https://liberty.ioCurated star highlights with fresh energy.en-usFri, 14 Aug 2026 11:31:42 -0400https://liberty.io/discriminant-of-elliptic-curves.htmlhttps://liberty.io/discriminant-of-elliptic-curves.htmlIn the study of elliptic curves, specifically in Weierstrass form, you have the equation $E : y^2 = x^3 +ax +b$. However I have found the discriminant comes in two different forms: $\Delta = ......Fri, 14 Aug 2026 15:29:58 -0400https://liberty.io/two-independent-events-a-and-b-find-b-given-only-a-and-a-cup-b.htmlhttps://liberty.io/two-independent-events-a-and-b-find-b-given-only-a-and-a-cup-b.htmlSorry to be a burden but my maths exam is tomorrow and I have no idea how to do this at all. Here is the exact question : The two independent events $A$ and $B$ are such that $P(A) = 0.2$, $P(A\cu......Fri, 14 Aug 2026 15:25:23 -0400https://liberty.io/find-the-3-angles-of-the-hyperbolic-triangle.htmlhttps://liberty.io/find-the-3-angles-of-the-hyperbolic-triangle.htmlA(0,5) B(0,2) C(4,2) In Euclidean geometry the three points given are the vertices of a right-angled triangle. Find the three angles of the hyperbolic triangle with vertices A,B,C. Find the hyperb......Fri, 14 Aug 2026 15:22:41 -0400https://liberty.io/definition-of-minor-for-a-general-m-times-n-matrix.htmlhttps://liberty.io/definition-of-minor-for-a-general-m-times-n-matrix.htmlFollowing is the definition of minor given in the text : For a matrix $A$, consider its $k × k$ submatrix, obtained by taking $k$ rows and $k$ columns. The determinant of this matrix is calle......Fri, 14 Aug 2026 15:20:51 -0400https://liberty.io/what-is-a-null-set.htmlhttps://liberty.io/what-is-a-null-set.htmlI am very confused with null sets. I get that a set which has no elements will be called a null set but I am not getting the examples given below. Please help me by explaining how $P,Q,R$ are all......Fri, 14 Aug 2026 15:15:26 -0400https://liberty.io/absolute-value-graph-sketching-x-1-1.htmlhttps://liberty.io/absolute-value-graph-sketching-x-1-1.htmlWhere would you start if you were told to plot: $$||x-1|-1|$$ I looked at just $f(x) = |x-1|$ and noticed that the two equations are: $\pm (x-1)$ for $x \geq 1$ and $x < 1$. Extrapolating then......Fri, 14 Aug 2026 15:12:11 -0400https://liberty.io/how-can-i-prove-that-the-angles-of-the-tetrahedral-structure-is-109-5-.htmlhttps://liberty.io/how-can-i-prove-that-the-angles-of-the-tetrahedral-structure-is-109-5-.htmlHow can I prove that the dihedral angles of the regular tetrahedral structure is $109.5^\circ$ with calculus or any other technique. I could do it with geometry by considering a cube....Fri, 14 Aug 2026 15:08:55 -0400https://liberty.io/what-is-the-vertex-of-y-x-x.htmlhttps://liberty.io/what-is-the-vertex-of-y-x-x.htmlFor $x > 0$, the graph takes on a parabola-like shape. Is there a method to algebraically and not graphically determine the vertex of the graph? Desmos rendering of $x^x$...Fri, 14 Aug 2026 14:44:53 -0400https://liberty.io/construct-triangle-given-angle-and-centroid.htmlhttps://liberty.io/construct-triangle-given-angle-and-centroid.htmlI am stumped by Euclidea problem 8.11: From a triangle are given angle A and the centroid G Construct the points B and C. Please only a hint...Fri, 14 Aug 2026 14:44:42 -0400https://liberty.io/what-is-physical-interpretation-of-dot-product-duplicate.htmlhttps://liberty.io/what-is-physical-interpretation-of-dot-product-duplicate.htmlConsider two vectors $V_1$ and $V_2$ in $\mathbb{R}^3$. When we take their dot product we get a real number. How is that number related to the vectors? Is there any way we can visualize it?...Fri, 14 Aug 2026 14:43:37 -0400https://liberty.io/solve-this-logical-puzzle-only-using-logical-equivalencies-is-there-a-.htmlhttps://liberty.io/solve-this-logical-puzzle-only-using-logical-equivalencies-is-there-a-.htmlHere's a puzzle: Four friends have been identified as suspects for an authorized access into a computer system. They have made statements to the investigating authorities. Alice said "Carlos......Fri, 14 Aug 2026 14:31:11 -0400https://liberty.io/how-to-prove-that-the-altitudes-of-the-triangle-are-concurrent.htmlhttps://liberty.io/how-to-prove-that-the-altitudes-of-the-triangle-are-concurrent.htmlGiven a triangle of sides $A$ $B$ $C$ .how to prove that altitudes from sides $A$ $B$ $C$ are concurrent . There is a theorem called ceva's theorem.but i dont know how to use that theorem in this p......Fri, 14 Aug 2026 14:25:07 -0400https://liberty.io/the-position-of-a-ladder-leaning-against-a-wall-and-touching-a-box-und.htmlhttps://liberty.io/the-position-of-a-ladder-leaning-against-a-wall-and-touching-a-box-und.htmlI was reading a newspaper and there was a little math riddle, I thought "how funny, that's gonna be easy, let's do it" and here am I... The problem goes as follow : in a barn, there is a 1 meter c......Fri, 14 Aug 2026 14:23:58 -0400https://liberty.io/what-defines-a-chaotic-system.htmlhttps://liberty.io/what-defines-a-chaotic-system.htmlSuppose I have an equation of motion of some sort. All I know about chaotic systems is that small changes in initial conditions will increase the outcome at $t$ by exceedingly large amounts. My que......Fri, 14 Aug 2026 14:19:22 -0400https://liberty.io/solving-the-equation-11x-2-6000x-27500-0-preferably-without-the-quadra.htmlhttps://liberty.io/solving-the-equation-11x-2-6000x-27500-0-preferably-without-the-quadra.htmlI obtained this form while solving an aptitude question. $$\frac{3000}{x-50} + \frac{3000}{x+50} = 11$$ I changed it into quadratic equation $$11x^2 -6000x - 27500 =0$$ but I don't know how to ......Fri, 14 Aug 2026 14:17:14 -0400https://liberty.io/simply-explain-the-general-number-field-sieve.htmlhttps://liberty.io/simply-explain-the-general-number-field-sieve.htmlAs a beginner to the world of integer factorization, my idea of factoring an integer is to generate a large list of prime numbers below this number and to repeatedly try to divide the integer by th......Fri, 14 Aug 2026 14:16:59 -0400https://liberty.io/how-does-one-orient-a-simplicial-complex.htmlhttps://liberty.io/how-does-one-orient-a-simplicial-complex.htmlI have a simplicial complex, built out of hyper-tetrahedra (5-cells) with the topology of $S_{4}$ and I would like to assign an ordering to it's vertices (some couple thousand), so that I can apply a ...Fri, 14 Aug 2026 14:13:30 -0400https://liberty.io/finding-the-area-of-an-ellipse-using-the-formula.htmlhttps://liberty.io/finding-the-area-of-an-ellipse-using-the-formula.htmlI'm preparing for the ACTM Sate contest, and I stumbled across this problem. I've never learned how to decipher equations like this, so could someone explain what these equations mean and how to so......Fri, 14 Aug 2026 14:13:14 -0400https://liberty.io/easiest-way-to-solve-this-integral-closed.htmlhttps://liberty.io/easiest-way-to-solve-this-integral-closed.htmlI was solving this problem from a calculus textbook and I got stuck at this particular problem. I tried to put it into Integral Calculator after I was unable to solve it, but now I wonder if there ......Fri, 14 Aug 2026 14:11:52 -0400https://liberty.io/trace-and-the-coefficients-of-the-characteristic-polynomial-of-a-matri.htmlhttps://liberty.io/trace-and-the-coefficients-of-the-characteristic-polynomial-of-a-matri.htmlLet $A\in M(\mathbb F)_{n \times n}$ Prove that the trace of A is minus the coefficient of $\lambda ^{n-1}$ in the characteristic polynomial of A. I had several ideas to approach this problem - ......Fri, 14 Aug 2026 14:07:18 -0400https://liberty.io/how-to-show-this-sequence-is-decreasing.htmlhttps://liberty.io/how-to-show-this-sequence-is-decreasing.htmlI was given a practice question in my math lecture today, it was to prove that the sequence below was decreasing: $a_n = \frac{2^n}{(n+1)!}$ using the condition that a sequence is decreasing if $......Fri, 14 Aug 2026 13:54:08 -0400https://liberty.io/probability-x-will-happen-out-of-two-tries.htmlhttps://liberty.io/probability-x-will-happen-out-of-two-tries.htmlSay you grab a card out a deck, check it, and put it back. If the probability of getting a non-club card is $\frac{3}{4}$, what is the probability it will happen at least once out of two tries? Y......Fri, 14 Aug 2026 13:49:51 -0400https://liberty.io/orthogonal-projection-onto-the-unit-simplex.htmlhttps://liberty.io/orthogonal-projection-onto-the-unit-simplex.htmlThe Unit Simplex is defined by: $$ \mathcal{S} = \left\{ x \in \mathbb{{R}^{n}} \mid x \succeq 0, \, \boldsymbol{1}^{T} x = 1 \right\} $$ Orthogonal Projection onto the Unit Simplex is defined by......Fri, 14 Aug 2026 13:48:20 -0400https://liberty.io/what-is-the-equation-of-a-half-sine-wave.htmlhttps://liberty.io/what-is-the-equation-of-a-half-sine-wave.htmlBasically, I am trying to generate a half sine wave using excel. The waveform that I am looking for is a wave which has a positive component from $0$ to $\pi$ and a zero component from $\pi$ to $2\......Fri, 14 Aug 2026 13:40:11 -0400https://liberty.io/find-the-partial-sum-of-the-series.htmlhttps://liberty.io/find-the-partial-sum-of-the-series.htmlCase: $$\sum _{n=1}^{\infty}(-1)^{n+1}(2n-1)$$ Question: $$S_n =\ ?$$ My partial solution and investigation: $$S_n = \sum _{k=1}^n (-1)^{n+1}(2n-1) = 1 -3+5-7+9-11+\cdots+(-1)^{n+1}(2n-1)$$ I see......Fri, 14 Aug 2026 13:38:23 -0400https://liberty.io/you-roll-a-6-sided-die-what-is-your-p-prime-closed.htmlhttps://liberty.io/you-roll-a-6-sided-die-what-is-your-p-prime-closed.htmlyou roll a 6 sided die. whqat is P(prime)? write your answer as a percentage...Fri, 14 Aug 2026 13:34:05 -0400https://liberty.io/proving-tan-20-circ-cdot-tan-30-circ-cdot-tan-40-circ-tan-10-circ.htmlhttps://liberty.io/proving-tan-20-circ-cdot-tan-30-circ-cdot-tan-40-circ-tan-10-circ.htmlhow to prove that : $\tan20^{\circ}.\tan30^{\circ}.\tan40^{\circ}=\tan10^{\circ}$? I know how to prove $ \frac{\tan 20^{0}\cdot\tan 30^{0}}{\tan 10^{0}}=\tan 50^{0}, $ in this way: $ \tan{20^......Fri, 14 Aug 2026 13:32:02 -0400https://liberty.io/determining-a-branch-of-logarithm.htmlhttps://liberty.io/determining-a-branch-of-logarithm.htmlThe question I have is that what is the explicit mapping that takes the value $-i \pi/2$ at $-i$ where the mapping is a branch of the logarithm in the slit plane $\mathbb{C}- [0,\infty)$? I'm famil......Fri, 14 Aug 2026 13:24:20 -0400https://liberty.io/is-it-possible-in-a-group-of-seven-people-for-each-to-be-friends-with-.htmlhttps://liberty.io/is-it-possible-in-a-group-of-seven-people-for-each-to-be-friends-with-.htmlIs it possible in a group of seven people for each to be friends with exactly 3 others? 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How can I find the average distance from point $C$ to every point o......Fri, 14 Aug 2026 13:19:16 -0400https://liberty.io/proof-of-the-ham-sandwich-theorem.htmlhttps://liberty.io/proof-of-the-ham-sandwich-theorem.htmlI have doubts about the proof of the Ham-Sandwich theorem descibed on planetmath (http://planetmath.org/proofofhamsandwichtheorem) and wikipedia (http://en.wikipedia.org/wiki/Ham_sandwich_theorem):......Fri, 14 Aug 2026 13:18:31 -0400https://liberty.io/can-a-039-closed-039-curve-be-a-function.htmlhttps://liberty.io/can-a-039-closed-039-curve-be-a-function.htmlFor a 'closed' curve (I don't know if there is such in mathematics called closed curve, but I mean a curve which is 'closed') e.g.1)It is well-known unit circle with equation$$x^2+y^2=1$$e.g.2)It i......Fri, 14 Aug 2026 13:12:36 -0400https://liberty.io/how-can-i-find-the-volume-of-intersection-between-two-cubes.htmlhttps://liberty.io/how-can-i-find-the-volume-of-intersection-between-two-cubes.htmlHow can I find the volume of the intersection of two cubes aligned to the axis? The cubes can have different sizes and positions. (I add a picture to show a simple example case of two cubes) After ......Fri, 14 Aug 2026 12:39:00 -0400https://liberty.io/fundamental-solution-of-heat-equation.htmlhttps://liberty.io/fundamental-solution-of-heat-equation.htmlFor every bounded function $u_0\in C(\mathbb R)$ the function $$u(t,x)=\frac{1}{\sqrt{4\pi t}}\int_{-\infty}^{\infty}e^{-\frac{-(x-y)^2}{4t}}u_0(y)dy$$ is a solution of $\dfrac{\partial v}{\partial......Fri, 14 Aug 2026 12:34:58 -0400https://liberty.io/giving-proof-by-counterexample.htmlhttps://liberty.io/giving-proof-by-counterexample.htmlI just started learning college mathematics and one of the things I don't like is giving proofs by counterexamples. My question is how is disproving by giving counterexample is seen by advanced '...Fri, 14 Aug 2026 12:31:24 -0400https://liberty.io/is-it-possible-to-convert-a-one-to-many-function-to-a-one-to-one-funct.htmlhttps://liberty.io/is-it-possible-to-convert-a-one-to-many-function-to-a-one-to-one-funct.htmlIs it possible to convert a one-to-many function to a one-to-one function. The input is going to be unique and hence the out of the function should also be unique. Repeating output would only creat......Fri, 14 Aug 2026 12:30:14 -0400https://liberty.io/minewsweeper-numbers-of-mines-to-board-size-probability-of-wins.htmlhttps://liberty.io/minewsweeper-numbers-of-mines-to-board-size-probability-of-wins.htmlIn a typical minesweeper setup, we have: Beginner: 10 mines in 9x9 Intermediate: 40 mines in 16x16 Expert: 99 mines in 16x30 That means mines occupy ~12.35%, ~15.63%, and ~20.63% of the board, ...Fri, 14 Aug 2026 12:18:08 -0400https://liberty.io/what-is-a-intersection-b-039.htmlhttps://liberty.io/what-is-a-intersection-b-039.htmlI was answering the questions in my book and came over the question: What is A intersection B complement? I thought the answer would be shading everything EXCEPT the middle part where A and B inte......Fri, 14 Aug 2026 12:13:46 -0400https://liberty.io/find-a-and-b-so-that-f-is-differentiable-everywhere.htmlhttps://liberty.io/find-a-and-b-so-that-f-is-differentiable-everywhere.htmlThe piecewise function is $$f(x)=\begin{cases} ax^3 & x\leq 2 \\ x^2+b, & x>2 \end{cases} $$ First I plug in the 2 into both functions and set them equal to each other $$ 2^2+b=a(2)^3......Fri, 14 Aug 2026 12:03:01 -0400https://liberty.io/is-this-batman-equation-for-real-closed.htmlhttps://liberty.io/is-this-batman-equation-for-real-closed.htmlHardOCP has an image with an equation which apparently draws the Batman logo. Is this for real? Batman Equation in text form: \begin{align} &\left(\left(\frac x7\right)^2\sqrt{\frac{||x|-3|}{......Fri, 14 Aug 2026 11:56:44 -0400https://liberty.io/how-to-split-large-numbers-easily-for-factoring-the-middle-term.htmlhttps://liberty.io/how-to-split-large-numbers-easily-for-factoring-the-middle-term.htmlI was doing a problem when I got suddenly stuck with a middle term factorisation. The equation is as follows: $$x^4-289x^2+14400=0.$$ I managed to somehow solve it, but it took a lot of time. Can you ...Fri, 14 Aug 2026 11:52:07 -0400https://liberty.io/vertical-asymptotes-of-a-given-non-rational-radical-funtion.htmlhttps://liberty.io/vertical-asymptotes-of-a-given-non-rational-radical-funtion.htmlWe have that $f$ is a function $f(x) = x\sqrt{x+4}$. Hence, $f'(x) = \dfrac{3x+8}{2\sqrt{x+4}}$. Then, $\lim_{x \to -4^+}f'(x) = -\infty$. This means that $f$ has a vertical slope at $f(-4)$. I......Fri, 14 Aug 2026 11:47:43 -0400https://liberty.io/how-do-i-find-the-mode-when-my-data-points-have-decimals.htmlhttps://liberty.io/how-do-i-find-the-mode-when-my-data-points-have-decimals.htmlMy data has numbers such as 70.35, 70.39, 70.1. None of the numbers are exactly the same. How do I find the mode?...Fri, 14 Aug 2026 11:33:25 -0400https://liberty.io/is-it-necessary-for-the-imaginary-axis-to-be-perpendicular-to-the-real.htmlhttps://liberty.io/is-it-necessary-for-the-imaginary-axis-to-be-perpendicular-to-the-real.htmlThe Real number line is in one dimension. If you want to map a complex number, you would have to add a second dimension to that number line- the Imaginary-axis. The Imaginary-axis is always ...Fri, 14 Aug 2026 11:32:36 -0400https://liberty.io/what-is-the-definition-of-binary-sequence.htmlhttps://liberty.io/what-is-the-definition-of-binary-sequence.htmlCan I write an infinite binary sequence like so: ...0111001001, ...10010 because I saw some people write infinite binary set from left to right like so: 1011000... , 101111... But I was not sure ......Fri, 14 Aug 2026 11:28:24 -0400https://liberty.io/find-critical-and-saddle-points-for-a-multivariable-function.htmlhttps://liberty.io/find-critical-and-saddle-points-for-a-multivariable-function.htmlThe function is $f(x, y) = x^2 + 2y^2 + 2xy - 4y + 15$ here's what I did, $$f_x = 2x + 2y$$ $$f_y = 4y + 2x - 4$$ $$f_{xx} = 2$$ $$f_{yy} = 4$$ $$f_{xy} = 2$$ Putting $f_x = 0$ and $f_y = 0$, w......Fri, 14 Aug 2026 11:05:06 -0400https://liberty.io/when-is-the-equicorrelation-matrix-positive-definite.htmlhttps://liberty.io/when-is-the-equicorrelation-matrix-positive-definite.htmlShow that the matrix, $$ A= \begin{bmatrix} 1 & \rho & \dots & \rho \\ \rho & 1 & \dots & \rho \\ \vdots & \rho & \ddots & \vdots \\ \rho & \dot......Fri, 14 Aug 2026 11:04:35 -0400https://liberty.io/alternate-definition-of-a-derivative-question.htmlhttps://liberty.io/alternate-definition-of-a-derivative-question.htmlThe alternate form of the derivative is $$f'(a)=\lim_\limits{x \to a}\frac{f(x)-f(a)}{x-a}$$ Can this be rewritten to this and still be true? $$f'(a)=\lim_\limits{x \to a}\frac{f(a)-f(x)}{a......Fri, 14 Aug 2026 10:48:50 -0400https://liberty.io/formula-for-calculating-total-yearly-payments-with-yearly-escalation-o.htmlhttps://liberty.io/formula-for-calculating-total-yearly-payments-with-yearly-escalation-o.htmlMy knowledge on math/terminology is pretty terrible so seeking some help with a formula (which I'll be converting to javascript to be used in a calculator). I'm trying to calculate the total amoun......Fri, 14 Aug 2026 10:18:46 -0400https://liberty.io/finding-sum-of-factors-of-a-number-using-prime-factorization.htmlhttps://liberty.io/finding-sum-of-factors-of-a-number-using-prime-factorization.htmlGiven a number, there is an algorithm described here to find it's sum and number of factors. For example, let us take the number $1225$ : It's factors are $1, 5, 7, 25, 35, 49, 175, 245, 1225 $ an......Fri, 14 Aug 2026 10:17:21 -0400https://liberty.io/how-to-integrate-with-respect-to-x-2.htmlhttps://liberty.io/how-to-integrate-with-respect-to-x-2.html$$\int x\,dx^2$$ I'm having trouble comprehending this question. I can perform substitution, but when I do I come out wrong. Here's how I've happened to handle this: $\int x\,dx^2$ $u=x^2$ $du=......Fri, 14 Aug 2026 10:01:32 -0400https://liberty.io/which-day-of-the-week-will-be-100-days-from-now.htmlhttps://liberty.io/which-day-of-the-week-will-be-100-days-from-now.htmlIf today is Wednesday, what day of the week will it be $100$ days from now? Going forward $1,8,15,\dots,92,99$ days all result in Wednesday. So the $100$th day will be Thursday. But the solution ......Fri, 14 Aug 2026 09:49:16 -0400https://liberty.io/notation-for-eigenvalues.htmlhttps://liberty.io/notation-for-eigenvalues.htmlIs there a specific notation for eigenvalues? specifically, I'd like to write: $$m\equiv \text{smallest eigenvaue of }H$$ I've seen some sources write this as: $H\succeq mI$, where "$\succeq0$" mea......Fri, 14 Aug 2026 09:48:50 -0400https://liberty.io/what-is-the-mean-of-outcomes-of-six-faced-fair-die-closed.htmlhttps://liberty.io/what-is-the-mean-of-outcomes-of-six-faced-fair-die-closed.htmlA six faced fair die is tossed a large number of times (tending to infinity). What is the mean of outcomes of the die? My try: let the die be tossed $n$ times then the mean of outcomes $$=\frac{......Fri, 14 Aug 2026 09:37:45 -0400https://liberty.io/can-a-function-be-of-class-c1-even-if-its-partial-derivatives-are-not-.htmlhttps://liberty.io/can-a-function-be-of-class-c1-even-if-its-partial-derivatives-are-not-.htmlI know that if all partial derivatives of a function f exist and are continuous then the function is said to be of class C1 (continuously differentiable). However, I was not able to find whether ......Fri, 14 Aug 2026 09:30:59 -0400https://liberty.io/visualizing-the-4th-dimension.htmlhttps://liberty.io/visualizing-the-4th-dimension.htmlIn a freshers lecture of 3-D geometry, our teacher said that 3-D objects can be viewed as projections of 4-D objects. How does this helps us visualize 4-D objects? I searched that we can at least see ...Fri, 14 Aug 2026 09:29:19 -0400https://liberty.io/deval-function-in-matlab.htmlhttps://liberty.io/deval-function-in-matlab.htmlI can't understand how the deval function works in matlab, it's hard for me to understand the code, are there any books or descriptions of numerical methods where I can read about numerical algorit......Fri, 14 Aug 2026 09:29:05 -0400https://liberty.io/how-to-find-the-closed-form-to-the-fibonacci-numbers-duplicate.htmlhttps://liberty.io/how-to-find-the-closed-form-to-the-fibonacci-numbers-duplicate.htmlPossible Duplicate: Prove this formula for the Fibonacci Sequence How to find the closed form to the fibonacci numbers? I have seen is possible calculate the fibonacci numbers without recursi......Fri, 14 Aug 2026 09:17:00 -0400https://liberty.io/help-explaining-structural-induction.htmlhttps://liberty.io/help-explaining-structural-induction.htmlI am trying to wrap my head around structural induction. Can someone break it down and explain it around this problem? Let S, a subset of $\mathbb{N}*\mathbb{N}$, be defined recursively by: Base ......Fri, 14 Aug 2026 09:13:55 -0400https://liberty.io/method-of-completing-squares-with-3-variables.htmlhttps://liberty.io/method-of-completing-squares-with-3-variables.htmlI want to use the method "completing squares" for this term: $x^2-2xy +y^2+z^2*a+2xz-2yz$ The result should be $(x-y+z)^2 +(a-1)*z^3$ Is there a "recipe" behind how to do this? Hope someone coul......Fri, 14 Aug 2026 09:02:32 -0400https://liberty.io/number-of-cycles-and-paths-in-a-wheel-graph-closed.htmlhttps://liberty.io/number-of-cycles-and-paths-in-a-wheel-graph-closed.htmlSay I have a wheel graph $W_n$ where $n$ represents the number of vertices and $n \geq 3$. Is there a general formula to determine the number of cycles there are in total or the number of cycles of a ...Fri, 14 Aug 2026 08:59:19 -0400https://liberty.io/why-is-the-surface-of-a-torus-flat.htmlhttps://liberty.io/why-is-the-surface-of-a-torus-flat.htmlWhy is the surface of a torus is said to be flat? If you consider the geometry of the torus, its surface has locally positive (spherical), negative (hyperbolic) and flat curvature....Fri, 14 Aug 2026 08:53:47 -0400https://liberty.io/can-something-be-statistically-impossible.htmlhttps://liberty.io/can-something-be-statistically-impossible.htmlDoes it make sense when people say "statistically impossible"?...Fri, 14 Aug 2026 08:53:22 -0400https://liberty.io/the-tits-cone-geometric-understanding.htmlhttps://liberty.io/the-tits-cone-geometric-understanding.htmlI know that the definition of the Tits cone is $Y=\bigcup_{w\in W}{wC}$ with W the Coxeter Groups and C the fundamental chamber. One theorem says that Y is the whole space if W is finite. But how c......Fri, 14 Aug 2026 08:48:23 -0400https://liberty.io/why-are-the-nonzero-integers-under-multiplication-not-a-group-closed.htmlhttps://liberty.io/why-are-the-nonzero-integers-under-multiplication-not-a-group-closed.htmlI was just checking out a GRE mathematics practise test. I have never had theories about groups, so could someone explain how to test if something is a group and why the 'nonzero integers under ...Fri, 14 Aug 2026 08:45:43 -0400https://liberty.io/is-there-a-better-notation-for-integrating-twice.htmlhttps://liberty.io/is-there-a-better-notation-for-integrating-twice.htmlI'm studying kinematics with varying acceleration. To calculate the displacement from the acceleration you have to integrate the acceleration with respect to t, then integrate that with respect to t, ...Fri, 14 Aug 2026 08:38:06 -0400https://liberty.io/what-is-d-in-equation-of-a-plane.htmlhttps://liberty.io/what-is-d-in-equation-of-a-plane.htmlIn a previous Quora post on this topic (https://www.quora.com/How-does-the-equation-ax-+-by-+-cz-d-define-a-plane), a respondent states that $D$ represents the distance from the origin to a plane. ......Fri, 14 Aug 2026 08:36:17 -0400https://liberty.io/probability-of-deck-of-cards-such-that-each-person-receives-one-ace.htmlhttps://liberty.io/probability-of-deck-of-cards-such-that-each-person-receives-one-ace.htmlSuppose that a deck of 52 cards containing four aces is shuffled thoroughly and the cards are then distributed among four players so that each player receives 13 cards. Determine the probability that ...Fri, 14 Aug 2026 08:33:09 -0400https://liberty.io/strong-induction-doesn-039-t-require-a-base-case.htmlhttps://liberty.io/strong-induction-doesn-039-t-require-a-base-case.htmlI'm considering the natural numbers to be the nonnegative integers. The principle of strong induction can be stated as follows, "If $P$ is a property such that for any $x$, if $P$ holds for all n......Fri, 14 Aug 2026 08:28:09 -0400https://liberty.io/derivative-of-frac-d-dx-ln-x-sqrt-x-2-1.htmlhttps://liberty.io/derivative-of-frac-d-dx-ln-x-sqrt-x-2-1.htmlShow that $\frac{d}{dx}\ln (x+\sqrt{x^2+1}) = \frac{1}{\sqrt{x^2+1}}$ So I've done this so far: $$\frac{d}{dx}\ln (x+\sqrt{x^2+1}) = \frac{1+x(x^2+1)^{-0.5}}{x+(x^2+1)^{0.5}}$$ I have tried a few ...Fri, 14 Aug 2026 08:19:26 -0400https://liberty.io/prove-that-f-x0-g-x0.htmlhttps://liberty.io/prove-that-f-x0-g-x0.htmlLet f and g be differentiable functions on an interval $(a,b)$ such that $a < b$. If there is $x0 ∈ (a,b)$ for which $f(x0) =g(x0)$ and $f(x) ≤ g(x)$ for all $x ∈ (a,b)$, prove that $f’(x0) =g’(......Fri, 14 Aug 2026 08:14:32 -0400https://liberty.io/what-is-the-purpose-of-finding-roots.htmlhttps://liberty.io/what-is-the-purpose-of-finding-roots.htmlI think I know what a root is. It's the value of elements in the domain of a function that map to 0 in the range of the function. At least that's how I see it. Do correct me if I'm wrong. However......Fri, 14 Aug 2026 08:13:17 -0400https://liberty.io/tricks-for-constructing-hilbert-style-proofs.htmlhttps://liberty.io/tricks-for-constructing-hilbert-style-proofs.htmlSeveral times in my studies, I've come across Hilbert-style proof systems for various systems of logic, and when an author says, "Theorem: $\varphi$ is provable in system $\cal H$," or "Theorem: the ...Fri, 14 Aug 2026 07:53:32 -0400https://liberty.io/is-mathbb-z-a-solvable-group.htmlhttps://liberty.io/is-mathbb-z-a-solvable-group.htmlOk, so abelian groups are solvable. And Thm II.8.5 of Hungerford says A group is solvable iff it has a solvable series. (The group may be finite or infinite.) However, I can't seem to find a solv......Fri, 14 Aug 2026 07:53:21 -0400https://liberty.io/what-is-the-solution-to-this-inequality.htmlhttps://liberty.io/what-is-the-solution-to-this-inequality.htmlThis is the given inequality I've been trying to solve $$1/6\leq \frac{1}{\mid x \mid} \leq 1/2$$ However the answer I get is $(0,6] \cup [2,6]$ which is not the answer given in my book. Could you ...Fri, 14 Aug 2026 07:53:13 -0400https://liberty.io/proofs-that-every-mathematician-should-know-closed.htmlhttps://liberty.io/proofs-that-every-mathematician-should-know-closed.htmlThere are mathematical proofs that have that "wow" factor in being elegant, simplifying one's view of mathematics, lifting one's perception into the light of knowledge, etc. So I'd like to know what ...Fri, 14 Aug 2026 07:51:22 -0400https://liberty.io/pythagorean-triple-where-a-b.htmlhttps://liberty.io/pythagorean-triple-where-a-b.htmlI cannot find even a single webpage mentioning this topic. I'm a programmer and I'm looking for a 45-45-90 triangle where all of the sides are whole numbers. In the video I am watching, they say to......Fri, 14 Aug 2026 07:50:35 -0400https://liberty.io/joint-entropy-of-2-independent-random-variables.htmlhttps://liberty.io/joint-entropy-of-2-independent-random-variables.htmlSay we have two independent random variables $X$ and $Y$. What is their joint entropy $H(X,Y)$? I worked this out, but I am not sure if the result I reached is correct. The definitions of entrop......Fri, 14 Aug 2026 07:39:42 -0400https://liberty.io/how-do-i-eliminate-the-parameter-to-find-a-cartesian-equation-closed.htmlhttps://liberty.io/how-do-i-eliminate-the-parameter-to-find-a-cartesian-equation-closed.htmlGiven the two parametric equations $$x=t^2+1$$ $$y=t+1 \iff t= y-1$$ How do I eliminate parameter $t$ to find a Cartesian equation? Do I substitute? This is confusing me, so I would appreciate......Fri, 14 Aug 2026 07:36:31 -0400https://liberty.io/exercise-3-4-in-rotman-039-s-an-introduction-to-algebraic-topology.htmlhttps://liberty.io/exercise-3-4-in-rotman-039-s-an-introduction-to-algebraic-topology.htmlI am self-learning algebraic topology by reading Rotman's An Introduction to Algebraic Topology. I am stuck on Exercise 3.4 on page 41. I'd be grateful for any hints or solution. Exercise 3.4: Le......Fri, 14 Aug 2026 07:28:54 -0400https://liberty.io/solve-sin-2x-sin-x-frac-pi-6.htmlhttps://liberty.io/solve-sin-2x-sin-x-frac-pi-6.htmlI got a this equation to solve and $\sin(2x) = \sin(x-\frac{\pi}{6})$ They was nice to give me for possible answers it could be. x = $\frac{\pi}{6}$ x = $\frac{\pi}{18}$ x = $\frac{7\pi}{18}$ ......Fri, 14 Aug 2026 07:27:31 -0400https://liberty.io/what-is-the-difference-between-an-array-and-a-vector.htmlhttps://liberty.io/what-is-the-difference-between-an-array-and-a-vector.htmlOkay so I'm doing a little bit of vector calculus at university (mainly with neural networks and the k-means clustering for cluster analysis in a 3 dimensional field or hyperplane) And from what I ...Fri, 14 Aug 2026 07:26:23 -0400https://liberty.io/prove-a-countable-intersection-of-open-and-dense-sets-in-a-compact-hau.htmlhttps://liberty.io/prove-a-countable-intersection-of-open-and-dense-sets-in-a-compact-hau.htmlLet $X$ be a compact Hausdorff space and let $\{U_n\}$ be a countable collection of subsets that are open and dense in $X$. Show that the intersection $$\bigcap\limits_{n=1}^\infty U_n$$ is dense. I ...Fri, 14 Aug 2026 07:24:49 -0400https://liberty.io/difference-between-postulates-axioms-and-theorems.htmlhttps://liberty.io/difference-between-postulates-axioms-and-theorems.htmlI'm trying to get an overarching understanding of the components of mathematical systems so that in my self study of each category of math I can break them down by their unique aspects, i.e. the ...Fri, 14 Aug 2026 07:10:23 -0400https://liberty.io/convolution-of-a-function-with-itself.htmlhttps://liberty.io/convolution-of-a-function-with-itself.htmlFunction $\phi (x)$ is defined as: $$\phi(x) = \begin{cases} 1 & \text{ if } 0 \leq x \leq 1\\0 & \text{otherwise} \end{cases} $$ How do I find the convolution of $\phi(x)$ with itself? I t......Fri, 14 Aug 2026 07:01:28 -0400https://liberty.io/graphing-the-surface-z-xy.htmlhttps://liberty.io/graphing-the-surface-z-xy.htmlLet the surface $S \subset \mathbb{R}^3$ be the graph of the function $f:\mathbb{R}^2 \to \mathbb{R}, f (x, y) = xy$. Let $U$ be the portion of $S$ for which $x^2 + y^2 ≤ 2$ and let $C$ be the boun......Fri, 14 Aug 2026 06:57:36 -0400https://liberty.io/probability-of-2-digit-appearing-in-random-3-digit-number.htmlhttps://liberty.io/probability-of-2-digit-appearing-in-random-3-digit-number.htmlLet's say I have {1, 2} what is the probability of them both appearing in three digit number? [1x2, 2x1, 12x, 21x, x12, x21] I tried the following but is not getting the correct answer 1 - P(Bo......Fri, 14 Aug 2026 06:45:57 -0400https://liberty.io/difference-between-a-reduced-residue-class-and-a-reduced-residue-syste.htmlhttps://liberty.io/difference-between-a-reduced-residue-class-and-a-reduced-residue-syste.htmlCurrently studying Dirichlets proof of Primes in arithmetic progressions. Is there a difference between a Reduced Residue Class and a Reduced Residue System? Any subset R of the integers is called a ...Fri, 14 Aug 2026 06:43:50 -0400https://liberty.io/finding-the-moment-generating-function-of-standard-normal-random-varia.htmlhttps://liberty.io/finding-the-moment-generating-function-of-standard-normal-random-varia.htmlGiven $Y\sim N(\mu, \sigma^2)$. I'm trying to find the moment generating function of $Z=\frac{Y-\mu}{\sigma}$ using the MGF transform method. Here's what I've tried: $$M_Y(t)=e^{\mu t+\frac{\sigma^......Fri, 14 Aug 2026 06:40:19 -0400https://liberty.io/what-technique-should-i-apply-to-find-the-derivative-of-a-ceiling-or-f.htmlhttps://liberty.io/what-technique-should-i-apply-to-find-the-derivative-of-a-ceiling-or-f.htmlMay I know the technique to apply to find the derivative, whenever I see a ceiling function of floor function. Thank You! e.g $$ \frac{d}{dx}(x*\lceil x \rceil )$$ and $$ \frac{d}{dx}(x*\lfloor......Fri, 14 Aug 2026 06:39:55 -0400https://liberty.io/subtracting-two-points-the-math.htmlhttps://liberty.io/subtracting-two-points-the-math.htmlI would like to confirm the math of subtracting two points. I've looked online but all of the sources I have seen want to end up with a vector, I need to end with a point. I am looking at this fu......Fri, 14 Aug 2026 06:38:55 -0400https://liberty.io/i-have-100-hundred-different-items-how-to-i-calculate-all-possible-com.htmlhttps://liberty.io/i-have-100-hundred-different-items-how-to-i-calculate-all-possible-com.htmlELI5 the formula to figure out all possible combinations without repeating any items but no upper or lower limit of the number to include. Order does not matter....Fri, 14 Aug 2026 06:38:54 -0400https://liberty.io/how-would-i-set-up-the-taylor-039-s-inequality-to-prove-that-the-funct.htmlhttps://liberty.io/how-would-i-set-up-the-taylor-039-s-inequality-to-prove-that-the-funct.htmlHow would I set up the Taylor's Inequality to prove that the function $f(x) = \frac{1}{x}$ is equal to its Taylor Series expansion centered at $x=1$? I've done the Taylor series expansion, but I'm......Fri, 14 Aug 2026 06:36:36 -0400https://liberty.io/find-area-of-the-shaded-portion.htmlhttps://liberty.io/find-area-of-the-shaded-portion.htmlThe distances between the dots in the figure are equal to 1, both horizontally and vertically. What is the measure of the common area of the triangle and the square which you can see shaded in the ......Fri, 14 Aug 2026 06:35:55 -0400https://liberty.io/infinite-row-of-squares-some-of-them-white-with-probability-p-what-is-.htmlhttps://liberty.io/infinite-row-of-squares-some-of-them-white-with-probability-p-what-is-.htmlSuppose that you have an infinite horizontal row of squares, each square of size $1\times1$ (like a row of squares in a grid of infinite size). The probability of square being white is $p$. Some wh......Fri, 14 Aug 2026 06:25:51 -0400https://liberty.io/difference-between-mathrm-rot-and-mathrm-rot-curl.htmlhttps://liberty.io/difference-between-mathrm-rot-and-mathrm-rot-curl.htmlI'm not sure whether you've heard of that difference at all and if it's of importance in English notations. In German at least you define: $$\mathrm{rot\,\textbf{V}} = \nabla \times \textbf{V}, \quad\...Fri, 14 Aug 2026 06:25:36 -0400https://liberty.io/does-sum-frac-4-n-n-2-2n-converge.htmlhttps://liberty.io/does-sum-frac-4-n-n-2-2n-converge.htmlDoes $\sum\frac{4^n(n!)^2}{(2n)!}$ converge? The ratio test is inconclusive. I know that $\sum\frac{2^n(n!)^2}{(2n)!}$ converges, if that's of any help....Fri, 14 Aug 2026 06:20:50 -0400https://liberty.io/number-of-directed-graph-with-n-vertices-closed.htmlhttps://liberty.io/number-of-directed-graph-with-n-vertices-closed.htmlIn terms of $n$, how many distinct digraphs are there whose vertex set is $X$, where $X$ is a nonempty finite set with $|X| = n$? How does the answer change if we allow loops, but not parallel ar......Fri, 14 Aug 2026 06:13:52 -0400https://liberty.io/if-x-in-mathbb-r-solve-4x-2-40-lfloor-x-rfloor-51-0.htmlhttps://liberty.io/if-x-in-mathbb-r-solve-4x-2-40-lfloor-x-rfloor-51-0.htmlIf $x\in\mathbb R$, solve $$4x^2-40\lfloor x\rfloor+51=0$$ where $\lfloor x\rfloor$ denotes the integer part of the number. $\lfloor x\rfloor\le x$ and $\lfloor x\rfloor=x-\{x\}$, where $\{x\}$ m......Fri, 14 Aug 2026 06:07:34 -0400https://liberty.io/using-the-inverse-matrix-to-solve-a-linear-system.htmlhttps://liberty.io/using-the-inverse-matrix-to-solve-a-linear-system.htmlI have the following question: $$x - 2y = 1$$ $$2x + 3y = 4$$ So let's say: $$A = \left[ \begin{matrix} 1 & -2 \\ 2 & 3 \end{matrix} \right]$$ So according to some formula to $2 \ti......Fri, 14 Aug 2026 06:01:23 -0400