https://liberty.ioCurated star highlights with fresh energy.en-usTue, 11 Aug 2026 10:50:29 -0400https://liberty.io/proving-that-pi-2x-2-pi-x.htmlhttps://liberty.io/proving-that-pi-2x-2-pi-x.htmlIn our analytic number theory class we were given the following problem as homework: prove rigorously that for large $x$ the number of primes in $(1,x]$ exceeds that in $(x,2x]$. In class we prove......Tue, 11 Aug 2026 14:46:14 -0400https://liberty.io/flipping-a-fraction-within-an-inequality.htmlhttps://liberty.io/flipping-a-fraction-within-an-inequality.htmlIs it possible to flip a faction within an inequality? Such that: $$\frac13 < x < \frac23$$ becomes, $$3 > \frac1x > \frac32$$...Tue, 11 Aug 2026 14:28:59 -0400https://liberty.io/tetrahedron-packing-in-cube.htmlhttps://liberty.io/tetrahedron-packing-in-cube.htmlI'm thinking about following solid geometry problem. Q: Suppose you have a box of "cube" shape with edge length 1. Then, How many regular tetrahedrons(with edge length 1) can be in the box? So, t......Tue, 11 Aug 2026 14:11:37 -0400https://liberty.io/how-can-you-visually-distinguish-between-a-parabola-and-hyperbola.htmlhttps://liberty.io/how-can-you-visually-distinguish-between-a-parabola-and-hyperbola.htmlI am fully aware of the mathematical definition of either and how they relate to conic sections. Still, I wonder whether there is an easy way to see that these are fundamentally different curves.......Tue, 11 Aug 2026 14:05:09 -0400https://liberty.io/solution-to-equations-in-3-variable-without-forming-a-cubic-polynomial.htmlhttps://liberty.io/solution-to-equations-in-3-variable-without-forming-a-cubic-polynomial.htmlCan we solve 3 equations:- $$a+b+c = x$$ $$abc = y$$ $$ab+bc+ac = z$$ to get the value of $a,b$ and $c$ where $x,y,z$ are constants As you can probably see I'm trying to find a solution to a cubic ...Tue, 11 Aug 2026 14:04:11 -0400https://liberty.io/symbol-for-kernel-and-range-of-a-linear-transformation.htmlhttps://liberty.io/symbol-for-kernel-and-range-of-a-linear-transformation.htmlI'm wondering what symbol is used for the kernel and the range of a linear transformation. I've seen them being written as such: $\ker(T)$ $\mathcal{R}(T)$ Are these the "correct" symbols for the ...Tue, 11 Aug 2026 13:58:40 -0400https://liberty.io/what-does-x-2-1-mean.htmlhttps://liberty.io/what-does-x-2-1-mean.htmlI am studying some inequality properties of absolute values and I bumped into some expressions like $|x-2| < 1$ that I just can't get the meaning of them. Lets say I have this expression $$ |......Tue, 11 Aug 2026 13:40:49 -0400https://liberty.io/is-there-a-geometric-interpretation-for-a-function-039-s-alpha-subleve.htmlhttps://liberty.io/is-there-a-geometric-interpretation-for-a-function-039-s-alpha-subleve.htmlIn Boyd and Vandenberghe's "Convex Optimization": The $\alpha$-sublevel set set of a function $f:\mathbb{R}^n \rightarrow \mathbb{R}$ is defined as $$C_\alpha=\{x \in \mathbf{dom} f|f(x)\leq\......Tue, 11 Aug 2026 13:40:07 -0400https://liberty.io/study-according-to-the-values-of-x-the-relative-position-of-line-curve.htmlhttps://liberty.io/study-according-to-the-values-of-x-the-relative-position-of-line-curve.htmlWe have $F(x) = x + \ln \left(\frac{1+2e^{-2x}}{1+e^{-x}}\right)$ represents curve $C$ and a line $d: y=x$ In an exercise it is required to study the relative position of these two functions, ......Tue, 11 Aug 2026 13:37:43 -0400https://liberty.io/finding-all-real-zeros-of-the-polynomial.htmlhttps://liberty.io/finding-all-real-zeros-of-the-polynomial.htmlOkay, so I need to find all real zeros in this polynomial... $$f(x) = 2x^3 + x^2 - 13x + 6$$ I know that the first step is to find the factors of 6 and 2, then see which when multiplied by the ot......Tue, 11 Aug 2026 13:36:04 -0400https://liberty.io/strategies-for-factoring-expressions-with-four-terms.htmlhttps://liberty.io/strategies-for-factoring-expressions-with-four-terms.htmlI'm trying to come up with a general strategy for factoring expressions with four terms on the basis of the symmetries of the expressions. One thought I had was the following: count up the number of ...Tue, 11 Aug 2026 13:33:22 -0400https://liberty.io/novikoff-039-s-proof-for-perceptron-convergence.htmlhttps://liberty.io/novikoff-039-s-proof-for-perceptron-convergence.htmlIn Machine Learning, the Perceptron algorithm converges on linearly separable data in a finite number of steps. One can prove that $(R/\gamma)^2$ is an upper bound for how many errors the algorithm......Tue, 11 Aug 2026 13:25:05 -0400https://liberty.io/simple-formula-for-a-sieries-like-1-2-5-10-20-50-100.htmlhttps://liberty.io/simple-formula-for-a-sieries-like-1-2-5-10-20-50-100.htmlI'm looking for a simple formula that will give a series that looks like this: $1; 2; 5; 10; 20; 50; 100; ...$ That means a function that will give this output: $f(1) = 1$ $f(2) = 2$ $f(3) = 5$ $f......Tue, 11 Aug 2026 13:20:43 -0400https://liberty.io/geometric-interpretation-of-eigendecomposition.htmlhttps://liberty.io/geometric-interpretation-of-eigendecomposition.htmlIf we have an orthogonal matrix $U$, then $U^Tx$ is essentially a rotation of the vector $x$. If we have a diagonal matrix $\Lambda$, then $\Lambda x$ is scaling the vector $x$ in each direction by......Tue, 11 Aug 2026 13:18:59 -0400https://liberty.io/how-to-determine-the-right-side-of-the-dice.htmlhttps://liberty.io/how-to-determine-the-right-side-of-the-dice.htmlsay I have a dice, with following numbering arrangements: top front right bottom back left 1 2 3 6 5 4 the dice can be rolled freely of course I think it is enough to give the ...Tue, 11 Aug 2026 12:57:09 -0400https://liberty.io/finding-partial-fractions-involving-complex-numbers-express-h-z-frac-2.htmlhttps://liberty.io/finding-partial-fractions-involving-complex-numbers-express-h-z-frac-2.htmlI'm attempting the following problem in the context of complex analysis: Express $h(z) = \dfrac{2z + 4i}{z^2 + 2z + 2}$ as partial fractions. I got $\dfrac{2z + 4i}{z^2 + 2z + 2} = \dfrac{2z + ......Tue, 11 Aug 2026 12:56:57 -0400https://liberty.io/cardinality-of-union-of-3-sets.htmlhttps://liberty.io/cardinality-of-union-of-3-sets.htmlI know there are answers to this specific question, but I'm trying to solve it through a specific line of thought which was used in a textbook on modern algebra for the case with 2 sets. It goes like ...Tue, 11 Aug 2026 12:52:28 -0400https://liberty.io/continuity-is-required-for-differentiability.htmlhttps://liberty.io/continuity-is-required-for-differentiability.htmlMy professor emphasized that: Differentiability implies continuity and Continuity is required for differentiability. Since a function like $\frac 1 x$ is differentiable but not continuous, I thou......Tue, 11 Aug 2026 12:48:10 -0400https://liberty.io/how-to-round-0-45-is-it-0-or-1.htmlhttps://liberty.io/how-to-round-0-45-is-it-0-or-1.htmlThis question is inspired by How to round 0.4999... ? Is it 0 or 1? I didn't quite understand the logic of the answer. It seems you round every decimal place no matter how far back it goes? In the ......Tue, 11 Aug 2026 12:47:14 -0400https://liberty.io/how-was-this-approximation-of-pi-involving-sqrt-5-arrived-at.htmlhttps://liberty.io/how-was-this-approximation-of-pi-involving-sqrt-5-arrived-at.htmlThe Wikipedia article for Approximations of $\pi$ contains this little gem: $$ \pi \approx \frac{63}{25}\times\frac{17 + 15\sqrt{5}}{7 + 15\sqrt{5}} $$ which is clearly in $\mathbb{Q[\sqrt{5}]}$. ...Tue, 11 Aug 2026 12:46:58 -0400https://liberty.io/measurable-space-vs-measure-space.htmlhttps://liberty.io/measurable-space-vs-measure-space.htmlFrom what I understood a Measurable Space is $(X,S)$ where $X$ is a set and $S\subset P(X)$ is a $\sigma$ algebra a Measure Space is $(X,S,\mu)$ where $X$ is a set and $S\subset P(X)$ is a $\sigma$ ...Tue, 11 Aug 2026 12:46:20 -0400https://liberty.io/using-conjugates-to-find-a-limit-with-a-cubic-root-lim-h-to-0-frac-sqr.htmlhttps://liberty.io/using-conjugates-to-find-a-limit-with-a-cubic-root-lim-h-to-0-frac-sqr.htmlI currently have $$\lim_{h\to 0}\frac{\sqrt[3]{h+1}-1}{h}$$ Now, I know that when you have square root instead of a cubic root it's easy. You just multiply by the conjugate and simplify afterwards......Tue, 11 Aug 2026 12:45:30 -0400https://liberty.io/039-imperfect-039-numbers.htmlhttps://liberty.io/039-imperfect-039-numbers.htmlA perfect number (integer) is equal to the sum of its divisors, including 1, and excluding itself. This has been around since Euclid. Recently, I noticed that at least for the initial integers, it is ...Tue, 11 Aug 2026 12:40:07 -0400https://liberty.io/proof-with-double-counting.htmlhttps://liberty.io/proof-with-double-counting.htmlUsing double counting (and without algebraic calculations), prove that $$ \sum_{k=r}^n \binom n k \binom k r = 2^{n-r} \binom{n}{n-r}. $$ This question was assigned to me on a quiz and I was not......Tue, 11 Aug 2026 12:29:16 -0400https://liberty.io/evans-pde-5-17.htmlhttps://liberty.io/evans-pde-5-17.htmlI'm working through Evans PDE $2$nd edition Chapter $5$, question $17$: Assume $F:\mathbb{R}\rightarrow\mathbb{R}$ is $C^1$ with $F'$ bounded. Suppose $U\in \mathbb{R}^n$ is bounded and $u\in W^......Tue, 11 Aug 2026 12:13:45 -0400https://liberty.io/eliminate-xy-term-of-conic.htmlhttps://liberty.io/eliminate-xy-term-of-conic.htmlFor the following problem, I am trying to eliminate xy, and I've tried numerous times to solve if with no luck. I need to find the general form of the equation rotated to eliminate the xy term. $ ......Tue, 11 Aug 2026 12:12:11 -0400https://liberty.io/proving-the-universal-mapping-property-for-polynomial-rings.htmlhttps://liberty.io/proving-the-universal-mapping-property-for-polynomial-rings.htmlI am studying from Patrick Morandi's Field and Galois Theory, and I am stuck on Problem 6 from Section 1 (on page 13). Verify the following universal mapping property for polynomial rings: ......Tue, 11 Aug 2026 12:09:12 -0400https://liberty.io/proof-verification-if-the-diagonals-of-a-parallelogram-are-congruent-t.htmlhttps://liberty.io/proof-verification-if-the-diagonals-of-a-parallelogram-are-congruent-t.htmlTo prove. If $|\overline{AC}| = |\overline{BD}|$, then $\square ABCD$ is a rectangle. $\overline{AC}=\sqrt{(c_1-a_1)^2+(d_2-0)^2}=\sqrt{(c_1-a_1)^2+d_{2}^2}$ $\overline{BD}=\sqrt{b_{1}^2+d_{2}^2}......Tue, 11 Aug 2026 12:07:36 -0400https://liberty.io/convolution-of-two-gaussians-is-a-gaussian.htmlhttps://liberty.io/convolution-of-two-gaussians-is-a-gaussian.htmlI know that the product of two Gaussians is a Gaussian, and I know that the convolution of two Gaussians is also a Gaussian. I guess I was just wondering if there's a proof out there to show that the ...Tue, 11 Aug 2026 11:58:58 -0400https://liberty.io/user-basanta-raj-pahari-mathematics-stack-exchange.htmlhttps://liberty.io/user-basanta-raj-pahari-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Tue, 11 Aug 2026 11:55:27 -0400https://liberty.io/intuition-behind-the-ilate-rule.htmlhttps://liberty.io/intuition-behind-the-ilate-rule.htmlOften I have wondered about this question, but today I had a chance to recollect it and hence I am posting it here. During high-school days one generally learns Integration and I still loving doing ...Tue, 11 Aug 2026 11:47:51 -0400https://liberty.io/how-is-the-gradient-of-exponential-functions-with-different-bases-in-t.htmlhttps://liberty.io/how-is-the-gradient-of-exponential-functions-with-different-bases-in-t.htmlI am studying exponential functions at the moment, and this table was presented in my textbook to show that for exponential functions with increasing 'bases' the gradient of the function increases.......Tue, 11 Aug 2026 11:39:34 -0400https://liberty.io/a-question-concerning-an-exercise-from-tao-vu.htmlhttps://liberty.io/a-question-concerning-an-exercise-from-tao-vu.htmlThis is the exercise 4.1.5 from Tao Vu Additive Combinatorics. $Z$ is a finite additive group with a fixed symmetric non-degenerate bilinear form $\cdot$ Define $e: \mathbb{R}/\mathbb{Z} \to \mat......Tue, 11 Aug 2026 11:30:34 -0400https://liberty.io/line-integral-vs-arc-length.htmlhttps://liberty.io/line-integral-vs-arc-length.htmlI am trying to understand when do to line integral and when to do arc length. So I know the formula for arc length varies based on $dx$ or $dy$ like so: $s=\int_a^b \sqrt{1+[f'(x)]^2} \, \mathrm{d}......Tue, 11 Aug 2026 11:12:57 -0400https://liberty.io/what-does-it-mean-for-a-matrix-to-be-nearly-singular.htmlhttps://liberty.io/what-does-it-mean-for-a-matrix-to-be-nearly-singular.htmlI am currently enrolled in numerical analysis course, and a term I have not heard of came up: nearly singular matrix. I know that a non-singular matrix is one where the column vectors are linearly ...Tue, 11 Aug 2026 11:01:03 -0400https://liberty.io/convert-rectangular-coordinate-3-0-to-polar-coordinate.htmlhttps://liberty.io/convert-rectangular-coordinate-3-0-to-polar-coordinate.htmlI'm trying to convert (-3,0) to polar coordinate. I can get r=$\sqrt {(-3)^2 +(0)^2}$ =3, but when computing for the angle $\theta$=$\tan^{-1} (\frac {0}{-3})$=0 but the answer for the a......Tue, 11 Aug 2026 11:00:44 -0400https://liberty.io/finding-stagnation-points.htmlhttps://liberty.io/finding-stagnation-points.htmlI am trying to find the stagnation point of a fluid flow from a complex potential. The complex potential is given by: $\Omega(z) = Uz + \cfrac{m}{2\pi}\ln z$. From this I found the streamfunction t......Tue, 11 Aug 2026 10:59:39 -0400https://liberty.io/how-to-calculate-the-surface-area-of-a-cube-given-a-diagonal.htmlhttps://liberty.io/how-to-calculate-the-surface-area-of-a-cube-given-a-diagonal.htmlI encountered this problem while practicing for a mathematics competition. A cube has a diagonal length of 10. What is the surface area of the cube? No Calculators Allowed. (Emphasis mine) I'......Tue, 11 Aug 2026 10:59:26 -0400https://liberty.io/how-do-i-get-the-square-root-of-a-complex-number.htmlhttps://liberty.io/how-do-i-get-the-square-root-of-a-complex-number.htmlIf I'm given a complex number (say $9 + 4i$), how do I calculate its square root?...Tue, 11 Aug 2026 10:57:10 -0400https://liberty.io/trigonometric-identities-like-a-sin-x-b-cos-y-cdots.htmlhttps://liberty.io/trigonometric-identities-like-a-sin-x-b-cos-y-cdots.htmlAre there any identities for trigonometric equations of the form: $$A\sin(x) + B\sin(y) = \cdots$$ $$A\sin(x) + B\cos(y) = \cdots$$ $$A\cos(x) + B\cos(y) = \cdots$$ I can't find any mention of them ...Tue, 11 Aug 2026 10:35:56 -0400https://liberty.io/platonic-solids-in-4d.htmlhttps://liberty.io/platonic-solids-in-4d.htmlI've noticed in this wikipedia article that each 4D figure is constructed of 3D platonic solids to get to these, just like using polygons on solids. In the hypercube, you have the cube in the cente......Tue, 11 Aug 2026 10:29:46 -0400https://liberty.io/find-the-cosine-of-the-angle-abc.htmlhttps://liberty.io/find-the-cosine-of-the-angle-abc.htmlI have problem with solving this task. I know that the answer might be A. But only with calculator by calculating the angles. can someone explain me or give me a hint to solve it. cos^-1 (5/13) =......Tue, 11 Aug 2026 10:25:27 -0400https://liberty.io/how-different-are-open-and-closed-intervals.htmlhttps://liberty.io/how-different-are-open-and-closed-intervals.htmlI am taking an online maths course and I got little confused on the concepts of open and closed intervals. In the lecture, professor says there is a big difference between $(a, b)$ and $[a, b]$. The ...Tue, 11 Aug 2026 10:24:59 -0400https://liberty.io/is-there-a-category-of-categories.htmlhttps://liberty.io/is-there-a-category-of-categories.htmlMy question is quite simple, I would like to know if we can define the category of the categories, unlike Cat which is the category of the small categories. By the way, are there any particular rea......Tue, 11 Aug 2026 10:16:49 -0400https://liberty.io/what-is-fog-x-f-g-x-f-x-fof-n-1-x-fog-2012-x-closed.htmlhttps://liberty.io/what-is-fog-x-f-g-x-f-x-fof-n-1-x-fog-2012-x-closed.htmlI found a math assuming fog(x)=f(g(x)), f"(x)=fof^(n-1)(x), fog^2012(x)=0. What do they mean actually? Please describe in a way that I understand....Tue, 11 Aug 2026 10:12:26 -0400https://liberty.io/int-1-sin-x-dx.htmlhttps://liberty.io/int-1-sin-x-dx.htmlI'm trying to find $\int 1/\sin(x)\ dx$, but I can't figure out how to do it. Also, what would be the value of $$\int\limits_0^{2\pi} \frac{1}{\sin x} dx\quad ?$$ Based on symmetry, I would try to......Tue, 11 Aug 2026 10:11:31 -0400https://liberty.io/winding-number-of-vector-field-on-surface.htmlhttps://liberty.io/winding-number-of-vector-field-on-surface.htmlI found a term "winding number of vector field with respect to another vector field" in a paper without definition. Because my paper I am reading is talking about the surface, so I don't know if I ......Tue, 11 Aug 2026 10:11:13 -0400https://liberty.io/average-area-of-the-shadow-of-a-convex-shape-closed.htmlhttps://liberty.io/average-area-of-the-shadow-of-a-convex-shape-closed.htmlWhat is the average area of the shadow of a convex shape taken over all possible orientations? If we take a sphere, its surface area is exactly 4 times the area of its shadow. How can it be genera......Tue, 11 Aug 2026 10:04:36 -0400https://liberty.io/ramsey-number-proof-r-3-3-3-3-leq-4-r-3-3-3-1-2.htmlhttps://liberty.io/ramsey-number-proof-r-3-3-3-3-leq-4-r-3-3-3-1-2.htmlI am trying to prove: $R(3,3,3,3)\leq 4(R(3,3,3)-1) + 2$ I am confused as to how one can go from a $4$ color problem to a $3$ color problem by multiplying and adding. edit: $R$ is the Ramsey ......Tue, 11 Aug 2026 10:02:57 -0400https://liberty.io/need-help-with-solution-for-problem-a5-from-imo-2015-closed.htmlhttps://liberty.io/need-help-with-solution-for-problem-a5-from-imo-2015-closed.htmlI'm going through solution for A5 from International Mathematical Olympiad 2015. Something's not right. Here's how it goes: First, we assume all functions below map integers to integers. Second, ......Tue, 11 Aug 2026 10:00:42 -0400https://liberty.io/find-the-general-solution-to-the-ode.htmlhttps://liberty.io/find-the-general-solution-to-the-ode.htmlI was asked to find the general solution to these ODEs. $$\frac{dy}{dx}=\frac{(-8x+3y-31)}{(-3x+y-11)}$$ I tried by rearranging to the form $M(x,y)+N(x,y)\frac{dy}{dx}=0$ and let $y=vx$, but what ......Tue, 11 Aug 2026 09:58:25 -0400https://liberty.io/how-to-prove-that-the-tangent-to-a-circle-is-perpendicular-to-the-radi.htmlhttps://liberty.io/how-to-prove-that-the-tangent-to-a-circle-is-perpendicular-to-the-radi.htmlI've tried drawing a parallel chord to the tangent but then how would you prove that the chord is perpendicular to the radius?...Tue, 11 Aug 2026 09:53:19 -0400https://liberty.io/relation-betwen-coefficients-and-roots-of-a-polynomial-duplicate.htmlhttps://liberty.io/relation-betwen-coefficients-and-roots-of-a-polynomial-duplicate.htmlPossible Duplicate: Create polynomial coefficients from its roots I am reading the first chapter titled Numerical Solutions Of Equations And Interpolation by K.A. Stroud (Advanced Engineering ......Tue, 11 Aug 2026 09:49:15 -0400https://liberty.io/determine-sqrt-1-50-cdot51-cdot52-cdot53-without-a-calculator.htmlhttps://liberty.io/determine-sqrt-1-50-cdot51-cdot52-cdot53-without-a-calculator.htmlI've tried a lot of things but failed to do it, I've calculated the result inside the square root which is $7027801$ using substitution and factoring but $\sqrt{7027801}$ isn't possible to simplify....Tue, 11 Aug 2026 09:49:11 -0400https://liberty.io/is-there-a-unique-minimal-expression-for-every-boolean-function.htmlhttps://liberty.io/is-there-a-unique-minimal-expression-for-every-boolean-function.htmlIs there a unique minimal expression for every boolean function? I've heard that there are some boolean expressions for which the minimal form is not unique. What are the characteristics of this k......Tue, 11 Aug 2026 09:42:22 -0400https://liberty.io/prove-in-full-detail-that-the-set-is-a-vector-space.htmlhttps://liberty.io/prove-in-full-detail-that-the-set-is-a-vector-space.htmlSo I'm doing a review test and I have this problem: Prove in full detail, with the standard operations in R2, that the set {(x,2x): x is a real number} is a vector space. Attempt: Given: $(x_1, 2x......Tue, 11 Aug 2026 09:31:48 -0400https://liberty.io/physical-meaning-of-one-dimensional-heat-equation.htmlhttps://liberty.io/physical-meaning-of-one-dimensional-heat-equation.htmlConsider heat conduction in a $1$-D material. The temperature distribution $u$ after sufficiently long time can be modeled by $$-(a(x)u_x)_x = f(x), \qquad x \in [0,1]$$ where $a$ is heat conduct......Tue, 11 Aug 2026 09:25:25 -0400https://liberty.io/is-the-absolute-value-function-a-linear-function.htmlhttps://liberty.io/is-the-absolute-value-function-a-linear-function.htmlI'm inclined to say yes, as it doesn't involve exponentiation, roots, logarithmic or trigonometric functions, but I watched a video where the teacher said that the absolute value function is "clear......Tue, 11 Aug 2026 09:10:18 -0400https://liberty.io/maximum-number-of-horizontal-asymptotes-of-a-rational-function.htmlhttps://liberty.io/maximum-number-of-horizontal-asymptotes-of-a-rational-function.htmlThe maximum number of horizontal asymptotes a Rational function can have is 1 according to Thomas calculus 14th edition. The only reasoning I could come up with is: given a function f(x)/g(x), if ......Tue, 11 Aug 2026 09:04:59 -0400https://liberty.io/image-of-intersection-of-sects-not-equal-to-intersection-of-images-of-.htmlhttps://liberty.io/image-of-intersection-of-sects-not-equal-to-intersection-of-images-of-.htmlHow to disprove, if $f$ is a function, $f(A \cap B) != f(A) \cap f(B)?$...Tue, 11 Aug 2026 09:02:14 -0400https://liberty.io/the-math-model-of-the-slot-machines-closed.htmlhttps://liberty.io/the-math-model-of-the-slot-machines-closed.htmlCould you please tell me or give me the sources - what we understand by a math model of the slot machine. What does it mean that the game is well or badly designed?...Tue, 11 Aug 2026 08:58:58 -0400https://liberty.io/convert-boolean-expression-into-sop-and-pos.htmlhttps://liberty.io/convert-boolean-expression-into-sop-and-pos.htmlConvert the following expression into SOP (sum of products) and POS (product of sums) canonical forms using boolean algebra method: $(ac + b)(a + b'c) + ac$ Attempt at solution: $(ac + b)(a + b......Tue, 11 Aug 2026 08:54:43 -0400https://liberty.io/do-negative-dimensions-make-sense-duplicate.htmlhttps://liberty.io/do-negative-dimensions-make-sense-duplicate.htmlSome time ago I read in a popular physics book that in M-theory, there are some "things" which can be said to have dimension $-1$. Probably, the author was vastly exaggerating, but this left me ...Tue, 11 Aug 2026 08:43:50 -0400https://liberty.io/what-is-milnor-fiber.htmlhttps://liberty.io/what-is-milnor-fiber.htmlLet $f:\mathbb{C}^{n+1}\to\mathbb{C}$ be a map with $0\in\mathbb{C}$ a critical value. Let $X_0=f^{-1}(0)$ and $x\in X_0$, then the map $F:\frac{f}{|f|}:S_{\varepsilon,x}\setminus(S_{\varepsilon,x}......Tue, 11 Aug 2026 08:41:57 -0400https://liberty.io/how-can-i-figure-out-what-size-shipping-box-i-need-based-on-multiple-d.htmlhttps://liberty.io/how-can-i-figure-out-what-size-shipping-box-i-need-based-on-multiple-d.htmlIs there a math formula I can use to provide a guideline of what length x width x height box I would need to buy based on smaller individual boxes that I want to put inside this larger box? For ex......Tue, 11 Aug 2026 08:39:06 -0400https://liberty.io/dividing-100-pens-among-5-children-everyone-gets-an-odd-number-of-pens.htmlhttps://liberty.io/dividing-100-pens-among-5-children-everyone-gets-an-odd-number-of-pens.htmlSo I want to divide 100 pens among 5 children in a way that each of them gets an odd number of pens. I think it's not possible but what is the answer and why? Tag's may be wrong, please feel free to ...Tue, 11 Aug 2026 08:35:41 -0400https://liberty.io/what-is-a-tensor-product-chebyshev-grid.htmlhttps://liberty.io/what-is-a-tensor-product-chebyshev-grid.htmlWhat is the difference between "Chebyshev grid" and "tensor-product Chebyshev grid"? Are they defined on a 2D vector?...Tue, 11 Aug 2026 08:34:52 -0400https://liberty.io/why-is-a-circle-1-dimensional.htmlhttps://liberty.io/why-is-a-circle-1-dimensional.htmlIn the textbook I am reading, it says a dimension is the number of independent parameters needed to specify a point. In order to make a circle, you need two points to specify the $x$ and $y$ positi......Tue, 11 Aug 2026 08:33:37 -0400https://liberty.io/independence-of-sigma-algebras-related-to-independence-of-random-varia.htmlhttps://liberty.io/independence-of-sigma-algebras-related-to-independence-of-random-varia.htmlIn the lecture about probability theory, we had the following defintion for independence of a finite number of sub-sigma-algebras on the probability space $(\Omega, \mathcal{A},P)$: The sub-sigma-...Tue, 11 Aug 2026 08:33:16 -0400https://liberty.io/explaining-the-physical-meaning-of-an-eigenvalue-in-a-real-world-probl.htmlhttps://liberty.io/explaining-the-physical-meaning-of-an-eigenvalue-in-a-real-world-probl.htmlContextual Problem A PhD student in Applied Mathematics is defending his dissertation and needs to make 10 gallon keg consisting of vodka and beer to placate his thesis committee. Suppose that all ...Tue, 11 Aug 2026 08:23:09 -0400https://liberty.io/correct-method-of-proving-raabe-039-s-test.htmlhttps://liberty.io/correct-method-of-proving-raabe-039-s-test.htmlI was wondering if my method of proof for Raabe's test was valid, since it is different from the normal method used with comparing to a sequence $\frac{1}{n^{p}}$ for some p > 1. Raabe's Test (As ...Tue, 11 Aug 2026 08:10:45 -0400https://liberty.io/solve-int-sqrt-7x-4-dx.htmlhttps://liberty.io/solve-int-sqrt-7x-4-dx.htmlI need to solve the following integral $$\int \sqrt{7x + 4}\,dx$$ I did the following steps: \begin{align} \text{Let} \, u &= 7x+4 \quad \text{Let} \, du = 7 \, dx \\ \int &\sqrt{u} \, ......Tue, 11 Aug 2026 07:28:39 -0400https://liberty.io/find-the-maclaurin-series-for-cos-2x-using-the-series-for-sin-2x.htmlhttps://liberty.io/find-the-maclaurin-series-for-cos-2x-using-the-series-for-sin-2x.htmlI know that $$\sin(2x)= 2x - \frac{8x^3}{3!} + \frac{32x^5}{5!} - \frac{128x^7}{7!} + \cdots $$ $$\sin(2x)= \sum_{n=0}^\infty (-1)^n {2^{2n+1}x^{2n+1} \over (2n+1)!}$$ But I don't see how I can us......Tue, 11 Aug 2026 07:26:34 -0400https://liberty.io/jacobi-symbol-of-quadratic-residues.htmlhttps://liberty.io/jacobi-symbol-of-quadratic-residues.htmlIn my textbook there's the following definition: Let $n$ be an odd number and $n = \prod_{i=1}^s q_i^{r_i}$ its prime factorization. Then the Jacobi symbol is defined as \begin{equation*} \l......Tue, 11 Aug 2026 07:19:39 -0400https://liberty.io/proof-that-tree-n-where-n-3-is-finite.htmlhttps://liberty.io/proof-that-tree-n-where-n-3-is-finite.htmlReading online, it generally seems accepted that TREE(n) where n >= 3 is a finite number, but large enough to be incomputable and only has extremely loose lower bounds today. TREE(n) is the function ...Tue, 11 Aug 2026 07:16:00 -0400https://liberty.io/how-do-i-find-the-period-of-the-sine-function-y-20-sin-left-frac-5-pi-.htmlhttps://liberty.io/how-do-i-find-the-period-of-the-sine-function-y-20-sin-left-frac-5-pi-.htmlUsing Desmos I can see the period is $0.8$ but how do I get there? I understand that the period is $2\pi/$co-efficient of $x$ but the $-2/5$ is throwing me off....Tue, 11 Aug 2026 07:07:52 -0400https://liberty.io/q-rotation-matrix-preserving-properties.htmlhttps://liberty.io/q-rotation-matrix-preserving-properties.htmlGiven a simple rotation matrix: $$\mathbf{R}(\theta) = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \text{cos}(\theta) & \text{sin}(\theta) \\ 0 & -\text{sin}(\theta)& \text{cos}(\thet......Tue, 11 Aug 2026 07:05:38 -0400https://liberty.io/calculate-local-maximum-and-minimum-of-f-x-2-sin-x-5-sin-x.htmlhttps://liberty.io/calculate-local-maximum-and-minimum-of-f-x-2-sin-x-5-sin-x.htmlI am working on my scholarship exam practice which assumes high school or pre-university math knowledge. Could you please have a look on my approach? The minimum of the function $f(x)=(2+\sin x)......Tue, 11 Aug 2026 07:01:25 -0400https://liberty.io/set-with-no-elements.htmlhttps://liberty.io/set-with-no-elements.htmlConsider a set defined on the universe of integers. It contains a large number of subsets. These subsets include the empty set and sets which contain multiple integer elements. However our set does......Tue, 11 Aug 2026 06:58:24 -0400https://liberty.io/possible-n-plus-r-question.htmlhttps://liberty.io/possible-n-plus-r-question.htmlI am having trouble with this question on how to solve. Can someone please help? Mark and Juanita own a sandwich shop. They offer 3 kinds of bread, 5 kinds of meat, and 3 kinds of cheese. Each typ......Tue, 11 Aug 2026 06:57:16 -0400https://liberty.io/derivative-rules.htmlhttps://liberty.io/derivative-rules.htmlI have some problems when I have to derive: I don't know where I have to begin and where I have to stop. For example, if I have $$f(x)=(x^2 +1)(x^2 +3)$$ I know that I have to use the product rule ......Tue, 11 Aug 2026 06:50:14 -0400https://liberty.io/understanding-arctan-with-the-leibniz-series.htmlhttps://liberty.io/understanding-arctan-with-the-leibniz-series.htmlI am a newbie at complex math, and formulas that uses complex functions ($\sum_{i=a}^B$, $\arctan \theta$, $\tan \theta$ , etc.). I mostly understand all functions that I said in the last sentance......Tue, 11 Aug 2026 06:43:22 -0400https://liberty.io/sum-of-two-pascal-random-variables.htmlhttps://liberty.io/sum-of-two-pascal-random-variables.htmlI'm having trouble finding the PMF of a sum of pascal random variables. The problem stated is: Let $X \sim Pascal(m,p)$ and $Y \sim Pascal(l,p)$ where X and Y are independent of each other. Let $......Tue, 11 Aug 2026 06:38:11 -0400https://liberty.io/determine-cycle-from-adjacency-matrix.htmlhttps://liberty.io/determine-cycle-from-adjacency-matrix.htmlIs there a way/algorithm to determine if there is a cycle in a graph if I only have the adjacency matrix and can not visualize the graph?...Tue, 11 Aug 2026 06:36:00 -0400https://liberty.io/why-is-e-eulers-constant-included-in-the-formula-for-continuous-intere.htmlhttps://liberty.io/why-is-e-eulers-constant-included-in-the-formula-for-continuous-intere.htmlLets start with the most basic continuous interest formula and the evaluation of e: Continuous interest: $\left(1+\frac{r}{n}\right)^n$ Evaluation of e: $\lim_{x\to\infty} \left(1+\frac{1}{x}\ri......Tue, 11 Aug 2026 06:34:10 -0400https://liberty.io/can-the-volume-be-non-positive.htmlhttps://liberty.io/can-the-volume-be-non-positive.htmlI have a question please I have asked my teacher if the area could be negative she said :no never But what about the volume ? E.x If $y=x+2$ and $y=-x-2$,$x=0$ about x-axis The volume become zer......Tue, 11 Aug 2026 06:27:38 -0400https://liberty.io/find-the-two-sided-p-value.htmlhttps://liberty.io/find-the-two-sided-p-value.htmlThe book has the following text(about finding a trust zone). For $H_0 : \pi = 0.20$, for instance, the score test statistic is $Z(s) = -2.50$, which has two-sided P-value $0.012 < 0.05$, so $0.20$ ...Tue, 11 Aug 2026 06:25:42 -0400https://liberty.io/calculus-integration-problem-int-sin-5-x-cos-2-x-dx.htmlhttps://liberty.io/calculus-integration-problem-int-sin-5-x-cos-2-x-dx.htmlWhat's the integration of $$\int \sin^5 (x) \cos^2 (x)\,dx?$$...Tue, 11 Aug 2026 06:25:29 -0400https://liberty.io/find-the-value-of-cos-2-pi-5-using-radicals-duplicate.htmlhttps://liberty.io/find-the-value-of-cos-2-pi-5-using-radicals-duplicate.htmlThis is homework so if there is another example that can illustrate the technique I would happily accept that as guidance. The only thing I have been able to find is a question asking about $\cos(2......Tue, 11 Aug 2026 06:20:22 -0400https://liberty.io/what-does-the-relaxation-function-do-in-dijkstra-039-s-algorithm.htmlhttps://liberty.io/what-does-the-relaxation-function-do-in-dijkstra-039-s-algorithm.htmlI'm not sure what the relaxation function in Dijkstra algorithm does. Here is my current understanding: The relaxation function is essentially deciding which edge to choose from different alternat......Tue, 11 Aug 2026 06:02:16 -0400https://liberty.io/is-12-mod-13-the-same-as-1-mod-13.htmlhttps://liberty.io/is-12-mod-13-the-same-as-1-mod-13.htmlI'm in the process of studying for an exam and this just popped into my head. Sorry if it's a dumb question...Tue, 11 Aug 2026 05:50:28 -0400https://liberty.io/we-have-a-box-with-100-screws-90-good-10-defective-what-is-the-prob-th.htmlhttps://liberty.io/we-have-a-box-with-100-screws-90-good-10-defective-what-is-the-prob-th.htmlProbability to choose a non-defective screw for the 1st time is $\dfrac{90}{100}$ 2nd time $\dfrac{89}{99}$ 3rd time $\dfrac{88}{98}$ ... 10th time $\dfrac{81}{91}$ The final result is a ...Tue, 11 Aug 2026 05:50:26 -0400https://liberty.io/why-is-an-image-called-an-image.htmlhttps://liberty.io/why-is-an-image-called-an-image.htmlGiven a function $f : A \to B$, the image, denoted by $\operatorname{Im}f$ is the set of all $f(x)$ where $x \in A$. Why do we call this set the image? When was it first used, and what motivated it......Tue, 11 Aug 2026 05:48:02 -0400https://liberty.io/how-to-reverse-this-bitwise-and-xor-encoding-algorithm.htmlhttps://liberty.io/how-to-reverse-this-bitwise-and-xor-encoding-algorithm.htmlI have been given an "encoding" algorithm that does bitwise XOR and bitwise AND. Originally it's a C code that operates on integers with bit-shifts, but I have translated it into a simpler pseudocode ...Tue, 11 Aug 2026 05:46:56 -0400https://liberty.io/graph-of-this-integral-is-it-a-half-circle-or-quarter-circle.htmlhttps://liberty.io/graph-of-this-integral-is-it-a-half-circle-or-quarter-circle.htmlI'm reading this text: $$\int_0^1 \sqrt{1 - x^2} \cdot dx$$ Why isn't the above integral the right half of a circle? Why did they show the equation of $y^2$? Was the point of that just to show th......Tue, 11 Aug 2026 05:46:51 -0400https://liberty.io/a-family-has-three-children-what-is-the-probability-that-at-least-one-.htmlhttps://liberty.io/a-family-has-three-children-what-is-the-probability-that-at-least-one-.htmlAccording to me there are $4$ possible outcomes: $$GGG \ \ BBB \ \ BGG \ \ BBG $$ Out of these four outcomes, $3$ are favorable. So the probability should be $\frac{3}{4}$. But should you take......Tue, 11 Aug 2026 05:43:39 -0400https://liberty.io/show-1-sqrt-5-is-irreducible-in-mathbb-z-sqrt-5.htmlhttps://liberty.io/show-1-sqrt-5-is-irreducible-in-mathbb-z-sqrt-5.htmlShow $1+\sqrt{5}$ is irreducible in $\mathbb{Z}[\sqrt{5}]$. Similar questions use the norm ( see e.g. How to show $1 + \sqrt{ 5 }$ is irreducible ), but I do not know yet about such a mapping, so I......Tue, 11 Aug 2026 05:36:23 -0400https://liberty.io/notation-for-summation-of-factorials.htmlhttps://liberty.io/notation-for-summation-of-factorials.html$1!+2!+3!...$ The sum of all the factorials up to a chosen positive integer $n$. This would be expressed using sigma notation. If you can show me this, that would be enough for me, for now. If you......Tue, 11 Aug 2026 05:30:33 -0400https://liberty.io/law-of-total-variance-intuition.htmlhttps://liberty.io/law-of-total-variance-intuition.htmlIntuitively, what's the difference between 2 following terms on the right hand side of the law of total variance? $$\text{Var}(Y) = \Bbb E\left[\text{Var}\left(Y|X\right)\right] + \text{Var}\left(\......Tue, 11 Aug 2026 05:27:24 -0400https://liberty.io/form-of-the-particular-solution-for-odes.htmlhttps://liberty.io/form-of-the-particular-solution-for-odes.htmlI have been working on finding the form of the particular solution for the following differential equations, using Paul's online notes as a reference, and every time I submit my answers, they are w......Tue, 11 Aug 2026 05:26:48 -0400