https://liberty.ioCurated star highlights with fresh energy.en-usFri, 28 Aug 2026 05:20:42 -0400https://liberty.io/expected-waiting-time-for-bus.htmlhttps://liberty.io/expected-waiting-time-for-bus.htmlSuppose that the inter-arrival time between consecutive buses is 15 minutes with probability 0.5 and 30 minutes with probability 0.5. You just arrived at a bus stop. What is your expected waiting t......Fri, 28 Aug 2026 09:19:59 -0400https://liberty.io/conditions-for-the-mean-value-theorem.htmlhttps://liberty.io/conditions-for-the-mean-value-theorem.htmlthe mean value theorem which most of us know starts with the conditions that $f$ is continuous on the closed interval $[a,b]$ and differentiable on the opened interval $(a,b)$, then there exists a ......Fri, 28 Aug 2026 09:13:34 -0400https://liberty.io/verification-of-proof-for-when-a-number-is-divisible-by-4.htmlhttps://liberty.io/verification-of-proof-for-when-a-number-is-divisible-by-4.htmlI have never taken a number theory course and so am only going off of the first few chapters in an introductory number theory book. The divisibility property I wish to prove is the following: Def......Fri, 28 Aug 2026 09:09:57 -0400https://liberty.io/root-test-applied-to-power-series.htmlhttps://liberty.io/root-test-applied-to-power-series.htmlWhen I apply the root test to power series I know that $$R= \frac{1}{\limsup_{n\rightarrow\infty}{\sqrt[n]{|c_n|}}}$$ So if I get $1$ as the result of the limit the radius of convergence of the p......Fri, 28 Aug 2026 08:57:29 -0400https://liberty.io/subtract-binary-numbers-with-1-s-complement.htmlhttps://liberty.io/subtract-binary-numbers-with-1-s-complement.htmlI'm trying to figure out how to subtract two binary numbers with a complementary one or two, When I need to address carrier and when not? Do I need to solve the problem in decimal numbers? And ...Fri, 28 Aug 2026 08:56:09 -0400https://liberty.io/product-of-banded-matrices.htmlhttps://liberty.io/product-of-banded-matrices.htmlHow can one show that the product of two banded matrices is a banded matrix with upper and lower bandwidths equal to the sum of the upper and lower bandwidths (respectively) of the multiplicands ? ......Fri, 28 Aug 2026 08:52:25 -0400https://liberty.io/norm-of-multiplication-and-multiplication-of-norms.htmlhttps://liberty.io/norm-of-multiplication-and-multiplication-of-norms.htmlIt is well known that $\|u \cdot v \|_2 \le \|u \|_2 \cdot \| v \|_2 $ for all $u, v \in \mathbb{R}^d$. Is the following true for all $p \in \mathbb{R}$: $$\|u \cdot v \|_p \le \|u \|_p \cdo......Fri, 28 Aug 2026 08:51:24 -0400https://liberty.io/difference-between-magnitude-of-gradient-vs-directional-derivative-of-.htmlhttps://liberty.io/difference-between-magnitude-of-gradient-vs-directional-derivative-of-.htmlI've read that the directional derivative is the rate of change of a function $f$ in a given direction $\mathbf{v}$, given as $\nabla f\cdot \mathbf{v}$. I've also read (perhaps incorrectly) that the ...Fri, 28 Aug 2026 08:50:31 -0400https://liberty.io/how-does-the-second-derivative-imply-concavity.htmlhttps://liberty.io/how-does-the-second-derivative-imply-concavity.htmlconsider this equation, $$y=x(400-x)$$the second derivative of this equation is $$y''=-2$$ As far as I know, a negative sign in the second derivative indicates the curve will concave down. As it is a ...Fri, 28 Aug 2026 08:49:33 -0400https://liberty.io/optimal-strategy-for-deal-or-no-deal.htmlhttps://liberty.io/optimal-strategy-for-deal-or-no-deal.htmlWhen I have watched Deal or No Deal (I try not to make a habit of it) I always do little sums in my head to work out if the banker is offering a good deal. Where odds drop below "evens" it's easy t......Fri, 28 Aug 2026 08:46:23 -0400https://liberty.io/finding-the-exact-value-of-tan-pi-5.htmlhttps://liberty.io/finding-the-exact-value-of-tan-pi-5.htmlHi, I realise there has been a question already asked regarding this particular exact value, but this question requires for it to be done under different conditions, which is the part I require hel......Fri, 28 Aug 2026 08:33:23 -0400https://liberty.io/puzzle-123456789-100-with-three-operations.htmlhttps://liberty.io/puzzle-123456789-100-with-three-operations.htmlGiven the sequence 123456789: You can insert three operations ($+$,$-$,$\times$,$/$) into this sequence to make the equation = 100. My question is: is there a way to solve this without brute forc......Fri, 28 Aug 2026 08:30:30 -0400https://liberty.io/actually-playable-games-based-on-graphs.htmlhttps://liberty.io/actually-playable-games-based-on-graphs.htmlIn computer science lessons, we have recently got the task to program something using graphs. Due to my enthusiasm for computer games, i would really prefer to implement a concept for a game. The ...Fri, 28 Aug 2026 08:29:01 -0400https://liberty.io/how-to-show-c-n-k-c-n-n-k.htmlhttps://liberty.io/how-to-show-c-n-k-c-n-n-k.htmlI am doing a question from the textbook "Calculus and Statistics" by Gemigani. If $n = 2m$ and $k= 1,2, \ldots ,m$ Prove that $C(n,k)= C(n,n-k)$ Ok so my approach begins with writing out the for......Fri, 28 Aug 2026 08:21:55 -0400https://liberty.io/find-a-second-degree-polynomial-that-goes-through-3-points.htmlhttps://liberty.io/find-a-second-degree-polynomial-that-goes-through-3-points.htmlI am having trouble calculating the quadratic curve $f(x)$ that goes through 3 points; for example a curve that goes through $A(1,3), B(-1,-5), and C(-2,12)$. I can only guess that the curve is upw......Fri, 28 Aug 2026 08:17:37 -0400https://liberty.io/why-is-there-no-online-record-of-pentagram-area-formula-closed.htmlhttps://liberty.io/why-is-there-no-online-record-of-pentagram-area-formula-closed.htmlI'm talking about the short version of the area of a pentagram inscribed in a circle when the radius is given and so Area $=1.123r^2$. I only got this formula from a book but I researched about it......Fri, 28 Aug 2026 08:14:30 -0400https://liberty.io/how-to-find-the-roots-of-x-4-1.htmlhttps://liberty.io/how-to-find-the-roots-of-x-4-1.htmlI'm trying to find the roots of $x^4+1$. I've already found in this site solutions for polynomials like this $x^n+a$, where $a$ is a negative term. I don't remember how to solve an equation when $a......Fri, 28 Aug 2026 08:14:18 -0400https://liberty.io/differential-geometry-and-transition-maps-as-communative-diagrams.htmlhttps://liberty.io/differential-geometry-and-transition-maps-as-communative-diagrams.htmlConsider the picture below. I know pictures are usually looked down upon on this site but I have no experience with TikZ. Anyway, I am learning about regular surfaces and transition maps in differ......Fri, 28 Aug 2026 08:10:19 -0400https://liberty.io/the-derivative-of-log-2-n-5.htmlhttps://liberty.io/the-derivative-of-log-2-n-5.htmlThe derivative of $ (\log_2 n)^5$? (log base 2) Hey everyone, I am not sure how to go about this question because I am not sure what to do with the power $5$ ( or any other power ) in the log? Shou......Fri, 28 Aug 2026 08:07:22 -0400https://liberty.io/finding-all-orthogonal-matrices-of-a-given-form.htmlhttps://liberty.io/finding-all-orthogonal-matrices-of-a-given-form.htmlI am given this problem. I need to find all $a,b,c,d \in \mathbb{R}$ for which the matrix $A$ is orthogonal. $$A:=\frac{1}{3} \begin{pmatrix} -1 & 2 & a \\ 2 & 2 & b \\ 2 & c ......Fri, 28 Aug 2026 08:03:27 -0400https://liberty.io/lusin-039-s-theorem-proof-verification-folland-chap-2-exercise-44.htmlhttps://liberty.io/lusin-039-s-theorem-proof-verification-folland-chap-2-exercise-44.htmlConisder Lusin's Theorem (Folland Chap 2 Ex 44): If $f: [a,b] \rightarrow \mathbb{C}$ is Lebesgue measurable and $\epsilon > 0$, there is a compact subset $E \in [a,b]$ s.t. $\mu(E^C) < \...Fri, 28 Aug 2026 08:00:38 -0400https://liberty.io/can-i-break-up-log-a-b.htmlhttps://liberty.io/can-i-break-up-log-a-b.htmlFor constants $a$ and $b$, I know that I can break up $\log(a/b)$ into $\log(a) - \log(b)$. Can I conveniently break up $\log(a - b)$ somehow into several terms?...Fri, 28 Aug 2026 07:52:45 -0400https://liberty.io/how-to-find-whether-equation-of-angle-bisector-represents-the-obtuse-o.htmlhttps://liberty.io/how-to-find-whether-equation-of-angle-bisector-represents-the-obtuse-o.htmlTwo lines: $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$ are given. I know that the equation of its bisectors is ${a_1x + b_1y + c_1 \over \sqrt{(a_1^2 + b_1^2)}} = \pm {a_2x + b_2y + c_2 \over\...Fri, 28 Aug 2026 07:48:41 -0400https://liberty.io/chance-on-throwing-a-six-with-6-dice.htmlhttps://liberty.io/chance-on-throwing-a-six-with-6-dice.htmlThe chance to throw a 6 with one die is 1/6 And 6 times 1/6 = 1 So, if I throw with 6 dice, the chance to throw at least 1 six should be 1. But when I throw 6 dice, I sometimes don't throw any 6......Fri, 28 Aug 2026 07:47:09 -0400https://liberty.io/why-is-gradient-the-direction-of-steepest-ascent.htmlhttps://liberty.io/why-is-gradient-the-direction-of-steepest-ascent.html$$f(x_1,x_2,\dots, x_n):\mathbb{R}^n \to \mathbb{R}$$ The definition of the gradient is $$ \frac{\partial f}{\partial x_1}\hat{e}_1 +\ \cdots +\frac{\partial f}{\partial x_n}\hat{e}_n$$ which is a ......Fri, 28 Aug 2026 07:42:25 -0400https://liberty.io/sequence-series-convergence-of-2-2n-3-1-n.htmlhttps://liberty.io/sequence-series-convergence-of-2-2n-3-1-n.htmlthis step in the proof is confusing me: $$\sum_1^\infty {\frac{4^{n}}{3^{n-1}}}\qquad \longrightarrow \qquad\sum_1^\infty 4\left(\frac43\right)^{n-1}$$ please explain how/why this happened? che......Fri, 28 Aug 2026 07:40:14 -0400https://liberty.io/vertex-and-edge-connectivity-of-a-graph.htmlhttps://liberty.io/vertex-and-edge-connectivity-of-a-graph.htmlI want to draw a graph of 8 vertices and 16 edges with maximum vertex connectivity and maximum edge connectivity and also draw a graph with minimum vertex connectivity and minimum edge ...Fri, 28 Aug 2026 07:40:00 -0400https://liberty.io/general-formula-for-a-diagonal-parabola.htmlhttps://liberty.io/general-formula-for-a-diagonal-parabola.htmlWhat is the general formula for a diagonal parabola facing a given direction with a given vertex (x,y)?...Fri, 28 Aug 2026 07:33:18 -0400https://liberty.io/natural-log-of-pi-to-sixty-digits-a-reference-needed.htmlhttps://liberty.io/natural-log-of-pi-to-sixty-digits-a-reference-needed.htmlI have found a table of the logs of gamma functions at basic fractions, accurate to 60 decimal digits. It omits $\ln(\Gamma(1/2)) = \ln(\pi)/2$ and I want that number. Where is a cit-able sourc......Fri, 28 Aug 2026 07:29:49 -0400https://liberty.io/function-with-an-asymptote-at-y-1-and-y-1-closed.htmlhttps://liberty.io/function-with-an-asymptote-at-y-1-and-y-1-closed.htmlI'm looking for a function that has two asymptotes parallel to the x-axis. Preferably it should also only cross the x-axis at (0,0) and be built without using any trigonometric functions. Mind you,......Fri, 28 Aug 2026 07:26:38 -0400https://liberty.io/calculating-linear-velocity.htmlhttps://liberty.io/calculating-linear-velocity.htmlI have the question Use the answer to question 1 to calculate the linear velocity of a point on the surface of the Earth (radius of Earth = $6.4 \times 10^6$ [m]). So the answer for question 1 ......Fri, 28 Aug 2026 07:25:03 -0400https://liberty.io/normalization-condition-and-phase-constant.htmlhttps://liberty.io/normalization-condition-and-phase-constant.htmlI have been given a wave function and been tasked with verifying it has been normalized. $$\psi(x, 0) = \left(\frac{2\alpha}{\pi}\right)^{1/4} e^{-ikx} e^{-\alpha x^2}$$. I've normalized......Fri, 28 Aug 2026 07:24:16 -0400https://liberty.io/extended-3-square-problem.htmlhttps://liberty.io/extended-3-square-problem.htmlWe know about the famous 3-square problem. Draw three adjacent squares with same size and any two adjacent square has a common side. This looks like a rectangle. Name all of the vertices of squ......Fri, 28 Aug 2026 07:21:56 -0400https://liberty.io/negation-of-the-definition-of-limit-of-a-function.htmlhttps://liberty.io/negation-of-the-definition-of-limit-of-a-function.htmlQuestion Suppose we are dealing with real valued functions $f(x)$ of one real variable $x$. We say that the limit of $f(x)$ at point $a$ exists and equals to $L$ if and only if $$\exists L:\lef......Fri, 28 Aug 2026 07:11:44 -0400https://liberty.io/analyticity-and-removable-singularity.htmlhttps://liberty.io/analyticity-and-removable-singularity.htmlIn Churchill's book of Complex Analysis there are two statements that I can't match them to be consistent: In one place it says that a function must be analytic at a removable singular point : why......Fri, 28 Aug 2026 07:09:32 -0400https://liberty.io/how-should-i-study-the-matrix-cookbook.htmlhttps://liberty.io/how-should-i-study-the-matrix-cookbook.htmlI use The Matrix Cookbook by Kaare Brandt Petersen and Michael Syskind Pedersen to solve many problems (mostly matrix derivatives). In most cases, I just map the problem to one of the formula and s......Fri, 28 Aug 2026 06:53:25 -0400https://liberty.io/area-of-a-circular-segment.htmlhttps://liberty.io/area-of-a-circular-segment.htmlSee the picture below: How can I calculate the area in black, using no handy formulas which will give me the answer if I plug in the right values? I had the idea to take $\displaystyle \int_{0.5r}......Fri, 28 Aug 2026 06:52:00 -0400https://liberty.io/proof-of-the-deduction-theorem-explanation.htmlhttps://liberty.io/proof-of-the-deduction-theorem-explanation.htmlI'm reading through this proof of the deduction theorem, and there are a few things I don't understand. The basic idea is to show that if $\Gamma\cup \{A\}\vdash B$, then we have a proof of $B$ wi......Fri, 28 Aug 2026 06:33:19 -0400https://liberty.io/how-are-c-0-c-1-norms-defined.htmlhttps://liberty.io/how-are-c-0-c-1-norms-defined.htmlHow are $C^0,C^1$ norms defined? I know $L_p,L_\infty$ norms but are the former defined....Fri, 28 Aug 2026 06:26:24 -0400https://liberty.io/can-you-disprove-this-rule-pedmsa-division-before-multiplication-subtr.htmlhttps://liberty.io/can-you-disprove-this-rule-pedmsa-division-before-multiplication-subtr.htmlI know the proper teaching of PEMDAS/PEDMAS/BODMAS/BOMDAS.. is that multiplication and division have equal priority and are evaluated going through the arithmetic expression left to right (So, doing ...Fri, 28 Aug 2026 06:25:58 -0400https://liberty.io/what-is-approximation-theory.htmlhttps://liberty.io/what-is-approximation-theory.htmlWhat exactly is "Approximation Theory"? If I read the wikipedia-article I doesn't get much clearer. Why are "pure" mathematicians interested in it? I see a lot of people that do harmonic analysis a......Fri, 28 Aug 2026 06:23:41 -0400https://liberty.io/what-is-an-imaginary-line.htmlhttps://liberty.io/what-is-an-imaginary-line.htmlAn imaginary point is defined as an ordered pair of values, at least one of which is complex. My text says that if $h^2 < ab$ in the equation : $$ ax^2 + 2hxy + by^2 $$ it represents two imagi......Fri, 28 Aug 2026 06:22:13 -0400https://liberty.io/nice-examples-of-groups-which-are-not-obviously-groups.htmlhttps://liberty.io/nice-examples-of-groups-which-are-not-obviously-groups.htmlI am searching for some groups, where it is not so obvious that they are groups. In the lecture's script there are only examples like $\mathbb{Z}$ under addition and other things like that. I don't ...Fri, 28 Aug 2026 06:11:50 -0400https://liberty.io/formula-for-continuously-dividing-by-2.htmlhttps://liberty.io/formula-for-continuously-dividing-by-2.htmlI was helping my daughter with her physics homework. The problem had a bee flying back and forth between two bikes moving toward each other. The question was to see how far the bee traveled. Turned......Fri, 28 Aug 2026 06:02:34 -0400https://liberty.io/dividing-matrices.htmlhttps://liberty.io/dividing-matrices.htmlHI Im learning about matrices in school and am curious about how to divide them so i can find out different identity matrices for refecting shapes over linear functions...Fri, 28 Aug 2026 05:55:20 -0400https://liberty.io/prove-that-a-x-b-y-0-where-x-sqrt-log-a-b-y-sqrt-log-b-a-a-0-b-0-a-b-n.htmlhttps://liberty.io/prove-that-a-x-b-y-0-where-x-sqrt-log-a-b-y-sqrt-log-b-a-a-0-b-0-a-b-n.htmlI'm solving logarithm questions. I got stuck in this question. Prove that $a^x - b^y = 0$ where $x = \sqrt{\log_a b}$ & $y = \sqrt{\log_b a}$ , $a > 0$, $b > 0$ & $a, b \ne1$ I've ......Fri, 28 Aug 2026 05:47:28 -0400https://liberty.io/eigenvalues-of-product-matrix.htmlhttps://liberty.io/eigenvalues-of-product-matrix.htmlI have two matrices, both positive definite, real symmetric and one is diagonal. What can I say about lower and upper bound of the eigenvalues of the product matrix in terms the of lower and upper ......Fri, 28 Aug 2026 05:46:21 -0400https://liberty.io/what-is-the-purpose-of-the-limit.htmlhttps://liberty.io/what-is-the-purpose-of-the-limit.htmlI haven't taken Calculas yet, but I see the use of limit (approaching zero or infinity) in other classes such as physics. I just wanted some intuitive explanation of the limit. I was thinking t......Fri, 28 Aug 2026 05:39:00 -0400https://liberty.io/prove-that-two-equations-are-equivalent.htmlhttps://liberty.io/prove-that-two-equations-are-equivalent.htmlEDIT: Missed something very important! Sorry! We have $x^4+1=2(2x-1)^{1/4}$ not $x^4+1=2\sqrt{2x-1}$. One friend of mine told me that the equation $x^4+1=2(2x-1)^{1/4}$, where $x\geq \frac{1}{2}$ is ...Fri, 28 Aug 2026 05:34:59 -0400https://liberty.io/finding-d-in-rsa.htmlhttps://liberty.io/finding-d-in-rsa.htmlSuppose your RSA modulus is $55 = 5 * 11$ and your encryption exponent is $e = 3$. Find the decryption modulus d. I know $d = 40-13 = 27$ However, I get $1$. $$40 = (P_1-1)(P_2-1)$$ extended ...Fri, 28 Aug 2026 05:33:27 -0400https://liberty.io/cartesian-product-with-three-sets.htmlhttps://liberty.io/cartesian-product-with-three-sets.htmlI was given a question that asks "Let A={1,2} ,B={x,3,y},C={2,y}. Find A x B x C and C x B x C " I have an example that I could go by but i'm not sure what they were multiplying together to get it.......Fri, 28 Aug 2026 05:28:43 -0400https://liberty.io/integration-with-respect-to-dx-dy-and-dz-more-than-one-variable.htmlhttps://liberty.io/integration-with-respect-to-dx-dy-and-dz-more-than-one-variable.htmlSorry if my title was vague but i was not entirely sure what its called. Anyways i was solving some work and energy problems and encountered this integration: $$\int_{2,1,4}^{2,-3,3} 2x\sin^2y \......Fri, 28 Aug 2026 05:22:33 -0400https://liberty.io/numbers-in-a-list-which-are-perfect-squares-and-perfect-cubes-of-numbe.htmlhttps://liberty.io/numbers-in-a-list-which-are-perfect-squares-and-perfect-cubes-of-numbe.htmlHow many numbers in the list $$1,2,3,...,2001$$ are perfect squares and perfect cubes of whole numbers? My progress: Well I do know $$1,4,9,16,25,36,...$$ are perfect squares and $$1,8,27,64,...$$......Fri, 28 Aug 2026 05:14:59 -0400https://liberty.io/how-do-i-find-the-value-of-a-partial-sum-without-calculator.htmlhttps://liberty.io/how-do-i-find-the-value-of-a-partial-sum-without-calculator.htmlWithout using a calculator, how do I find: $$\sum_{k=0}^7 \left(\tfrac{2}{3}\right)^k $$ I know about the formula for the sum of a geometric series $\frac{a\cdot(1-r^n)}{(1-r)}$ but im not sure h......Fri, 28 Aug 2026 04:53:57 -0400https://liberty.io/does-anyone-know-a-simple-proof-for-the-fundamental-theorem-of-finitel.htmlhttps://liberty.io/does-anyone-know-a-simple-proof-for-the-fundamental-theorem-of-finitel.htmlI am looking for a simple/short proof for the fundamental theorem of finitely generated abelian group. Let $(G,\ +)$ be an abelian group. Assume $G$ is finitely generated. Prove that there exist ......Fri, 28 Aug 2026 04:25:46 -0400https://liberty.io/wedge-sum-example.htmlhttps://liberty.io/wedge-sum-example.htmlI am trying to understand the concept of wedge sum in the context of Algebraic Topology. Say I am given two sets, $A=[1, 2, 3]$ and $B=[4, 5]$ Now, disjoint union $A \sqcup B$ = $[(1,0), (2,0), (......Fri, 28 Aug 2026 04:23:11 -0400https://liberty.io/using-fsolve-to-solve-non-linear-equations-in-matlab.htmlhttps://liberty.io/using-fsolve-to-solve-non-linear-equations-in-matlab.htmlI'm trying to solve this system equations in matlab syms y z i x=[29.6,29.4,34.4,35.1,34.3,36.1,31.6,32.9,31.5,28.3,37.1,28.4,27.3,33.1,31.3,33.1] n = 16 eqns = sym(zeros(1, 2 * n)); for i = 1:n......Fri, 28 Aug 2026 04:21:10 -0400https://liberty.io/is-this-the-correct-notation.htmlhttps://liberty.io/is-this-the-correct-notation.html$\left [ \left \{ 1,2,3,4 \right \}! \right]^{-1}$Is this the correct notation if one wanted to obtain the factorial for each number in a sequence and then take the sequence and inverse each number......Fri, 28 Aug 2026 04:20:49 -0400https://liberty.io/why-does-the-sum-of-inverse-squares-equal-pi-2-6-duplicate.htmlhttps://liberty.io/why-does-the-sum-of-inverse-squares-equal-pi-2-6-duplicate.htmlI've heard that $1+1/4+1/9+1/16+1/25+...$ converges to $\pi^2/6$. This was very surprising to me and I was wondering if there was a reason that it converges to this number? I am also confused why......Fri, 28 Aug 2026 04:00:35 -0400https://liberty.io/inverse-of-a-step-function.htmlhttps://liberty.io/inverse-of-a-step-function.htmlHow can I write the function below $$f(x)=\left\{\begin{array}{ll} 1, & 0\leq t\leq 1, \\ 0, & t>1 \end{array}\right.$$ using the unit step function? I mean, I don't know how could I wr......Fri, 28 Aug 2026 03:58:29 -0400https://liberty.io/how-to-calculate-recurring-probability.htmlhttps://liberty.io/how-to-calculate-recurring-probability.htmlI have a very basic understanding of statistics and probability. I just passed an intro college course on it. The problem is, there's one type of problem that I'd like to know how to do, and I don'......Fri, 28 Aug 2026 03:41:12 -0400https://liberty.io/how-do-the-sets-emptyset-times-b-a-times-emptyset-emptyset-times-empty.htmlhttps://liberty.io/how-do-the-sets-emptyset-times-b-a-times-emptyset-emptyset-times-empty.htmlIf we have a function $f:A \rightarrow B$, then one way to give meaning, I think, to this function, in terms of set theory, is to say, that $f$ is actually a binary relation $f=(A,B,G_f)$, where $G......Fri, 28 Aug 2026 03:30:25 -0400https://liberty.io/find-the-12th-term-and-the-sum-of-the-first-12-terms-of-a-geometric-se.htmlhttps://liberty.io/find-the-12th-term-and-the-sum-of-the-first-12-terms-of-a-geometric-se.htmlA geometric series has a first term $\sqrt{2}$ and a second term $\sqrt{6}$ . Find the 12th term and the sum of the first 12 terms. I can get to the answers as irrational numbers using a calculato......Fri, 28 Aug 2026 03:27:47 -0400https://liberty.io/intuition-for-dense-sets-real-analysis.htmlhttps://liberty.io/intuition-for-dense-sets-real-analysis.htmlI have been having problems with dense sets as my lecturer didn't really develop an intuition for dense sets in my class. So can any of you please help me with that? And can you please tell me (the ...Fri, 28 Aug 2026 03:25:35 -0400https://liberty.io/finite-and-infinite-speed-of-propagation-for-wave-and-heat-equation.htmlhttps://liberty.io/finite-and-infinite-speed-of-propagation-for-wave-and-heat-equation.htmlWhat is the formal definition of Finite and infinite speed of propagation? I have searched for it, is the finite one means the solution is only determined by a bounded region? Also I do not under......Fri, 28 Aug 2026 03:13:44 -0400https://liberty.io/why-are-direct-proofs-often-considered-better-than-indirect-proofs-clo.htmlhttps://liberty.io/why-are-direct-proofs-often-considered-better-than-indirect-proofs-clo.htmlAs the title indicates, I'm curious why direct proofs are often more preferable than indirect proofs. I can see the appeal of a direct proof, for it often provides more insight into why and how the ...Fri, 28 Aug 2026 03:12:37 -0400https://liberty.io/finding-all-complex-z-that-satisfy-z-4-4z-8-0.htmlhttps://liberty.io/finding-all-complex-z-that-satisfy-z-4-4z-8-0.htmlI was given this task: $$z^4+4z+8=0$$ And I am totally stuck at solving this. The university handed out some solutions that are the following:$$z=−2±2i$$ And well... plugging this into the funct......Fri, 28 Aug 2026 03:03:55 -0400https://liberty.io/what-is-the-most-unusual-proof-you-know-that-sqrt-2-is-irrational.htmlhttps://liberty.io/what-is-the-most-unusual-proof-you-know-that-sqrt-2-is-irrational.htmlWhat is the most unusual proof you know that $\sqrt{2}$ is irrational? Here is my favorite: Theorem: $\sqrt{2}$ is irrational. Proof: $3^2-2\cdot 2^2 = 1$. (That's it) That is a corol......Fri, 28 Aug 2026 03:03:45 -0400https://liberty.io/solve-the-equation-x-2-frac-9x-2-x-3-2-27.htmlhttps://liberty.io/solve-the-equation-x-2-frac-9x-2-x-3-2-27.htmlProblem Statement:- Solve the equation $$x^2+\dfrac{9x^2}{(x+3)^2}=27$$ I have tried to turn it into a quadratic equation so as to be saved from solving a quartic equation, but have not been able to ...Fri, 28 Aug 2026 02:55:04 -0400https://liberty.io/4-x-4-x-1-24-what-is-the-value-of-2x-x.htmlhttps://liberty.io/4-x-4-x-1-24-what-is-the-value-of-2x-x.htmlSo, every week our teacher gives us a very difficult question worth 1 point and I can never get them right, so I would highly appreciate the person who tells and explains, in good detail, why this ...Fri, 28 Aug 2026 02:52:27 -0400https://liberty.io/logarithmic-equation-solve-for-x.htmlhttps://liberty.io/logarithmic-equation-solve-for-x.html$$\log_{3x}81 = 2$$ How would I go about solving this? This is what I tried: $$\log_{3x}81 = 2$$ $$\frac{\log81}{\log 3 + \log x }= 2$$ Where do I go from here? If I isolate $\log x$ on one si......Fri, 28 Aug 2026 02:44:48 -0400https://liberty.io/how-can-a-solid-with-semi-circular-cross-sections-be-perpendicular-to-.htmlhttps://liberty.io/how-can-a-solid-with-semi-circular-cross-sections-be-perpendicular-to-.htmlLet S be the solid with flat base, whose base is the region in the xy plane defined by the curves $$y=e^x$$,$$y=−3$$,$$x=0$$ and $$x=1$$, and whose cross-sections perpendicular to the x axis are semi-...Fri, 28 Aug 2026 02:27:33 -0400https://liberty.io/simplest-known-solution-to-world-039-s-hardest-easy-geometry-problem.htmlhttps://liberty.io/simplest-known-solution-to-world-039-s-hardest-easy-geometry-problem.htmlI've seen multiple solutions to this well-known problem (posed, among other sites, at http://thinkzone.wlonk.com/MathFun/Triangle.htm). None of the solutions are particularly simple. I was wonderin......Fri, 28 Aug 2026 02:21:49 -0400https://liberty.io/how-to-show-that-pi-pi-is-not-a-integer.htmlhttps://liberty.io/how-to-show-that-pi-pi-is-not-a-integer.htmlThis goes without saying, but, I can't use a calculator to evaluate $\pi^\pi$. I think we need to find a integer $x$ such that $$x<\pi^\pi < x+1. \tag{1}$$ However, since I have no ideia wha......Fri, 28 Aug 2026 02:19:37 -0400https://liberty.io/proving-cos-x-2-sin-x-2-1.htmlhttps://liberty.io/proving-cos-x-2-sin-x-2-1.htmlI need to prove that $\cos(x)^2+\sin(x)^2=1$ Here's how I started (using the Cauchy product): \begin{align} \cos(x)^2+\sin(x)^2 &=\sum_{k=0}^{\infty}\sum_{l=0}^k(-1)^l\frac{x^{2l}}{(2l)!}(-1)......Fri, 28 Aug 2026 02:09:52 -0400https://liberty.io/converting-the-equation-of-the-line-to-the-vector-and-symmetric-forms-.htmlhttps://liberty.io/converting-the-equation-of-the-line-to-the-vector-and-symmetric-forms-.htmlDoes anyone know how to convert the equation of the line to the vector and symmetric forms: $x=-1-2t$ $y=1+3t$ $z=2+t$...Fri, 28 Aug 2026 02:04:10 -0400https://liberty.io/is-the-number-8-special-in-turning-a-sphere-inside-out.htmlhttps://liberty.io/is-the-number-8-special-in-turning-a-sphere-inside-out.htmlSo after watching the famous video on youtube How to turn a sphere inside out I noticed that the sphere is deformed into 8 bulges in the process. Is there something special about the number 8 here?......Fri, 28 Aug 2026 02:01:58 -0400https://liberty.io/finding-the-equation-of-the-tangent-in-slope-intercept-form-at-a-parti.htmlhttps://liberty.io/finding-the-equation-of-the-tangent-in-slope-intercept-form-at-a-parti.htmlOne of the practice problems in my Calculus book is as follows: The graph of y=$8/(x^2-4)$ is called the Witch of Agnesi. (a) Find y' d/dx (u/v) = (v du/dx - u dv/dx)/($v^2$) $=((x^2+4)(0)-(8)(......Fri, 28 Aug 2026 02:01:41 -0400https://liberty.io/confusion-about-expectation-moment-generating-function-distributation.htmlhttps://liberty.io/confusion-about-expectation-moment-generating-function-distributation.html(I am currently studying a high dimensional probability course with very little background knowledge in probability theory as a whole, so I hope it's not annoying that I seem to be oblivious about ......Fri, 28 Aug 2026 02:00:43 -0400https://liberty.io/related-rates-of-change-cylinder-question.htmlhttps://liberty.io/related-rates-of-change-cylinder-question.htmlA cylindrical tank with radius 5 cm is being filled with water at rate of 3 cm^3 per min. how fast is the height of the water increasing? I dont want this question solved, but please help me corre......Fri, 28 Aug 2026 01:56:19 -0400https://liberty.io/coordinates-rotation-by-120-degree.htmlhttps://liberty.io/coordinates-rotation-by-120-degree.htmlIf I have a point on a standard grid with coordinates say: $A_1=(1000,0)$ $A_2=(707,707)$ Is there a easy way to transfer this points to $\pm 120$ degrees from the origin $(0,0)$, and kee......Fri, 28 Aug 2026 01:53:19 -0400https://liberty.io/difference-between-simplicial-and-singular-homology.htmlhttps://liberty.io/difference-between-simplicial-and-singular-homology.htmlI am having some difficulties understanding the difference between simplicial and singular homology. I am aware of the fact that they are isomorphic, i.e. the homology groups are in fact the same (......Fri, 28 Aug 2026 01:43:28 -0400https://liberty.io/relationship-between-simple-interest-and-arithmetic-progression.htmlhttps://liberty.io/relationship-between-simple-interest-and-arithmetic-progression.htmlDescribe the relationship between the mathematics of calculating simple interest, and the mathematics behind analyzing arithmetic sequences. How might simple interest be described as a problem of a......Fri, 28 Aug 2026 01:36:16 -0400https://liberty.io/definition-of-multidimensional-wiener-process.htmlhttps://liberty.io/definition-of-multidimensional-wiener-process.htmlI've been reading through Evans' excellent Introduction to Stochastic Differential Equations (used to be lecture notes available on his website a few years back, and became this book), and got some......Fri, 28 Aug 2026 01:30:10 -0400https://liberty.io/what-is-the-latest-verified-research-on-the-3x-1-problem-closed.htmlhttps://liberty.io/what-is-the-latest-verified-research-on-the-3x-1-problem-closed.htmlWikipedia : Collatz Conjecture Take any positive integer n. If n is even, divide it by $2$ to get $n / 2$. If n is odd, multiply it by $3$ and add $1$ to obtain $3n + 1$. Repeat the process (which......Fri, 28 Aug 2026 01:25:29 -0400https://liberty.io/understanding-the-determinant-of-an-infinite-matrix.htmlhttps://liberty.io/understanding-the-determinant-of-an-infinite-matrix.htmlThe fredholm determinant of an infinite matrix $A$ is defined by $$ \det(I+A) = \exp(\text{tr}(\log(I+A))). $$ I'm trying to understand the formula and how to use it. To be concrete here are my ...Fri, 28 Aug 2026 01:24:19 -0400https://liberty.io/what-are-differences-between-affine-space-and-vector-space.htmlhttps://liberty.io/what-are-differences-between-affine-space-and-vector-space.htmlI know smilar questions have been asked and I have looked at them but none of them seems to have satisfactory answer. I am reading the book a course in mathematics for student of physics vol. 1 by ......Fri, 28 Aug 2026 01:16:49 -0400https://liberty.io/solving-the-equation-ln-x-x.htmlhttps://liberty.io/solving-the-equation-ln-x-x.htmlI tried solving this equation for a long time but did not succeed. Any help is appreciated. $$\ln x=-x$$ I am not sure the tag is correct, I am not familiar with English mathematical terms. Please ...Fri, 28 Aug 2026 01:15:39 -0400https://liberty.io/most-general-solution-for-ode.htmlhttps://liberty.io/most-general-solution-for-ode.htmlI am facing some difficulties understanding the difference between a general solution and the most general solution of a 2nd order ODE for example for homogeneous constant coefficients case with 2 ......Fri, 28 Aug 2026 01:09:54 -0400https://liberty.io/need-help-determining-the-pairs-of-quaternions-that-anticommute.htmlhttps://liberty.io/need-help-determining-the-pairs-of-quaternions-that-anticommute.htmlI tried to solve another exercise and I would be grateful if someone could tell me if my answer is right. This is the exercise: Characterize the pairs $p,q \in \mathbb H$ such that $pq = -qp$. ......Fri, 28 Aug 2026 01:09:19 -0400https://liberty.io/deriving-the-rayleigh-distribution-from-the-gaussian.htmlhttps://liberty.io/deriving-the-rayleigh-distribution-from-the-gaussian.htmlTake a complex number $z=u+iv$ and call its magnitude $x$. Consider the real and imaginary parts to be Gaussian distributed: $$ f_{U}(u)=\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp\left(\frac{-u^{2}}{2\s......Fri, 28 Aug 2026 01:03:46 -0400https://liberty.io/solving-modulus-equation-systems.htmlhttps://liberty.io/solving-modulus-equation-systems.htmlI am studying for a test in discrete math and I created my own question but I cannot seem to solve it. Is it possible to solve the following equation system (without brainless testing), and if so......Fri, 28 Aug 2026 01:00:53 -0400https://liberty.io/why-does-0-75-times-1-25-equal-0-93-and-not-1.htmlhttps://liberty.io/why-does-0-75-times-1-25-equal-0-93-and-not-1.htmlIf percentage points can equal decimal points, then $1.25$ should equal $125$% when multiplying $1.25$ by any number. Therefore $125$% of $0.75$ should equal $1$ if $0.75$ is also $75$% of $1$. ......Fri, 28 Aug 2026 01:00:35 -0400https://liberty.io/find-the-point-of-intersection-of-the-straight-line-frac-x-1-4-frac-y-.htmlhttps://liberty.io/find-the-point-of-intersection-of-the-straight-line-frac-x-1-4-frac-y-.htmlFind the point of intersection of the straight line $$\frac{X+1}{4}=\frac{Y-2}{-2}=\frac{Z+6}{7}$$ and plane $3X+8Y-9Z=0$ the point of the line is $M(-1,2,-6)$ and direction vector of the line is......Fri, 28 Aug 2026 00:52:23 -0400https://liberty.io/find-the-value-of-c-such-that-the-system-has-a-solution-other-than-0-0.htmlhttps://liberty.io/find-the-value-of-c-such-that-the-system-has-a-solution-other-than-0-0.htmlI am asked to the find the value $c$ such that the system has a solution other than $(0, 0, 0)$. This is the linear system: \begin{array}{rcrcrcl} cx & & & + & 4z& = &am......Fri, 28 Aug 2026 00:46:37 -0400https://liberty.io/sard-039-s-theorem-proof-detail-question-regarding-partial-derivatives.htmlhttps://liberty.io/sard-039-s-theorem-proof-detail-question-regarding-partial-derivatives.htmlSard's Theorem: Suppose $M$ and $N$ are smooth manifolds with or without boundary and $F:M\to N$ is a smooth map. Then the set of critical values of $F$ has measure zero in $N$. In what appears to......Fri, 28 Aug 2026 00:37:41 -0400https://liberty.io/intuition-on-the-curl-formula.htmlhttps://liberty.io/intuition-on-the-curl-formula.htmlI'm trying to understand the formula for the curl: $$\operatorname{Curl}(F) = \nabla \times F $$ Why the vector product? What means the vector product of a vector and a operator? What is the mean......Fri, 28 Aug 2026 00:34:49 -0400https://liberty.io/equation-for-x-axis-3d-paraboloid.htmlhttps://liberty.io/equation-for-x-axis-3d-paraboloid.htmlI've been playing around with plenty of variants of paraboloid equations. However I couldn't come up with the equation for a x-axis parallel 3D paraboloid of revolution. For a 2D parabola the equ......Fri, 28 Aug 2026 00:34:02 -0400https://liberty.io/given-log2-a-log7-b-find-log-56.htmlhttps://liberty.io/given-log2-a-log7-b-find-log-56.htmlI don't know how to solve this. Can someone help me? How do I use the information above to help me find Log 56?...Fri, 28 Aug 2026 00:27:38 -0400https://liberty.io/are-there-infinitely-many-pythagorean-triples.htmlhttps://liberty.io/are-there-infinitely-many-pythagorean-triples.htmlI believe these questions are all asking different things, but: Are there infinitely many (integer) solutions to the pythagorean theorem? Is every positive integer part of a solution to the pytha......Fri, 28 Aug 2026 00:20:50 -0400