https://liberty.ioCurated star highlights with fresh energy.en-usSun, 16 Aug 2026 10:18:08 -0400https://liberty.io/cofinite-topology.htmlhttps://liberty.io/cofinite-topology.htmlIf we have topological space $\mathbb{R}$ equipped with the co finite topology. If we have finite subsets in consideration they are definitely closed because open sets are of the form complement of ...Sun, 16 Aug 2026 14:10:25 -0400https://liberty.io/please-help-how-to-work-the-inequation-x-1-x-5-0.htmlhttps://liberty.io/please-help-how-to-work-the-inequation-x-1-x-5-0.htmlI am studying for college finals and I cannot solve the inequation (x-1)/(x-5)<0 using the method the professor taught us, which I have to use in the exam. This is another inequation using said ...Sun, 16 Aug 2026 14:06:06 -0400https://liberty.io/trace-of-a-matrix-when-to-use-what-is-trace-trick.htmlhttps://liberty.io/trace-of-a-matrix-when-to-use-what-is-trace-trick.htmlOn calculating log-likelihood function for some multivariate distributions, such as multivariate Normal, I see some examples where the matrices are suddenly changed to trace, even when the matrix i......Sun, 16 Aug 2026 14:00:56 -0400https://liberty.io/relation-between-the-roots-of-a-cubic-equation-and-the-coefficients.htmlhttps://liberty.io/relation-between-the-roots-of-a-cubic-equation-and-the-coefficients.html$ax^3 +bx^2 + cx + d= 0$ If the roots are $\alpha$ $\beta$ and $\gamma$, Is there any relationship between the sum of the squares of the roots and the coefficients of the quadratic equation.. In ...Sun, 16 Aug 2026 14:00:17 -0400https://liberty.io/generators-of-the-symmetric-group-s-3.htmlhttps://liberty.io/generators-of-the-symmetric-group-s-3.htmlI am trying to find the generator(s) of the Symmetric Group $S_3$ and I have attempted this via brute force by listing the permutations of $S_3$ and composing and repeating them but I have not foun......Sun, 16 Aug 2026 13:59:25 -0400https://liberty.io/how-is-logistic-loss-and-cross-entropy-related.htmlhttps://liberty.io/how-is-logistic-loss-and-cross-entropy-related.htmlI found that Kullback-Leibler loss, log-loss or cross-entropy is the same loss function. Is the logistic-loss function used in logistic regression equivalent to the cross-entropy function? If yes, ......Sun, 16 Aug 2026 13:56:53 -0400https://liberty.io/limit-lim-x-to0-ln-x.htmlhttps://liberty.io/limit-lim-x-to0-ln-x.htmlCan i ask for the limit as $x$ approaches $\lim_{x \to 0-}\ln x$? Please explain. Its because since the limit of a function only exists if the lim as $x$ approaches some number $n$ from both the po......Sun, 16 Aug 2026 13:56:01 -0400https://liberty.io/what-is-the-definition-of-finitely-generated.htmlhttps://liberty.io/what-is-the-definition-of-finitely-generated.htmlTo be specific, here is an example. Note that $(\mathbb{Z},+)$ is a group. Definition 1. (Lang) Let $G$ be a group and $S\subset G$. If $G$ is the intersection of all subgroups $H$ of $G$ containi......Sun, 16 Aug 2026 13:40:53 -0400https://liberty.io/how-to-solve-the-limit-lim-limits-x-to-infty-x-arctan-x-frac-x-pi-2.htmlhttps://liberty.io/how-to-solve-the-limit-lim-limits-x-to-infty-x-arctan-x-frac-x-pi-2.htmlNext week I have a math exam. While I was doing some exercises I came across this interesting limit: $\lim\limits_{x\to \infty} (x \arctan x - \frac{x\pi}{2})$ After struggling a lot, I decided to ...Sun, 16 Aug 2026 13:36:22 -0400https://liberty.io/in-a-30-60-right-triangle-the-side-opposite-the-30-degree-angle-is-hal.htmlhttps://liberty.io/in-a-30-60-right-triangle-the-side-opposite-the-30-degree-angle-is-hal.htmlIn a 30-60 right triangle the side opposite the 30 degree angle is half the length of the hypotenuse. A statement from the trigonometry section of Simmons' Precalculus in a nutshell. Please explain....Sun, 16 Aug 2026 13:27:06 -0400https://liberty.io/what-are-the-odds-are-getting-your-6-digit-birth-date-in-your-10-digit.htmlhttps://liberty.io/what-are-the-odds-are-getting-your-6-digit-birth-date-in-your-10-digit.htmlI was issued a cellphone number that has my 6-digit birth date in it. Does anyone know how to calculate the odds of that happening? If there are 10 billion possible 10-digit phone numbers, and a 6-......Sun, 16 Aug 2026 13:25:42 -0400https://liberty.io/solving-x-d-equiv-a-pmod-29-using-primitive-roots.htmlhttps://liberty.io/solving-x-d-equiv-a-pmod-29-using-primitive-roots.htmlCan anyone please detail the general approach to questions of the form $x^d \equiv a \pmod{29}$? For example, Wolfram Alpha states that $x^5 \equiv 8 \pmod{29}$ has one solution, $x^4 \equiv 4 \pmo......Sun, 16 Aug 2026 13:25:36 -0400https://liberty.io/how-to-prove-this-limit-does-not-exist.htmlhttps://liberty.io/how-to-prove-this-limit-does-not-exist.htmlI'm a relative novice to epsilon-delta proofs. My professor assigned this practice problem and I'm having terrible trouble understanding the answer he gave. Moreover, I can't find a good account fo......Sun, 16 Aug 2026 13:18:42 -0400https://liberty.io/if-z-is-a-complex-number-whose-imaginary-part-is-non-zero-and-z-1-z-is.htmlhttps://liberty.io/if-z-is-a-complex-number-whose-imaginary-part-is-non-zero-and-z-1-z-is.htmlTwo questions from Grade Twelve class on complex numbers If $z$ is a complex number whose imaginary part is non zero, and $z + 1/z$ is real, what is $z$? How do you solve graphically given the mod......Sun, 16 Aug 2026 13:15:41 -0400https://liberty.io/write-the-piecewise-function-in-terms-of-unit-step-functions.htmlhttps://liberty.io/write-the-piecewise-function-in-terms-of-unit-step-functions.htmlWrite the piecewise function $f(t) = \begin{cases} 2t, & 0\leq t < 3 \\ 6, & 3 \le t < 5 \\ 2t, & t \ge 5 \\ \end{cases} $ in terms of unit step ...Sun, 16 Aug 2026 13:09:43 -0400https://liberty.io/what-039-s-the-name-for-this-weird-shape-why-is-there-a-kite-inside-of.htmlhttps://liberty.io/what-039-s-the-name-for-this-weird-shape-why-is-there-a-kite-inside-of.htmlA triangle, but I drew lines in the corners going out, and did the same with the mini-shapes formed: I have no idea what to call this thing. For a square, it forms a diamond inside of it. For a h......Sun, 16 Aug 2026 13:07:45 -0400https://liberty.io/example-of-a-function-that-is-not-continuous-at-0-0-but-continuous-whe.htmlhttps://liberty.io/example-of-a-function-that-is-not-continuous-at-0-0-but-continuous-whe.htmlI am looking for a continuous function $f: \mathbb{R}^2-\{0, 0\} \to \mathbb{R}$ satisfying the following condition: For any $g, h: \mathbb{R}\to\mathbb{R}$ continuous functions with $g(0)=h(0)=0$,......Sun, 16 Aug 2026 13:05:21 -0400https://liberty.io/the-mean-of-a-deterministic-sequence.htmlhttps://liberty.io/the-mean-of-a-deterministic-sequence.htmlcould someone explain to me why the expected value of $y(n)$ is the following: $\operatorname{E}(y(n)) = f(n)$ when $y(n) = x(n) + f(n)$ and $x(n)$ has zero mean. But why is the expected value o......Sun, 16 Aug 2026 12:59:33 -0400https://liberty.io/hexagon-maze-algorithm.htmlhttps://liberty.io/hexagon-maze-algorithm.htmlCan anyone suggest a good algorithm to create structures like this? Note that what I what I am asking for is not a true maze with one start and one solution. Rather, it's for a video game, so like......Sun, 16 Aug 2026 12:52:16 -0400https://liberty.io/the-rate-of-change-of-distance-between-two-airplanes-closed.htmlhttps://liberty.io/the-rate-of-change-of-distance-between-two-airplanes-closed.htmlAn airplane passes over an airport at noon traveling $540$ mi/hr due north. At 1:00pm, another airplane passes over the same airport at the same elevation traveling due east at $580$ mi/hr. Assumin......Sun, 16 Aug 2026 12:50:41 -0400https://liberty.io/find-a-one-parameter-family-of-solutions.htmlhttps://liberty.io/find-a-one-parameter-family-of-solutions.htmlIm stuck on this problem. Am I suppose to get the anti-derivative function of the differential and then plug in the initial value?...Sun, 16 Aug 2026 12:47:31 -0400https://liberty.io/solve-x-x-5-5-for-x-duplicate.htmlhttps://liberty.io/solve-x-x-5-5-for-x-duplicate.htmlA friend who tutors high school math showed me this equation. I could not solve it. But by trial & error in Python I discovered $\sqrt[5]{5}$ is the answer. Further realized that for any intege......Sun, 16 Aug 2026 12:42:56 -0400https://liberty.io/pronunciation-of-the-symbol-varnothing-of-the-empty-set.htmlhttps://liberty.io/pronunciation-of-the-symbol-varnothing-of-the-empty-set.htmlThe symbol $\varnothing$ for the empty set was introduced by Bourbaki, inspired by the Norwegian alphabet $\varnothing.$ It has no relation with the Greek letter $\phi.$ From my schooldays, when the ...Sun, 16 Aug 2026 12:26:43 -0400https://liberty.io/herstein-or-herstein.htmlhttps://liberty.io/herstein-or-herstein.htmlI read somewhere that Herstein would prepare me nicely (abstract algebra-wise) for grad studies. I don't remember where I read it and now that I was about to buy the book I found out that there two ...Sun, 16 Aug 2026 12:23:50 -0400https://liberty.io/consistency-of-peano-axioms-hilbert-039-s-second-problem.htmlhttps://liberty.io/consistency-of-peano-axioms-hilbert-039-s-second-problem.html(Putting aside for the moment that Wikipedia might not be the best source of knowledge.) I just came across this Wikipedia paragraph on the Peano-Axioms: The vast majority of contemporary ...Sun, 16 Aug 2026 12:14:46 -0400https://liberty.io/fourier-transform-of-even-odd-function.htmlhttps://liberty.io/fourier-transform-of-even-odd-function.htmlHow can I show that the Fourier transform of an even integrable function $f\colon \mathbb{R}\to\mathbb{R}$ is even real-valued function? And the Fourier transform of an odd integrable function $f\c......Sun, 16 Aug 2026 12:04:56 -0400https://liberty.io/solve-sqrt-x-2-4-x-2.htmlhttps://liberty.io/solve-sqrt-x-2-4-x-2.htmlSolve for x $$\sqrt{x^2-4}=x-2$$ My try: $$\sqrt{(x-2)(x+2)}=x-2$$ $$\sqrt{x+2}=\sqrt{x-2}$$ $$x+2=x-2$$ $$0x=4$$ This is not correct $x={4\over 0}$ How else can I solve this equation?...Sun, 16 Aug 2026 12:00:20 -0400https://liberty.io/difference-between-maximum-and-minimum.htmlhttps://liberty.io/difference-between-maximum-and-minimum.htmlIf I have a problem such as this: We need to enclose a field with a fence. We have 500m of fencing material and a building is on one side of the field and so won’t need any fencing. Determine the ...Sun, 16 Aug 2026 11:55:49 -0400https://liberty.io/find-exact-value-of-tan-when-given-cos.htmlhttps://liberty.io/find-exact-value-of-tan-when-given-cos.htmlGiven $\cos30 = \frac{\sqrt3}{2}$ use trigonometric identities to find the exact value of $\tan\frac{\pi}{3}$ I understand that $\cos30 = \frac{\sqrt3}{2}$ from the standard trig values chart and ......Sun, 16 Aug 2026 11:53:18 -0400https://liberty.io/simplify-this-problem-from-spivak-039-s-calculis.htmlhttps://liberty.io/simplify-this-problem-from-spivak-039-s-calculis.htmlI'm sorry I don't have the book with me (don't know the problem number) since I moved, but I still remember the problem. It said something about finding $\int x^n\ dx$ by letting $p_n = \int_0^1 x......Sun, 16 Aug 2026 11:49:20 -0400https://liberty.io/what-039-s-the-geometrical-meaning-of-immersion.htmlhttps://liberty.io/what-039-s-the-geometrical-meaning-of-immersion.htmlDoes that just mean to different tangent vectors, their images are different tangent vectors?...Sun, 16 Aug 2026 11:45:21 -0400https://liberty.io/solve-for-x-in-e-2-ln-x-ln-x-2-x-3-1.htmlhttps://liberty.io/solve-for-x-in-e-2-ln-x-ln-x-2-x-3-1.htmlSo the question is to solve for x in: $$e^{[2\ln(x)-\ln(x^2+x-3)]} = 1$$ I took the natural log of both sides, and simplified. Here is what I've gotten it down to: $$2\ln(x) = \ln(x^2+x-3)$$ And ......Sun, 16 Aug 2026 11:43:21 -0400https://liberty.io/establishing-a-covering-relation.htmlhttps://liberty.io/establishing-a-covering-relation.htmlThe problem I am working on is, "What is the covering relation of the partial ordering $\{(A,B)|A⊆B\}$ on the power set of $S$, where $S=\{a, b, c\}$?" I am reading the answer key, and I can follo......Sun, 16 Aug 2026 11:41:29 -0400https://liberty.io/a-curiosity-how-do-we-prove-mathbb-r-is-closed-under-addition-and-mult.htmlhttps://liberty.io/a-curiosity-how-do-we-prove-mathbb-r-is-closed-under-addition-and-mult.htmlSo I tried looking around for this question, but I didn't find much of anything - mostly unrelated-but-similarly-worded stuff. So either I suck at Googling or whatever but I'll get to the point. S......Sun, 16 Aug 2026 11:38:11 -0400https://liberty.io/what-do-you-call-the-product-of-a-matrix-039-s-diagonal-elements.htmlhttps://liberty.io/what-do-you-call-the-product-of-a-matrix-039-s-diagonal-elements.htmlThe trace of $A$, an $N\times N$ matrix, is $\displaystyle\sum_{i=1}^N A_{ii}$. What do you call $\displaystyle\prod_{i=1}^N A_{ii}$?...Sun, 16 Aug 2026 11:32:17 -0400https://liberty.io/what-does-a-complex-root-signify.htmlhttps://liberty.io/what-does-a-complex-root-signify.htmlWhat does it tell me when I find that a polynomial has complex roots, except for the obvious fact that it crosses zero for these values? To me it seems that the existance of complex roots must hav......Sun, 16 Aug 2026 11:30:31 -0400https://liberty.io/multiplicative-property-of-trace.htmlhttps://liberty.io/multiplicative-property-of-trace.htmlLet $T_1 \in \mathcal{L}(V)$ and $T_2 \in \mathcal{L}(V)$ be positive operators. Prove that the trace of their product is non-negative i.e., tr($T_1 T_2) \geq 0$ Attempt 1: Obviously, a positive op......Sun, 16 Aug 2026 11:24:20 -0400https://liberty.io/why-is-the-approximation-of-hessian-j-tj-reasonable.htmlhttps://liberty.io/why-is-the-approximation-of-hessian-j-tj-reasonable.htmlI met this equation frequently in Guass-Newton optimizations. But I dont understand why the left and right side of the equation can be equal. Lets say the Jacobian is $2$ by $2$ and Hessian is $$\...Sun, 16 Aug 2026 11:23:41 -0400https://liberty.io/convergence-in-distribution-of-differences-sums-of-random-variables.htmlhttps://liberty.io/convergence-in-distribution-of-differences-sums-of-random-variables.htmlSuppose $X_n$ and $Y_n$ are sequences of random variables, defined on a common probability space, and such that $X_n-Y_n \to C$ converges in distribution to the constant r.v. $C\equiv c$ as $n \to ......Sun, 16 Aug 2026 11:05:31 -0400https://liberty.io/jordan-decomposition-of-idempotent-matrix.htmlhttps://liberty.io/jordan-decomposition-of-idempotent-matrix.htmlMatrix A $\in$ $\mathbb{R}^{n\times n}$ is idempotent if $A^{2} = A$. Describe the Jordan form of A. How do I do this? I am able to decompose a matrix to its Jordan form given that the matrix conta......Sun, 16 Aug 2026 10:57:11 -0400https://liberty.io/is-every-vector-field-the-gradient-of-a-scalar-field.htmlhttps://liberty.io/is-every-vector-field-the-gradient-of-a-scalar-field.htmlI was wondering whether every vector field is the gradient of a certain scalar field. If not, could you provide a counterexample? Thanks....Sun, 16 Aug 2026 10:55:55 -0400https://liberty.io/can-a-connected-eulerian-graph-have-an-even-number-of-vertices-and-an-.htmlhttps://liberty.io/can-a-connected-eulerian-graph-have-an-even-number-of-vertices-and-an-.htmlI know that the requirements for a Eulerian graph are that all vertex degrees are even and that it is connected. But I am not sure how that would work with the amount of vertices and edges. So, if......Sun, 16 Aug 2026 10:53:52 -0400https://liberty.io/laplace-transform-of-derivatives.htmlhttps://liberty.io/laplace-transform-of-derivatives.htmlIn an example in the book we are given the following ODE: $x''-x'-6x=0$. In addition we are given the following definitions: $L{f'(t)}=sF(s)-f(0)$ and $L{f''}=s^2-sf(0)-f'(0)$ The part I am con......Sun, 16 Aug 2026 10:42:53 -0400https://liberty.io/why-isn-039-t-1900-a-leap-year-closed.htmlhttps://liberty.io/why-isn-039-t-1900-a-leap-year-closed.htmlI searched leap years online and found that 1900 is not, contrary to what I thought, a leap year. But, why is it not if 1900 is divisible by 4: $\frac{1900}{4} = 475$ My brother was working on his......Sun, 16 Aug 2026 10:41:48 -0400https://liberty.io/how-do-i-find-the-taylor-series-of-f-x-5cos-x-a-11pi.htmlhttps://liberty.io/how-do-i-find-the-taylor-series-of-f-x-5cos-x-a-11pi.htmlI started the problem but I am confused how to set up. f(x) = 5cos(x) f(11pi) = -5 f'(x) = -5sin(x) f'(11pi) = 0 f''(x) = -5cos(x) f''(11pi) = -5 f'''(x) = 5sin(x) f'''......Sun, 16 Aug 2026 10:39:26 -0400https://liberty.io/are-there-any-examples-of-non-computable-real-numbers.htmlhttps://liberty.io/are-there-any-examples-of-non-computable-real-numbers.htmlIs this true, that if we can describe any (real) number somehow, then it is computable? For example, $\pi$ is computable although it is irrational, i.e. endless decimal fraction. It was just a lu......Sun, 16 Aug 2026 10:19:50 -0400https://liberty.io/prove-that-the-values-of-x-for-which-x-frac-x-2-1-198-lie-between-frac.htmlhttps://liberty.io/prove-that-the-values-of-x-for-which-x-frac-x-2-1-198-lie-between-frac.htmlWithout using a calculator, prove that the values of $x$ for which $x = \dfrac{x^2+1}{198}$ lie between $\dfrac{1}{198}$ and $199.99 \overline{49}$. Then use this result to prove that $\sqrt{2} &lt......Sun, 16 Aug 2026 10:08:11 -0400https://liberty.io/showing-a-function-does-not-have-a-local-max-or-min.htmlhttps://liberty.io/showing-a-function-does-not-have-a-local-max-or-min.htmlThe function is $f(x) = \begin{cases} x^2\sin(1/x) & \text{if $x \neq 0$} \\ 0 & \text{if $x = 0$} \end{cases} $ I have proved that $f$ is differentiable at $0$, and $f^{'}(0) = 0$. ......Sun, 16 Aug 2026 09:58:10 -0400https://liberty.io/show-that-the-square-of-any-prime-number-is-the-factof-of-some-integer.htmlhttps://liberty.io/show-that-the-square-of-any-prime-number-is-the-factof-of-some-integer.htmlThis is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with prime numbers, but other than that, the textbook gave no hints really a......Sun, 16 Aug 2026 09:54:33 -0400https://liberty.io/complementary-subspaces.htmlhttps://liberty.io/complementary-subspaces.htmlHow do I go about proving this? Two subspaces $V$ and $W$ of $\mathbb{R}^n$ are called complements if any vector $x$ in $\mathbb{R}^n$ can be expressed uniquely as $x = v+ w$ , where $v\in V$ and......Sun, 16 Aug 2026 09:26:41 -0400https://liberty.io/how-does-this-definition-of-the-exponential-function-have-e-0-1.htmlhttps://liberty.io/how-does-this-definition-of-the-exponential-function-have-e-0-1.htmlRudin defines a power series for the exponential function and says that $E(0)=1$. But when $z=0$, the first term in the series is $0^0$, which is undefined. Why is it equal to $1$?...Sun, 16 Aug 2026 09:24:41 -0400https://liberty.io/vector-analysis-calculate-the-flux-of-the-vector-field-s.htmlhttps://liberty.io/vector-analysis-calculate-the-flux-of-the-vector-field-s.html$\mathbf{A}(x,y,z)= (\frac{-6x}{x^2+y^2},{-6y}{x^2+y^2},z+1)$ $S: x^2+4y^2=4, 0 \leq z \leq 1$ The task is to calculate the flux away from the z-axis. Solution: The divergence theorem obviously......Sun, 16 Aug 2026 09:15:41 -0400https://liberty.io/how-to-represent-a-decimal-number-in-ternary-or-base-3-number-system.htmlhttps://liberty.io/how-to-represent-a-decimal-number-in-ternary-or-base-3-number-system.htmlI need to convert a decimal number into base $3$ number. $47_{10} ~~\mbox{is}~~ 1202_{3}$. But how do you represent a negative number in base $3$ notation like $-297$? please include a example...Sun, 16 Aug 2026 09:13:47 -0400https://liberty.io/arithmetic-mean-why-does-it-work.htmlhttps://liberty.io/arithmetic-mean-why-does-it-work.htmlI've been using the formula for the arithmetic mean all my life, but I'm not sure why it works. My current intuition is this one: The arithmetic mean is a number that when multiplied by the numbe......Sun, 16 Aug 2026 09:08:41 -0400https://liberty.io/user-ashvin-swaminathan-mathematics-stack-exchange.htmlhttps://liberty.io/user-ashvin-swaminathan-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Sun, 16 Aug 2026 09:06:51 -0400https://liberty.io/second-partial-derivative-test-failing-when-determinant-is-0.htmlhttps://liberty.io/second-partial-derivative-test-failing-when-determinant-is-0.htmlI am studying the second partial derivative test and it says when determinant of the hessian matrix is $0$ then there is no conclusion. However I saw on a website a method that tells you how to wor......Sun, 16 Aug 2026 09:05:20 -0400https://liberty.io/describing-asymptotic-behaviour-of-a-function.htmlhttps://liberty.io/describing-asymptotic-behaviour-of-a-function.htmlFor question B! x^2+x+1/x^2 = 1+ [x+1/x^2] shouldnt the answer be asymptote at x=0 and y=1 ?? i dont understand the textbook solution...Sun, 16 Aug 2026 09:03:31 -0400https://liberty.io/which-geometric-figure-polyhedron-has-15-quadrilateral-faces.htmlhttps://liberty.io/which-geometric-figure-polyhedron-has-15-quadrilateral-faces.htmlI am looking for a polyhedron which consists only out of 15 quadrilateral faces? Does such a thing exist?...Sun, 16 Aug 2026 09:02:27 -0400https://liberty.io/evaluating-2x-3-9x-2-10x-13-for-x-3-sqrt-5-is-there-an-efficient-way-t.htmlhttps://liberty.io/evaluating-2x-3-9x-2-10x-13-for-x-3-sqrt-5-is-there-an-efficient-way-t.htmlI was helping my sibling with a math problem from a past year paper for a competitive exam. It requires us to evaluate this cubic expression for a given value of $x$ which has an $a+b$ form where $......Sun, 16 Aug 2026 08:56:38 -0400https://liberty.io/example-of-distinctions-between-multiple-integral-and-iterated-integra.htmlhttps://liberty.io/example-of-distinctions-between-multiple-integral-and-iterated-integra.htmlThe following question is asked in the book of Analysis On Manifolds by Munkres, and given at page 103 as question 3. $Q=A\times B$,where $A$ is a rectangle in $\mathbb{R^k}$ and $B$ is a rectangl......Sun, 16 Aug 2026 08:45:30 -0400https://liberty.io/cardioid-in-coffee-mug.htmlhttps://liberty.io/cardioid-in-coffee-mug.htmlI've been learning about polar curves in my Calc class and the other day I saw this suspiciously $r=1-\cos \theta$ looking thing in my coffee cup (well actually $r=1-\sin \theta$ if we're being pe......Sun, 16 Aug 2026 08:44:14 -0400https://liberty.io/the-expected-value-of-the-square-of-sample-variance.htmlhttps://liberty.io/the-expected-value-of-the-square-of-sample-variance.htmlSuppose $X_1, \cdots, X_n$ are i.d.d. samples from population $X \sim N(\mu,\sigma^2)$, and the sample variance is denoted by $T = \sum_{i = 1}^n \frac{(X_i - \overline{X})^2}{n}$. I am curious abo......Sun, 16 Aug 2026 08:43:46 -0400https://liberty.io/show-that-the-volume-of-a-sphere-of-radius-r-is-v-frac-4-3-pi-r-3.htmlhttps://liberty.io/show-that-the-volume-of-a-sphere-of-radius-r-is-v-frac-4-3-pi-r-3.htmlIf we place the sphere so that its center is at the origin, then the plane $P_x$ intersects the sphere in a circle whose radius is $y = \sqrt{r^2-x^2}$.So the cross-sectional area is $$A(x) = \pi y......Sun, 16 Aug 2026 08:37:32 -0400https://liberty.io/fourier-transform-of-e-iwt.htmlhttps://liberty.io/fourier-transform-of-e-iwt.htmlFirst, I have searched the internet as well as here for an answer to my question but did not find one (or anything close enough to lead me to the answer on my own). I have a basic understanding o......Sun, 16 Aug 2026 08:37:09 -0400https://liberty.io/does-convergence-in-distribution-implies-convergence-of-characteristic.htmlhttps://liberty.io/does-convergence-in-distribution-implies-convergence-of-characteristic.htmlLevy's continuity theorem says that if a sequence of random variables $X_{i}$ has characteristic functions $\phi_{X_i}$ converging pointwise to a characteristic function $\phi_{X}$ of other random ...Sun, 16 Aug 2026 08:34:13 -0400https://liberty.io/how-do-i-solve-this-quadratic-equation-with-a-fraction.htmlhttps://liberty.io/how-do-i-solve-this-quadratic-equation-with-a-fraction.htmlI seem to have trouble with quadratic equations when it comes to fractions and square roots. $$ \frac{1}{x}+2x=3 $$ How do I solve this equation?...Sun, 16 Aug 2026 08:30:21 -0400https://liberty.io/when-the-population-variance-is-unknown-we-should-use-t-distribution.htmlhttps://liberty.io/when-the-population-variance-is-unknown-we-should-use-t-distribution.htmlIt is clear that T-distribution should be used when the sample size is small and the population variance is unknown. My question is Why? Why we use t-distribution in this case? Anybody give me spec......Sun, 16 Aug 2026 08:29:18 -0400https://liberty.io/sigma-algebra-generated-by-two-sigma-algebras-is-generated-by-the-inte.htmlhttps://liberty.io/sigma-algebra-generated-by-two-sigma-algebras-is-generated-by-the-inte.htmlLet $\mathcal{G}$ and $\mathcal{H}$ be two $\sigma$-algebras on a set $\Omega$. Does it hold that $$\sigma(\mathcal{H},\mathcal{G}) = \sigma\{G\cap H: G\in\mathcal{G}, H\in \mathcal{H}\}?$$ How to ......Sun, 16 Aug 2026 08:27:02 -0400https://liberty.io/how-to-find-dy-dx-fx-fy.htmlhttps://liberty.io/how-to-find-dy-dx-fx-fy.htmlI need some walkthrough in solving the following question: find dy/dx = - fx/fy? 3x^2 - y^2 + x^3 = 0. I need to know the method to solve this question. According to my understanding what I have ...Sun, 16 Aug 2026 08:06:18 -0400https://liberty.io/difference-between-power-law-distribution-and-exponential-decay.htmlhttps://liberty.io/difference-between-power-law-distribution-and-exponential-decay.htmlThis is probably a silly one, I've read in Wikipedia about power law and exponential decay. I really don't see any difference between them. For example, if I have a histogram or a plot that looks l......Sun, 16 Aug 2026 08:03:11 -0400https://liberty.io/probability-a-flaw-in-logic-the-emperor-039-s-proposition-with-marbles.htmlhttps://liberty.io/probability-a-flaw-in-logic-the-emperor-039-s-proposition-with-marbles.htmlI've tried searching for this question but couldn't find it on stackexchange. This is a common type of interview question; I ran into it doing brain teasers on a probability puzzles app, and if you......Sun, 16 Aug 2026 08:02:11 -0400https://liberty.io/prove-that-two-equations-are-the-same-if-they-have-the-same-solution-s.htmlhttps://liberty.io/prove-that-two-equations-are-the-same-if-they-have-the-same-solution-s.htmlI'm having trouble understanding the logic underlying the proof for this problem (from Hefferon's Linear Algebra): Prove that, where $a, b, c, d, e$ are real numbers with $a \ne 0$, if this linear ...Sun, 16 Aug 2026 08:01:43 -0400https://liberty.io/directional-derivative-finding-a-line-equation.htmlhttps://liberty.io/directional-derivative-finding-a-line-equation.htmlGiven the function $h(x,y,z)=5+x^2-3yz$ represents the temperature. find the directional derivative on the point $P(1,2,-1)$ that does the same angles in the positive direction of $x,y,z$ axis is ......Sun, 16 Aug 2026 07:55:57 -0400https://liberty.io/using-the-pumping-lemma-to-show-that-a-language-is-not-regular-compute.htmlhttps://liberty.io/using-the-pumping-lemma-to-show-that-a-language-is-not-regular-compute.htmlShow that $L=\{a^{n^2} | n \ge 0\}$ is not regular Hey guys. I'm taking a CS class and this stuff is really new to me so bear with me. I tried to look if I get some contradiction by using the pump......Sun, 16 Aug 2026 07:50:05 -0400https://liberty.io/solving-for-x-with-exponents-algebra.htmlhttps://liberty.io/solving-for-x-with-exponents-algebra.htmlSo I am trying to help a friend do her homework and I am a bit stuck. $$8x+3 = 3x^2$$ I can look at this and see that the answer is $3$, but I am having a hard time remembering how to solve for ......Sun, 16 Aug 2026 07:47:51 -0400https://liberty.io/calculate-the-minimum-number-of-moves-required-to-solve-the-tower-of-h.htmlhttps://liberty.io/calculate-the-minimum-number-of-moves-required-to-solve-the-tower-of-h.htmlLet's say you have a variant of The Tower of Hanoi, where you have $4$ pegs and $4$ discs. The formula for $3$ pegs and $n$ discs is $2^n-1$ but how would you work out the minimum number of moves in ...Sun, 16 Aug 2026 07:35:02 -0400https://liberty.io/what-039-s-the-difference-between-a-dependent-function-type-pi-types-a.htmlhttps://liberty.io/what-039-s-the-difference-between-a-dependent-function-type-pi-types-a.htmlI was reading the book "Homotopy Type Theory", and got confused about the distinction between a family of types $B : A \to \mathcal U$ and a dependent function type $\Pi_{(x:A)} B(x) : \to \mathcal......Sun, 16 Aug 2026 07:33:18 -0400https://liberty.io/distance-between-the-two-points-p-and-q.htmlhttps://liberty.io/distance-between-the-two-points-p-and-q.htmlLet $d(x,y)$ denote the distance between two points, $x$ and $y$, on the plane. 1) $P(2,9),\quad Q(-1,13)\Rightarrow d(P,Q) = 5.$ 2) $P(1,-2),\quad Q(2,10)\Rightarrow d(P,Q) = \sqrt{145}.$ 3) $P......Sun, 16 Aug 2026 07:33:00 -0400https://liberty.io/how-to-determine-concavity-without-inflection-point.htmlhttps://liberty.io/how-to-determine-concavity-without-inflection-point.htmlI'm trying to determine this function concavity: 2=0 is a contradiction, so we have no inflection points on this function, so ¿How could I determine the concavity if I have no inflection points? ......Sun, 16 Aug 2026 07:29:28 -0400https://liberty.io/discrete-math-inclusive-or-vs-exclusive-or.htmlhttps://liberty.io/discrete-math-inclusive-or-vs-exclusive-or.htmlQuestion: Are inclusive or and exclusive or both considered dis-junction ? My own thinking: I believe yes, because both inclusive and exclusive or are using or. They just happen to have different ...Sun, 16 Aug 2026 07:28:42 -0400https://liberty.io/how-to-solve-equations-with-casio-fx-85wa-calculator.htmlhttps://liberty.io/how-to-solve-equations-with-casio-fx-85wa-calculator.htmlI am testing the casio FX-85WA calculator to solve equations, so I am trying to solve a very simple one: $2x-1=3$. But I am unable to input the equal sign, as it puts the result. I've seen others ...Sun, 16 Aug 2026 07:24:24 -0400https://liberty.io/is-blitzer-precalculus-my-best-option-considering-my-circumstances.htmlhttps://liberty.io/is-blitzer-precalculus-my-best-option-considering-my-circumstances.htmlSo I am wrapping up a College Algebra course from my local community college and am scheduled to start the first semester of a two semester precalculus course next month. The text book I worked out......Sun, 16 Aug 2026 07:18:03 -0400https://liberty.io/weighted-probability-problem.htmlhttps://liberty.io/weighted-probability-problem.htmlLets say, few independent events are happening. $E_1, E_2,...,E_n$. The probability of each of these events happening is given as $P_1,P_2,...,P_n$. Each of these events carry weightage, say $W_1,W......Sun, 16 Aug 2026 07:18:00 -0400https://liberty.io/area-of-a-regular-hexagon-via-area-of-triangles.htmlhttps://liberty.io/area-of-a-regular-hexagon-via-area-of-triangles.htmlProblem: Find the area of a regular hexagon whose sides measures 5 cm Sol'n 1: I can cut the hexagon into 6 small triangles, so the area of triangle times 6 will be equal to the area of the polygon. ...Sun, 16 Aug 2026 06:55:41 -0400https://liberty.io/determinant-of-a-very-large-matrix-in-matlab.htmlhttps://liberty.io/determinant-of-a-very-large-matrix-in-matlab.htmlI have a very large random matrix which its elements are either $0$ or $1$ randomly. The size of the matrix is $5000$, however when I want to calculate the determinant of the matrix, it is either $......Sun, 16 Aug 2026 06:47:40 -0400https://liberty.io/why-does-fixed-point-iteration-work.htmlhttps://liberty.io/why-does-fixed-point-iteration-work.htmlI have searched online for an answer, but everyone gave the method, and no one explained why is it working. I'll first write what I do understand. Let $f(x)$ be a continuous function at $[a,b]$. ...Sun, 16 Aug 2026 06:39:02 -0400https://liberty.io/derivative-of-sin-2x-cos-3x.htmlhttps://liberty.io/derivative-of-sin-2x-cos-3x.htmlWhat is the derivative of $\sin (2x)\cos(3x)?$ Is it $\cos (2x) \cos(3x)-\sin (2x) \sin(3x)$ or $2\cos (2x) \cos(3x)-3\sin (2x) \sin(3x)?$ The answer is given the latter but I only get the form......Sun, 16 Aug 2026 06:33:23 -0400https://liberty.io/identifying-the-formula-for-a-quartic-graphic.htmlhttps://liberty.io/identifying-the-formula-for-a-quartic-graphic.htmlI am attempting to help someone with their homework and these concepts are a bit above me. I apologize for the terrible graph drawing. I am using a surface pro 3 and it has an awful camera so I can't ...Sun, 16 Aug 2026 06:30:56 -0400https://liberty.io/what-is-the-kernel-of-a-homomorphism.htmlhttps://liberty.io/what-is-the-kernel-of-a-homomorphism.htmlI've done a lot of problems before but I am trying to get a really basic definition of kernel so that I may apply to any possible given question that I may be presented with. Would I be corre......Sun, 16 Aug 2026 06:29:31 -0400https://liberty.io/intuition-for-class-numbers.htmlhttps://liberty.io/intuition-for-class-numbers.htmlSo I've been thinking about the analytic class number formula lately, and class numbers in general and I'm trying to develop a good intuition for them. My basic question, which may be too general/...Sun, 16 Aug 2026 06:24:08 -0400https://liberty.io/implicit-derivative-with-three-variable.htmlhttps://liberty.io/implicit-derivative-with-three-variable.htmlFind $\frac{\partial z}{\partial x}$ from these equation: $xyz-2xz+3yz-4xy=0$ How $\frac{\partial z}{\partial x}$ can be done using manually method? Thanks... I stuck at variable whose contains '......Sun, 16 Aug 2026 06:19:06 -0400https://liberty.io/why-do-we-say-the-harmonic-series-is-divergent-duplicate.htmlhttps://liberty.io/why-do-we-say-the-harmonic-series-is-divergent-duplicate.htmlIf we have $\Sigma\frac{1}{n}$, why do we say it is divergent? Yes, it is constantly increasing, but after a certain point, $n$ will be so large that we will be certain of millions of digits. If we ...Sun, 16 Aug 2026 06:18:30 -0400https://liberty.io/how-to-understand-compactness-duplicate.htmlhttps://liberty.io/how-to-understand-compactness-duplicate.htmlHow to understand the compactness in topology space in intuitive way?...Sun, 16 Aug 2026 06:11:14 -0400https://liberty.io/why-is-it-possible-to-factor-a-limit.htmlhttps://liberty.io/why-is-it-possible-to-factor-a-limit.htmlI am confused by the examples in calculus textbook where they factor a polynomial to find the limit. I don't understand how the limits of $\frac{1}{x-3}$ and $\frac{x+3}{x^2-9} $ are the same. Th......Sun, 16 Aug 2026 06:08:14 -0400https://liberty.io/finding-the-splitting-field-of-a-polynomial.htmlhttps://liberty.io/finding-the-splitting-field-of-a-polynomial.htmlI am confused on how to find the splitting field of a polynomial. For example, consider the polynomial $$p(x)=x^2+2.$$ I know that the splitting field is the smallest field that contains the roots ......Sun, 16 Aug 2026 06:06:14 -0400https://liberty.io/slope-of-secant-and-tangent-lines-supported-by-mvt.htmlhttps://liberty.io/slope-of-secant-and-tangent-lines-supported-by-mvt.htmlFor the function $f(x) = x^{1/3}$ on the interval $[1,8]$ find the point $(c,f(c))$ guaranteed by the Mean Value Theorem, at which the slope of the tangent line is equal to the slope of the secant ......Sun, 16 Aug 2026 06:00:00 -0400https://liberty.io/where-was-my-mistake-integration-by-trig-substitution-problem.htmlhttps://liberty.io/where-was-my-mistake-integration-by-trig-substitution-problem.htmlI am attempting to solve the problem $$\int \frac{dx}{x^2+x+1}$$ First, I complete the square, then factor out a $\frac {3}{4}$: $$\int \frac{dx}{\frac{3}{4}(\frac{4}{3}(x+\frac{1}{2})^2+1)}$$ ......Sun, 16 Aug 2026 05:59:58 -0400https://liberty.io/how-to-calculate-modulo-negative-number-duplicate.htmlhttps://liberty.io/how-to-calculate-modulo-negative-number-duplicate.htmlWhy 134 mod -26 is equal 22? I found an explanation, which says: 134 = (-26 * -6) - 22 However, 134 = (-26*-5)+4, so why 134 mod -26 is not equal 4 if 134=(26*5)+4 => 134 mod 26=4 All i could f......Sun, 16 Aug 2026 05:59:21 -0400https://liberty.io/is-there-any-equation-for-triangle.htmlhttps://liberty.io/is-there-any-equation-for-triangle.htmlLike there's an equation of a circle, is there any equation of a triangle? I've been trying to build one and the closest thing I've managed to do is to create an equation of 2 lines and use the $x......Sun, 16 Aug 2026 05:57:33 -0400https://liberty.io/if-capital-letters-are-supposed-to-be-sets-why-is-n-used-as-a-number.htmlhttps://liberty.io/if-capital-letters-are-supposed-to-be-sets-why-is-n-used-as-a-number.htmlIf capital letters are supposed to be sets, why is $N$ used as a number? For example, in this definition of a Cauchy Sequence it says: A sequence ${p_n}$ in a metric space $X$ is said to be a C......Sun, 16 Aug 2026 05:45:39 -0400