https://liberty.ioCurated star highlights with fresh energy.en-usFri, 21 Aug 2026 03:15:04 -0400https://liberty.io/given-a-steady-state-vector-is-it-possible-to-calculate-the-correspond.htmlhttps://liberty.io/given-a-steady-state-vector-is-it-possible-to-calculate-the-correspond.htmlKnowing that there is a probability matrix M (where all columns add to 1) which when applied to a given vector P produces the same vector P, what is the best solution to find M? I can get my head a......Fri, 21 Aug 2026 07:14:35 -0400https://liberty.io/3d-projection-on-a-2d-plane-weak-maths-ressources.htmlhttps://liberty.io/3d-projection-on-a-2d-plane-weak-maths-ressources.htmlAs the title says, i want to project 3D points with known (x, y, z) coordinates into a 2D plane with (x', y') coordinates, knowing that the x and y axes are respectively identical to the x' and y' ......Fri, 21 Aug 2026 07:12:41 -0400https://liberty.io/expansion-of-complex-equation.htmlhttps://liberty.io/expansion-of-complex-equation.htmlFind the value of $$\left(\frac{-1+\sqrt 3i}{2}\right)^{15} + \left(\frac{-1-\sqrt 3i}{2}\right)^{15}.$$ In general, how do we find the value of expansion of equation of high orders other than bi......Fri, 21 Aug 2026 07:09:16 -0400https://liberty.io/how-do-i-calculate-gf-2-galois-field-without-using-a-tool.htmlhttps://liberty.io/how-do-i-calculate-gf-2-galois-field-without-using-a-tool.htmlI am stuck and can't find anything on Google or SO. I have: 1+1+1+0 and I am asked to calculate the answer over GF(2). My question: what answer? I have no clue what to do. Can someone help me by ...Fri, 21 Aug 2026 07:04:44 -0400https://liberty.io/if-the-product-and-the-sum-of-two-complex-numbers-are-real-what-can-we.htmlhttps://liberty.io/if-the-product-and-the-sum-of-two-complex-numbers-are-real-what-can-we.htmlIf the product and the sum of two complex numbers are real, what can we say about the numbers? Prove it. Explanation will be very helpful. Actually the question asks for a proof. Hence, a proof is......Fri, 21 Aug 2026 07:00:52 -0400https://liberty.io/geometric-intuition-of-graph-laplacian-matrices.htmlhttps://liberty.io/geometric-intuition-of-graph-laplacian-matrices.htmlI am reading about Laplacian matrices for the first time and struggling to gain intuition as to why they are so useful. Could anyone provide insight as to the geometric significance of the Laplacia......Fri, 21 Aug 2026 06:52:03 -0400https://liberty.io/if-sin-x-cos-x-1-3-then-determine-sin-x-cos-x.htmlhttps://liberty.io/if-sin-x-cos-x-1-3-then-determine-sin-x-cos-x.htmlIf $$\sin(x) - \cos(x) = \frac{1}{3}$$ then determine $$\sin(x)\cos(x)$$ I know that the expected solution is squaring both sides of equation and solving it this way: \begin{gather} \sin^2(x......Fri, 21 Aug 2026 06:51:56 -0400https://liberty.io/graph-with-both-slant-and-horizontal-asymptotes.htmlhttps://liberty.io/graph-with-both-slant-and-horizontal-asymptotes.htmlIs there such a graph? A graph that increases at a decreasing rate with the graph approaching a slant asymptote as x decreases to negative infinity while the graph approaching a horizontal asymptot......Fri, 21 Aug 2026 06:51:07 -0400https://liberty.io/is-zfc-arithmetically-sound.htmlhttps://liberty.io/is-zfc-arithmetically-sound.htmlI apologize that this question is fairly philosophical and not purely mathematical. For the purposes of this question, I would like to take the point of view that that natural numbers are "rea......Fri, 21 Aug 2026 06:41:25 -0400https://liberty.io/how-to-solve-log-x-1-log-x-2-2.htmlhttps://liberty.io/how-to-solve-log-x-1-log-x-2-2.htmlI'm doing this exercise: $$\log(x - 1) + \log(x - 2) = 2$$ My steps: Step 1: $$\log(x-1)(x - 2) = 2$$ Step 2: $$(x - 1)(x - 2) = 10^2$$ Step 3: $$x^2 - 3x + 2 = 100$$ Step 4: $$x^2 = 98 ......Fri, 21 Aug 2026 06:32:46 -0400https://liberty.io/clearing-a-manufacturing-backlog.htmlhttps://liberty.io/clearing-a-manufacturing-backlog.htmlI have a simple problem that I think I may be over complicating, but I'm not certain. I work in a manufacturing environment as a process engineer and I'm setting up a calculator to determine how l......Fri, 21 Aug 2026 06:30:30 -0400https://liberty.io/given-two-square-matrices-a-and-b-is-ab-2-a-2b-2-true-or-false.htmlhttps://liberty.io/given-two-square-matrices-a-and-b-is-ab-2-a-2b-2-true-or-false.htmlI tried with a counter example and it came out that this claim is false. I took a matrix $$ A= \left[ {\begin{array}{cc} 2 & 1\\ 3 & 2\\ \end{array} } \right] $$ and a matri......Fri, 21 Aug 2026 06:29:36 -0400https://liberty.io/definition-of-pi-in-terms-of-cos.htmlhttps://liberty.io/definition-of-pi-in-terms-of-cos.htmlI have to define $\pi$ in terms of cos and to show why the definition is well-defined. I think that I haven't understood what it is asking because $\pi$ would be the angle that divided for $2$ giv......Fri, 21 Aug 2026 06:26:46 -0400https://liberty.io/find-max-of-total-area-of-solenoid.htmlhttps://liberty.io/find-max-of-total-area-of-solenoid.htmlI'm having problems findind the best way to maximize the total area of my solenoid, of course given $l$ as the costant value of the total lenght of my wire. I can't find it on the internet, so I'm ...Fri, 21 Aug 2026 06:21:53 -0400https://liberty.io/partial-derivative-of-x-is-quotient-rule-necessary.htmlhttps://liberty.io/partial-derivative-of-x-is-quotient-rule-necessary.htmlLet $$u(x,y)=\frac{x}{x^2+y^2}$$ I'm trying to determine if the given function is harmonic. I know that the 2nd partial derivative with respect to $x$ should, when added to the 2nd partial deriva......Fri, 21 Aug 2026 06:00:52 -0400https://liberty.io/understanding-cardano-039-s-formula.htmlhttps://liberty.io/understanding-cardano-039-s-formula.htmlIn deriving his formula, Cardano arrives at the equation $y^3+py+q=0$. By substituting $y=\sqrt[3]{u}+\sqrt[3]{v}$, he gets the equation $(u+v+q)+(\sqrt[3]{u}\sqrt[3]{v})(3\sqrt[3]{u} \sqrt[3]{v} +......Fri, 21 Aug 2026 05:56:28 -0400https://liberty.io/transforming-the-laplace-operator-from-polar-to-cartesian-coordinates.htmlhttps://liberty.io/transforming-the-laplace-operator-from-polar-to-cartesian-coordinates.htmlI'm trying to find the error in my logic here. Let's say we are given the Laplace operator in polar coordinates: $$ \frac{\partial^2}{\partial r^2} + \frac{1}{r}\frac{\partial}{\partial r} + \fr......Fri, 21 Aug 2026 05:38:18 -0400https://liberty.io/how-can-i-plot-f-x-y-x-2-y-2.htmlhttps://liberty.io/how-can-i-plot-f-x-y-x-2-y-2.htmlI want to plot $f(x, y) = x^2 + y^2$? I can plot functions of a single variable but I don't know how to plot multivariable function....Fri, 21 Aug 2026 05:27:42 -0400https://liberty.io/find-general-solution-in-radians-of-cos-x-cos-3x-cos-5x-0.htmlhttps://liberty.io/find-general-solution-in-radians-of-cos-x-cos-3x-cos-5x-0.htmlI study maths as a hobby. I have this problem: Find in radians the general solution of: $$\cos x + \cos 3x + \cos 5x = 0$$ I have said: $\cos 3x + \cos 5x = 2\cos\left( \frac{1}{2}(3x + 5x)\right)\......Fri, 21 Aug 2026 05:21:47 -0400https://liberty.io/find-the-exponential-value-of-a-number.htmlhttps://liberty.io/find-the-exponential-value-of-a-number.htmlI'm wondering if there is a way to add two numbers with the same base but different exponents without converting the base of the numbers. For example lets say your adding 2^2 and 2^4 the value of 2......Fri, 21 Aug 2026 05:14:11 -0400https://liberty.io/2-sphere-surface-coordinate-dimension.htmlhttps://liberty.io/2-sphere-surface-coordinate-dimension.htmlOrdinary sphere in $\mathbb{R}^3$ is two-dimensional object (2-sphere), i.e. it requires at least two coordinates to define point on a surface. As I notice, however, there is a catch. If we use ...Fri, 21 Aug 2026 05:12:45 -0400https://liberty.io/what-does-rigorous-proof-mean.htmlhttps://liberty.io/what-does-rigorous-proof-mean.htmlI have heard several times that some mathematician has given another and more rigorous for an established theorem, but I don't know what does it really mean and what differences makes it to be more '...Fri, 21 Aug 2026 05:08:11 -0400https://liberty.io/what-are-the-smallest-and-largest-values-of-v-w-what-are-the-smallest-.htmlhttps://liberty.io/what-are-the-smallest-and-largest-values-of-v-w-what-are-the-smallest-.htmlIf $||v|| = 5$ and $||w|| = 3$, what are the smallest and largest values of $||v - w||$? What are the smallest and largest values of $v \cdot w$? How can I solve these two problems? For $||v - w||......Fri, 21 Aug 2026 05:06:12 -0400https://liberty.io/what-is-the-difference-between-variable-argument-and-parameter.htmlhttps://liberty.io/what-is-the-difference-between-variable-argument-and-parameter.htmlI'm sure that these terms should be different since there exists a difference between parameter and argument in computer science but I'm not sure about their differences in math....Fri, 21 Aug 2026 05:05:54 -0400https://liberty.io/definition-statistics-calculations-for-evenly-spaced-numbers.htmlhttps://liberty.io/definition-statistics-calculations-for-evenly-spaced-numbers.htmlIt seems like this ought to have a name like "smoothness" or something similar, but smoothness relates to something entirely different, and I can't find terminology that applies to the ...Fri, 21 Aug 2026 04:49:30 -0400https://liberty.io/understanding-the-giry-monad.htmlhttps://liberty.io/understanding-the-giry-monad.htmlI would like to understand the Giry monad, which is used to reason about probability in category theory. The issue is that I'm hitting a stumbling block understanding monads in general, in the cat......Fri, 21 Aug 2026 04:23:21 -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 -0400