https://liberty.ioCurated star highlights with fresh energy.en-usFri, 14 Aug 2026 01:29:51 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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$$ ......Fri, 14 Aug 2026 01:28:46 -0400https://liberty.io/cummulative-probability-5-chance-per-iteration.htmlhttps://liberty.io/cummulative-probability-5-chance-per-iteration.htmlLet's say you eat at some very cheap restaurant and every time you do there's a 5% chance you'll get food poisoning. How do you calculate the probability of getting food poisoning if you eat there x ...Fri, 14 Aug 2026 01:20:32 -0400https://liberty.io/express-the-following-invertible-matrix-a-as-a-product-of-elementary-m.htmlhttps://liberty.io/express-the-following-invertible-matrix-a-as-a-product-of-elementary-m.htmlI've been at this for a while... I tried to the inverse method but it keeps on saying I'm getting it wrong... Can anyone show me a step-by-step solution? The matrix I have is a $3\times 3$ square one(...Fri, 14 Aug 2026 01:17:28 -0400https://liberty.io/heat-equation-for-a-cylinder-in-cylindrical-coordinates.htmlhttps://liberty.io/heat-equation-for-a-cylinder-in-cylindrical-coordinates.htmlSo I have a description of a Partial differential equation given here. $\text{Consider a long pipe of inner radius a}=2.5cm \text{ and outer radius b}=3 cm, \text{made of copper for which the ther......Fri, 14 Aug 2026 01:14:36 -0400https://liberty.io/how-prove-this-tan-frac-2-pi-13-4-sin-frac-6-pi-13-sqrt-13-2-sqrt-13.htmlhttps://liberty.io/how-prove-this-tan-frac-2-pi-13-4-sin-frac-6-pi-13-sqrt-13-2-sqrt-13.htmlNice Question: show that: The follow nice trigonometry $$\tan{\dfrac{2\pi}{13}}+4\sin{\dfrac{6\pi}{13}}=\sqrt{13+2\sqrt{13}}$$ This problem I have ugly solution, maybe someone have nice meth......Fri, 14 Aug 2026 01:13:46 -0400https://liberty.io/i-have-3-quarters-2-dimes-and-2-pennies-if-i-select-at-least-one-coin-.htmlhttps://liberty.io/i-have-3-quarters-2-dimes-and-2-pennies-if-i-select-at-least-one-coin-.htmlI have 3 quarters, 2 dimes, and 2 pennies in my pocket. If I select at least one coin from my pocket, how many different amounts of money can I make? I checked out some of the other coin questions. ...Fri, 14 Aug 2026 01:10:33 -0400https://liberty.io/from-a-vector-to-a-skew-symmetric-matrix.htmlhttps://liberty.io/from-a-vector-to-a-skew-symmetric-matrix.htmlIs there an existing linear mapping that maps a 3-dimensional vector: $$\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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://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 -0400https://liberty.io/tiling-a-square-problem.htmlhttps://liberty.io/tiling-a-square-problem.htmlQuestion: A 2018 x 2018 square is tiles with some 4 x 1 rectangles and some 2 x 2 squares. One of the 4 x 1 rectangles is removes and replaced with a 2 x 2 square. Is it still possible to tile the ...Thu, 13 Aug 2026 22:53:33 -0400https://liberty.io/dividing-divergent-and-convergent-sequences.htmlhttps://liberty.io/dividing-divergent-and-convergent-sequences.htmlLet $a_n$ be a diverging sequence and $b_n$ be a converging sequence. With that in mind does $\frac{a_n}{b_n}$ always diverge? If so, what is the proof for this?...Thu, 13 Aug 2026 22:51:21 -0400https://liberty.io/how-far-does-a-free-falling-object-fall-in-the-3-rd-second.htmlhttps://liberty.io/how-far-does-a-free-falling-object-fall-in-the-3-rd-second.htmlThe question: An object is dropped from a cliff. How far does the object fall in the 3rd second?" I calculated that a ball dropped from rest from a cliff will fall $45\text{ m}$ in $3 \text{ s}$, ...Thu, 13 Aug 2026 22:45:50 -0400https://liberty.io/ellipse-word-problem.htmlhttps://liberty.io/ellipse-word-problem.htmlThe ellipse is 5 meters across and 8 meters long with decorative fountains located at the foci. How far from the center should the fountains be located? (Rounded to the nearest hundredth). How far ......Thu, 13 Aug 2026 22:43:58 -0400https://liberty.io/changing-from-quadratic-formula-to-standard-form.htmlhttps://liberty.io/changing-from-quadratic-formula-to-standard-form.htmlThe graph of a quadratic function has $x$-intercepts $-1$ and $3$ and a range consisting of all numbers less than or equal to $4$. Determine an expression for the function. This is my problem. I k......Thu, 13 Aug 2026 22:39:45 -0400https://liberty.io/where-does-r-1-2-cos-theta-have-tangents.htmlhttps://liberty.io/where-does-r-1-2-cos-theta-have-tangents.htmlWhere does: $$r = 1 + 2\cos(\theta)$$ Have horizontal and vertical tangent lines? $x = r\cos(\theta) = \cos(\theta) + 2\cos^2(\theta)$ $y = r\sin(\theta) = \sin(\theta) + 2\sin(\theta)\cos(\the......Thu, 13 Aug 2026 22:32:28 -0400https://liberty.io/how-should-i-approach-finding-the-derivative-using-the-limit-process-f.htmlhttps://liberty.io/how-should-i-approach-finding-the-derivative-using-the-limit-process-f.htmlI am setting up the problem to find the limit as delta x goes to zero, using the definition in the correct format. No matter what I've tried, I am stuck without finding the right cancellations to ......Thu, 13 Aug 2026 22:29:18 -0400https://liberty.io/proof-of-the-least-upper-bound-or-completeness-axiom-corollary-greates.htmlhttps://liberty.io/proof-of-the-least-upper-bound-or-completeness-axiom-corollary-greates.htmlFirst off, I don't think this is a duplicate. Sorry if it is! Second this is not a homework question, so completeness is much appreciated. I am trying to prove the corollary of the least upper bound ...Thu, 13 Aug 2026 22:25:51 -0400https://liberty.io/how-to-show-that-the-sequence-frac-n-2-n-1-2n-2-4n-1-converges-to-frac.htmlhttps://liberty.io/how-to-show-that-the-sequence-frac-n-2-n-1-2n-2-4n-1-converges-to-frac.htmlGood day, I was wondering how I could prove that $$\lim_{n\to \infty}\frac{n^2+n+1}{2n^2-4n+1}=1/2$$ By definition....Thu, 13 Aug 2026 22:25:35 -0400https://liberty.io/proof-by-induction-cup-n-1-infty-a-n-is-countable-for-countable-a-n.htmlhttps://liberty.io/proof-by-induction-cup-n-1-infty-a-n-is-countable-for-countable-a-n.htmlHere is Theorem 1.5.8 in "Understanding Analysis" by S. Abbott, 2nd edition. Theorem 1.5.8 (i) If $A_1, A_2, ..., A_n$ are each countable sets, then the union $A_1\cup A_2 \cup ... \cup A_n$ is ...Thu, 13 Aug 2026 22:20:53 -0400https://liberty.io/evaluation-of-surface-integral-over-the-unit-disk-and-plane-in-xy-axis.htmlhttps://liberty.io/evaluation-of-surface-integral-over-the-unit-disk-and-plane-in-xy-axis.htmlEvaluation of $$\int_{S}\int(x^2+y^2+z^2)dS$$ Where $S:z=x+y,x^2+y^2\leq 1$ What i try: Let $g(x,y,z)=x^2+y^2+z^2$ and $-1\leq x,y\leq 1$ Let $S(x,y)=<x,y,x+y>$ Then $S_{x}(x,y)=<1,0,1......Thu, 13 Aug 2026 22:11:18 -0400https://liberty.io/question-about-bearings-and-trig.htmlhttps://liberty.io/question-about-bearings-and-trig.htmlA plane travelled for 600 km from airport A, on a bearing of 210°, before changing direction and travelling a further 250 km on a bearing of 300° to land at airport B. Calculate the three-figure be......Thu, 13 Aug 2026 22:08:05 -0400https://liberty.io/finding-the-turning-point-of-a-parabola.htmlhttps://liberty.io/finding-the-turning-point-of-a-parabola.htmlI know how to find the turning point of a parabola in most equations but I'm not sure how to solve it in this form, if anyone can help me please do! $$y=x^2+4x-5$$ Thanks...Thu, 13 Aug 2026 22:04:09 -0400https://liberty.io/what-is-the-difference-between-closed-form-expression-and-analytic-exp.htmlhttps://liberty.io/what-is-the-difference-between-closed-form-expression-and-analytic-exp.htmlWhat is the difference between closed-form expression and analytic expression? I often see them get referenced in settings where (in my opinion) they are essentially interchangeable. What is a ta......Thu, 13 Aug 2026 21:57:28 -0400https://liberty.io/series-expansion-for-e-x-2.htmlhttps://liberty.io/series-expansion-for-e-x-2.htmlI was interested in one of the topics on this website, while I noticed a notation that I haven't been able to understand. This is the original question. One of the answers to the question states th......Thu, 13 Aug 2026 21:52:22 -0400https://liberty.io/how-to-find-a-minimal-polynomial-field-theory.htmlhttps://liberty.io/how-to-find-a-minimal-polynomial-field-theory.htmlI was asked to find a minimal polynomial of $$\alpha = \frac{3\sqrt{5} - 2\sqrt{7} + \sqrt{35}}{1 - \sqrt{5} + \sqrt{7}}$$ over Q. I'm not able to find it without the help of WolframAlpha, which s......Thu, 13 Aug 2026 21:48:23 -0400https://liberty.io/is-a-matrix-a-with-an-eigenvalue-of-0-invertible.htmlhttps://liberty.io/is-a-matrix-a-with-an-eigenvalue-of-0-invertible.htmlJust wanted some input to see if my proof is satisfactory or if it needs some cleaning up. Here is what I have. Proof Suppose $A$ is square matrix and invertible and, for the sake of contradic......Thu, 13 Aug 2026 21:45:56 -0400https://liberty.io/why-is-the-sum-over-all-positive-integers-equal-to-1-12-duplicate.htmlhttps://liberty.io/why-is-the-sum-over-all-positive-integers-equal-to-1-12-duplicate.htmlRecently, sources for mathematical infotainment, for example numberphile, have given some information on the interpretation of divergent series as real numbers, for example $\sum_{i=0}^\infty i = ......Thu, 13 Aug 2026 21:37:51 -0400https://liberty.io/direct-sum-vs-direct-product-vs-tensor-product-closed.htmlhttps://liberty.io/direct-sum-vs-direct-product-vs-tensor-product-closed.htmlThere are a lot of questions like this all over the site, but I cannot find one that resolved my confusion- what are the formal definitions of direct sums, direct products, and tensor products (in ......Thu, 13 Aug 2026 21:35:01 -0400https://liberty.io/mississippi-problem.htmlhttps://liberty.io/mississippi-problem.htmlHow many arrangement of the letters in MISSISSIPPI have at least 2 adjacent S's? I was thinking that I can glue two of the S's together, so there will be 9 letters plus the special letter SS, and ......Thu, 13 Aug 2026 21:33:32 -0400https://liberty.io/cosine-integral.htmlhttps://liberty.io/cosine-integral.htmlShow that $$\int_0^x \frac{1-\cos(t)}{t}=\gamma+\ln(x)-\operatorname{Ci}(x)$$ where $$\operatorname{Ci}(x)=-\int_x^\infty \frac{\cos(t)}{t} \, dt$$ and gamma is an euler-mascheroni constant. I did......Thu, 13 Aug 2026 21:27:14 -0400https://liberty.io/imo-1988-problem-6.htmlhttps://liberty.io/imo-1988-problem-6.htmlIn 1988, IMO presented a problem, to prove that $k$ must be a square if $a^2+b^2=k(1+ab)$, for positive integers $a$, $b$ and $k$. I am wondering about the solutions, not obvious from the proof. B......Thu, 13 Aug 2026 21:25:47 -0400https://liberty.io/using-the-definition-of-derivative-to-find-tan-2x.htmlhttps://liberty.io/using-the-definition-of-derivative-to-find-tan-2x.htmlThe instructions: Use the definition of derivative to find $f'(x)$ if $f(x)=\tan^2(x)$. I've been working on this problem, trying every way I can think of. At first I tried this method: $$\lim_{h......Thu, 13 Aug 2026 21:24:15 -0400https://liberty.io/understanding-kleene-star-in-relation-to-regular-expressions.htmlhttps://liberty.io/understanding-kleene-star-in-relation-to-regular-expressions.htmlI have a homework problem for my intro computation class where I have to write a regular expression for an NFA. In order to help understand if my language is correct- I want to post some example re......Thu, 13 Aug 2026 20:59:25 -0400