https://liberty.ioCurated star highlights with fresh energy.en-usSun, 16 Aug 2026 06:17:16 -0400https://liberty.io/prove-that-the-values-of-x-for-which-x-frac-x-2-1-198-lie-between-frac.htmlhttps://liberty.io/prove-that-the-values-of-x-for-which-x-frac-x-2-1-198-lie-between-frac.htmlWithout using a calculator, prove that the values of $x$ for which $x = \dfrac{x^2+1}{198}$ lie between $\dfrac{1}{198}$ and $199.99 \overline{49}$. Then use this result to prove that $\sqrt{2} &lt......Sun, 16 Aug 2026 10:08:11 -0400https://liberty.io/showing-a-function-does-not-have-a-local-max-or-min.htmlhttps://liberty.io/showing-a-function-does-not-have-a-local-max-or-min.htmlThe function is $f(x) = \begin{cases} x^2\sin(1/x) & \text{if $x \neq 0$} \\ 0 & \text{if $x = 0$} \end{cases} $ I have proved that $f$ is differentiable at $0$, and $f^{'}(0) = 0$. ......Sun, 16 Aug 2026 09:58:10 -0400https://liberty.io/show-that-the-square-of-any-prime-number-is-the-factof-of-some-integer.htmlhttps://liberty.io/show-that-the-square-of-any-prime-number-is-the-factof-of-some-integer.htmlThis is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with prime numbers, but other than that, the textbook gave no hints really a......Sun, 16 Aug 2026 09:54:33 -0400https://liberty.io/complementary-subspaces.htmlhttps://liberty.io/complementary-subspaces.htmlHow do I go about proving this? Two subspaces $V$ and $W$ of $\mathbb{R}^n$ are called complements if any vector $x$ in $\mathbb{R}^n$ can be expressed uniquely as $x = v+ w$ , where $v\in V$ and......Sun, 16 Aug 2026 09:26:41 -0400https://liberty.io/how-does-this-definition-of-the-exponential-function-have-e-0-1.htmlhttps://liberty.io/how-does-this-definition-of-the-exponential-function-have-e-0-1.htmlRudin defines a power series for the exponential function and says that $E(0)=1$. But when $z=0$, the first term in the series is $0^0$, which is undefined. Why is it equal to $1$?...Sun, 16 Aug 2026 09:24:41 -0400https://liberty.io/vector-analysis-calculate-the-flux-of-the-vector-field-s.htmlhttps://liberty.io/vector-analysis-calculate-the-flux-of-the-vector-field-s.html$\mathbf{A}(x,y,z)= (\frac{-6x}{x^2+y^2},{-6y}{x^2+y^2},z+1)$ $S: x^2+4y^2=4, 0 \leq z \leq 1$ The task is to calculate the flux away from the z-axis. Solution: The divergence theorem obviously......Sun, 16 Aug 2026 09:15:41 -0400https://liberty.io/how-to-represent-a-decimal-number-in-ternary-or-base-3-number-system.htmlhttps://liberty.io/how-to-represent-a-decimal-number-in-ternary-or-base-3-number-system.htmlI need to convert a decimal number into base $3$ number. $47_{10} ~~\mbox{is}~~ 1202_{3}$. But how do you represent a negative number in base $3$ notation like $-297$? please include a example...Sun, 16 Aug 2026 09:13:47 -0400https://liberty.io/arithmetic-mean-why-does-it-work.htmlhttps://liberty.io/arithmetic-mean-why-does-it-work.htmlI've been using the formula for the arithmetic mean all my life, but I'm not sure why it works. My current intuition is this one: The arithmetic mean is a number that when multiplied by the numbe......Sun, 16 Aug 2026 09:08:41 -0400https://liberty.io/user-ashvin-swaminathan-mathematics-stack-exchange.htmlhttps://liberty.io/user-ashvin-swaminathan-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Sun, 16 Aug 2026 09:06:51 -0400https://liberty.io/second-partial-derivative-test-failing-when-determinant-is-0.htmlhttps://liberty.io/second-partial-derivative-test-failing-when-determinant-is-0.htmlI am studying the second partial derivative test and it says when determinant of the hessian matrix is $0$ then there is no conclusion. However I saw on a website a method that tells you how to wor......Sun, 16 Aug 2026 09:05:20 -0400https://liberty.io/describing-asymptotic-behaviour-of-a-function.htmlhttps://liberty.io/describing-asymptotic-behaviour-of-a-function.htmlFor question B! x^2+x+1/x^2 = 1+ [x+1/x^2] shouldnt the answer be asymptote at x=0 and y=1 ?? i dont understand the textbook solution...Sun, 16 Aug 2026 09:03:31 -0400https://liberty.io/which-geometric-figure-polyhedron-has-15-quadrilateral-faces.htmlhttps://liberty.io/which-geometric-figure-polyhedron-has-15-quadrilateral-faces.htmlI am looking for a polyhedron which consists only out of 15 quadrilateral faces? Does such a thing exist?...Sun, 16 Aug 2026 09:02:27 -0400https://liberty.io/evaluating-2x-3-9x-2-10x-13-for-x-3-sqrt-5-is-there-an-efficient-way-t.htmlhttps://liberty.io/evaluating-2x-3-9x-2-10x-13-for-x-3-sqrt-5-is-there-an-efficient-way-t.htmlI was helping my sibling with a math problem from a past year paper for a competitive exam. It requires us to evaluate this cubic expression for a given value of $x$ which has an $a+b$ form where $......Sun, 16 Aug 2026 08:56:38 -0400https://liberty.io/example-of-distinctions-between-multiple-integral-and-iterated-integra.htmlhttps://liberty.io/example-of-distinctions-between-multiple-integral-and-iterated-integra.htmlThe following question is asked in the book of Analysis On Manifolds by Munkres, and given at page 103 as question 3. $Q=A\times B$,where $A$ is a rectangle in $\mathbb{R^k}$ and $B$ is a rectangl......Sun, 16 Aug 2026 08:45:30 -0400https://liberty.io/cardioid-in-coffee-mug.htmlhttps://liberty.io/cardioid-in-coffee-mug.htmlI've been learning about polar curves in my Calc class and the other day I saw this suspiciously $r=1-\cos \theta$ looking thing in my coffee cup (well actually $r=1-\sin \theta$ if we're being pe......Sun, 16 Aug 2026 08:44:14 -0400https://liberty.io/the-expected-value-of-the-square-of-sample-variance.htmlhttps://liberty.io/the-expected-value-of-the-square-of-sample-variance.htmlSuppose $X_1, \cdots, X_n$ are i.d.d. samples from population $X \sim N(\mu,\sigma^2)$, and the sample variance is denoted by $T = \sum_{i = 1}^n \frac{(X_i - \overline{X})^2}{n}$. I am curious abo......Sun, 16 Aug 2026 08:43:46 -0400https://liberty.io/show-that-the-volume-of-a-sphere-of-radius-r-is-v-frac-4-3-pi-r-3.htmlhttps://liberty.io/show-that-the-volume-of-a-sphere-of-radius-r-is-v-frac-4-3-pi-r-3.htmlIf we place the sphere so that its center is at the origin, then the plane $P_x$ intersects the sphere in a circle whose radius is $y = \sqrt{r^2-x^2}$.So the cross-sectional area is $$A(x) = \pi y......Sun, 16 Aug 2026 08:37:32 -0400https://liberty.io/fourier-transform-of-e-iwt.htmlhttps://liberty.io/fourier-transform-of-e-iwt.htmlFirst, I have searched the internet as well as here for an answer to my question but did not find one (or anything close enough to lead me to the answer on my own). I have a basic understanding o......Sun, 16 Aug 2026 08:37:09 -0400https://liberty.io/does-convergence-in-distribution-implies-convergence-of-characteristic.htmlhttps://liberty.io/does-convergence-in-distribution-implies-convergence-of-characteristic.htmlLevy's continuity theorem says that if a sequence of random variables $X_{i}$ has characteristic functions $\phi_{X_i}$ converging pointwise to a characteristic function $\phi_{X}$ of other random ...Sun, 16 Aug 2026 08:34:13 -0400https://liberty.io/how-do-i-solve-this-quadratic-equation-with-a-fraction.htmlhttps://liberty.io/how-do-i-solve-this-quadratic-equation-with-a-fraction.htmlI seem to have trouble with quadratic equations when it comes to fractions and square roots. $$ \frac{1}{x}+2x=3 $$ How do I solve this equation?...Sun, 16 Aug 2026 08:30:21 -0400https://liberty.io/when-the-population-variance-is-unknown-we-should-use-t-distribution.htmlhttps://liberty.io/when-the-population-variance-is-unknown-we-should-use-t-distribution.htmlIt is clear that T-distribution should be used when the sample size is small and the population variance is unknown. My question is Why? Why we use t-distribution in this case? Anybody give me spec......Sun, 16 Aug 2026 08:29:18 -0400https://liberty.io/sigma-algebra-generated-by-two-sigma-algebras-is-generated-by-the-inte.htmlhttps://liberty.io/sigma-algebra-generated-by-two-sigma-algebras-is-generated-by-the-inte.htmlLet $\mathcal{G}$ and $\mathcal{H}$ be two $\sigma$-algebras on a set $\Omega$. Does it hold that $$\sigma(\mathcal{H},\mathcal{G}) = \sigma\{G\cap H: G\in\mathcal{G}, H\in \mathcal{H}\}?$$ How to ......Sun, 16 Aug 2026 08:27:02 -0400https://liberty.io/how-to-find-dy-dx-fx-fy.htmlhttps://liberty.io/how-to-find-dy-dx-fx-fy.htmlI need some walkthrough in solving the following question: find dy/dx = - fx/fy? 3x^2 - y^2 + x^3 = 0. I need to know the method to solve this question. According to my understanding what I have ...Sun, 16 Aug 2026 08:06:18 -0400https://liberty.io/difference-between-power-law-distribution-and-exponential-decay.htmlhttps://liberty.io/difference-between-power-law-distribution-and-exponential-decay.htmlThis is probably a silly one, I've read in Wikipedia about power law and exponential decay. I really don't see any difference between them. For example, if I have a histogram or a plot that looks l......Sun, 16 Aug 2026 08:03:11 -0400https://liberty.io/probability-a-flaw-in-logic-the-emperor-039-s-proposition-with-marbles.htmlhttps://liberty.io/probability-a-flaw-in-logic-the-emperor-039-s-proposition-with-marbles.htmlI've tried searching for this question but couldn't find it on stackexchange. This is a common type of interview question; I ran into it doing brain teasers on a probability puzzles app, and if you......Sun, 16 Aug 2026 08:02:11 -0400https://liberty.io/prove-that-two-equations-are-the-same-if-they-have-the-same-solution-s.htmlhttps://liberty.io/prove-that-two-equations-are-the-same-if-they-have-the-same-solution-s.htmlI'm having trouble understanding the logic underlying the proof for this problem (from Hefferon's Linear Algebra): Prove that, where $a, b, c, d, e$ are real numbers with $a \ne 0$, if this linear ...Sun, 16 Aug 2026 08:01:43 -0400https://liberty.io/directional-derivative-finding-a-line-equation.htmlhttps://liberty.io/directional-derivative-finding-a-line-equation.htmlGiven the function $h(x,y,z)=5+x^2-3yz$ represents the temperature. find the directional derivative on the point $P(1,2,-1)$ that does the same angles in the positive direction of $x,y,z$ axis is ......Sun, 16 Aug 2026 07:55:57 -0400https://liberty.io/using-the-pumping-lemma-to-show-that-a-language-is-not-regular-compute.htmlhttps://liberty.io/using-the-pumping-lemma-to-show-that-a-language-is-not-regular-compute.htmlShow that $L=\{a^{n^2} | n \ge 0\}$ is not regular Hey guys. I'm taking a CS class and this stuff is really new to me so bear with me. I tried to look if I get some contradiction by using the pump......Sun, 16 Aug 2026 07:50:05 -0400https://liberty.io/solving-for-x-with-exponents-algebra.htmlhttps://liberty.io/solving-for-x-with-exponents-algebra.htmlSo I am trying to help a friend do her homework and I am a bit stuck. $$8x+3 = 3x^2$$ I can look at this and see that the answer is $3$, but I am having a hard time remembering how to solve for ......Sun, 16 Aug 2026 07:47:51 -0400https://liberty.io/calculate-the-minimum-number-of-moves-required-to-solve-the-tower-of-h.htmlhttps://liberty.io/calculate-the-minimum-number-of-moves-required-to-solve-the-tower-of-h.htmlLet's say you have a variant of The Tower of Hanoi, where you have $4$ pegs and $4$ discs. The formula for $3$ pegs and $n$ discs is $2^n-1$ but how would you work out the minimum number of moves in ...Sun, 16 Aug 2026 07:35:02 -0400https://liberty.io/what-039-s-the-difference-between-a-dependent-function-type-pi-types-a.htmlhttps://liberty.io/what-039-s-the-difference-between-a-dependent-function-type-pi-types-a.htmlI was reading the book "Homotopy Type Theory", and got confused about the distinction between a family of types $B : A \to \mathcal U$ and a dependent function type $\Pi_{(x:A)} B(x) : \to \mathcal......Sun, 16 Aug 2026 07:33:18 -0400https://liberty.io/distance-between-the-two-points-p-and-q.htmlhttps://liberty.io/distance-between-the-two-points-p-and-q.htmlLet $d(x,y)$ denote the distance between two points, $x$ and $y$, on the plane. 1) $P(2,9),\quad Q(-1,13)\Rightarrow d(P,Q) = 5.$ 2) $P(1,-2),\quad Q(2,10)\Rightarrow d(P,Q) = \sqrt{145}.$ 3) $P......Sun, 16 Aug 2026 07:33:00 -0400https://liberty.io/how-to-determine-concavity-without-inflection-point.htmlhttps://liberty.io/how-to-determine-concavity-without-inflection-point.htmlI'm trying to determine this function concavity: 2=0 is a contradiction, so we have no inflection points on this function, so ¿How could I determine the concavity if I have no inflection points? ......Sun, 16 Aug 2026 07:29:28 -0400https://liberty.io/discrete-math-inclusive-or-vs-exclusive-or.htmlhttps://liberty.io/discrete-math-inclusive-or-vs-exclusive-or.htmlQuestion: Are inclusive or and exclusive or both considered dis-junction ? My own thinking: I believe yes, because both inclusive and exclusive or are using or. They just happen to have different ...Sun, 16 Aug 2026 07:28:42 -0400https://liberty.io/how-to-solve-equations-with-casio-fx-85wa-calculator.htmlhttps://liberty.io/how-to-solve-equations-with-casio-fx-85wa-calculator.htmlI am testing the casio FX-85WA calculator to solve equations, so I am trying to solve a very simple one: $2x-1=3$. But I am unable to input the equal sign, as it puts the result. I've seen others ...Sun, 16 Aug 2026 07:24:24 -0400https://liberty.io/is-blitzer-precalculus-my-best-option-considering-my-circumstances.htmlhttps://liberty.io/is-blitzer-precalculus-my-best-option-considering-my-circumstances.htmlSo I am wrapping up a College Algebra course from my local community college and am scheduled to start the first semester of a two semester precalculus course next month. The text book I worked out......Sun, 16 Aug 2026 07:18:03 -0400https://liberty.io/weighted-probability-problem.htmlhttps://liberty.io/weighted-probability-problem.htmlLets say, few independent events are happening. $E_1, E_2,...,E_n$. The probability of each of these events happening is given as $P_1,P_2,...,P_n$. Each of these events carry weightage, say $W_1,W......Sun, 16 Aug 2026 07:18:00 -0400https://liberty.io/area-of-a-regular-hexagon-via-area-of-triangles.htmlhttps://liberty.io/area-of-a-regular-hexagon-via-area-of-triangles.htmlProblem: Find the area of a regular hexagon whose sides measures 5 cm Sol'n 1: I can cut the hexagon into 6 small triangles, so the area of triangle times 6 will be equal to the area of the polygon. ...Sun, 16 Aug 2026 06:55:41 -0400https://liberty.io/determinant-of-a-very-large-matrix-in-matlab.htmlhttps://liberty.io/determinant-of-a-very-large-matrix-in-matlab.htmlI have a very large random matrix which its elements are either $0$ or $1$ randomly. The size of the matrix is $5000$, however when I want to calculate the determinant of the matrix, it is either $......Sun, 16 Aug 2026 06:47:40 -0400https://liberty.io/why-does-fixed-point-iteration-work.htmlhttps://liberty.io/why-does-fixed-point-iteration-work.htmlI have searched online for an answer, but everyone gave the method, and no one explained why is it working. I'll first write what I do understand. Let $f(x)$ be a continuous function at $[a,b]$. ...Sun, 16 Aug 2026 06:39:02 -0400https://liberty.io/derivative-of-sin-2x-cos-3x.htmlhttps://liberty.io/derivative-of-sin-2x-cos-3x.htmlWhat is the derivative of $\sin (2x)\cos(3x)?$ Is it $\cos (2x) \cos(3x)-\sin (2x) \sin(3x)$ or $2\cos (2x) \cos(3x)-3\sin (2x) \sin(3x)?$ The answer is given the latter but I only get the form......Sun, 16 Aug 2026 06:33:23 -0400https://liberty.io/identifying-the-formula-for-a-quartic-graphic.htmlhttps://liberty.io/identifying-the-formula-for-a-quartic-graphic.htmlI am attempting to help someone with their homework and these concepts are a bit above me. I apologize for the terrible graph drawing. I am using a surface pro 3 and it has an awful camera so I can't ...Sun, 16 Aug 2026 06:30:56 -0400https://liberty.io/what-is-the-kernel-of-a-homomorphism.htmlhttps://liberty.io/what-is-the-kernel-of-a-homomorphism.htmlI've done a lot of problems before but I am trying to get a really basic definition of kernel so that I may apply to any possible given question that I may be presented with. Would I be corre......Sun, 16 Aug 2026 06:29:31 -0400https://liberty.io/intuition-for-class-numbers.htmlhttps://liberty.io/intuition-for-class-numbers.htmlSo I've been thinking about the analytic class number formula lately, and class numbers in general and I'm trying to develop a good intuition for them. My basic question, which may be too general/...Sun, 16 Aug 2026 06:24:08 -0400https://liberty.io/implicit-derivative-with-three-variable.htmlhttps://liberty.io/implicit-derivative-with-three-variable.htmlFind $\frac{\partial z}{\partial x}$ from these equation: $xyz-2xz+3yz-4xy=0$ How $\frac{\partial z}{\partial x}$ can be done using manually method? Thanks... I stuck at variable whose contains '......Sun, 16 Aug 2026 06:19:06 -0400https://liberty.io/why-do-we-say-the-harmonic-series-is-divergent-duplicate.htmlhttps://liberty.io/why-do-we-say-the-harmonic-series-is-divergent-duplicate.htmlIf we have $\Sigma\frac{1}{n}$, why do we say it is divergent? Yes, it is constantly increasing, but after a certain point, $n$ will be so large that we will be certain of millions of digits. If we ...Sun, 16 Aug 2026 06:18:30 -0400https://liberty.io/how-to-understand-compactness-duplicate.htmlhttps://liberty.io/how-to-understand-compactness-duplicate.htmlHow to understand the compactness in topology space in intuitive way?...Sun, 16 Aug 2026 06:11:14 -0400https://liberty.io/why-is-it-possible-to-factor-a-limit.htmlhttps://liberty.io/why-is-it-possible-to-factor-a-limit.htmlI am confused by the examples in calculus textbook where they factor a polynomial to find the limit. I don't understand how the limits of $\frac{1}{x-3}$ and $\frac{x+3}{x^2-9} $ are the same. Th......Sun, 16 Aug 2026 06:08:14 -0400https://liberty.io/finding-the-splitting-field-of-a-polynomial.htmlhttps://liberty.io/finding-the-splitting-field-of-a-polynomial.htmlI am confused on how to find the splitting field of a polynomial. For example, consider the polynomial $$p(x)=x^2+2.$$ I know that the splitting field is the smallest field that contains the roots ......Sun, 16 Aug 2026 06:06:14 -0400https://liberty.io/slope-of-secant-and-tangent-lines-supported-by-mvt.htmlhttps://liberty.io/slope-of-secant-and-tangent-lines-supported-by-mvt.htmlFor the function $f(x) = x^{1/3}$ on the interval $[1,8]$ find the point $(c,f(c))$ guaranteed by the Mean Value Theorem, at which the slope of the tangent line is equal to the slope of the secant ......Sun, 16 Aug 2026 06:00:00 -0400https://liberty.io/where-was-my-mistake-integration-by-trig-substitution-problem.htmlhttps://liberty.io/where-was-my-mistake-integration-by-trig-substitution-problem.htmlI am attempting to solve the problem $$\int \frac{dx}{x^2+x+1}$$ First, I complete the square, then factor out a $\frac {3}{4}$: $$\int \frac{dx}{\frac{3}{4}(\frac{4}{3}(x+\frac{1}{2})^2+1)}$$ ......Sun, 16 Aug 2026 05:59:58 -0400https://liberty.io/how-to-calculate-modulo-negative-number-duplicate.htmlhttps://liberty.io/how-to-calculate-modulo-negative-number-duplicate.htmlWhy 134 mod -26 is equal 22? I found an explanation, which says: 134 = (-26 * -6) - 22 However, 134 = (-26*-5)+4, so why 134 mod -26 is not equal 4 if 134=(26*5)+4 => 134 mod 26=4 All i could f......Sun, 16 Aug 2026 05:59:21 -0400https://liberty.io/is-there-any-equation-for-triangle.htmlhttps://liberty.io/is-there-any-equation-for-triangle.htmlLike there's an equation of a circle, is there any equation of a triangle? I've been trying to build one and the closest thing I've managed to do is to create an equation of 2 lines and use the $x......Sun, 16 Aug 2026 05:57:33 -0400https://liberty.io/if-capital-letters-are-supposed-to-be-sets-why-is-n-used-as-a-number.htmlhttps://liberty.io/if-capital-letters-are-supposed-to-be-sets-why-is-n-used-as-a-number.htmlIf capital letters are supposed to be sets, why is $N$ used as a number? For example, in this definition of a Cauchy Sequence it says: A sequence ${p_n}$ in a metric space $X$ is said to be a C......Sun, 16 Aug 2026 05:45:39 -0400https://liberty.io/a-line-connecting-an-interior-and-an-exterior-point-of-a-circle-should.htmlhttps://liberty.io/a-line-connecting-an-interior-and-an-exterior-point-of-a-circle-should.htmlHow can someone prove in Euclidean geometry that the statement "A line connecting an interior and an exterior point of a circle should intersect the circle at some point" follows from the axiom......Sun, 16 Aug 2026 05:24:37 -0400https://liberty.io/information-gain-calculation-for-decision-tree-when-choosing-root-node.htmlhttps://liberty.io/information-gain-calculation-for-decision-tree-when-choosing-root-node.htmlI want to know if my calculation is wrong or correct, because i got a different result when i use an online calculator. Here is the dataset: A;B;C;Class y;y;y;group1 n;y;y;group2 y;n;n;group3 y;y;n;...Sun, 16 Aug 2026 05:09:17 -0400https://liberty.io/can-you-integrate-without-a-dx-duplicate.htmlhttps://liberty.io/can-you-integrate-without-a-dx-duplicate.htmlLong ago I realized that manipulation of derivatives was possible using algebraic quantities. One could take a differential instead of derivatives $$ d[\sin x]=\cos x\ dx $$ $$ \frac{d[\sin x]}{dx}......Sun, 16 Aug 2026 05:07:23 -0400https://liberty.io/fisher-yates-algorithm-proof.htmlhttps://liberty.io/fisher-yates-algorithm-proof.htmlI have an array of numbers $[a_0, \ldots, a_n]$ and want to shuffle it fairly. The Fisher-Yates algorithm proposes the following. from $i = 0$ to $n$: $j$ = randomly pick an index in $(0, i)$, swa......Sun, 16 Aug 2026 05:05:12 -0400https://liberty.io/what-does-p-a-u-b-mean-in-terms-of-real-values.htmlhttps://liberty.io/what-does-p-a-u-b-mean-in-terms-of-real-values.htmlI can't find a proper summary or reference of how to translate formulas in probability notation to arithmetic notation (i.e. when using real values). For example, if $P(A) = .7$ and $P(B)=.35$, what ...Sun, 16 Aug 2026 05:02:40 -0400https://liberty.io/prove-det-i-m-ab-det-i-n-ba-duplicate.htmlhttps://liberty.io/prove-det-i-m-ab-det-i-n-ba-duplicate.html$A$ is an $m\times n$ matrix and $B$ is an $n\times m$ matrix. $$ \det(I_m + AB) = \det(I_n + BA) $$ Solution: I found this guys: $$ \det\begin{pmatrix}I&-B\\\\A&I\end{pmatrix} \det\begin{...Sun, 16 Aug 2026 04:42:35 -0400https://liberty.io/in-matrix-notation-what-does-it-mean-x-039-x.htmlhttps://liberty.io/in-matrix-notation-what-does-it-mean-x-039-x.htmlIn matrix notation what does it mean X 'X I see it in this academic paper, I can't find it in matrix operation and can't search with google. http://papers.ssrn.com/sol3/papers.cfm?abstract_id=24......Sun, 16 Aug 2026 04:37:23 -0400https://liberty.io/help-me-to-find-a-volume-of-the-ring-shaped-solid.htmlhttps://liberty.io/help-me-to-find-a-volume-of-the-ring-shaped-solid.htmlA cylindrical drill with radius 5 is used to bore a hole through the center of a sphere of radius 8. Find the volume of the ring shaped solid that remains. Alright, my thing is that i did not unde......Sun, 16 Aug 2026 04:24:26 -0400https://liberty.io/inverse-of-a-1-b-1.htmlhttps://liberty.io/inverse-of-a-1-b-1.htmlIf $A, B$ and $A + B$ are all $n × n$ invertible matrices. Prove that $A^{−1} + B^{−1}$ is invertible and the inverse is $A(A + B) ^{−1}B$. I am afraid I am really stuck on this one, and I haven't ...Sun, 16 Aug 2026 04:20:15 -0400https://liberty.io/find-limit-of-a-n-ln-1-a-n-1.htmlhttps://liberty.io/find-limit-of-a-n-ln-1-a-n-1.htmlSequence $a_n$ is defined as $a_n = \ln (1 + a_{n-1})$, where $a_0 > -1$. Find all $a_0$ for which the sequence converges. Using arithmetic properties of limits, I've found that if the sequence ...Sun, 16 Aug 2026 04:12:04 -0400https://liberty.io/jacobson-039-s-density-theorem-for-semisimple-algebras.htmlhttps://liberty.io/jacobson-039-s-density-theorem-for-semisimple-algebras.htmlI am trying to follow the proof of the following: Let $A$ be an algebra over an algebraically closed field $k$. Let $V=V_1\oplus \ldots \oplus V_r$, where $V_i$ are pairwise non-isomorphic irred......Sun, 16 Aug 2026 04:06:22 -0400https://liberty.io/mod-remainder-operation-and-relation-name-and-meaning.htmlhttps://liberty.io/mod-remainder-operation-and-relation-name-and-meaning.htmlI am trying to write the Euclidean algorithm in the following way: $A = \lfloor A \div B \rfloor \times B + (\text{remainder of}) \: A \div B $ Now is there any symbol I can use to say "remaind......Sun, 16 Aug 2026 04:04:12 -0400https://liberty.io/duffing-oscillator.htmlhttps://liberty.io/duffing-oscillator.htmlI have to draw the phase space of this modification of Duffing oscillator: \begin{align} \dot x &= y,\\ \dot y &= x - x^3 - ay - (x^2)y \quad \text{ when } \quad a=1 \; \text{ and } \;......Sun, 16 Aug 2026 03:58:59 -0400https://liberty.io/meaning-of-the-word-039-linear-039-in-mathematics.htmlhttps://liberty.io/meaning-of-the-word-039-linear-039-in-mathematics.htmlWhat exactly does 'linear' mean. For example, Hilbert space is a linear space. What is the difference between a linear and a 'non-linear' space. I am starting a self-study of linear algebra so this ...Sun, 16 Aug 2026 03:54:54 -0400https://liberty.io/solve-the-system-of-equations-in-the-set-of-real-numbers.htmlhttps://liberty.io/solve-the-system-of-equations-in-the-set-of-real-numbers.htmlSolve the system of equations in the set of real numbers: $$\begin{cases} \frac1x + \frac1{y+z} = \frac13 \\ \frac1y + \frac1{x+z} = \frac15 \\ \frac1z + \frac1{x+y} = \frac17 \end{cases}$$ I go......Sun, 16 Aug 2026 03:53:27 -0400https://liberty.io/solve-a-system-of-linear-inequalities-without-graphing-i-e-using-pure-.htmlhttps://liberty.io/solve-a-system-of-linear-inequalities-without-graphing-i-e-using-pure-.htmlI have a system of linear inequalities such as: $y\geq 2x+1$ $y>(x/2)-1$ I know how to solve this system of inequalities (in order to find possible values that will satisfy it) by graphing th......Sun, 16 Aug 2026 03:47:10 -0400https://liberty.io/beads-on-a-bracelet.htmlhttps://liberty.io/beads-on-a-bracelet.htmlHow many ways can 1 blue bead, 7 red beads and 12 green beads be arranged on a bracelet or turnover necklace? There are 20 beads so to my mind the answer is $\frac{19!}{1!\cdot7!\cdot12!\cdot2} = ......Sun, 16 Aug 2026 03:45:08 -0400https://liberty.io/when-is-the-number-11111-cdots1-a-prime-number.htmlhttps://liberty.io/when-is-the-number-11111-cdots1-a-prime-number.htmlFor which $n$ is the sum: $$\sum_{k=0}^{n}10^k$$ a prime number? Are they finite?...Sun, 16 Aug 2026 03:40:43 -0400https://liberty.io/about-definition-of-superset.htmlhttps://liberty.io/about-definition-of-superset.htmlI have read definition of superset somewhere as "a set containing all elements of a smaller set.". This implies that for set A to be a superset of B, B must be a proper subset of A, that is, A must ...Sun, 16 Aug 2026 03:37:11 -0400https://liberty.io/is-wolfram-alpha-wrong-with-a-simple-derivative.htmlhttps://liberty.io/is-wolfram-alpha-wrong-with-a-simple-derivative.htmlLet $f=\frac{x^2w(y-z)t}{18l}$ then (imho): $\frac{\partial{f(x,y,z)}}{\partial{w}}=\frac{x^2t(y-z)}{18l}$ However Wolfram Alpha produces a quite different result: Wolfram alpha result http://www....Sun, 16 Aug 2026 03:31:40 -0400https://liberty.io/self-affinity-vs-self-similarity.htmlhttps://liberty.io/self-affinity-vs-self-similarity.htmlI'm a student in Engineering school without abundant math knowledge. And I am sorry if this is a silly question. I know a little about ''self-similar'', and recently also encountered the concept of "...Sun, 16 Aug 2026 03:22:51 -0400https://liberty.io/factoring-x-5-x-1-closed.htmlhttps://liberty.io/factoring-x-5-x-1-closed.htmlFactor. $\displaystyle x^5 + x + 1$. I'm trying my hand on these types of expressions. In the book, it mentions that for these types you should rewrite the $x$ term such as $2x-x$. But I don't kno......Sun, 16 Aug 2026 03:16:04 -0400https://liberty.io/derivative-of-legendre-polynomial.htmlhttps://liberty.io/derivative-of-legendre-polynomial.htmlI am given with a relation \begin{equation} \frac{d^2}{dx^2}(P_l(x))=\frac{1}{2}\sum_{n=(0,1),2}^{l-2}(l-n)(l+n+1)(2n+1)P_n(x). \end{equation} The above sum starts with 0 for even end point, and 1 ......Sun, 16 Aug 2026 03:13:07 -0400https://liberty.io/product-of-permutation-matrices.htmlhttps://liberty.io/product-of-permutation-matrices.htmlI want to prove that the product of two permutation matrices is itself a permutation matrix. But I don't know how. Please help!...Sun, 16 Aug 2026 03:07:47 -0400https://liberty.io/finding-the-derivative-without-using-product-or-quotient-rule.htmlhttps://liberty.io/finding-the-derivative-without-using-product-or-quotient-rule.htmlI have a math problem where I am required to find the derivative of a function with the limitations of not being allowed to use the Product or Quotient Rule of Differentiation. The problem looks l......Sun, 16 Aug 2026 03:07:30 -0400https://liberty.io/recursive-vs-inductive-definition.htmlhttps://liberty.io/recursive-vs-inductive-definition.htmlI once had an argument with a professor of mine, if the following definition was a recursive or inductive definition: Suppose you have sequence of real numbers. Define $a_0:=2$ and $a_{i+1}:=\frac......Sun, 16 Aug 2026 02:30:44 -0400https://liberty.io/use-depth-first-search-to-produce-a-spanning-tree-for-the-given-simple.htmlhttps://liberty.io/use-depth-first-search-to-produce-a-spanning-tree-for-the-given-simple.htmlUse depth-first search to produce a spanning tree for the given simple graph. Choose $a$ as the root of this spanning tree. Here is answer and I am not sure whether is correct or not Can anyone ......Sun, 16 Aug 2026 02:22:20 -0400https://liberty.io/notation-is-there-a-symbol-for-not-a-function-of.htmlhttps://liberty.io/notation-is-there-a-symbol-for-not-a-function-of.htmlFor example, let's say a term $A(x,y)$, a function of two random variables $x$ and $y$, is the argument of an expectation over $y$. The resulting term is no longer a function of $y$. Is there a ...Sun, 16 Aug 2026 02:19:26 -0400https://liberty.io/integrate-x-e-x-2-from-infty-to-infty.htmlhttps://liberty.io/integrate-x-e-x-2-from-infty-to-infty.htmlI suppose this question is very simple, but I still don't get where I am wrong. I try to proof that the integral: $$\int_{-\infty}^{\infty} x e^{-x^2} dx$$ is equal to zero. I suppose that by ...Sun, 16 Aug 2026 02:14:48 -0400https://liberty.io/moment-of-inertia-of-a-disc.htmlhttps://liberty.io/moment-of-inertia-of-a-disc.htmlIn my mechanics textbook there is a derivation of the moment of inertia of a disc of mass $m$ and radius $r$ about an axis through its centre and perpendicular to its plane surface, which goes some......Sun, 16 Aug 2026 02:10:07 -0400https://liberty.io/how-to-properly-take-derivatives-in-calculus-of-variations-euler-lagra.htmlhttps://liberty.io/how-to-properly-take-derivatives-in-calculus-of-variations-euler-lagra.htmlWhy is it that, in calculus of variations (specifically Euler-Lagrange), we can take the derivative of a function with respect to a function $f$ and set this derivative to $0$ if only $f'$ appears ......Sun, 16 Aug 2026 02:10:04 -0400https://liberty.io/derivative-of-e-ln-1-x.htmlhttps://liberty.io/derivative-of-e-ln-1-x.htmlThis question looks so simple, yet it confused me. If $f(x) = e^{\ln(1/x)}$, then $f'(x) =$ ? I got $e^{\ln(1/x)} \cdot \ln(1/x) \cdot (-1/x^2)$. And the correct answer is just the plain $-1/x......Sun, 16 Aug 2026 02:06:47 -0400https://liberty.io/what-is-formally-to-define-a-function-by-induction.htmlhttps://liberty.io/what-is-formally-to-define-a-function-by-induction.htmlLet me be more specific: Given the context of a function as a relation between 2 sets with specific properties, what is formally happening when I define a function by induction? EXAMPLE: (from La......Sun, 16 Aug 2026 02:01:08 -0400https://liberty.io/square-root-of-height-of-a-paraboloid-equal-to-radius.htmlhttps://liberty.io/square-root-of-height-of-a-paraboloid-equal-to-radius.htmlI'm reading the book Mathematics: It's Content, Methods, and Meanings and I'm unsure as to how one of the variables in an example was derived. The question is about the volume of a paraboloid and h......Sun, 16 Aug 2026 01:42:42 -0400https://liberty.io/numbers-that-cannot-be-expressed-as-fractions.htmlhttps://liberty.io/numbers-that-cannot-be-expressed-as-fractions.htmlWhat are Numbers that cannot be expressed as Fractions called?...Sun, 16 Aug 2026 01:27:25 -0400https://liberty.io/is-kruskal-rank-and-rank-of-a-matrix-the-same.htmlhttps://liberty.io/is-kruskal-rank-and-rank-of-a-matrix-the-same.htmlThe definition of Kruskal rank : maximum value of $k$ such that any $k$ columns of a matrix $\textbf{A}$ are linearly independent, then $k$ is the Kruskal rank of matrix $\textbf{A}$. How is it ...Sun, 16 Aug 2026 01:25:34 -0400https://liberty.io/infinite-amount-of-vertical-asymptotes.htmlhttps://liberty.io/infinite-amount-of-vertical-asymptotes.htmlIs it possible that the graph of function has infinitely many vertical asymptotes? I suppose, that it is not possible, because such function would not exist. But I need to prove it in a math-fashi......Sun, 16 Aug 2026 01:15:22 -0400https://liberty.io/the-most-common-theorems-taught-in-abstract-algebra.htmlhttps://liberty.io/the-most-common-theorems-taught-in-abstract-algebra.htmlI am self learning abstract algebra. I want to know which theorems are a must to understand. Now these are limits I have to deal with (please consider when answering): I have limited inter......Sun, 16 Aug 2026 01:15:19 -0400https://liberty.io/using-chain-rule-to-find-frac-dz-dt.htmlhttps://liberty.io/using-chain-rule-to-find-frac-dz-dt.htmlI know what the chain rule is but i am unsure how to apply it to this case. $$z=x^3\cdot e^{2y}$$ $$x=2t$$ $$y=t^2$$ Edit: Fixed title to $\frac{dz}{dt}$....Sun, 16 Aug 2026 01:11:47 -0400https://liberty.io/cos-2-cos-1-x-x-2-how.htmlhttps://liberty.io/cos-2-cos-1-x-x-2-how.html$$\cos^2 (\cos^{-1} x)$$ $$=x^2$$ Which trigonometric rule was used here?...Sun, 16 Aug 2026 01:09:04 -0400https://liberty.io/prove-that-sinh-2x-2sinh-x-cosh-x.htmlhttps://liberty.io/prove-that-sinh-2x-2sinh-x-cosh-x.htmlWhen trying to solve a textbook question I'm getting a wrong result but I'm pretty sure I'm not doing anything wrong. Can somebody spot my mistake? Question: Prove that $$sinh(2x) = 2sinh(x)cosh......Sun, 16 Aug 2026 01:08:48 -0400https://liberty.io/term-for-sum-difference-product-quotient.htmlhttps://liberty.io/term-for-sum-difference-product-quotient.htmlFeel free to let me know if this is better suited for: https://english.stackexchange.com/ https://stackoverflow.com/ That said, I'm wondering if there's a math term for the nature of "sum", "...Sun, 16 Aug 2026 01:03:09 -0400https://liberty.io/what-sets-does-the-shades-on-the-venn-diagram-represent.htmlhttps://liberty.io/what-sets-does-the-shades-on-the-venn-diagram-represent.htmlI need help with this Venn Diagram. I think I got the first (please check if they are right). The last one is the one that I'm missing. Would appreciate your help. a. $A'∩B$ b. $(A∪B)'=A'∩B'$ c.......Sun, 16 Aug 2026 00:55:16 -0400https://liberty.io/how-many-essentially-different-drive-ya-nuts-puzzles-exist.htmlhttps://liberty.io/how-many-essentially-different-drive-ya-nuts-puzzles-exist.htmlIn the Drive Ya Nuts puzzle, we are given $7$ hexagonal nuts, whose edges have been numbered from $1$ to $6$. Each nut uses all $6$ numbers. The aim of the puzzle is to place all the nuts in a ...Sun, 16 Aug 2026 00:54:37 -0400https://liberty.io/finding-the-lengths-of-the-sides-of-a-triangle-given-3-angles-only.htmlhttps://liberty.io/finding-the-lengths-of-the-sides-of-a-triangle-given-3-angles-only.htmlIf a right $\triangle ABC$ has $\angle A= 90^\circ$, $\angle B=45^\circ$, $\angle C=45^\circ$. Is there a way of finding the lengths of the sides $a$, $b$ & $c$ without knowing any of their len......Sun, 16 Aug 2026 00:49:52 -0400https://liberty.io/references-to-supplement-goldblatt-039-s-topoi.htmlhttps://liberty.io/references-to-supplement-goldblatt-039-s-topoi.htmlI've been using the book "Topoi: A Categorial Analysis of Logic" by Robert Goldblatt. I found it intuitive and fun, but I feel like some parts are missing. For instance, I struggled to prove $a^1 \......Sun, 16 Aug 2026 00:39:45 -0400