwise or counter-clockwise. Given the X,Y coordinates for the start point, center point, and end point I first ...Fri, 14 Aug 2026 01:46:39 -0400How is $A\sin\theta +B\cos\theta = C\sin(\theta + \phi)$ derived?https://liberty.io/how-is-a-sin-theta-b-cos-theta-c-sin-theta-phi-derived.htmlhttps://liberty.io/how-is-a-sin-theta-b-cos-theta-c-sin-theta-phi-derived.htmlI have come across this trig identity and I want to understand how it was derived. I have never seen it before, nor have I seen it in any of the online resources including the many trig identity ch......Fri, 14 Aug 2026 01:44:49 -0400Gaussian elimination with $4x-3y=11$ and $x-3y=-1$https://liberty.io/gaussian-elimination-with-4x-3y-11-and-x-3y-1.htmlhttps://liberty.io/gaussian-elimination-with-4x-3y-11-and-x-3y-1.html(Meta: What's the notation for showing two equations on two lines within a set of square braces []?) Excuse syntax. I would like to use Gaussian elimination to solve the system $[4x+3y=11]$ and $[x......Fri, 14 Aug 2026 01:44:43 -0400Why do we call it a $\sigma$-algebra?https://liberty.io/why-do-we-call-it-a-sigma-algebra.htmlhttps://liberty.io/why-do-we-call-it-a-sigma-algebra.htmlIn simple terms, a $\sigma$-algebra is the collection of all of the things we know how to measure. Why don't we call it something that more directly suggests this, for example a 'measure space?'...Fri, 14 Aug 2026 01:35:42 -0400Trace of a matrix times outer producthttps://liberty.io/trace-of-a-matrix-times-outer-product.htmlhttps://liberty.io/trace-of-a-matrix-times-outer-product.html$\DeclareMathOperator{\trace}{tr}$Is there any relationship between $\trace(Sxx^T)$ and $x^TSx$? Is there a nice way to write the set of quadratic functions of the components of a vector $x$ given ...Fri, 14 Aug 2026 01:30:01 -0400Quadruple Integral $\int\limits_0^1\!\!\int\limits_0^1\!\!\int\limits_0^1\!\!\int\limits_0^1\frac1{1-xyzw}\,dw\,dz\,dy\,dx$https://liberty.io/quadruple-integral-int-limits-0-1-int-limits-0-1-int-limits-0-1-int-li.htmlhttps://liberty.io/quadruple-integral-int-limits-0-1-int-limits-0-1-int-limits-0-1-int-li.htmlIn page 122 of the book Topics in Number Theory (1956) by William J. LeVeque, there is an exercise for evaluating the following integral in two ways. $$\int_0^1\!\!\!\int_0^1\frac1{1-xy}\,dy\,dx$$ ......Fri, 14 Aug 2026 01:28:46 -0400Cummulative Probability 5% chance per iterationhttps://liberty.io/cummulative-probability-5-chance-per-iteration.htmlhttps://liberty.io/cummulative-probability-5-chance-per-iteration.htmlLet's say you eat at some very cheap restaurant and every time you do there's a 5% chance you'll get food poisoning. How do you calculate the probability of getting food poisoning if you eat there x ...Fri, 14 Aug 2026 01:20:32 -0400Express the following invertible matrix A as a product of elementary matriceshttps://liberty.io/express-the-following-invertible-matrix-a-as-a-product-of-elementary-m.htmlhttps://liberty.io/express-the-following-invertible-matrix-a-as-a-product-of-elementary-m.htmlI've been at this for a while... I tried to the inverse method but it keeps on saying I'm getting it wrong... Can anyone show me a step-by-step solution? The matrix I have is a $3\times 3$ square one(...Fri, 14 Aug 2026 01:17:28 -0400Heat equation for a cylinder in cylindrical coordinateshttps://liberty.io/heat-equation-for-a-cylinder-in-cylindrical-coordinates.htmlhttps://liberty.io/heat-equation-for-a-cylinder-in-cylindrical-coordinates.htmlSo I have a description of a Partial differential equation given here. $\text{Consider a long pipe of inner radius a}=2.5cm \text{ and outer radius b}=3 cm, \text{made of copper for which the ther......Fri, 14 Aug 2026 01:14:36 -0400How prove this $\tan{\frac{2\pi}{13}}+4\sin{\frac{6\pi}{13}}=\sqrt{13+2\sqrt{13}}$https://liberty.io/how-prove-this-tan-frac-2-pi-13-4-sin-frac-6-pi-13-sqrt-13-2-sqrt-13.htmlhttps://liberty.io/how-prove-this-tan-frac-2-pi-13-4-sin-frac-6-pi-13-sqrt-13-2-sqrt-13.htmlNice Question: show that: The follow nice trigonometry $$\tan{\dfrac{2\pi}{13}}+4\sin{\dfrac{6\pi}{13}}=\sqrt{13+2\sqrt{13}}$$ This problem I have ugly solution, maybe someone have nice meth......Fri, 14 Aug 2026 01:13:46 -0400I have 3 quarters, 2 dimes, and 2 pennies. If I select at least one coin, how many different amounts of money can I make?https://liberty.io/i-have-3-quarters-2-dimes-and-2-pennies-if-i-select-at-least-one-coin-.htmlhttps://liberty.io/i-have-3-quarters-2-dimes-and-2-pennies-if-i-select-at-least-one-coin-.htmlI have 3 quarters, 2 dimes, and 2 pennies in my pocket. If I select at least one coin from my pocket, how many different amounts of money can I make? I checked out some of the other coin questions. ...Fri, 14 Aug 2026 01:10:33 -0400From a vector to a skew symmetric matrixhttps://liberty.io/from-a-vector-to-a-skew-symmetric-matrix.htmlhttps://liberty.io/from-a-vector-to-a-skew-symmetric-matrix.htmlIs there an existing linear mapping that maps a 3-dimensional vector: $$\mathbf{v}=\begin{pmatrix} v_1\\v_2\\v_3 \end{pmatrix}$$ to a corresponding skew-symmetric matrix: $$\mathbf{V}=\begin{pmatri......Fri, 14 Aug 2026 01:06:30 -0400let $L$ be a linear form, is $L$ continuous?https://liberty.io/let-l-be-a-linear-form-is-l-continuous.htmlhttps://liberty.io/let-l-be-a-linear-form-is-l-continuous.htmlLet $L$ be a linear form, goes from $E$ normed vectorial space, in $K=\mathbb R$ or $\mathbb C$. Is it continuous? My idea, is that $K$ is a banach space, so the space of linear forms is a banach......Fri, 14 Aug 2026 01:03:25 -0400Is $xy=0$ a function?https://liberty.io/is-xy-0-a-function.htmlhttps://liberty.io/is-xy-0-a-function.htmlGraphically, it passes the vertical line test. However, I am confused because for $x=0$, there are infinite $y$ values that satisfy $xy=0$. Can someone help me understand this?...Fri, 14 Aug 2026 00:58:32 -0400Induction Help: prove $2n+1< 2^n$ for all $n$ greater than or equal to $3$.https://liberty.io/induction-help-prove-2n-1-2-n-for-all-n-greater-than-or-equal-to-3.htmlhttps://liberty.io/induction-help-prove-2n-1-2-n-for-all-n-greater-than-or-equal-to-3.htmlI understand that he expanded the left side but I'm having trouble figuring out what he did on the right side of the inequality. Where the did the $2$ (in circle) come from?...Fri, 14 Aug 2026 00:46:41 -0400Regarding understanding of the following SVD code in matlabhttps://liberty.io/regarding-understanding-of-the-following-svd-code-in-matlab.htmlhttps://liberty.io/regarding-understanding-of-the-following-svd-code-in-matlab.htmlI earlier had a doubt regarding pseudo-inverse of a matrix, and earlier I was using pinv function. But one of the answers gave an alternative technique that is singular value decomposition method. ......Fri, 14 Aug 2026 00:45:47 -0400$a$ and $b$ are solutions of $ \frac{1}{x^{2} - 10x-29} + \frac{1}{x^{2} - 10x-45} - \frac{2}{x^{2} - 10x-69} = 0 $, $a+b=?$https://liberty.io/a-and-b-are-solutions-of-frac-1-x-2-10x-29-frac-1-x-2-10x-45-frac-2-x-.htmlhttps://liberty.io/a-and-b-are-solutions-of-frac-1-x-2-10x-29-frac-1-x-2-10x-45-frac-2-x-.html$a$ and $b$ are solutions of $$ \frac{1}{x^{2} - 10x-29} + \frac{1}{x^{2} - 10x-45} - \frac{2}{x^{2} - 10x-69} = 0 $$ What is $a+b=?$ $$ $$ Are there better approaches than the one below? Solution......Fri, 14 Aug 2026 00:42:14 -0400Representation of null set or empty set?https://liberty.io/representation-of-null-set-or-empty-set.htmlhttps://liberty.io/representation-of-null-set-or-empty-set.htmlI am having confusion that how to represent null set. Yes it seems like I ask silly question. Well I know null set can be represent either $\{\}$ or $\emptyset$. But can I write null set like $\{\...Fri, 14 Aug 2026 00:30:08 -0400How many options are there for 15 student to divide into 3 equal sized groups?https://liberty.io/how-many-options-are-there-for-15-student-to-divide-into-3-equal-sized.htmlhttps://liberty.io/how-many-options-are-there-for-15-student-to-divide-into-3-equal-sized.htmlHow many options are there for 15 students to divide into 3 equal-sized groups? Now I know the solution is $\;\dfrac {15!}{5!5!5!3!}\;$ but I can't understand why. Can anyone please enlighten me?...Fri, 14 Aug 2026 00:28:26 -0400interpretation of SVD for text mining topic analysishttps://liberty.io/interpretation-of-svd-for-text-mining-topic-analysis.htmlhttps://liberty.io/interpretation-of-svd-for-text-mining-topic-analysis.htmlBackground I'm learning about text mining by building my own text mining toolkit from scratch - the best way to learn! SVD The Singular Value Decomposition is often cited as a good way to: Visu......Fri, 14 Aug 2026 00:23:01 -0400Find the point on the line $y = 2x + 1$ that is closest to the point $(5, 2)$?https://liberty.io/find-the-point-on-the-line-y-2x-1-that-is-closest-to-the-point-5-2.htmlhttps://liberty.io/find-the-point-on-the-line-y-2x-1-that-is-closest-to-the-point-5-2.htmlThe question is: Find the point on the line $y = 2x + 1$ that is closest to the point $(5, 2)$. I know the high (or middle) school method. Create a line that is perpendicular to this line, so the s......Fri, 14 Aug 2026 00:14:03 -0400How to Prove these Product to Sum Trig Identities [closed]https://liberty.io/how-to-prove-these-product-to-sum-trig-identities-closed.htmlhttps://liberty.io/how-to-prove-these-product-to-sum-trig-identities-closed.html$$ \sin(a)\cos(b) = \frac{1}{2}(\sin(a-b) + \sin(a+b)) $$ $$ \sin(a)\sin(b) = \frac{1}{2}(\cos(a-b) - \cos(a+b)) $$ $$ \cos(a)\cos(b) = \frac{1}{2}(\cos(a-b) + \cos(a+b)) $$ How do you derive these ...Fri, 14 Aug 2026 00:12:20 -0400How to derive the volume and surface area formula for truncated cylinder without calculus?https://liberty.io/how-to-derive-the-volume-and-surface-area-formula-for-truncated-cylind.htmlhttps://liberty.io/how-to-derive-the-volume-and-surface-area-formula-for-truncated-cylind.htmlseveral hours ago, I saw the problem involving geometry. The problem is tell about truncated cylinder. I want to know how to derive the formula of volume and its surface area without calculus, but......Fri, 14 Aug 2026 00:12:10 -0400Sequence that converges to 0 but its function divergeshttps://liberty.io/sequence-that-converges-to-0-but-its-function-diverges.htmlhttps://liberty.io/sequence-that-converges-to-0-but-its-function-diverges.htmlLet f(x)=-2x+1 if x<0 and f(x)=$x^2$+x if x>0. Give a sequence {$x_n$} in $\mathbb R$\ {0} such that {$x_n$} converges to zero but {f($x_n$)} diverges. I've tried with the sequence {1/n} but it......Fri, 14 Aug 2026 00:12:04 -0400Suppose we draw two cards without replacement out of a standard deck of 52 cardshttps://liberty.io/suppose-we-draw-two-cards-without-replacement-out-of-a-standard-deck-o.htmlhttps://liberty.io/suppose-we-draw-two-cards-without-replacement-out-of-a-standard-deck-o.htmlSuppose we draw two cards without replacement out of a standard deck of 52 cards, while each time a card is drawn randomly with the (remaining) cards well-shuffled. Let A be the event that the firs......Fri, 14 Aug 2026 00:06:59 -0400Change of basis for a polynomialhttps://liberty.io/change-of-basis-for-a-polynomial.htmlhttps://liberty.io/change-of-basis-for-a-polynomial.htmlSo I am given the following. $B=\{1+t+t^2,2-t^2,t+t^2\}$ and $C=\{1+t,1+t^2,1-t+t^2\}$ are two bases for $\mathbb P_2$. I have to find a) The standard basis for $\mathbb P_2$ is $S=\{1,t,t^2\}$. ......Thu, 13 Aug 2026 23:59:57 -0400For what values of t, the solution for this equation existhttps://liberty.io/for-what-values-of-t-the-solution-for-this-equation-exist.htmlhttps://liberty.io/for-what-values-of-t-the-solution-for-this-equation-exist.htmlI need help in finding maximal solution for the problem: $$ \cases {{\dot{x} = x^2+t}\\{x(0)=0}}$$ I know that because $x(1) \geq \frac{1}{2}$ and that every solution $x(t)$ of the problem is grea......Thu, 13 Aug 2026 23:54:51 -0400The description of abelian Lie groupshttps://liberty.io/the-description-of-abelian-lie-groups.htmlhttps://liberty.io/the-description-of-abelian-lie-groups.htmlQuestion. Classify all abelian connected Lie groups. There is a problem in my problem sheet which asks me to describe all abelian connected Lie groups (moreover this is the first problem so it sho......Thu, 13 Aug 2026 23:53:41 -0400What is the antiderivative of $e^{-x^2}$https://liberty.io/what-is-the-antiderivative-of-e-x-2.htmlhttps://liberty.io/what-is-the-antiderivative-of-e-x-2.htmlI was wondering what the antiderivative of $e^{-x^2}$ was, and when I wolfram alpha'd it I got $$\displaystyle \int e^{-x^2} \textrm{d}x = \dfrac{1}{2} \sqrt{\pi} \space \text{erf} (x) + C$$ So, ......Thu, 13 Aug 2026 23:47:31 -0400What is the difference of pure strategic equilibria and nash equilibria?https://liberty.io/what-is-the-difference-of-pure-strategic-equilibria-and-nash-equilibri.htmlhttps://liberty.io/what-is-the-difference-of-pure-strategic-equilibria-and-nash-equilibri.htmlAre they the same thing just named differently or with minor differences?...Thu, 13 Aug 2026 23:38:09 -0400How to change the limits of a summation when the index $k$ is replaced by $-k$?https://liberty.io/how-to-change-the-limits-of-a-summation-when-the-index-k-is-replaced-b.htmlhttps://liberty.io/how-to-change-the-limits-of-a-summation-when-the-index-k-is-replaced-b.htmlIs what I am doing below correct, please assist. $$\sum_{k=-\infty}^{-1}\frac{e^{kt}}{1-kt} = \sum_{k=1}^{\infty}\frac{e^{-{kt}}}{1-kt}$$ Is this the rule on how to "invert" the limits, and doe......Thu, 13 Aug 2026 23:23:54 -0400What exactly is in the standart atlas of $R^n$?https://liberty.io/what-exactly-is-in-the-standart-atlas-of-r-n.htmlhttps://liberty.io/what-exactly-is-in-the-standart-atlas-of-r-n.htmlI saw a theorem saying that any atlas on a manifold can be extended to a unique maximal atlas. Concerning the standard atlas of $\mathbb{R}^n$, we saw that this was the atlas generated by the char......Thu, 13 Aug 2026 23:11:42 -0400Can Navier-Stokes equation be derived from Cauchy momentum equation?https://liberty.io/can-navier-stokes-equation-be-derived-from-cauchy-momentum-equation.htmlhttps://liberty.io/can-navier-stokes-equation-be-derived-from-cauchy-momentum-equation.html$$\frac{{\partial (\rho {v}_x})}{\partial t}+\nabla\cdot{(\rho {v}_x\mathbf{v}) = -(\nabla P)_x + \rho g_x+\nabla\cdot\mathbf{\tau_x}}$$ This is the $x$ component of the Cauchy momentum equation, w......Thu, 13 Aug 2026 23:08:43 -0400Let G be a group and $a\in Z(G)$. Prove that $\langle a\rangle$ is a normal subgroup of G.https://liberty.io/let-g-be-a-group-and-a-in-z-g-prove-that-langle-a-rangle-is-a-normal-s.htmlhttps://liberty.io/let-g-be-a-group-and-a-in-z-g-prove-that-langle-a-rangle-is-a-normal-s.htmlLet $G$ be a group and $a\in Z(G)$, where $$ Z(G) =\lbrace x \in G\mid xg=gx~~ \forall g\in G\}.$$ Prove that $H=\langle a\rangle$ is a normal subgroup of $G$. To show that $H$ is normal in $G$, I ...Thu, 13 Aug 2026 23:00:10 -0400How to find a conserved quantity in this differential equation.https://liberty.io/how-to-find-a-conserved-quantity-in-this-differential-equation.htmlhttps://liberty.io/how-to-find-a-conserved-quantity-in-this-differential-equation.htmlConsider the system: $$\ddot x = x^3 -x$$ What is the method to follow to find a conserved quantity for this system? So far what I have is: $\dot x = y$ and $\dot y = x^3 - x$ and I can find the ...Thu, 13 Aug 2026 22:58:23 -0400Confusion regarding $\log(x)$ and $\ln(x)$https://liberty.io/confusion-regarding-log-x-and-ln-x.htmlhttps://liberty.io/confusion-regarding-log-x-and-ln-x.htmlI was solving an integral and I encountered in some question $$\displaystyle \int_{2}^{4}\frac{1}{x} \, \mathrm dx$$ I know its integration is $\log(x)$. But my answer comes correct when I use $\......Thu, 13 Aug 2026 22:55:23 -0400