https://liberty.ioCurated star highlights with fresh energy.en-usSun, 06 Sep 2026 03:38:35 -0400https://liberty.io/is-it-viable-to-ask-in-an-infinite-set-about-the-cardinality.htmlhttps://liberty.io/is-it-viable-to-ask-in-an-infinite-set-about-the-cardinality.htmlCan you ask given an infinite set about its cardinality? Does an infinite set have a cardinality? So, for example, what would be the cardinality of $+\infty$?...Sun, 06 Sep 2026 07:38:34 -0400https://liberty.io/computational-complexity-of-computing-the-determinant.htmlhttps://liberty.io/computational-complexity-of-computing-the-determinant.htmlThe formula for the determinant of an $n$ by $ n$ matrix given by expansion of minors involves $n!$ terms. As such, computing the determinant of a given matrix of with integer entries via expansion......Sun, 06 Sep 2026 07:34:28 -0400https://liberty.io/how-to-solve-23-2020-2020-bmod-37-please-see-the-body-of-the-question.htmlhttps://liberty.io/how-to-solve-23-2020-2020-bmod-37-please-see-the-body-of-the-question.htmlHow to solve $$23^{{2020}^{2020}} \mod 37.$$ Below given is my understanding of trying to solve the problem. From $$x^{p-1} = 1 \mod p$$ I deduce that $$23^{2020} \mod 37$$ would be $$23^{56.36+4}......Sun, 06 Sep 2026 07:32:11 -0400https://liberty.io/books-like-terence-tao-039-s-an-introduction-to-measure-theory.htmlhttps://liberty.io/books-like-terence-tao-039-s-an-introduction-to-measure-theory.htmlThis is an attempt to get a list going of books in mathematics (algebra, topology, geometry etc.) that in the spirit of Terence Tao's An Introduction to Measure Theory: According to my viewpoint, the ...Sun, 06 Sep 2026 07:29:43 -0400https://liberty.io/determine-a-in-the-system-such-that-the-system-is-consistent.htmlhttps://liberty.io/determine-a-in-the-system-such-that-the-system-is-consistent.htmlThe system is: $x_1 + 2x_2 - x_3 = 2, $ $2x_1 - x_2 + x_3 = 1 $, $-x_1 + 4x_2 -2x_3 = a $ To begin solving the system, I did Row 2 - 2(Row 1). Row 3 + Row 1, then Row 3 + Row 2. This left me with \......Sun, 06 Sep 2026 07:28:53 -0400https://liberty.io/finding-derivative-of-three-variables.htmlhttps://liberty.io/finding-derivative-of-three-variables.htmlConsider a box with dimensions x, y, and z. x is changing at a rate of 1 m/s, y at -2 m/s and z at 1 m/s. Find the rate that the volume, surface area and diagonal length ($s = \sqrt{x^2+y^2+z^2}$) ......Sun, 06 Sep 2026 07:19:14 -0400https://liberty.io/how-do-i-find-out-the-value-of-p-for-sin-p-cos-p-0.htmlhttps://liberty.io/how-do-i-find-out-the-value-of-p-for-sin-p-cos-p-0.htmlI tried doing this by putting $$\frac{ \sin p}{\sqrt2} +\frac{ \cos p}{\sqrt2} = 0 $$ which implies $$ \sin(p + π/4) = \sin 0 $$ which implies $p+ π/4 = nπ $. Now according to the question I'm solv......Sun, 06 Sep 2026 07:06:48 -0400https://liberty.io/why-does-unary-minus-operator-sometimes-take-precedence-over-exponenti.htmlhttps://liberty.io/why-does-unary-minus-operator-sometimes-take-precedence-over-exponenti.htmlHow should I evaluate 2*-2^3? Which one of these two is the correct one? 2*((-2)^3) 2*(-(2^3)) I was wondering what was the correct operator precedence. Everyone was taught at school to evaluate ...Sun, 06 Sep 2026 07:02:18 -0400https://liberty.io/an-intuitive-approach-to-the-jordan-normal-form.htmlhttps://liberty.io/an-intuitive-approach-to-the-jordan-normal-form.htmlI want to understand the meaning behind the Jordan Normal form, as I think this is crucial for a mathematician. As far as I understand this, the idea is to get the closest representation of an arbi......Sun, 06 Sep 2026 07:00:42 -0400https://liberty.io/how-many-ways-can-seven-people-sit-around-a-circular-table.htmlhttps://liberty.io/how-many-ways-can-seven-people-sit-around-a-circular-table.htmlHow many ways seven people can sit around a circular table? For first, I thought it was $7!$ (the number of ways of sitting in seven chairs), but the answer is $(7-1)!$. I don't understand how si......Sun, 06 Sep 2026 06:58:44 -0400https://liberty.io/how-to-find-the-number-of-arbitrary-constants-in-solution-of-a-given-d.htmlhttps://liberty.io/how-to-find-the-number-of-arbitrary-constants-in-solution-of-a-given-d.htmlHow to find the number of arbitrary constants in solution of a given differential equation ?...Sun, 06 Sep 2026 06:52:38 -0400https://liberty.io/redudant-condition-for-wiener-process-continuity.htmlhttps://liberty.io/redudant-condition-for-wiener-process-continuity.htmlIn my book, a Wiener process is defined as a process which has, amongst others, the conditions $W_t - W_s \sim \mathcal{N}(0,t - s), t > s,$ and that the process has a "continuous trajectory". ......Sun, 06 Sep 2026 06:51:01 -0400https://liberty.io/how-to-compute-the-volume-of-intersection-between-two-hyperspheres.htmlhttps://liberty.io/how-to-compute-the-volume-of-intersection-between-two-hyperspheres.htmlLet's say I have two n-spheres and I've no prior knowledge about the spheres (such as one of the sphere might be inside the other one) and I need to compute the volume of the intersection of the two ...Sun, 06 Sep 2026 06:44:10 -0400https://liberty.io/matrix-addition-can-a-3x2-matrix-a-2x3-matrix-be-added-together.htmlhttps://liberty.io/matrix-addition-can-a-3x2-matrix-a-2x3-matrix-be-added-together.htmlI have two matrices; $$ 1) \left( \begin{matrix} 3 & -2 \\ 1 & 2 \\ 0 & -3\\ \end{matrix} \right) $$ And $$ 2) \left( \begin{matrix} -2 & 1 & 3 \\ 4 & ......Sun, 06 Sep 2026 06:39:23 -0400https://liberty.io/how-to-calclulate-1-17-mod-60-closed.htmlhttps://liberty.io/how-to-calclulate-1-17-mod-60-closed.htmlhow can I calculate $ d = 17^{-1} (\text{mod} ~ 60) $ ? I was reading this article and then I wrote down this steps: 60 = 3 * 17 + 9 17 = 1 * 9 + 8 9 = 1 * 8 + 1 In the end (using EEA) I get x = ......Sun, 06 Sep 2026 06:31:44 -0400https://liberty.io/how-to-solve-the-inequality-of-x-2-2x-4-0.htmlhttps://liberty.io/how-to-solve-the-inequality-of-x-2-2x-4-0.htmlI'm reading Calculus with Analytic Geometry and in a problem in the first chapter (page 6) Solve the following inequalities $x^2 + 2x + 4 > 0$ Apparently, $x^2 + 2x + 4$ has no solution in real ...Sun, 06 Sep 2026 06:30:41 -0400https://liberty.io/how-to-simplify-sqrt-3-2.htmlhttps://liberty.io/how-to-simplify-sqrt-3-2.htmlMy textbook says to do the following steps to simplify the $\sqrt{3/2}$: $\sqrt{3/2}=\sqrt{((3/2)\times (2/2))} =\sqrt{6/4} = \sqrt{6}/\sqrt{4}=\sqrt{6}/2.$ However, if the fourth step is valid (1. ...Sun, 06 Sep 2026 06:21:07 -0400https://liberty.io/what-is-prime-gaps-relationship-with-number-6.htmlhttps://liberty.io/what-is-prime-gaps-relationship-with-number-6.htmlOut of the 78499 prime number under 1 million. There are 32821 prime gaps (difference between two consecutive prime numbers) of a multiple 6. A bar chart of differences and frequency of occurrence ......Sun, 06 Sep 2026 06:12:45 -0400https://liberty.io/find-the-number-of-real-values-of-x-such-that-x-2-10000-x-10000x-is-th.htmlhttps://liberty.io/find-the-number-of-real-values-of-x-such-that-x-2-10000-x-10000x-is-th.htmlFind the number of real values of $x$ such that $x^2 + 10000[x] = 10000x$ ($[]$ is the floor function) What I Tried: If $x$ is an integer then :- $$x^2 = 0$$ $$\rightarrow x = 0$$ So $x = 0$ is th......Sun, 06 Sep 2026 06:10:58 -0400https://liberty.io/why-do-we-say-pi-is-the-ratio-of-the-circumference-to-the-diameter-and.htmlhttps://liberty.io/why-do-we-say-pi-is-the-ratio-of-the-circumference-to-the-diameter-and.htmlEverywhere I see it written, $\pi$ is described as "the ratio of the circumference to the diameter". I know $\pi = C/d$, but the ratio $3.14...:1$ is $3.14$... diameters to $1$ circumference. So ...Sun, 06 Sep 2026 06:06:38 -0400https://liberty.io/cross-products-and-unit-circle.htmlhttps://liberty.io/cross-products-and-unit-circle.htmlI am first year maths student at university. I am currently studying cross products. For this question, I managed to solve the first 2 parts but am struggling with the general question at the e......Sun, 06 Sep 2026 06:00:25 -0400https://liberty.io/how-to-find-the-height-of-a-rectangle-given-its-area-and-the-differenc.htmlhttps://liberty.io/how-to-find-the-height-of-a-rectangle-given-its-area-and-the-differenc.htmlThe rectangular base is $4.50 \text{ cm}$ shorter than the height and the rectangle has a surface area of $135 \text{ cm}$. Solve the rectangle height with an equation. I know and feel that this is ...Sun, 06 Sep 2026 05:55:09 -0400https://liberty.io/velocity-increase-while-acceleration-decrease.htmlhttps://liberty.io/velocity-increase-while-acceleration-decrease.htmlI'm having trouble understanding velocity and acceleration. Is it possible for velocity to be increasing while acceleration decreases and vice versa?...Sun, 06 Sep 2026 05:46:25 -0400https://liberty.io/find-the-exact-value-of-the-trigonometric-function-sin-7-pi-6.htmlhttps://liberty.io/find-the-exact-value-of-the-trigonometric-function-sin-7-pi-6.htmlI am finding it a little difficult to solve this problem. The reference angle for $\sin 7\pi/6$ is sin 30 degrees (I think) which is sine 1/2. But that is not the answer. How do I sove this problem?...Sun, 06 Sep 2026 05:28:51 -0400https://liberty.io/integration-by-parts-when-can-you-not-use-the-table-method-why.htmlhttps://liberty.io/integration-by-parts-when-can-you-not-use-the-table-method-why.htmlI'm learning about integration by parts, primarily from Stewart's text (7th edition). In a supplemental book I have it brings up something called the Table Method. I really find this method appealing ...Sun, 06 Sep 2026 05:27:01 -0400https://liberty.io/why-do-we-have-to-calculate-1-binomcdf-for-p-x-11.htmlhttps://liberty.io/why-do-we-have-to-calculate-1-binomcdf-for-p-x-11.htmlWhy do we have to calculate 1-binomcdf for P(X>11)? Can't we just calculate binomcdf with left and right bound like the normalcdf?...Sun, 06 Sep 2026 05:26:40 -0400https://liberty.io/the-infinitely-nested-radicals-problem-and-ramanujan-039-s-wondrous-fo.htmlhttps://liberty.io/the-infinitely-nested-radicals-problem-and-ramanujan-039-s-wondrous-fo.htmlIn mathematics, a nested-radical is any expression where a radical (or root sign) is nested inside another radical, eg. $\sqrt{2 + \sqrt{3}}$. By extension, an Infinitely nested radical (aka, a Con......Sun, 06 Sep 2026 05:23:01 -0400https://liberty.io/writing-parametric-curve-in-cartesian-form-closed.htmlhttps://liberty.io/writing-parametric-curve-in-cartesian-form-closed.html$$x=4 \ln(t), \qquad y=2t$$ Write it in Cartesian form, giving $y$ explicitly in terms of $x$....Sun, 06 Sep 2026 05:22:06 -0400https://liberty.io/what-do-mean-and-what-does-it-mean-if-it-is-used-in-an-equation.htmlhttps://liberty.io/what-do-mean-and-what-does-it-mean-if-it-is-used-in-an-equation.htmlWhat do the square bracket symbols mean? Are they what I hear are "sets"? And when it is in an equation, how is it interpreted? Here is an example: $$\dfrac{dy}{dx}[2x^{2}+y(x)^{2}]=50x+2y(dy/dx)......Sun, 06 Sep 2026 05:20:53 -0400https://liberty.io/evaluate-the-angle-x-in-terms-of-angles-and-y-and-z.htmlhttps://liberty.io/evaluate-the-angle-x-in-terms-of-angles-and-y-and-z.htmlI need help with this problem and problems like this. I know that to solve this problem I have to find which angles are the same and etc, but how do I know that. How can I see that two angles are the ...Sun, 06 Sep 2026 05:11:41 -0400https://liberty.io/are-the-product-and-sum-of-two-householder-matrices-also-householder-m.htmlhttps://liberty.io/are-the-product-and-sum-of-two-householder-matrices-also-householder-m.htmlSuppose we have two Householder matrices $$H_i=I_n-2v_iv_i^T$$ where $\|v_i\|_2 = 1$ for $i \in \{1,2\}$. Is it true that $H_1 H_2$ is also a Householder matrix? Is it true that $H_1 + H_2$ is......Sun, 06 Sep 2026 05:09:35 -0400https://liberty.io/evaluate-the-indefinite-integral.htmlhttps://liberty.io/evaluate-the-indefinite-integral.html$ I = \int (x^2 + 2x)\cos(x) dx $ Integration by Parts, choose $u$: $$\begin{align} u &= \cos(x) \\ dv &= (x^2 + 2x)dx \\ du &= -\sin(x) \\ v &= \frac{1}{3}x^3 + x^2 \end{align} $$ ...Sun, 06 Sep 2026 04:53:37 -0400https://liberty.io/what-is-the-meaning-of-a-space-time-white.htmlhttps://liberty.io/what-is-the-meaning-of-a-space-time-white.htmlFor reference this is from page 27 here Consider the stochastic heat equation $$ \begin{cases} \dfrac{\partial u}{\partial t}(x,t)=\dfrac{\partial ^2 u}{\partial x^2}(x,t) +f(u(x,t))\dot{W}(x,t) &a......Sun, 06 Sep 2026 04:53:34 -0400https://liberty.io/ramanujan-nagell-theorem-proof-question.htmlhttps://liberty.io/ramanujan-nagell-theorem-proof-question.htmlI'm currently working through Stewart and Tall's Algebraic Number Theory. In particular, section 4.9 of this book provides a proof of the Ramanujan-Nagell Theorem, which states that the only integer ...Sun, 06 Sep 2026 04:53:27 -0400https://liberty.io/how-to-find-the-inverse-of-a-rational-function-when-one-or-both-of-the.htmlhttps://liberty.io/how-to-find-the-inverse-of-a-rational-function-when-one-or-both-of-the.htmlIf rational functions have the form $f(x)=\frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials, how do you find the inverse of $f(x)$ when $P(x)$ and/or $Q(x)$ are polynomials of degree $2$ or ...Sun, 06 Sep 2026 04:48:54 -0400https://liberty.io/about-graph-symmetric-with-respect-to-the-origin.htmlhttps://liberty.io/about-graph-symmetric-with-respect-to-the-origin.htmlWhy it is called a reflection about the y-axis followed by a reflection about the x-axis?...Sun, 06 Sep 2026 04:46:56 -0400https://liberty.io/how-many-factors-of-10-in-100-duplicate.htmlhttps://liberty.io/how-many-factors-of-10-in-100-duplicate.htmlPossible Duplicate: Highest power of a prime $p$ dividing $N!$ How many factors of 10 are there in $100!$ (IIT Question)? Is it 26,25,24 or any other value Please tell how you have done it...Sun, 06 Sep 2026 04:45:59 -0400https://liberty.io/geometry-prove-that-these-3-points-don-039-t-lie-on-the-same-line-clos.htmlhttps://liberty.io/geometry-prove-that-these-3-points-don-039-t-lie-on-the-same-line-clos.htmlRectangular coordinate system in the plane The Points are given - $A(5,0) B(1,3) C(-2,0)$. Prove that these $3$ points don't lie on the same line and find the surface of the triangle $ABC$....Sun, 06 Sep 2026 04:42:17 -0400https://liberty.io/combinatorial-proofs-having-a-difficult-time-understanding-how-to-writ.htmlhttps://liberty.io/combinatorial-proofs-having-a-difficult-time-understanding-how-to-writ.htmlCan someone explain how combinatorial proofs work? I've included an example questions that's been giving me a hard time. Any insight on the topic would be great. $$\sum_{k=1}^{n}k{n \choose k} =......Sun, 06 Sep 2026 04:36:37 -0400https://liberty.io/what-is-the-geometric-meaning-of-singular-matrix.htmlhttps://liberty.io/what-is-the-geometric-meaning-of-singular-matrix.htmlCould anyone help explain what is the geometric meaning of singular matrix ? What's the difference between singular and non-singular matrix ? I know the definition, but couldn't understand it very ......Sun, 06 Sep 2026 04:33:11 -0400https://liberty.io/what-is-the-correct-way-to-read-f-circ-g.htmlhttps://liberty.io/what-is-the-correct-way-to-read-f-circ-g.htmlLet $f : X \to Y$ and $g : Y \to Z$ be functions. We define the composition $g \circ f : X \to Z$ by $g \circ f(x) = g(f(x))$ for each $x \in X$. I have also heard the composition read out like th......Sun, 06 Sep 2026 04:32:23 -0400https://liberty.io/what-is-the-definition-of-a-positive-integer.htmlhttps://liberty.io/what-is-the-definition-of-a-positive-integer.htmlI am reading the book "The Number-System of Algebra (2nd edition)". At the starting of page-4 the author writes: A positive integer is a symbol for the number of things in a group of distinct ...Sun, 06 Sep 2026 04:27:08 -0400https://liberty.io/should-i-use-ds-or-dr.htmlhttps://liberty.io/should-i-use-ds-or-dr.htmlThe question states "Integrate $f(x, y, z) = x + \sqrt y -z^2$ over the patch $C = C_1 + C_2$ and they proceed to give $C_1$ and $C_2$ as $r_1(t)$ and $r_1(t)$. I've been dealing with 2 forms of line ...Sun, 06 Sep 2026 04:25:50 -0400https://liberty.io/gre-math-subject-test-closed.htmlhttps://liberty.io/gre-math-subject-test-closed.htmlI am studying for GRE Math. I am looking for specific tips. What types of questions usually come up? Does anyone know any tricks (e.g. integration tricks) that might be helpful? Which theorems are ...Sun, 06 Sep 2026 04:17:09 -0400https://liberty.io/why-is-x-5-6x-3-irreducible.htmlhttps://liberty.io/why-is-x-5-6x-3-irreducible.htmlIn my math course, it is written that since over $\mathbb{F}_5[X]$, $P$ has no root in $\mathbb{F}_{25}$, it is irreducible over $\mathbb{Q}[X]$ by Gauss Lemma. I don't really understand… If we p......Sun, 06 Sep 2026 04:14:58 -0400https://liberty.io/the-introduction-rule-for-implication.htmlhttps://liberty.io/the-introduction-rule-for-implication.htmlThe introduction rule for implication says that you can prove a statement of the form $P \implies Q$ by assuming $P$ and deducing $Q$. (Btw. I don't know why it's called an introduction rule - you ......Sun, 06 Sep 2026 04:14:02 -0400https://liberty.io/indefinite-integral-of-absolute-value-of-x-is-there-a-closed-form-solu.htmlhttps://liberty.io/indefinite-integral-of-absolute-value-of-x-is-there-a-closed-form-solu.htmlBeen searching the net for awhile and everything just comes back about doing the definite integral. So just thought to ask here. Title says it all. Is there a closed form solution for the indefin......Sun, 06 Sep 2026 04:11:30 -0400https://liberty.io/what-is-null-quantification.htmlhttps://liberty.io/what-is-null-quantification.htmlI want to know the definition of null quantification and when and why we need to use it. In a book I was learning, it says: "establish rules for null quantification that we can use when a ...Sun, 06 Sep 2026 03:58:59 -0400https://liberty.io/what-exactly-happens-when-i-do-a-bit-rotation.htmlhttps://liberty.io/what-exactly-happens-when-i-do-a-bit-rotation.htmlI'm sitting in my office and having difficulties to get to know that exactly happens, when I do a bit rotation of a binary number. An example: I have the binary number 1110. By doing bit rotatio......Sun, 06 Sep 2026 03:56:15 -0400https://liberty.io/odds-of-drawing-an-ace-on-the-first-draw-or-a-two-on-the-second.htmlhttps://liberty.io/odds-of-drawing-an-ace-on-the-first-draw-or-a-two-on-the-second.htmlI'm working on a puzzle which describes a casino game where you draw 13 cards. You win if your first card is an ace, or your second is a two, or... all the way up to the 13th draw being a king. In ...Sun, 06 Sep 2026 03:51:52 -0400https://liberty.io/applications-of-the-mean-value-theorem.htmlhttps://liberty.io/applications-of-the-mean-value-theorem.htmlWhat are some interesting applications of the Mean Value Theorem for derivatives? Both the 'extended' or 'non-extended' versions as seen here are of interest. So far I've seen some trivial applica......Sun, 06 Sep 2026 03:24:47 -0400https://liberty.io/interview-question-rolling-dice-of-posterior-probability.htmlhttps://liberty.io/interview-question-rolling-dice-of-posterior-probability.htmlYou have two 6-sided dice. You roll them and get a total of 10. You roll again and get a total of 9. If you get paid 1$ for each 5 that is rolled what is your expected winnings？ I guess it is the ...Sun, 06 Sep 2026 03:05:23 -0400https://liberty.io/sketch-the-graph-of-x-3-27.htmlhttps://liberty.io/sketch-the-graph-of-x-3-27.htmlNot sure if this is the right way to solve. The question ask to sketch the graph of each function by transforming the graph of an appropriate function of the form $y=x^n$. Indicate all $x$- and $......Sun, 06 Sep 2026 03:01:25 -0400https://liberty.io/peterson-graph-non-hamiltonian-proof-explanation.htmlhttps://liberty.io/peterson-graph-non-hamiltonian-proof-explanation.htmlI'm working on graph theory and I'm trying to find a generalised elegant proof to non-Hamiltonian graphs. I stumbled onto this proof from D. West, which is simple, but I'm having trouble understand......Sun, 06 Sep 2026 02:55:11 -0400https://liberty.io/how-can-i-find-the-gradient-vector-for-this-function.htmlhttps://liberty.io/how-can-i-find-the-gradient-vector-for-this-function.html$ x^4 + y^4 + z^4 = x^2 + y^2 +z^2$ How can I find the gradient vector of $f(x,y,z)$ at $(1,1,1)$? Can I rewrite the equation by just moving everything to one side? So: $x^4 - x^2 + y^4 -y^2 +z^......Sun, 06 Sep 2026 02:54:40 -0400https://liberty.io/differentiability-implies-continuity-a-question-about-the-proof.htmlhttps://liberty.io/differentiability-implies-continuity-a-question-about-the-proof.htmlI have a question, to aid my understanding, about the proof that differentiability implies continutity.$\mathstrut$ Differentiability Definition When we say a function is differentiable at $x_0......Sun, 06 Sep 2026 02:48:18 -0400https://liberty.io/the-jacobian-of-x-y-mapsto-x-y-2-y-x-2-under-the-substitution-u-x-y-2-.htmlhttps://liberty.io/the-jacobian-of-x-y-mapsto-x-y-2-y-x-2-under-the-substitution-u-x-y-2-.htmlI am given the map $(x,y)\mapsto (x+y^2,y+x^2)$. I am unable to find the Jacobian by making the substitution $u=x+y^2$ and $v=y+x^2$. Any hints would be appreciated. (I am trying to find whether t......Sun, 06 Sep 2026 02:47:00 -0400https://liberty.io/how-many-elements-of-a-3x3-rotation-matrix-are-redundant.htmlhttps://liberty.io/how-many-elements-of-a-3x3-rotation-matrix-are-redundant.htmlI read the following and got curious: A rotation matrix is an array of nine numbers. These are subject to the six norm and orthogonality constraints, so only three degrees of freedom are lef......Sun, 06 Sep 2026 02:42:02 -0400https://liberty.io/how-to-change-the-sign-of-the-dirac-delta-function-argument.htmlhttps://liberty.io/how-to-change-the-sign-of-the-dirac-delta-function-argument.htmlHow to proof the latter equality of, $$ f(a) = \int f(x)\,\delta(x-a)\,dx =\int f(x)\,\delta(a-x)\,dx. $$...Sun, 06 Sep 2026 02:30:05 -0400https://liberty.io/moment-generating-function-of-sample-mean-of-bernoulli-random-variable.htmlhttps://liberty.io/moment-generating-function-of-sample-mean-of-bernoulli-random-variable.htmlLet $p \in (0,1)$ and $n \in \mathbb{N}$. We consider a sample of $n$ i.i.d. Bernoulli variables $X_1,\dots,X_n$ with parameter p. Computer $E[e^{\lambda\bar{X_n}}]$ such that $\bar{X_n}= \frac{1}......Sun, 06 Sep 2026 02:30:01 -0400https://liberty.io/proper-normal-subgroup.htmlhttps://liberty.io/proper-normal-subgroup.htmlDefinition: A subgroup $N$ is a proper normal subgroup of a group $G$ IFF $Ng=gN \; \; \forall g \in G$ A proper normal subgroup $N$ of $G$ is a normal subgroup $N$ of $G$ that is not the whole of......Sun, 06 Sep 2026 02:29:04 -0400https://liberty.io/calculus-percentage-of-an-interval-for-which-a-function-is-increasing.htmlhttps://liberty.io/calculus-percentage-of-an-interval-for-which-a-function-is-increasing.htmlSo I have a question that asks, Given $f(x) = 7 + 10x ^ 3 - 5x ^ 4 - 2x ^ 5$ calculate the percentage of the interval $[-6, 4]$ on which $f(x)$ is increasing. So from this I determined that I m......Sun, 06 Sep 2026 02:28:40 -0400https://liberty.io/what-are-all-the-elements-of-a-5.htmlhttps://liberty.io/what-are-all-the-elements-of-a-5.htmlWhat are all the elements of $A_5?$ $$\{1,(1,2,3),(1,3,2),(1,2,4),(1,4,2),(1,2,5),(1,5,2),(1,3,4),(1,4,3),(1,3,5),(1,5,3),(1,4,5),(1,5,4),(2,3,4),(2,4,3),(2,3,5),(2,5,3),(2,4,5),(2,5,4)(3,4,5),(3,......Sun, 06 Sep 2026 02:08:57 -0400https://liberty.io/quad-to-quad-transformation.htmlhttps://liberty.io/quad-to-quad-transformation.htmlI am trying to find a transformation that maps each internal point of my first quadrilateral domain into each point of my second quadrilateral domain. Please see the following image: Each domain is ...Sun, 06 Sep 2026 02:04:27 -0400https://liberty.io/inversely-proportional-and-inversely-related.htmlhttps://liberty.io/inversely-proportional-and-inversely-related.htmlSuppose that we have a formula like $y=\frac{1}{\sqrt{x}}$. Is it correct to say that y is inversely proportional to x? what about y is inversely related to x? If not, what other phrase should we use?...Sun, 06 Sep 2026 02:04:01 -0400https://liberty.io/double-integral-over-area-bounded-by-ellipse-closed.htmlhttps://liberty.io/double-integral-over-area-bounded-by-ellipse-closed.htmlEvaluate the double integral of $\int \int(x+y)^{2} dxdy$ over the area bounded by the ellipse $(\frac{x}{a})^2+(\frac{y}{b})^2=1$ Please check answer = $\frac{ab\pi}{4}(a^{2}+b^{2})$...Sun, 06 Sep 2026 02:03:46 -0400https://liberty.io/quadratic-equation-has-integral-roots.htmlhttps://liberty.io/quadratic-equation-has-integral-roots.htmlFind all positive numbers $p$ for which the equation $x^2+px+3p = 0$ has integral roots. We have by the quadratic formula $$x = \dfrac{-p \pm \sqrt{p^2-12p}}{2}.$$ Thus, $p^2-12p = p(p-12)$ must b......Sun, 06 Sep 2026 01:59:57 -0400https://liberty.io/vectors-representing-the-principal-axes-of-an-ellipse.htmlhttps://liberty.io/vectors-representing-the-principal-axes-of-an-ellipse.htmlThe principal axes of the ellipse given by the equation $ 9x^2+6y^2-4xy=4$ are along the directions of the vectors a) $2i+3j$ and $3i-2j$ b) $2i+j$ and $i-2j$ c) $3i+2j$ and $2i-3j$ d) $i+2j$ and ......Sun, 06 Sep 2026 01:57:41 -0400https://liberty.io/how-to-make-a-given-matrix-symmetric.htmlhttps://liberty.io/how-to-make-a-given-matrix-symmetric.htmlGiven the real square block matrix $ M = \begin{pmatrix} A & D^t & 0\\ D & c E & c F\\ 0 & H & J \end{pmatrix} $ how can I make it symmetric? Note that the dimension of $......Sun, 06 Sep 2026 01:46:28 -0400https://liberty.io/experimental-analysis-of-an-algorithm-how-to-prove-that-the-graph-is-o.htmlhttps://liberty.io/experimental-analysis-of-an-algorithm-how-to-prove-that-the-graph-is-o.htmlThis question is probably stupid, but I've been trying to figure this out for hours and I still couldn't find anything about it. Probably I'm just too lost. So basically, I'm analysing an algorithm......Sun, 06 Sep 2026 01:36:05 -0400https://liberty.io/probability-of-not-occurring-either-a-or-b.htmlhttps://liberty.io/probability-of-not-occurring-either-a-or-b.htmlIn an event, $P(A)=0.6$, $P(B)=0.2$, $P(A ∩ B)=0.1$. what is the probability of not occurring either A or B. I am facing problem with the language. I don't understand what "not occurring either ......Sun, 06 Sep 2026 01:24:53 -0400https://liberty.io/find-the-ratio-of-area-to-volume-of-a-partial-cylinder.htmlhttps://liberty.io/find-the-ratio-of-area-to-volume-of-a-partial-cylinder.htmlConsider a cylindrical container of negligible thickness with radius $r$ and height $h$, placed sideways containing some liquid (with volume $V$) inside as shown below. Define a function $f(r,h,V)......Sun, 06 Sep 2026 01:21:58 -0400https://liberty.io/how-to-find-the-determinant-of-a-5x5-matrix.htmlhttps://liberty.io/how-to-find-the-determinant-of-a-5x5-matrix.htmlHow do I find the determinant of this? $$\begin{bmatrix} 0& 6& −2& −1& 5\\ 0& 0& 0& −9& −7\\ 0& 15& 35& 0& 0\\ 0 &−1 &−11& −2& 1\\ −2......Sun, 06 Sep 2026 01:21:12 -0400https://liberty.io/proving-that-the-cone-is-not-a-smooth-manifold.htmlhttps://liberty.io/proving-that-the-cone-is-not-a-smooth-manifold.htmlShow that the cone given by $C = \{(x, y, z) \in \mathbb{R}^3 \mid z = \sqrt{x^2 + y^2}\}$ is not a smooth manifold In the definition I'm using of a smooth manifold, each point $x \in M \subseteq \...Sun, 06 Sep 2026 01:19:11 -0400https://liberty.io/show-that-the-graph-of-y-x-3-sin-pi-x-extends-to-a-smooth-arc.htmlhttps://liberty.io/show-that-the-graph-of-y-x-3-sin-pi-x-extends-to-a-smooth-arc.htmlHere's the problem: Let $y(x)$ be a real-valued function defined on the interval $x\in [0,1]$ by means of the equation $$y(x)= \left\{ \begin{array}{lr} x^3\sin(\frac{\pi}{x}) \;\;\;\ma......Sun, 06 Sep 2026 01:17:25 -0400https://liberty.io/pythagorean-triples-and-perfect-squares.htmlhttps://liberty.io/pythagorean-triples-and-perfect-squares.htmlThis problem is giving me difficulty: Show that in any Pythagorean triple there exist at most a single perfect square So far I've been working with the equations for primitive Pythagorean Tripl......Sun, 06 Sep 2026 01:16:15 -0400https://liberty.io/trigonometric-functions-sin-cos-tan-differences-actual-applications.htmlhttps://liberty.io/trigonometric-functions-sin-cos-tan-differences-actual-applications.htmlBeforehand, I want to point out that I'm studying some basic geometric function and I'm pretty behind with this topic, hence may write something not too clever or my question might be silly; I ask ......Sun, 06 Sep 2026 01:10:03 -0400https://liberty.io/derivative-of-softmax-loss-function.htmlhttps://liberty.io/derivative-of-softmax-loss-function.htmlI am trying to wrap my head around back-propagation in a neural network with a Softmax classifier, which uses the Softmax function: \begin{equation} p_j = \frac{e^{o_j}}{\sum_k e^{o_k}} \end{equat......Sun, 06 Sep 2026 01:02:36 -0400https://liberty.io/solve-f-x-f-2x-2-f-2x-3-x.htmlhttps://liberty.io/solve-f-x-f-2x-2-f-2x-3-x.htmlThe full question is to solve $f(x)f(2x^2) = f(2x^3+x)$, prove there are at most one solution per degree of $f(x)$ and then find all the solutions. (Here $f(x)$ is assumed to be polynomial.) So fa......Sun, 06 Sep 2026 01:00:16 -0400https://liberty.io/connecting-midpoints-of-sides-of-a-triangle.htmlhttps://liberty.io/connecting-midpoints-of-sides-of-a-triangle.htmlIn triangle $\triangle ABC$, $AB=8$, $BC=14$ and $CA=10$. Points $M$, $N$, and $P $ are the midpoints of sides $AB$, $BC$, and $CA$, respectively. If $M$, $N$, and $P$ are connected to form a trian......Sun, 06 Sep 2026 00:58:34 -0400https://liberty.io/brownian-motion-characteristic-function.htmlhttps://liberty.io/brownian-motion-characteristic-function.htmlLet me remind first the construction of Brownian motion. Fix a vector $x \in \mathbb{R}^n$ and define $p(t,x,y) := (2\pi t )^{-\frac{n}{2}} \cdot \exp{\left( - \frac{|x-y|^2}{2t} \right)},$ for $y......Sun, 06 Sep 2026 00:36:49 -0400https://liberty.io/determine-all-functions-f-satisfying-the-functional-relation-f-x-f-lef.htmlhttps://liberty.io/determine-all-functions-f-satisfying-the-functional-relation-f-x-f-lef.htmlThe question is : Determine all functions f satisfying the functional relation $f(x)+f\left(\frac{1}{1-x}\right)=\frac{2(1-2x)}{x(1-x)}$ where $x$ is a real number . Also $x$ is not equal to $0$ or......Sun, 06 Sep 2026 00:35:02 -0400https://liberty.io/evaluating-int-0-2-tan-1-pi-x-tan-1-x-mathrm-d-x.htmlhttps://liberty.io/evaluating-int-0-2-tan-1-pi-x-tan-1-x-mathrm-d-x.htmlHint given: Write the integrand as an integral. I'm supposed to do this as double integration. My attempt: $$\int_0^2 [\tan^{-1}y]^{\pi x}_{x}$$ $$= \int_0^2 \int_x^{\pi x} \frac { \mathrm{d}y \...Sun, 06 Sep 2026 00:35:00 -0400https://liberty.io/what-does-the-long-039-039-mean-duplicate.htmlhttps://liberty.io/what-does-the-long-039-039-mean-duplicate.htmlI have often seen the long '|' symbol with a subscript expression afterward. What does this mean in mathematics? Here is an example I found from Wikipedia: $$\large\left.\frac{dy}{dx}\right|_{x=......Sun, 06 Sep 2026 00:28:54 -0400https://liberty.io/what-does-mathrm-re-x-mean.htmlhttps://liberty.io/what-does-mathrm-re-x-mean.htmlI see this all the time in Mathematica output as well as in text, such as near the top of the Wikipedia Beta function page....Sun, 06 Sep 2026 00:27:28 -0400https://liberty.io/how-to-find-a-data-value-using-only-standard-deviation.htmlhttps://liberty.io/how-to-find-a-data-value-using-only-standard-deviation.htmlI have a homework problem with the following data values: x, 17, 24, 13, 21. I need to find x, but I do not have the mean. I only have the standard deviation, which is square root of 14, or 3.745. ......Sun, 06 Sep 2026 00:20:07 -0400https://liberty.io/calculus-of-angular-and-linear-velocity.htmlhttps://liberty.io/calculus-of-angular-and-linear-velocity.htmlI have a question regarding the following forms in the case that both angular velocity and radius change per $dt$. Wiki defines angular velocity in two dimensions as: $$ \omega = \frac {d \phi} {d......Sun, 06 Sep 2026 00:11:32 -0400https://liberty.io/does-set-difference-distribute-over-set-intersection.htmlhttps://liberty.io/does-set-difference-distribute-over-set-intersection.htmlI am asked to prove that, if $A, B$ and $C$ are sets, then $$A-(B\cap C)=(A-B)\cap(A-C).$$ However, I think that either I have made an error in my working, or the wording of the problem contains a ...Sun, 06 Sep 2026 00:07:39 -0400https://liberty.io/if-two-one-sided-limits-exist-the-two-sided-limit-exists-duplicate.htmlhttps://liberty.io/if-two-one-sided-limits-exist-the-two-sided-limit-exists-duplicate.htmlprove that if the limit as x approaches a from the positive side of the x axis, f(x) = L, and if the limit as x approaches a from the negative side of the x axis, f(x) = L, then the limit as x appr......Sun, 06 Sep 2026 00:05:50 -0400https://liberty.io/gossip-problem-proof-by-induction.htmlhttps://liberty.io/gossip-problem-proof-by-induction.htmlQuestion Suppose there are $n$ people in a group, each aware of a scandal no one else in the group knows about. These people communicate by telephone; when two people in the group talk, they ...Sun, 06 Sep 2026 00:03:54 -0400https://liberty.io/how-to-find-the-arc-measure-of-the-arc-cut-by-one-side-the-of-a-circum.htmlhttps://liberty.io/how-to-find-the-arc-measure-of-the-arc-cut-by-one-side-the-of-a-circum.htmlFirst, circumscribed means inside a circle, right? What does it exactly mean by cut by one side? A regular hexagon has side angles each $120 ^o$ so it has $240^o$ arc measure. But why is the answer......Sat, 05 Sep 2026 23:54:44 -0400https://liberty.io/a-confusion-about-the-second-connection-axiom-of-euclidean-geometry.htmlhttps://liberty.io/a-confusion-about-the-second-connection-axiom-of-euclidean-geometry.htmlIn the book of Foundations of Geometry by Hilbert, at page 2, it is stated that I, 1. Two distinct points A and B always completely determine a straight line a. We write AB = a or BA = a. ......Sat, 05 Sep 2026 23:42:45 -0400https://liberty.io/asymptotic-expansion-of-tanh-at-infinity.htmlhttps://liberty.io/asymptotic-expansion-of-tanh-at-infinity.htmlDoes $\tanh(x)$ have an asymptotic expansion for $x \rightarrow \infty$?...Sat, 05 Sep 2026 23:40:20 -0400https://liberty.io/the-foci-of-an-ellipse.htmlhttps://liberty.io/the-foci-of-an-ellipse.htmlIn Wikipedia the definition of and ellipse is the following. An ellipse is a set of points, such that for any point $P$ of the set, the sum of the distances $|PF_{1}|, |PF_{2}|$ to two fixed ...Sat, 05 Sep 2026 23:39:32 -0400https://liberty.io/the-high-power-integral.htmlhttps://liberty.io/the-high-power-integral.htmlIm trying to solve the indefinite integral $$\int\frac{x}{(x^2+4)^3} \, \mathrm{d}x $$ I tried applying polynimial division and breaking to partial fractions but it didnt help...are there any ot......Sat, 05 Sep 2026 23:33:15 -0400https://liberty.io/solving-a-fractional-quadratic-equation-problem-by-completing-the-squa.htmlhttps://liberty.io/solving-a-fractional-quadratic-equation-problem-by-completing-the-squa.htmlI have the following problem to solve using the method of completing the square. $$2x^2-3x-1 = 0$$ Here is where I've gotten to so far on this problem. $$2x^2-3x = 1$$ $$x^2-\frac{3}{2}x = \frac{1......Sat, 05 Sep 2026 23:23:22 -0400https://liberty.io/if-you-know-all-angles-and-the-area-of-the-triangle-how-do-you-find-th.htmlhttps://liberty.io/if-you-know-all-angles-and-the-area-of-the-triangle-how-do-you-find-th.htmlIf we know the angles and area of the triangle, what would be the formula to find the side lengths? For example, we know the area is 400 ft and the angles are 40, 30, and 110 degrees, how could we ......Sat, 05 Sep 2026 23:18:15 -0400https://liberty.io/why-the-inverse-of-a-matrix-involves-division-by-the-determinant.htmlhttps://liberty.io/why-the-inverse-of-a-matrix-involves-division-by-the-determinant.htmlI was studying inverse matrix. Suddenly I stumbled on the inverse of 3×3 K. And it involved a division by the determinant (Well, only with numbers it was). And it was also said, about involving a ...Sat, 05 Sep 2026 22:57:22 -0400https://liberty.io/integral-center-of-mass-of-a-right-angled-triangle.htmlhttps://liberty.io/integral-center-of-mass-of-a-right-angled-triangle.htmlHow to find the center of mass in the $x$ direction? (answer $h/3$) $$dm=\sigma (d-y)\,\mathrm dx$$ $$\frac{hx}{d}=y$$ $$\begin{eqnarray} x_{cm}&=&\frac{2}{dh\sigma} \int x \sigma\left(d-......Sat, 05 Sep 2026 22:44:30 -0400https://liberty.io/what-does-the-associative-property-actually-mean.htmlhttps://liberty.io/what-does-the-associative-property-actually-mean.htmlI look at $(a+b)+c = a+(b+c)$ for $a,b,c \in \mathbb{R}$ and think this tells me if I see three addends and two of them are in parentheses I can shift them without changing the sum. It obvious, at ......Sat, 05 Sep 2026 22:43:31 -0400