https://liberty.ioCurated star highlights with fresh energy.en-usFri, 21 Aug 2026 00:19:49 -0400https://liberty.io/bernstein-inequality-for-non-centered-random-variables-is-there-a-coun.htmlhttps://liberty.io/bernstein-inequality-for-non-centered-random-variables-is-there-a-coun.htmlThe usual Bernstein inequality (see e.g. Rauhut + Fourcat, A Mathematical Introduction to Compressive Sensing, Theorem 7.30) states that if $X_1, \dots, X_m$ are independent mean zero random variab......Fri, 21 Aug 2026 03:58:58 -0400https://liberty.io/taylor-series-of-an-integral-function.htmlhttps://liberty.io/taylor-series-of-an-integral-function.htmlProblem $$I(x) = \int_{1}^x \frac{e^t - 1}{t}$$ Find $I'( \sqrt{x} )$. Solution We know that $F'(x) = f(x)$ by the fundamental theorem of calculus so $$I'(x) = \frac{e^t -1}{t}$$ And so $$I'( \......Fri, 21 Aug 2026 03:56:06 -0400https://liberty.io/chances-of-same-9-digit-number-repeating-in-90-000-attempts.htmlhttps://liberty.io/chances-of-same-9-digit-number-repeating-in-90-000-attempts.htmlI am here asking a specific question as it relates to a programming project at work. Specifically, I have a file that a user can create which contains a large number of fields. Four of these fields......Fri, 21 Aug 2026 03:53:33 -0400https://liberty.io/tribonacci-sequence-term.htmlhttps://liberty.io/tribonacci-sequence-term.htmlA tribonacci sequence is a sequence of numbers such that each term from the fourth onward is the sum of the previous three terms. The first three terms in a tribonacci sequence are called its seeds......Fri, 21 Aug 2026 03:53:02 -0400https://liberty.io/two-vertices-of-a-square.htmlhttps://liberty.io/two-vertices-of-a-square.htmlBeing (-2, 4) and B (3, -1) consecutive vertices of a square, determine the other two vertices How to solve this only algebraically? Would point-to-point distance (relative position between points......Fri, 21 Aug 2026 03:52:39 -0400https://liberty.io/how-to-factor-a-fourth-degree-polynomial.htmlhttps://liberty.io/how-to-factor-a-fourth-degree-polynomial.htmlI'm working on a math problem but I am having a hard time figuring out the method used by my textbook to make this factorization: $$x^4 + 10x^3 + 39x^2 + 70x + 50 = (x^2 + 4x + 5)(x^2 + 6x + 10)$$......Fri, 21 Aug 2026 03:48:17 -0400https://liberty.io/prove-that-the-sum-of-all-simple-roots-is-a-root.htmlhttps://liberty.io/prove-that-the-sum-of-all-simple-roots-is-a-root.htmlLet $\Delta$ be an indecomposable root system in a real inner product space $E$, and suppose that $\Phi$ is a simple system of roots in $\Delta$, with respect to an ordering of $E$. If $\Phi = \{\a......Fri, 21 Aug 2026 03:46:54 -0400https://liberty.io/equation-for-a-sinusoidal-wave-with-changing-frequency.htmlhttps://liberty.io/equation-for-a-sinusoidal-wave-with-changing-frequency.htmlI would like to find a sinusoidal wave whose period or frequency change to half (or double) with every step, someting like this But I cant find the precise coefficients for the period to decrease ......Fri, 21 Aug 2026 03:44:45 -0400https://liberty.io/what-are-the-names-of-numbers-in-the-binary-system.htmlhttps://liberty.io/what-are-the-names-of-numbers-in-the-binary-system.htmlThe names we use are very much related to the radix we use $0 - 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9$ zero - one - two - three - four - five - six - seven - eight -nine We repeat the names $21$ twenty......Fri, 21 Aug 2026 03:43:16 -0400https://liberty.io/what-is-the-geometric-meaning-of-the-inner-product-of-two-functions.htmlhttps://liberty.io/what-is-the-geometric-meaning-of-the-inner-product-of-two-functions.htmlWhen it comes to inner product I have thus far only dealt with vectors, and so the concept is very intuitive because one can easily visualize two vectors and how they get multiplied, and it is clea......Fri, 21 Aug 2026 03:40:03 -0400https://liberty.io/increasing-decreasing-sequence-of-events-intuition.htmlhttps://liberty.io/increasing-decreasing-sequence-of-events-intuition.htmlhaving trouble moving from the concept of increasing/decreasing sequence of real numbers to increasing/decreasing sequence of events. I have no problem with this concept regarding real numbers but ...Fri, 21 Aug 2026 03:39:48 -0400https://liberty.io/correctly-using-lotus-in-this-situation.htmlhttps://liberty.io/correctly-using-lotus-in-this-situation.htmlLet $X$ be a random variable with $E(X^2) < \infty$, and $Y$ be a binary random variable taking values $\{0,1\}$. The Odds Ratio (OR) between the odds of an outcome at $X=a+\epsilon$ (where $\ep......Fri, 21 Aug 2026 03:34:16 -0400https://liberty.io/proof-of-cauchy-039-s-theorem-for-abelian-groups.htmlhttps://liberty.io/proof-of-cauchy-039-s-theorem-for-abelian-groups.htmlI am reading the Dummit and Foote's proof of Cauchy's theorem for abelian groups. On the highlighted line, how do you conclude from $y\notin N$ and $y^p\in N$ that $\langle y^p \rangle \neq \langle......Fri, 21 Aug 2026 03:26:48 -0400https://liberty.io/why-are-restrictions-important.htmlhttps://liberty.io/why-are-restrictions-important.htmlWhen simplifying expressions, why do we add on restrictions for the simplified form if the original form was undefined at a certain point? The simplified form is defined at those points, so why sho......Fri, 21 Aug 2026 03:23:34 -0400https://liberty.io/second-order-taylor-series-terms-in-gradient-descent.htmlhttps://liberty.io/second-order-taylor-series-terms-in-gradient-descent.htmlMy machine learning textbook states the following when discussing second-order Taylor series approximations in the context of Gradient descent: The (directional) second derivative tells us how wel......Fri, 21 Aug 2026 03:21:45 -0400https://liberty.io/how-do-i-determine-the-length-of-the-shorter-base-of-a-trapezoid-from-.htmlhttps://liberty.io/how-do-i-determine-the-length-of-the-shorter-base-of-a-trapezoid-from-.htmlHow do I determine the length of the shorter base of a trapezoid from the longer base length, height, and only two angles? An example would be 24" longer base, with 45 deg angles at both ends wit......Fri, 21 Aug 2026 03:20:51 -0400https://liberty.io/how-to-prove-that-sqrt-3-2-sqrt-3-4-is-irrational-duplicate.htmlhttps://liberty.io/how-to-prove-that-sqrt-3-2-sqrt-3-4-is-irrational-duplicate.htmlSo while doing all sorts of proving and disproving statements regarding irrational numbers, I ran into this one and it quite stumped me: Prove that $\sqrt[3]{2} + \sqrt[3]{4}$ is irrational. I tr......Fri, 21 Aug 2026 03:19:32 -0400https://liberty.io/are-all-prime-numbers-greater-than-2-odd.htmlhttps://liberty.io/are-all-prime-numbers-greater-than-2-odd.htmlAre all prime numbers greater than 2 odd? I wasn't allowed to assume this was true. AFSOC that there exists an even prime $n$. Then by definition of even, $n = 2q$, where $q$ is a positive integ......Fri, 21 Aug 2026 03:15:21 -0400https://liberty.io/find-derivation-db-decade-for-given-amplitude-characteristic-of-low-pa.htmlhttps://liberty.io/find-derivation-db-decade-for-given-amplitude-characteristic-of-low-pa.htmlI am trying to find derivation (differential attenuation) for frequency's 600 and 2000 Hz for given amplitude characteristic of low pass filter, which look like this: I assume, that I should tran......Fri, 21 Aug 2026 03:13:16 -0400https://liberty.io/proof-by-induction-that-2-4-6-cdots-2n-n-n-1.htmlhttps://liberty.io/proof-by-induction-that-2-4-6-cdots-2n-n-n-1.htmlProving by induction. We'd like to show that $2 + 4 + 6 + \cdots+ 2n = n(n + 1)$. A nice way to do this is by induction. Let $S(n)$ be the statement above. An inductive proof would have the follow......Fri, 21 Aug 2026 03:10:33 -0400https://liberty.io/guess-distribution-from-mean-and-variance.htmlhttps://liberty.io/guess-distribution-from-mean-and-variance.htmlthis is maybe a silly question but I do not have any idea how to solve it. Given an unknown distribution with mean $E(x) = 0$ and variance $Var(x) = \sigma^2$, can I derive from these two parameter......Fri, 21 Aug 2026 03:09:18 -0400https://liberty.io/difference-between-surjections-injections-and-bijections.htmlhttps://liberty.io/difference-between-surjections-injections-and-bijections.htmlI am a little confused by the definitions of these different types of functions: I think the definition of a surjection is pretty clear in that each element of x is mapped to some value of y. But......Fri, 21 Aug 2026 03:09:16 -0400https://liberty.io/why-is-mathrm-arctan-0-not-infinity.htmlhttps://liberty.io/why-is-mathrm-arctan-0-not-infinity.html$\arctan x$ is defined as: $$\arctan x = \frac{1}{\tan(x)} = \frac{1}{\frac{\sin(x)}{\cos(x)}}$$ if I now have $x = 0$ I should get: $$\frac{1}{\frac{\sin(0)}{\cos(0)}} = \frac{1}{\frac{0}{1}} =......Fri, 21 Aug 2026 03:07:33 -0400https://liberty.io/prove-petersen-graph-is-not-hamiltonian-using-deduction-and-no-fancy-t.htmlhttps://liberty.io/prove-petersen-graph-is-not-hamiltonian-using-deduction-and-no-fancy-t.htmlProve Petersen graph is not Hamiltonian using basic terminology and deductions. I'm looking for an explanation without k-colouring or anything fancy like that since I haven't covered that in class. ...Fri, 21 Aug 2026 03:07:27 -0400https://liberty.io/is-there-a-known-optimal-strategy-for-go-fish.htmlhttps://liberty.io/is-there-a-known-optimal-strategy-for-go-fish.htmlIn the two-player game of Go Fish, using a standard desk of 52 cards, where a set consists of all four cards of a given rank, is there a known optimal strategy? When I play, if I can ask for a ran......Fri, 21 Aug 2026 03:06:24 -0400https://liberty.io/what-is-the-formal-negation-of-the-statement-there-is-much-x-in-y-clos.htmlhttps://liberty.io/what-is-the-formal-negation-of-the-statement-there-is-much-x-in-y-clos.htmlWhat is the formal negation of the statement "There is much X in Y"? The answer to me is that "It is not the case that there is much X in Y" But I want a more useful negation. Can I say that its ...Fri, 21 Aug 2026 03:06:03 -0400https://liberty.io/why-is-pi-2-g-where-g-is-the-gravitational-constant.htmlhttps://liberty.io/why-is-pi-2-g-where-g-is-the-gravitational-constant.htmlSome months ago a professor of mine showed us a 'proof' of why $g\approx 9.8 ~\text{m}/\text{s}^2$ (the gravitational acceleration at the surface of the Earth) is 'equal' to $\pi^2\approx9.86\dots$......Fri, 21 Aug 2026 03:03:57 -0400https://liberty.io/is-valid-the-brioschi-formula-of-gaussian-curvature-in-r-n.htmlhttps://liberty.io/is-valid-the-brioschi-formula-of-gaussian-curvature-in-r-n.htmlIs the Brioschi formula, for Gaussian Curvature of a Surface, valid only in codimension 1, or is valid for a surface in $R^n$ with $n>3$?...Fri, 21 Aug 2026 03:03:52 -0400https://liberty.io/lyapunov-function-for-nonlinear-systems.htmlhttps://liberty.io/lyapunov-function-for-nonlinear-systems.htmlSince there is no systematic method to find Lyapunov functions, how should I approach the question shown below to find corresponding Lyapunov function? With $\alpha<0$ $$\begin{align} x_1'&=-...Fri, 21 Aug 2026 03:01:15 -0400https://liberty.io/how-to-find-the-value-of-x-in-x-5-32.htmlhttps://liberty.io/how-to-find-the-value-of-x-in-x-5-32.htmlI understand that $2^5=32$ but how would one go about finding it without doing any guessing (what if the numbers were much greater)?...Fri, 21 Aug 2026 02:59:44 -0400https://liberty.io/proving-sum-k-0-n-k-choose-a-n-k-choose-b-n-1-choose-a-b-1.htmlhttps://liberty.io/proving-sum-k-0-n-k-choose-a-n-k-choose-b-n-1-choose-a-b-1.htmlGiven two positive integers $a$ and $b$, prove that $\sum_{k=0}^{n} {k \choose a} {n-k \choose b} = {n+1\choose a+b+1}$ I think there should be a good combinatorial proof for this, given the simp......Fri, 21 Aug 2026 02:50:22 -0400https://liberty.io/advanced-integration-techniques-quadratic-expressions-and-u-substituti.htmlhttps://liberty.io/advanced-integration-techniques-quadratic-expressions-and-u-substituti.htmlFind $$\int \frac{2x-1}{x^2-6x+13}dx $$ In the final steps after a u-substitution, one arrives at $$\int \frac{2u}{u^2+4}du + \int\frac{5}{ u^2+4}du$$ The next step is arriving at $$\ln(u^2+4) + 5\...Fri, 21 Aug 2026 02:31:41 -0400https://liberty.io/a-summation-identity.htmlhttps://liberty.io/a-summation-identity.htmlI have encountered this identity in Page 616 of Mathematical Methods for Students of Physics and Related Fields (Second Edition) by Sadri Hassani: $$ \sum_{m = 0}^{n}\left(-1\right)^{m}\, {\left(\,......Fri, 21 Aug 2026 02:31:39 -0400https://liberty.io/3-2-9-correct-syntax-for-a-negative-number-with-an-exponent.htmlhttps://liberty.io/3-2-9-correct-syntax-for-a-negative-number-with-an-exponent.htmlA friend is taking a college algebra class and they are teaching him that $$-3^2 = -9$$ Their explanation is: $$-3^2 = -(3^2) = -9.$$ It has been a long time for me but I thought that in the a......Fri, 21 Aug 2026 02:19:41 -0400https://liberty.io/repeated-linear-factors-in-partial-fractions.htmlhttps://liberty.io/repeated-linear-factors-in-partial-fractions.htmlI have a question about the following partial fraction: $$\frac{x^4+2x^3+6x^2+20x+6}{x^3+2x^2+x}$$ After long division you get: $$x+\frac{5x^2+20x+6}{x^3+2x^2+x}$$ So the factored form of the ...Fri, 21 Aug 2026 02:19:32 -0400https://liberty.io/why-the-fourier-and-laplace-transforms-of-the-heaviside-unit-step-func.htmlhttps://liberty.io/why-the-fourier-and-laplace-transforms-of-the-heaviside-unit-step-func.htmlThe Fourier transform of the Heaviside step function $u(t)$ is $\dfrac{1}{iω} + π δ(ω)$. The Laplace transform of the same function is $\dfrac{1}{s}$. (Edit: This was my mistake, see my answer.) I ...Fri, 21 Aug 2026 02:15:12 -0400https://liberty.io/ladder-method-for-lcm-and-gcd.htmlhttps://liberty.io/ladder-method-for-lcm-and-gcd.htmlI think the easiest method for finding lcm and gcd is the ladder method, which can even be extended to more than two integers. For example, $$ \text{lcm}(48,72,108)=2*2*3*3*2*2*1*3=432\\\text{gcd}......Fri, 21 Aug 2026 02:14:21 -0400https://liberty.io/floor-function-properties-2x-x-x-frac12-and-nx-sum-k-0-n-1-x-frac-k-n.htmlhttps://liberty.io/floor-function-properties-2x-x-x-frac12-and-nx-sum-k-0-n-1-x-frac-k-n.htmlI'm doing some exercises on Apostol's calculus, on the floor function. Now, he doesn't give an explicit definition of $[x]$, so I'm going with this one: DEFINITION Given $x\in \Bbb R$, the inte......Fri, 21 Aug 2026 02:09:46 -0400https://liberty.io/what-is-the-definition-of-quantum-tori.htmlhttps://liberty.io/what-is-the-definition-of-quantum-tori.htmlI have learned that the non-commutative torus (2-dimensional) is $\mathcal{A}_\theta$. But now I'm going to read the paper written by Mania, which shows the equivalence of $\mathcal{QT}$ and $\math......Fri, 21 Aug 2026 02:09:14 -0400https://liberty.io/large-database-of-mathematical-constants.htmlhttps://liberty.io/large-database-of-mathematical-constants.htmlIs there a repository of billions of mathematical constants, computed say with 15 digits of accuracy, so that when you find a number, say $5.859874482$, it tells you that it matches the first $9$ d......Fri, 21 Aug 2026 02:06:20 -0400https://liberty.io/find-the-area-of-the-curved-shape.htmlhttps://liberty.io/find-the-area-of-the-curved-shape.htmlHow to find area of this curved shape?...Fri, 21 Aug 2026 02:05:14 -0400https://liberty.io/why-are-the-formulas-for-finding-area-for-a-square-vs-a-rectangle-diff.htmlhttps://liberty.io/why-are-the-formulas-for-finding-area-for-a-square-vs-a-rectangle-diff.htmlI'm puzzled about why the formula for area of a square is side squared, but for a rectangle it's length times height. Why wouldn't it be for a square, side times side?...Fri, 21 Aug 2026 01:54:41 -0400https://liberty.io/center-of-mass-of-a-right-circular-cone.htmlhttps://liberty.io/center-of-mass-of-a-right-circular-cone.htmlHow can one find the center of a right circular cone with height $h$ and radius $r$? I've found these formulas: $$M_{xy} = \iiint\limits_V z \rho (x,y,z) \, dx \, dy \, dz$$ $$M_{yz} = \iiint\...Fri, 21 Aug 2026 01:44:08 -0400https://liberty.io/what-is-the-difference-between-np-and-conp.htmlhttps://liberty.io/what-is-the-difference-between-np-and-conp.htmlWhat is the difference between NP and coNP? Is coNP a subset of NP ? or vice versa? What is the significance of NP $\cap$ coNP when it comes to mathematics?...Fri, 21 Aug 2026 01:41:51 -0400https://liberty.io/how-to-symbolically-define-set-of-all-real-numbers-r-in-set-builder-no.htmlhttps://liberty.io/how-to-symbolically-define-set-of-all-real-numbers-r-in-set-builder-no.htmlCan we define R using set-builder notation without language semantics (purely in math symbols)? Is it valid to do the following: $\{x\in\mathbb{R}|x\}$...Fri, 21 Aug 2026 01:38:42 -0400https://liberty.io/find-tangent-plane-to-ellipsoid-x-2-2y-2-z-2-4-can-039-t-find-where-i-.htmlhttps://liberty.io/find-tangent-plane-to-ellipsoid-x-2-2y-2-z-2-4-can-039-t-find-where-i-.htmlSo I'm struggling on a pretty simple problem on Brilliant.org. Find the unit normal vector to the surface of the ellipsoid defined by : $x^2+2y^2+z^2=4$ at point : $(1;\frac{1}{\sqrt2};\sqrt2)$ ......Fri, 21 Aug 2026 01:35:02 -0400https://liberty.io/hyperbolic-functions-why-are-they-named-with-trig-functions.htmlhttps://liberty.io/hyperbolic-functions-why-are-they-named-with-trig-functions.htmlI don't really get why these hyperbolic functions are named after trig functions. Can someone enlighten me?...Fri, 21 Aug 2026 01:26:44 -0400https://liberty.io/knowing-when-to-use-factoring-how-to-factor-rational-zeroes.htmlhttps://liberty.io/knowing-when-to-use-factoring-how-to-factor-rational-zeroes.htmlHow do I know where I would be using factoring as opposed to rational zero theorem? Do I do Descartes rule of signs to get how many positive/negative and then attempt RZT to get rational zeroes, th......Fri, 21 Aug 2026 01:26:14 -0400https://liberty.io/the-divided-polynomial-algebra-over-a-field.htmlhttps://liberty.io/the-divided-polynomial-algebra-over-a-field.htmlLet $\Gamma_R[\alpha]$ denote the divided polynomial algebra over $R$; that is, the quotient of the free $R$-algebra $R\langle \alpha_1,\alpha_2,\cdots \rangle$ by the relations $$\alpha_n \cdot \a......Fri, 21 Aug 2026 01:26:04 -0400https://liberty.io/how-do-you-find-the-derivative-of-2-sin-pi-x.htmlhttps://liberty.io/how-do-you-find-the-derivative-of-2-sin-pi-x.htmlI don't understand how to take the derivative of this expression. $$y=2^{\sin (\pi x)}$$...Fri, 21 Aug 2026 01:24:54 -0400https://liberty.io/derivatives-of-trig-functions.htmlhttps://liberty.io/derivatives-of-trig-functions.htmlI am trying to figure out the derivative of $y= \frac{t\sin t}{1+t}$ I know the quotient rules are needed but I think what is confusing is some fairly simple math. Here is what I did. (1+t)(tcost)......Fri, 21 Aug 2026 01:24:49 -0400https://liberty.io/transition-function-of-a-nfa.htmlhttps://liberty.io/transition-function-of-a-nfa.htmlFor a DFA, the definition of the transition function for a string is: $$ \widehat{\delta}:Q\times\Sigma^\star\to Q $$ The first part (before the arrow) defines all combinations of all strings wit......Fri, 21 Aug 2026 01:15:24 -0400https://liberty.io/ratio-test-when-checking-the-convergence-of-series.htmlhttps://liberty.io/ratio-test-when-checking-the-convergence-of-series.htmlSuppose we have the series $\sum a_n$. Define, $$ L=\lim_{n\to\infty}\frac{a_{n+1}}{a_n} $$ Then, if $L<1$ the series is absolutely convergent (and hence convergent). if $L>1$ the series is ...Fri, 21 Aug 2026 01:15:04 -0400https://liberty.io/find-the-probability-of-n-different-people-having-the-same-birthday-mo.htmlhttps://liberty.io/find-the-probability-of-n-different-people-having-the-same-birthday-mo.htmlFind the probability of n different people having the same birthday month. Is the following the right way to do: Choose any one of the n people. Then the rest must have the same birthday month as......Fri, 21 Aug 2026 00:54:24 -0400https://liberty.io/types-of-polynomial-functions-quadratic-cubic-quartic-quintic.htmlhttps://liberty.io/types-of-polynomial-functions-quadratic-cubic-quartic-quintic.htmlI would very much like to have a complete list of the types of polynomial functions. I know that theres: Quadratic : (AX^2 + BX + C) Cubic : (AX^3 + BX^2 + C......Fri, 21 Aug 2026 00:45:18 -0400https://liberty.io/calculation-of-c-1-2-n.htmlhttps://liberty.io/calculation-of-c-1-2-n.htmlI have to represent the function on the left as a power series, and this is the solution to it but I don't know how to calculate this for example when n=1?...Fri, 21 Aug 2026 00:45:12 -0400https://liberty.io/is-there-a-planar-graph-calculator.htmlhttps://liberty.io/is-there-a-planar-graph-calculator.htmlI just took a exam on graph theory and there was one question that I'm not sure if I answered correctly. Is there any tool I can use to check if a graph is planar or not? The graph was a $K_6$ graph ...Fri, 21 Aug 2026 00:43:17 -0400https://liberty.io/definition-of-maximal-atlas.htmlhttps://liberty.io/definition-of-maximal-atlas.htmlI some how could not find the definition of maximal atlas on a manifold. What I see is that an atlas is said to be maximal atlas if it is not contained in any other atlas. What does this containm......Fri, 21 Aug 2026 00:31:36 -0400https://liberty.io/echelon-form-of-a-system-of-equations.htmlhttps://liberty.io/echelon-form-of-a-system-of-equations.htmlMy prof gave us this definition of an Echelon system: A system of m linear equations in n variables is called an echelon system if m ≤ n. Every variable is the leading variable of at m......Fri, 21 Aug 2026 00:21:31 -0400https://liberty.io/why-eliminate-radicals-in-the-denominator-rationalizing-the-denominato.htmlhttps://liberty.io/why-eliminate-radicals-in-the-denominator-rationalizing-the-denominato.htmlWhy do all school algebra texts define simplest form for expressions with radicals to not allow a radical in the denominator. For the classic example, $1/\sqrt{3}$ needs to be "simplified" to $\sqr......Thu, 20 Aug 2026 23:58:55 -0400https://liberty.io/sqrt-40-9x-2-sqrt-7-x-sqrt-x.htmlhttps://liberty.io/sqrt-40-9x-2-sqrt-7-x-sqrt-x.htmlIn MAO 1991, Find $2x+5$ if $x$ satisfies $\sqrt{40-9x}-2\sqrt{7-x}=\sqrt{-x}$ My attempt, I squared the the equation then I got $144x^2+1648x+4480=144x^2-1632x+4624$, which results $x=-9$, an......Thu, 20 Aug 2026 23:54:46 -0400https://liberty.io/intuition-for-probability-density-function-as-a-radon-nikodym-derivati.htmlhttps://liberty.io/intuition-for-probability-density-function-as-a-radon-nikodym-derivati.htmlIf someone asked me what it meant for $X$ to be standard normally distributed, I would tell them it means $X$ has probability density function $f(x) = \frac{1}{\sqrt{2\pi}}\mathrm e^{-x^2/2}$ for a......Thu, 20 Aug 2026 23:52:20 -0400https://liberty.io/converting-4-4-sqrt3-6-from-rectangular-to-cylindrical-coordinates.htmlhttps://liberty.io/converting-4-4-sqrt3-6-from-rectangular-to-cylindrical-coordinates.htmlI'm getting the wrong answer for $\theta$. What am I doing wrong? $$(4,-4{\sqrt3}, 6)$$ $$r =\sqrt{4^4 + (-4\sqrt{3})^2} = 8$$ $$z = 6$$ $$\tan\theta=\frac{-4\sqrt{3}}{4}$$ $$\tan\theta=-\sqrt{3......Thu, 20 Aug 2026 23:46:18 -0400https://liberty.io/should-coprime-numbers-both-be-prime.htmlhttps://liberty.io/should-coprime-numbers-both-be-prime.htmlTwo numbers $a$ and $b$ are coprime if and only if $(a, b) = 1$. $(4, 5) = 1$, are $4$ and $5$ coprime?...Thu, 20 Aug 2026 23:36:30 -0400https://liberty.io/ap-statistics-practice-question-about-linear-combinations.htmlhttps://liberty.io/ap-statistics-practice-question-about-linear-combinations.htmlThe question is as follows: A company ships gift baskets that contain apples and pears. The distributions of weight for the apples, the pears, and the baskets are each approximately normal. The me......Thu, 20 Aug 2026 23:36:28 -0400https://liberty.io/in-how-many-ways-can-12-distinct-pens-be-divided-equally-given-the-fol.htmlhttps://liberty.io/in-how-many-ways-can-12-distinct-pens-be-divided-equally-given-the-fol.htmlIn how many ways can $12$ distinct pens be divided equally 1) among $3$ children. options a) 12!/$(4!)^3$ b) 12!/$(4!)^3$ . 3! c) 12!/$(5!)^3$ d) 11!/$(4!)^3$ 2) into $3$ parcels. options a)......Thu, 20 Aug 2026 23:34:24 -0400https://liberty.io/calculating-mod-7.htmlhttps://liberty.io/calculating-mod-7.htmlProblem: Calculate $10^{10^{10}} \pmod 7$ According to Fermat's little theorem: $a^{p-1}\equiv1 \pmod p$, so $10^6\equiv1 \pmod 7$ and $10^n\equiv4 \pmod 6$, $n$ being any integer, why can we writ......Thu, 20 Aug 2026 23:33:50 -0400https://liberty.io/what-is-the-probability-that-none-of-the-events-happens.htmlhttps://liberty.io/what-is-the-probability-that-none-of-the-events-happens.htmlSuppose there are 3 events such that: P(A)=0.20 P(B)=0.10 P(C)=0.40 P(A ∩ B)=0.05 P(A ∩ C)=0.10 P(B ∩ C)=0.03 P(A ∩ B ∩ C)=0.01 What is the probability that none of the events happen......Thu, 20 Aug 2026 23:21:57 -0400https://liberty.io/how-can-we-prove-that-unitary-transformation-is-isometric.htmlhttps://liberty.io/how-can-we-prove-that-unitary-transformation-is-isometric.htmlI am studying Image Processing in which unitary transforms play an important role, one reason that I found for their use in image transformation is isometry(they preserve distance), I found a relev......Thu, 20 Aug 2026 23:15:52 -0400https://liberty.io/in-the-triangle-shown-what-is-the-degree-measure-of-angle-adb-duplicat.htmlhttps://liberty.io/in-the-triangle-shown-what-is-the-degree-measure-of-angle-adb-duplicat.htmlIn the triangle shown, what is the degree measure of $\angle ADB$ I have mainly tried angle-chasing, but it turns out that through plain angle-chasing you cannot find $\angle ADB$ or $\angle EBD$. I ...Thu, 20 Aug 2026 23:14:34 -0400https://liberty.io/distance-of-a-vector-from-a-subspace-linear-algebra.htmlhttps://liberty.io/distance-of-a-vector-from-a-subspace-linear-algebra.htmlI want to calculate the distance of the vector $x=(1,1,1,1)$ to the subspace $\{(1,0,2,0) , (0,1,0,2)\}$ I have solved this in 2 ways that I know of but the thing is, the results are different. ......Thu, 20 Aug 2026 23:13:06 -0400https://liberty.io/is-apple-ipad-tablet-good-for-mathematics-students-closed.htmlhttps://liberty.io/is-apple-ipad-tablet-good-for-mathematics-students-closed.htmlI am a math student. I'd like to find out if tablets (iPads, Galaxy Note 10.1) are worth the cost. How good are tablets for the purposes of reading textbooks as PDF and writing mathematics with a ......Thu, 20 Aug 2026 23:09:54 -0400https://liberty.io/handshaking-lemma-and-existence-of-the-graph.htmlhttps://liberty.io/handshaking-lemma-and-existence-of-the-graph.htmlHandshaking lemma in graph theory basically says that The degree sum is equal to twice the number of edges. The doubt I have is, does this condition enough to prove the existence of the graph. T......Thu, 20 Aug 2026 23:07:47 -0400https://liberty.io/cleverest-construction-of-a-dodecahedron-icosahedron.htmlhttps://liberty.io/cleverest-construction-of-a-dodecahedron-icosahedron.htmlOne can show, as an elementary application of Euler's formula, that there are at most five regular convex polytopes in 3-space. The tetrahedron, cube, and octahedron all admit very intuitive ...Thu, 20 Aug 2026 23:07:25 -0400https://liberty.io/the-product-or-composition-of-permutation-groups-in-two-line-notation.htmlhttps://liberty.io/the-product-or-composition-of-permutation-groups-in-two-line-notation.htmlThis question has been asked before, I know: Product of Permutations However, his did not resolve my problem. Here's an example I've been looking at, which is to find the product of two permutat......Thu, 20 Aug 2026 22:56:14 -0400https://liberty.io/in-classical-logic-why-is-p-rightarrow-q-true-if-both-p-and-q-are-fals.htmlhttps://liberty.io/in-classical-logic-why-is-p-rightarrow-q-true-if-both-p-and-q-are-fals.htmlI am studying entailment in classical first-order logic. The Truth Table we have been presented with for the statement $(p \Rightarrow q)\;$ (a.k.a. '$p$ implies $q$') is: $$\begin{array}{|c|c|c|} \...Thu, 20 Aug 2026 22:44:46 -0400https://liberty.io/when-is-it-possible-to-lift-a-function-to-its-covering-space.htmlhttps://liberty.io/when-is-it-possible-to-lift-a-function-to-its-covering-space.htmlGiven a continuous function $f: X \to Y$ a covering $p: E \to Y$, what are there conditions for me to say that there exists a lift $g: X \to E$ s.t. $p \circ g = f$? Under what conditions is this l......Thu, 20 Aug 2026 22:39:43 -0400https://liberty.io/curl-of-a-vector-in-spherical-coordinates.htmlhttps://liberty.io/curl-of-a-vector-in-spherical-coordinates.htmlThe curl of a Vector function in curvilinear coordinate system is given by $$ \nabla \times A = \frac 1 {h_1 h_2 h_3} \begin{vmatrix} h_1 \hat e_1 & h_2 \hat e_2 & h_3 \hat e_3\\ \partial \...Thu, 20 Aug 2026 22:35:55 -0400https://liberty.io/how-to-show-that-a-function-is-unbounded.htmlhttps://liberty.io/how-to-show-that-a-function-is-unbounded.htmlHow to prove that the function $f:(0,2)\to\mathbb{R}, f(x)=\frac{1}{x}$ is unbounded. I know for a function is unbounded if: $\forall M>0 \exists x\text{ such that }|f(x)|>M$...Thu, 20 Aug 2026 22:34:06 -0400https://liberty.io/what-039-s-the-limit-of-a-square-root-function.htmlhttps://liberty.io/what-039-s-the-limit-of-a-square-root-function.html$$\lim_{x \to \sqrt{3}^{-}} \sqrt{x^2-3}$$ What's the answer of this limit? There are two hyppothesis: $0$ and undefined. Undefined Because the square root of a negative number is undefined, $0$ ...Thu, 20 Aug 2026 22:29:35 -0400https://liberty.io/a-matrix-defines-a-self-adjoint-operator-if-and-only-if-it-is-symmetri.htmlhttps://liberty.io/a-matrix-defines-a-self-adjoint-operator-if-and-only-if-it-is-symmetri.htmlI know this is probably a silly question, but I'm stuck with it. $\forall x,y \in \mathbb{R}^{n}$, and $A$ real matrix ($n\times n$), then $\langle Ax,y\rangle=\langle x,Ay\rangle \iff A^{T}=A$ ......Thu, 20 Aug 2026 22:19:01 -0400https://liberty.io/express-arcsin-x-in-terms-of-arccos-x-solve-the-equation-2-arctan-x-ar.htmlhttps://liberty.io/express-arcsin-x-in-terms-of-arccos-x-solve-the-equation-2-arctan-x-ar.htmlExpress $\arcsin(x)$ in terms of $\arccos(x)$. Using the same, solve the equation $$ 2\,\tan^{-1}x = \sin^{-1} x + \cos^{-1} x $$ I'm not sure if I am on the right track, but here is what i di......Thu, 20 Aug 2026 22:18:01 -0400https://liberty.io/why-the-emphasis-on-projective-space-in-algebraic-geometry.htmlhttps://liberty.io/why-the-emphasis-on-projective-space-in-algebraic-geometry.htmlI have no doubt this is a basic question. However, I am working through Miranda's book on Riemann surfaces and algebraic curves, and it has yet to be addressed. Why does Miranda (and from what lit......Thu, 20 Aug 2026 22:14:44 -0400https://liberty.io/equation-of-a-3d-curve-shaped-like-a-logarithmic-spiral.htmlhttps://liberty.io/equation-of-a-3d-curve-shaped-like-a-logarithmic-spiral.htmlI'm not exactly certain of the mathematical description of this surface (if I were I wouldn't have a question), but I basically want to make a "3D spiral" which is basically a sine wave "wrapped" a......Thu, 20 Aug 2026 22:13:52 -0400https://liberty.io/how-to-find-the-focus-of-a-cassini-oval.htmlhttps://liberty.io/how-to-find-the-focus-of-a-cassini-oval.htmlYou are given some Cassini oval with foci somewhere on the x-axis. How to construct the foci using a compass and straightedge if only the coordinate axes and the oval are given? The foci are the same ...Thu, 20 Aug 2026 22:11:51 -0400https://liberty.io/what-is-the-difference-between-the-frobenius-norm-and-the-2-norm-of-a-.htmlhttps://liberty.io/what-is-the-difference-between-the-frobenius-norm-and-the-2-norm-of-a-.htmlGiven a matrix, is the Frobenius norm of that matrix always equal to the 2-norm of it, or are there certain matrices where these two norm methods would produce different results? If they are ident......Thu, 20 Aug 2026 22:06:20 -0400https://liberty.io/repeated-squaring-techniques.htmlhttps://liberty.io/repeated-squaring-techniques.htmlHere is the problem and the solution my text gives me. When I first approached this problem, I looked up "repeated squaring" online and tried the following algorithm from khan academy. Find the b......Thu, 20 Aug 2026 22:06:07 -0400https://liberty.io/epsilon-delta-limit-proofs-at-and-going-to-infinity.htmlhttps://liberty.io/epsilon-delta-limit-proofs-at-and-going-to-infinity.htmlSo I understand the concept of epsilon delta limit proofs with linear functions, easy enough, and I am still shaky about doing it with non linear but I am slowly understanding that. I don't quite ...Thu, 20 Aug 2026 21:49:14 -0400https://liberty.io/finding-a-parameterization-of-the-paraboloid-900z-25x-2-36y-2.htmlhttps://liberty.io/finding-a-parameterization-of-the-paraboloid-900z-25x-2-36y-2.htmlQuestion: Find a parameterization of the paraboloid $900z = 25x^2 + 36y^2$. My Work $25x^2 + 36y^2 = 900z$ $\implies (5x)^2 + (6y)^2 = (30\sqrt{z})^2$ Can we can represent this equation using ...Thu, 20 Aug 2026 21:48:09 -0400https://liberty.io/infinitite-products-in-complex-analysis.htmlhttps://liberty.io/infinitite-products-in-complex-analysis.htmlIn the text, "Functions of a Complex Variable" The notion of infinite products were discussed which are defined in $(1.)$. What makes an infinite product representation of holomorphic function so ...Thu, 20 Aug 2026 21:36:47 -0400https://liberty.io/ln-x-ln-y-kt-make-x-the-subject.htmlhttps://liberty.io/ln-x-ln-y-kt-make-x-the-subject.htmlSo I thought you take the inverse function of the whole expression getting: $x - y = e^{kt}$ and so your final answer would be $x = e^{kt} + y$ but according to the answers in the book $x = ye^{kt}$. ...Thu, 20 Aug 2026 21:27:38 -0400https://liberty.io/factoring-quadratic-trinomials.htmlhttps://liberty.io/factoring-quadratic-trinomials.htmlI'm currently doing some homework, but I'm COMPLETELY stuck on one problem. I need to factor the following trinomial: $$5x^2+7xy+2y^2$$ How can I solve this problem? I have no idea what to do bec......Thu, 20 Aug 2026 21:19:05 -0400https://liberty.io/what-will-be-the-remainder-when-2-31-is-divided-by-5.htmlhttps://liberty.io/what-will-be-the-remainder-when-2-31-is-divided-by-5.htmlThe question is given in the title: Find the remainder when $2^{31}$ is divided by $5$. My friend explained me this way: $2^2$ gives $-1$ remainder. So, any power of $2^2$ will give $-1$ ...Thu, 20 Aug 2026 21:18:44 -0400https://liberty.io/proof-of-the-usual-topology-on-r.htmlhttps://liberty.io/proof-of-the-usual-topology-on-r.htmlI'm trying to prove that the usual topology on R is in fact a topology for R. I've read through a few books on topology and most of them seem to skip over the proof and list it as "trivial". My math ...Thu, 20 Aug 2026 21:12:35 -0400https://liberty.io/how-many-bits-do-i-need-to-store-a-given-fraction.htmlhttps://liberty.io/how-many-bits-do-i-need-to-store-a-given-fraction.htmlUnlike integers, decimal fractions cannot be directly represented in binary. Therefore, what is the procedure to find how many bits to use to express a given binary fraction are sufficient? Basica......Thu, 20 Aug 2026 21:08:22 -0400https://liberty.io/motivation-for-definition-of-quotient-map-and-passing-to-the-quotient.htmlhttps://liberty.io/motivation-for-definition-of-quotient-map-and-passing-to-the-quotient.htmlI'm a bit confused by the definition of of a quotient map between topological spaces. What is the reason for constructing a topology in this way? I am familiar with quotients from algebra and even so ...Thu, 20 Aug 2026 21:07:34 -0400https://liberty.io/projection-matrices.htmlhttps://liberty.io/projection-matrices.htmlI have found these two apparently contradicting remarks about projection matrices: A matrix $P$ is idempotent if $PP = P$. An idempotent matrix that is also Hermitian is called a projection matrix......Thu, 20 Aug 2026 21:02:17 -0400https://liberty.io/how-many-minutes-in-1-day.htmlhttps://liberty.io/how-many-minutes-in-1-day.htmlThere are 24*60 minutes in a day (ignoring the imperfections of the natural world, the Earth and Sun). So there are 24*60 valid 24 hour times (excluding seconds) on a digital clock. Each of these......Thu, 20 Aug 2026 20:54:42 -0400https://liberty.io/things-to-check-for-with-a-orthogonal-matrix-or-vector.htmlhttps://liberty.io/things-to-check-for-with-a-orthogonal-matrix-or-vector.htmlWhat are there different rules you need to check for with an orthogonal matrix or vector? Why does an orthogonal matrix need to nxn but a orthogonal vector does not? $$\begin{bmatrix} 1\\ 0\\ 1......Thu, 20 Aug 2026 20:52:00 -0400https://liberty.io/confused-in-the-term-closed-in-closed-subgroup.htmlhttps://liberty.io/confused-in-the-term-closed-in-closed-subgroup.htmlwell, In Brian C Halls Book, I am not getting the definition of Matrix Lie group, as he says : A matrix Lie Group is any subgroup $G$ of $GL_n(\mathbb{C})$ with the following property: If $A_m$ is ......Thu, 20 Aug 2026 20:49:00 -0400