https://liberty.ioCurated star highlights with fresh energy.en-usThu, 13 Aug 2026 04:26:23 -0400https://liberty.io/finding-a-logarithmic-function-from-a-graph.htmlhttps://liberty.io/finding-a-logarithmic-function-from-a-graph.htmlHere's the graph. When I use the points $(-1,1)$ or $(-3,2)$ to use in the equation $a\log(-x-1)+k$, I can't find a finite value for k. Any ideas?...Thu, 13 Aug 2026 08:25:19 -0400https://liberty.io/writing-an-objective-function-such-that-the-problem-has-an-optimal-sol.htmlhttps://liberty.io/writing-an-objective-function-such-that-the-problem-has-an-optimal-sol.htmli know it's basic but I'm struggling with how to find the objective function of a linear program. How do i write an objective function given only two constraints of a linear program in three variab......Thu, 13 Aug 2026 08:24:18 -0400https://liberty.io/what-are-the-formulas-for-topological-transformations-how-to-obtain-th.htmlhttps://liberty.io/what-are-the-formulas-for-topological-transformations-how-to-obtain-th.htmlI'm reading Flegg's From Geometry to Topology, the author says that in Euclidean geometry, translation and rotation are: $$T:(x,y)\to(x+a,y+b)$$ $$R:(x,y)\to(x \cos \phi - y \sin \phi, x \sin \phi......Thu, 13 Aug 2026 08:23:24 -0400https://liberty.io/what-is-the-probability-of-selecting-a-red-marble-then-a-blue-marble-i.htmlhttps://liberty.io/what-is-the-probability-of-selecting-a-red-marble-then-a-blue-marble-i.htmlA bag contains 4 green marbles, 9 red marbles, 10 orange marbles, 9 brown marbles, and 11 blue marbles. You choose a marble, replace it, and choose again. What is P (red, then blue) This is my que......Thu, 13 Aug 2026 08:20:23 -0400https://liberty.io/when-to-use-normal-approximation.htmlhttps://liberty.io/when-to-use-normal-approximation.htmlDo we use normal approximation when discrete distributions are hard to solve? For example, $P(X\ge 7000)$ where is $X\sim\operatorname{Binomial}(13000, 0.7)$. Obviously, summing each case manually ...Thu, 13 Aug 2026 08:14:27 -0400https://liberty.io/finding-c-in-a-definite-integral-with-no-reference-point.htmlhttps://liberty.io/finding-c-in-a-definite-integral-with-no-reference-point.htmlHow to find C in this equation? $$\int_0^x f(t)~dt=\int_x^1t^2f(t)~dt + x^2/4 + x^4/8 +C$$ I attempted to move over $\int_0^x f(t)~dt$ to the right side, so that I could solve a definite integral......Thu, 13 Aug 2026 07:51:39 -0400https://liberty.io/problem-comprehending-the-representation-of-complex-numbers-by-a-unit-.htmlhttps://liberty.io/problem-comprehending-the-representation-of-complex-numbers-by-a-unit-.htmlI'm having a very basic problem with understanding the representation of complex numbers using the unit circle. An example of what I'm struggling with is shown in the initial diagram on this page ......Thu, 13 Aug 2026 07:45:34 -0400https://liberty.io/how-do-i-evaluate-the-following-derivative.htmlhttps://liberty.io/how-do-i-evaluate-the-following-derivative.htmlHow can I evaluate the derivative $$\left. \frac{d}{dx} \left( x \, \ln(x^5) \right)\right|_{x=1} \, ?$$ I have this math problem, and the biggest thing I do not understand, is which derivative ru......Thu, 13 Aug 2026 07:23:48 -0400https://liberty.io/describe-the-galois-group-of-a-field-extension.htmlhttps://liberty.io/describe-the-galois-group-of-a-field-extension.htmlI'm struggling to understand the basics of Galois theory. One of the things I don't understand is how to actually derive automorphisms of a field extension. Let's say you had a simple problem: $x^......Thu, 13 Aug 2026 07:17:45 -0400https://liberty.io/linear-transformations-and-polynomials.htmlhttps://liberty.io/linear-transformations-and-polynomials.htmlI've been having a really hard time recently understanding linear transformations. I would really like to know conceptually what exactly is going on. Let $T: P_3 \rightarrow P_2$ be defined by ......Thu, 13 Aug 2026 07:16:02 -0400https://liberty.io/user-christopher-d-long-mathematics-stack-exchange.htmlhttps://liberty.io/user-christopher-d-long-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Thu, 13 Aug 2026 07:14:02 -0400https://liberty.io/definition-of-finite-elements-ciarlet.htmlhttps://liberty.io/definition-of-finite-elements-ciarlet.htmlI dont understand this definition of finite elements: The triple $(K,P,\Sigma)$ with $K \subset \mathbb{R}^n$ connected lipschitz domain with non-empty interior $P$ finite dimensional vector spa......Thu, 13 Aug 2026 07:03:01 -0400https://liberty.io/what-is-the-approximate-area-of-the-shaded-region-of-the-given-figure.htmlhttps://liberty.io/what-is-the-approximate-area-of-the-shaded-region-of-the-given-figure.htmlHow do i find the area of the black shaded portion of the circle? I noticed the 4 so i think that's the radius. The formula to find the area $$A=πr^2$$ so I thought of using that to find the area ......Thu, 13 Aug 2026 07:00:12 -0400https://liberty.io/calculus-i-optimization-problem-maximization-of-profit.htmlhttps://liberty.io/calculus-i-optimization-problem-maximization-of-profit.htmlCalculus problem that I've been trying to get my head around. Problem: A company can sell 20 products if it charges $40 per product. For each dollar decrease or increase in the price, the compan......Thu, 13 Aug 2026 06:54:32 -0400https://liberty.io/integral-test-proof.htmlhttps://liberty.io/integral-test-proof.htmlCould someone explain what does it mean by "(and hence $\sum a_n$)"? Isn't $a_{n+1} \leq a_n$ for all $n$? How does the convergence of $\sum a_{n+1}$ imply $\sum a_n$ converges when we have $\sum ......Thu, 13 Aug 2026 06:48:48 -0400https://liberty.io/why-is-the-gradient-always-perpendicular-to-level-curves.htmlhttps://liberty.io/why-is-the-gradient-always-perpendicular-to-level-curves.htmlLet $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be a function. A level set is a set of points: $$L(c) = \{x \in \mathbb{R}^n | f(x) = c\}$$ Two vectors $a, b \in \mathbb{R}^n$ are perpendicular, when ...Thu, 13 Aug 2026 06:47:46 -0400https://liberty.io/why-is-the-area-of-an-equilateral-triangle-frac-sqrt-3-4-b-2.htmlhttps://liberty.io/why-is-the-area-of-an-equilateral-triangle-frac-sqrt-3-4-b-2.htmlWhy is the area of an equilateral triangle $$\frac{\sqrt{3}}{4}b^2$$ and not $$\frac12b^2$$ or $$bh$$...Thu, 13 Aug 2026 06:40:40 -0400https://liberty.io/question-on-applying-jensen-inequality-on-logarithm-of-a-sum.htmlhttps://liberty.io/question-on-applying-jensen-inequality-on-logarithm-of-a-sum.htmlI am confused by the meaning of the $t \in (0,1)$ parameter in Jensen's inequality $$ f( tx_1+(1-t)x_2) \le tf(x_1)+(1-t)f(x_2) $$ When I apply this to the logarithm $$ \log( tx_1+(1-t)x_2) \ge t \......Thu, 13 Aug 2026 06:39:30 -0400https://liberty.io/the-range-of-arc-cotangent-function-arccot-1.htmlhttps://liberty.io/the-range-of-arc-cotangent-function-arccot-1.htmlWe know that the range of arc-cotangent function is $(0,π)$ and we I calculate the value of $cot^{-1}(-1)$ by a calculator, I get ($-π/4$) Which is clearly not included in the range !! Why isn't......Thu, 13 Aug 2026 06:37:21 -0400https://liberty.io/explanation-of-proof-of-logarithm-power-rule.htmlhttps://liberty.io/explanation-of-proof-of-logarithm-power-rule.htmlCan someone explain how to get from steps $(1)$ to $(2)$ in this proof of the logarithm power rule? How is the "equivalent exponential expression found? The power rule: For any positive $x$, ......Thu, 13 Aug 2026 06:35:08 -0400https://liberty.io/imaginary-and-complex-dimensions.htmlhttps://liberty.io/imaginary-and-complex-dimensions.htmlSo I saw a really intriguing question here, and I thought, "Why stop at integers? Why stop at rationals? Why even stop at real numbers?" So I'm asking this: Is there such a thing as imaginary and c......Thu, 13 Aug 2026 06:33:04 -0400https://liberty.io/to-evaluate-the-limit-as-h-approaches-0-how-do-i-rewrite-the-limit-in-.htmlhttps://liberty.io/to-evaluate-the-limit-as-h-approaches-0-how-do-i-rewrite-the-limit-in-.htmlI found a Question from the textbook which is not well explained...or not at all explained. And it would be interesting to know how to solve it. Or perhaps I am simply that bad at understanding squ......Thu, 13 Aug 2026 06:32:40 -0400https://liberty.io/is-v-tav-diagonal-in-general.htmlhttps://liberty.io/is-v-tav-diagonal-in-general.htmlI want to know if the following is a true statement: If $A$ is real a symmetric and $V$ is orthogonal then $V^TAV=D$ where $D$ is a diagonal matrix. To prove it I can say $$(V^TAV)^T=V^TA^T......Thu, 13 Aug 2026 06:32:25 -0400https://liberty.io/how-do-i-plot-the-curve-of-a-conical-spiral-on-an-unwrapped-2d-conic-s.htmlhttps://liberty.io/how-do-i-plot-the-curve-of-a-conical-spiral-on-an-unwrapped-2d-conic-s.htmlExample images I need a real world solution to a reconstructive surgery problem! A fairly commonly encountered clinical scenario results in the need to create a tube (from skin) in the shape of an ...Thu, 13 Aug 2026 06:30:00 -0400https://liberty.io/is-a-single-line-line-segment-or-ray-a-valid-angle.htmlhttps://liberty.io/is-a-single-line-line-segment-or-ray-a-valid-angle.htmlAfter online research and several definitions of the term angle it seems they all fit a short set of restrictions for what defines an angle, "a shape formed by two rays that intersect at a point ca......Thu, 13 Aug 2026 06:16:05 -0400https://liberty.io/any-shortcut-to-calculate-factorial-of-a-number-without-calculator-or-.htmlhttps://liberty.io/any-shortcut-to-calculate-factorial-of-a-number-without-calculator-or-.htmlI've been searching the internet for quite a while now to find anything useful that could help me to figure out how to calculate factorial of a certain number without using calculator but no luck ...Thu, 13 Aug 2026 05:59:32 -0400https://liberty.io/work-done-pulling-a-leaky-bucket.htmlhttps://liberty.io/work-done-pulling-a-leaky-bucket.htmlA bucket that weighs $4.6$ pounds and a rope of negligible weight are used to draw water from a well that is $80$ feet deep. The bucket is filled with $35$ pounds of water and is pulled up at a rat......Thu, 13 Aug 2026 05:59:22 -0400https://liberty.io/why-does-10-35-2-7.htmlhttps://liberty.io/why-does-10-35-2-7.htmlMy question is not meant to be dumb. I'm working on a much more complicated problem, but I forgot some of the basics. Often we perform steps in math without really knowing why we can do a certain......Thu, 13 Aug 2026 05:53:53 -0400https://liberty.io/let-x-and-y-be-two-vectors-in-mathbb-r-d-with-mid-x-mid-mid-y-mid-find.htmlhttps://liberty.io/let-x-and-y-be-two-vectors-in-mathbb-r-d-with-mid-x-mid-mid-y-mid-find.htmlLet $u \in \mathbb{R}^d = V$ be a unit vector and set $W = \text{span}(u) ^{\bot}$ (with respect to the dot product). The reflector across W is $P_u = I_d - 2uu^T$. Let $x$ and $y$ be two vectors ......Thu, 13 Aug 2026 05:52:55 -0400https://liberty.io/why-is-a-maclaurin-series-the-function-itself-for-a-polynomial-f-x.htmlhttps://liberty.io/why-is-a-maclaurin-series-the-function-itself-for-a-polynomial-f-x.htmlIt's been about two years since I last took Calc 2, and I'm trying to brush up on some of the old material using Khanacademy. I watched a couple intro videos on the subject and understand Taylor se......Thu, 13 Aug 2026 05:27:08 -0400https://liberty.io/inscribed-angle-theorem-to-prove-an-angle-inscribed-in-a-semicircle-is.htmlhttps://liberty.io/inscribed-angle-theorem-to-prove-an-angle-inscribed-in-a-semicircle-is.htmlI came across a question in my HW book: Prove that an angle inscribed in a semicircle is a right angle. My proof was relatively simple: Proof: As the measure of an inscribed angle is equal to ......Thu, 13 Aug 2026 05:24:34 -0400https://liberty.io/homer-simpson-039-s-fermat-equation.htmlhttps://liberty.io/homer-simpson-039-s-fermat-equation.htmlDisprove the following equation $3987^{12} + 4365^{12} = 4472^{12}$ First, since both the two numbers on the LHS were odd and the RHS was even, I tried dividing by 3 and found $3 \mid 3987......Thu, 13 Aug 2026 05:20:40 -0400https://liberty.io/using-first-principles-find-derivative-of-ln-sec-x.htmlhttps://liberty.io/using-first-principles-find-derivative-of-ln-sec-x.htmlThe question is to use first principles only. Thus I started with the same and got $$ y = \ln(\sec(x)) $$ $$ \frac{dy}{dx} = \lim_{h\to 0} \frac{\ln(\sec(x+h)) - \ln(\sec(x))}{h} $$ after this I do......Thu, 13 Aug 2026 05:20:36 -0400https://liberty.io/questions-tagged-multinomial-theorem.htmlhttps://liberty.io/questions-tagged-multinomial-theorem.htmlQ&A for people studying math at any level and professionals in related fields...Thu, 13 Aug 2026 05:08:55 -0400https://liberty.io/the-derivative-of-a-power-series-has-the-same-radius-of-convergence.htmlhttps://liberty.io/the-derivative-of-a-power-series-has-the-same-radius-of-convergence.htmlI am trying to understand a proof regarding the derivative of a power function. Let ${ \sum_{k=0}^{ \infty }{a_k(x-x_0)^k} }$ be a real power series with a positive radius of convergence ${ R \le \......Thu, 13 Aug 2026 05:08:43 -0400https://liberty.io/use-of-trivially-true-in-math.htmlhttps://liberty.io/use-of-trivially-true-in-math.htmlTrivial, according to Google dictionary is an adjective with the meaning "of little value or importance." Some synonyms include: unimportant, insignificant, inconsequential, etc. It appea......Thu, 13 Aug 2026 05:05:18 -0400https://liberty.io/rsa-what-is-k-in-the-formula-to-calculate-d-closed.htmlhttps://liberty.io/rsa-what-is-k-in-the-formula-to-calculate-d-closed.htmlIn RSA, to calculate $d$, when given $\phi(n)$ and $e$, I stumbled upon this formula: $$d = \dfrac{k \phi(n) + 1}{e}$$ But what does $k$ stand for? How to obtain the value for $k$? Thank you, Chris...Thu, 13 Aug 2026 04:57:58 -0400https://liberty.io/what-does-a-space-mean.htmlhttps://liberty.io/what-does-a-space-mean.htmlIn Wikipedia, they say that a space is a set with some added structure. But what do they mean by "with some added structure"?...Thu, 13 Aug 2026 04:53:18 -0400https://liberty.io/a-circle-graph-is-not-a-function.htmlhttps://liberty.io/a-circle-graph-is-not-a-function.htmlI'm a little confused by the rule: If you draw a vertical line that intersects the graph at more than 1 point then it is not a function. Because then a circle like $y^2 + x^2 = 1$ is not a functio......Thu, 13 Aug 2026 04:42:52 -0400https://liberty.io/can-t-we-use-vector-product-to-find-the-angle-between-two-vectors.htmlhttps://liberty.io/can-t-we-use-vector-product-to-find-the-angle-between-two-vectors.htmlThere are two vectors : $A = (\hat i + j + k)$ and $B = (\hat i - \hat j - \hat k)$, where $\hat i$, $\hat j$, and $\hat k$ are unit vectors along $x$, $y$, and $z$ axis respectively. We have to fi......Thu, 13 Aug 2026 04:26:51 -0400https://liberty.io/wolfram-alpha-step-by-step-for-systems-of-differential-equation.htmlhttps://liberty.io/wolfram-alpha-step-by-step-for-systems-of-differential-equation.htmlDoes anyone know if wolfram alpha has step by step solutions for systems of differential equations? When I input them, it comes up with an answer but it does not give me the step by step solution. I ...Thu, 13 Aug 2026 04:12:51 -0400https://liberty.io/why-is-the-cross-product-of-two-vectors-orthogonal.htmlhttps://liberty.io/why-is-the-cross-product-of-two-vectors-orthogonal.htmlWhat is the intuition behind the fact that the cross product of two vectors is orthogonal? Every video I've seen just says it is orthogonal but they do not explain why. Since I have terrible memory......Thu, 13 Aug 2026 04:07:04 -0400https://liberty.io/how-to-calculate-matrix-raised-to-a-high-power.htmlhttps://liberty.io/how-to-calculate-matrix-raised-to-a-high-power.htmlHow would I go about calculating: $ \left( \begin{array}{cc} 3 & -\sqrt{3} \\ \sqrt{3} & -1 \end{array} \right)^{13}$ I have already attempted to find eigenvalues/eigenvectors, but I beli......Thu, 13 Aug 2026 04:01:35 -0400https://liberty.io/how-do-we-deal-with-arctan-or-other-inverse-functions-of-undefined-val.htmlhttps://liberty.io/how-do-we-deal-with-arctan-or-other-inverse-functions-of-undefined-val.htmlWhen converting coordinates from rectangular to cylindrical, to spherical, etc. we will eventually come across having to use $ArcTan(y/x)$, and/or $ArcCos(z/\rho)$ to derive $\theta$ and/or $\phi$. ...Thu, 13 Aug 2026 03:51:32 -0400https://liberty.io/how-do-you-pronounce-hat-n.htmlhttps://liberty.io/how-do-you-pronounce-hat-n.htmlI've found a symbol $\hat{n}$ in an equation, and I have no idea how to pronounce it, and how to write it phonetic. Edit: I've found it here: https://en.wikipedia.org/wiki/Quaternion#Exponential....Thu, 13 Aug 2026 03:46:08 -0400https://liberty.io/transitive-property-of-equality-and-the-fundamental-nature-of-algrebra.htmlhttps://liberty.io/transitive-property-of-equality-and-the-fundamental-nature-of-algrebra.htmlThe fundamental nature of algebra rests on the basic rule that whenever two numbers, variables, or expressions are equal, either one can be replaced at any time by the other one. For example, if we ...Thu, 13 Aug 2026 03:43:43 -0400https://liberty.io/height-of-vertical-ropes-of-golden-gate-bridge.htmlhttps://liberty.io/height-of-vertical-ropes-of-golden-gate-bridge.htmlThe prompt is to find the parabola equation for the main span of Golden Gate bridge. Based on the $3$ points on the parabola: $(0,227)$; vertex $(640,75)$; and $(1280,227)$ I found the equation: $......Thu, 13 Aug 2026 03:37:27 -0400https://liberty.io/user-john-d-cook-mathematics-stack-exchange.htmlhttps://liberty.io/user-john-d-cook-mathematics-stack-exchange.htmlQ&A for people studying math at any level and professionals in related fields...Thu, 13 Aug 2026 03:36:41 -0400https://liberty.io/what-is-the-best-book-for-pre-calculus.htmlhttps://liberty.io/what-is-the-best-book-for-pre-calculus.htmli have missed pre-calculus knowledge in my school but i was good at maths, and now i am a computer science student, i am feeling bad being bad in maths, so i am looking for the best Pre-Calculus bo......Thu, 13 Aug 2026 03:35:58 -0400https://liberty.io/exterior-algebra-as-quotient.htmlhttps://liberty.io/exterior-algebra-as-quotient.htmlGiven a vector space $W$, I understand what the tensor algebra $T(W)$ is, and I understand that the exterior algebra $\bigwedge W$ is defined as $\bigwedge W := T(W)/N$ where $N$ is the two-sided a......Thu, 13 Aug 2026 03:32:42 -0400https://liberty.io/two-integrating-factors-same-solution-to-differential-equation.htmlhttps://liberty.io/two-integrating-factors-same-solution-to-differential-equation.htmlIf we have to valid integrating factors to a differential equation we must get the same general solution using each of them although we get two new equations by multiplying the original for each of......Thu, 13 Aug 2026 03:23:39 -0400https://liberty.io/intuitive-understanding-of-the-operator-norm.htmlhttps://liberty.io/intuitive-understanding-of-the-operator-norm.htmlI understand various vector norms, but I don't understand operator norms. Specifically, norms on linear operators. Can anyone explain them?...Thu, 13 Aug 2026 03:20:15 -0400https://liberty.io/using-right-hand-rule-for-a-current-running-in-a-loop.htmlhttps://liberty.io/using-right-hand-rule-for-a-current-running-in-a-loop.htmlHello. Using the right hand rule, shouldn't the magnetic force be going into the paper, since velocity is to the left and the force caused by centripetal acceleration is downward? The answer key says ...Thu, 13 Aug 2026 03:19:14 -0400https://liberty.io/how-do-i-write-this-basic-recursive-formula-into-desmos.htmlhttps://liberty.io/how-do-i-write-this-basic-recursive-formula-into-desmos.htmlI want to graph a simple equation $f(x)$ which begins at $(0,1)$, then for every increasing $x$ integer increment, $f(x) = f(x-1) - (c * f(x-1))$. So in other words each time you go up by one $x$ ...Thu, 13 Aug 2026 03:18:26 -0400https://liberty.io/graph-of-the-function-2x-2-2-x-2-1.htmlhttps://liberty.io/graph-of-the-function-2x-2-2-x-2-1.htmlI am learning precalculus and my precalculus book gives this equation: for this graph: But when I enter that equation into some online graph tool like Symbolab ( https://www.symbolab.com/graphing-...Thu, 13 Aug 2026 03:09:19 -0400https://liberty.io/scalar-multiplication-of-vectors-on-left-or-right.htmlhttps://liberty.io/scalar-multiplication-of-vectors-on-left-or-right.htmlVectors are often written column-wise as if they were $n\times 1$ matrices: $$ \mathbf{v} := \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} $$ This notation implicitly identifies the vector $\mathbf{v......Thu, 13 Aug 2026 03:04:09 -0400https://liberty.io/multiplication-of-ordered-pairs.htmlhttps://liberty.io/multiplication-of-ordered-pairs.htmlFrom what I've seen, ordered pair multiplication is $(a,b) \times (c,d) = (ac-bd,ad+bc)$. Also, $(0,1)$ represents $i$. So if I wanted to take $(0,-1) \times (0.-1)$, that would be $= (-1,0)$, rig......Thu, 13 Aug 2026 02:53:31 -0400https://liberty.io/find-all-solutions-to-x-9-equiv-25-pmod-29.htmlhttps://liberty.io/find-all-solutions-to-x-9-equiv-25-pmod-29.htmlFind all solutions to $x^9 \equiv 25 \pmod {29}$ I referred to a table of indices in the back of my book for help but am still a bit confused, any help/hints are greatly appreciated....Thu, 13 Aug 2026 02:47:36 -0400https://liberty.io/approximation-of-sqrt2-using-euler-039-s-method.htmlhttps://liberty.io/approximation-of-sqrt2-using-euler-039-s-method.htmlConsider the differential equation $$\frac{dx}{dt}=\frac1{2x}.$$ This is a separable O.D.E. so we know how to find all of its solutions: they are of the form $$x(t)=\sqrt{t+C}$$ where $C$ is a ...Thu, 13 Aug 2026 02:39:44 -0400https://liberty.io/calculating-the-mean-of-a-folded-normal-distribution.htmlhttps://liberty.io/calculating-the-mean-of-a-folded-normal-distribution.htmlI have some data that I know is distributed according to the folded normal distribution. I am looking to calculate the mean and variance of the underyling "unfolded" normal distribution. Let's sa......Thu, 13 Aug 2026 02:38:15 -0400https://liberty.io/give-the-bayesian-posterior-mode.htmlhttps://liberty.io/give-the-bayesian-posterior-mode.htmlSuppose that $X_1, X_2, \ldots, X_n$ are IID Bernoulli random variables with success probability equal to an unknown parameter $\theta \in [0,1]$. Let $A$ and $B$ be nonnegative constants. If we i......Thu, 13 Aug 2026 02:36:11 -0400https://liberty.io/is-p-vs-np-problem-solved.htmlhttps://liberty.io/is-p-vs-np-problem-solved.htmlMany people have tried to solve the very famous problem "P vs NP" and a lot of solutions are proposed. (e.g. A. D. Plotnikov, On the Relationship between Classes P and NP). But I couldn't find any ...Thu, 13 Aug 2026 02:36:01 -0400https://liberty.io/factors-of-zero.htmlhttps://liberty.io/factors-of-zero.htmlHow many integer factors of 0 are there, and what are they? I'm just curious, but what counts as a factor of 0? My guess is that there are an infinite number of factors of 0, but is there a proof?...Thu, 13 Aug 2026 02:35:23 -0400https://liberty.io/how-to-prove-that-lim-x-to-pi-2-tan-x-tan2x-1.htmlhttps://liberty.io/how-to-prove-that-lim-x-to-pi-2-tan-x-tan2x-1.htmlHow do I prove that the limit of the following function is $1$? $\displaystyle\lim_{x\to \pi/2} (\tan x)^{\tan2x}$ I shouldn't use l'Hôpital's rule. I have proven that limit is greater than $......Thu, 13 Aug 2026 02:25:25 -0400https://liberty.io/what-are-the-4-fourth-roots-of-16-complex-numbers-included-closed.htmlhttps://liberty.io/what-are-the-4-fourth-roots-of-16-complex-numbers-included-closed.htmlI'm trying to find the 4 fourth roots of 16, so far I have -2 and 2, but I have no idea how to find the other 2 complex numbers....Thu, 13 Aug 2026 02:23:28 -0400https://liberty.io/rate-of-change-with-f-x-4x-2-7-on-1-b.htmlhttps://liberty.io/rate-of-change-with-f-x-4x-2-7-on-1-b.htmlAs part of a textbook exercise I am to find the rate of change of $f(x)=4x^2-7$ on inputs $[1,b]$. The solution provided is $4(b+1)$ and I am unable to arrive at this solution. Tried: $f(x_2)=4b......Thu, 13 Aug 2026 02:21:13 -0400https://liberty.io/checking-if-a-vector-field-is-conservative.htmlhttps://liberty.io/checking-if-a-vector-field-is-conservative.htmlI have three different vector fields and I want to check if they are conservative: $$1)\space \space\vec{f}(\vec{x}):=\frac{1}{||\vec{x}||}\vec{a}, \space \space D=\mathbb R^2 / (\vec{0}), \space ......Thu, 13 Aug 2026 02:04:20 -0400https://liberty.io/good-software-for-linear-integer-programming.htmlhttps://liberty.io/good-software-for-linear-integer-programming.htmlI never did any linear/integer programming so I am wondering the following two things What are some efficient free linear programming solvers? What are some efficient commercial linear programming ...Thu, 13 Aug 2026 01:59:29 -0400https://liberty.io/conditional-probability-all-heads-given-at-least-one-heads.htmlhttps://liberty.io/conditional-probability-all-heads-given-at-least-one-heads.htmlThree coins are flipped. If at least one of them comes up heads, then what is the probability that they all come up heads? The answer to this is 1/7, as it is the #(ways to have all heads)/#(ways t......Thu, 13 Aug 2026 01:44:59 -0400https://liberty.io/show-that-lim-2n-frac-1-n-1.htmlhttps://liberty.io/show-that-lim-2n-frac-1-n-1.htmlProve that $\lim ((2n)^\frac{1}{n}) = 1$. I have obtained the following: $$(2n)^\frac{1}{n} = 1 + k_{n}; n > 1$$ $$(2n) = (1 + k_{n})^n$$ By the Binomial Theorem: $$(1 + k_{n})^n = 1 + k_{n} + \...Thu, 13 Aug 2026 01:36:49 -0400https://liberty.io/why-is-the-slope-of-a-line-defined-as-the-change-in-y-over-the-change-.htmlhttps://liberty.io/why-is-the-slope-of-a-line-defined-as-the-change-in-y-over-the-change-.htmlI understand that this is a reasonable definition and it shows how "fast" $y$ values change corresponding to $x$ values because it's a ratio, but what I'm asking is, couldn't the slope have been de......Thu, 13 Aug 2026 01:22:53 -0400https://liberty.io/taylor-series-with-point-of-expansion.htmlhttps://liberty.io/taylor-series-with-point-of-expansion.htmlWrite the Taylor series for the function $f(x)= x^2-3x+1$ using $x=2$ as the point of expansion , i.e. write a formula for $f(2+h)$ Even though I know what Taylor Expansion is I am really confused......Thu, 13 Aug 2026 01:16:14 -0400https://liberty.io/if-g-is-non-abelian-show-that-aut-g-is-not-cyclic.htmlhttps://liberty.io/if-g-is-non-abelian-show-that-aut-g-is-not-cyclic.htmlIf $G$ is non-Abelian, show that $Aut(G)$ is not cyclic. I can see that $G$ non-Abelian $\Rightarrow$ $G / Z(G)$ is not cyclic. And $G / Z(G)$ is isomorphic to $Inn(G) \Rightarrow$ $Inn(G)$ is not ...Thu, 13 Aug 2026 01:15:38 -0400https://liberty.io/limits-involving-greatest-integer-function.htmlhttps://liberty.io/limits-involving-greatest-integer-function.htmlWhat can you say about the following limit: $$\lim_{x\rightarrow 0}\left(\left[\dfrac{100x}{\sin x}\right]+\left[\dfrac{99\sin x}{x}\right]\right)$$ where [.] represents the greatest integer funct......Thu, 13 Aug 2026 00:55:22 -0400https://liberty.io/is-pi-over-pi-rational.htmlhttps://liberty.io/is-pi-over-pi-rational.htmlA student asked this question in class today, and I wasn't sure of the answer. On the one hand, since Pi is irrational itself, Pi/Pi doesn't fit the definition of a rational number (namely a numbe......Thu, 13 Aug 2026 00:54:24 -0400https://liberty.io/numerically-equivalent-sets-power-set.htmlhttps://liberty.io/numerically-equivalent-sets-power-set.htmlWhat I am struggling with is understanding the part where we prove that the function is onto. Letting f be a element of $2^A$ means that in this particular example f could be either of the $f_1 f_......Thu, 13 Aug 2026 00:52:09 -0400https://liberty.io/difference-between-cauchy-theorem-and-cauchy-goursat-theorem.htmlhttps://liberty.io/difference-between-cauchy-theorem-and-cauchy-goursat-theorem.htmlIn some books it is given that Cauchy theorem is equivalent to Cauchy Goursat theorem.And in some other it is given to be different. In my class , our teacher has given it differently. Cauchy theorem ...Thu, 13 Aug 2026 00:42:35 -0400https://liberty.io/green-039-s-theorem-conditions.htmlhttps://liberty.io/green-039-s-theorem-conditions.htmlWhy must the following condition be satisfied before we can use Green's Theorem: L and M are functions of (x, y) defined on an open region containing D and have continuous partial derivatives there....Thu, 13 Aug 2026 00:34:52 -0400https://liberty.io/why-does-a-basis-of-a-vector-space-need-to-have-the-same-number-of-vec.htmlhttps://liberty.io/why-does-a-basis-of-a-vector-space-need-to-have-the-same-number-of-vec.htmlAs an example, in $\mathbb{R^3}$, the canonical basis is:\begin{align} B &= \left(\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix},\begin{bmatrix} ......Thu, 13 Aug 2026 00:28:42 -0400https://liberty.io/can-the-square-root-of-a-real-number-be-negative-duplicate.htmlhttps://liberty.io/can-the-square-root-of-a-real-number-be-negative-duplicate.htmlCan the square root of a real number be negative? Dealing with the questions of functions in eleventh class my maths teacher says that square root of a real number is always positive. How is it po......Thu, 13 Aug 2026 00:27:34 -0400https://liberty.io/the-most-powerful-test-for-variance-of-normal-distribution.htmlhttps://liberty.io/the-most-powerful-test-for-variance-of-normal-distribution.htmlLet $ X_1, \ldots, X_{10} $ be simple samples from normal distribution with expected value${}= 0$ and variance${}= \sigma^2 $. Write the most powerful test for $ \alpha = 0.05 $ to verify the hypo......Thu, 13 Aug 2026 00:07:11 -0400https://liberty.io/why-does-convex-function-mean-concave-up.htmlhttps://liberty.io/why-does-convex-function-mean-concave-up.htmlA function $f : \mathbb{R} \to \mathbb{R}$ is convex (or "concave up") provided that for all $x,y \in \mathbb{R}$ and $t \in [0,1]$, $$f(tx + (1-t)y) \le tf(x) + (1-t)f(y).$$ Equivalently, a line s......Thu, 13 Aug 2026 00:06:20 -0400https://liberty.io/a-limit-wrong-using-wolfram-alpha.htmlhttps://liberty.io/a-limit-wrong-using-wolfram-alpha.htmlI want to calculate the following limit: $$\displaystyle{\lim_{x \to 0} \cfrac{\displaystyle{\int_1^{x^2+1} \cfrac{e^{-t}}{t} \; dt}}{3x^2}}$$ For that, I use L'Hopital and the Fundamental Theore......Thu, 13 Aug 2026 00:05:59 -0400https://liberty.io/is-every-subset-of-a-set-a-set.htmlhttps://liberty.io/is-every-subset-of-a-set-a-set.htmlIn $\mathsf{ZF}$ we have the Power Set Axiom: if $x$ is a set, then $\mathcal P(x)$ is a set (see here for a nice listing of the axioms of $\mathsf{ZF}$). Yesterday I was explaining to some student......Thu, 13 Aug 2026 00:04:35 -0400https://liberty.io/how-to-find-the-period-of-the-sum-of-two-trigonometric-functions.htmlhttps://liberty.io/how-to-find-the-period-of-the-sum-of-two-trigonometric-functions.htmlI want to know if there exists a general method to find the period of the sum of two periodic trigonometric function. Example: $$f(x)=\cos(x/3)+\cos(x/4).$$...Thu, 13 Aug 2026 00:01:24 -0400https://liberty.io/what-is-usually-meant-by-logit-scale-or-log-scale.htmlhttps://liberty.io/what-is-usually-meant-by-logit-scale-or-log-scale.htmlThis question is more about the math terminology than about the math itself. Say we have x = logit(p). If one says "logit scale" does he mean: the scale of p, or the scale of x, i.e. the scale of ......Thu, 13 Aug 2026 00:00:15 -0400https://liberty.io/find-the-radius-of-convergence-r-of-the-series-and-find-the-interval-i.htmlhttps://liberty.io/find-the-radius-of-convergence-r-of-the-series-and-find-the-interval-i.html$$ \sum_{n=1}^{\infty} 3^n x^n / n^2.$$ I got $R=1/3$ Find the interval, I, of convergence of the series. (Enter your answer using interval notation.) I got $I= (-1/3, 1/3)$ i'm fairly certa......Thu, 13 Aug 2026 00:00:01 -0400https://liberty.io/let-m-and-n-be-two-integers-prove-that-mn-m-is-odd-if-and-only-if-m-is.htmlhttps://liberty.io/let-m-and-n-be-two-integers-prove-that-mn-m-is-odd-if-and-only-if-m-is.html---The statement: Let $m$ and $n$ be two integers. Prove that $mn+m$ is odd if and only if $m$ is odd and $n$ is even. ---Solution: First, we show that if $m$ is odd and $n$ is even, then $mn+m$ is......Wed, 12 Aug 2026 23:59:44 -0400https://liberty.io/cover-up-method-on-1-to-infinity.htmlhttps://liberty.io/cover-up-method-on-1-to-infinity.html$$\sum_{n=1}^{\infty}\frac{7}{n(n+3)}$$ I know this is convergent but I do not know how to apply "cover-up method" to solve this can I get some help pls? The answer I got is $77/18$...Wed, 12 Aug 2026 23:44:59 -0400https://liberty.io/evaluate-int-sin-3-x-cos-2-x-dx-duplicate.htmlhttps://liberty.io/evaluate-int-sin-3-x-cos-2-x-dx-duplicate.htmlI'm trying to solve $\int{\sin^3(x)\cos^2(x)}dx$. I got $-\frac{1}{2}\cos(x)+C$, but the memo says $\frac{1}{5}\cos^5(x)-\frac{1}{3}\cos^3(x)+C$ This is my working: Your help is appreciated!...Wed, 12 Aug 2026 23:40:54 -0400https://liberty.io/using-the-snowflake-method-to-factor-trinomials.htmlhttps://liberty.io/using-the-snowflake-method-to-factor-trinomials.htmlIn school, students may be taught different ways to factor the trinomial $$ax^2+bx+c$$ where $a \neq 1,0$. Possible methods include the classic Guess and Check Method, Grouping, Box Method, and the ...Wed, 12 Aug 2026 23:38:14 -0400https://liberty.io/eigenvalue-with-algebraic-multiplicity-1.htmlhttps://liberty.io/eigenvalue-with-algebraic-multiplicity-1.htmlIf I have some $\lambda_1$ as one of my eigenvalues, and it has algebraic multiplicity of $n$ with $n > 1$. When I try to get some eigenvectors associated with this $\lambda_1$, assuming that the ...Wed, 12 Aug 2026 23:27:31 -0400https://liberty.io/polar-to-cartesian-equation-conversion.htmlhttps://liberty.io/polar-to-cartesian-equation-conversion.htmlI have a polar equation defined as: $r = ae^{θ \tan m}$ where, $a$ and $m$ are constants, $θ$ is the angle between the horizontal axis from the origin $(x_c,y_c)$ to the coordinate. $e$ refers to ...Wed, 12 Aug 2026 23:27:22 -0400https://liberty.io/recommended-books-for-aime-usamo-preparation.htmlhttps://liberty.io/recommended-books-for-aime-usamo-preparation.htmlWhat books would you guys recommend to learn number theory, geometry, combinatorics, and algebra at a level appropriate for the AIME and/or USAMO? With the month of February approaching, the month ......Wed, 12 Aug 2026 23:19:59 -0400https://liberty.io/the-vertex-covering-of-a-graph-that-has-isolated-vertices.htmlhttps://liberty.io/the-vertex-covering-of-a-graph-that-has-isolated-vertices.htmlI know the theorem that in a graph G=(V,E) which has no isolated vertex, a set $C\subset V$ is a vertex covering in G if and only if the set V‐C is an independent set in G. I want to know whether t......Wed, 12 Aug 2026 23:16:33 -0400https://liberty.io/how-to-find-no-of-complex-root-of-any-polynomial.htmlhttps://liberty.io/how-to-find-no-of-complex-root-of-any-polynomial.htmlFor $f(x)=x^5+x^3-2x+1$, we know there should be $5$ roots . We can also find real roots of this polynomial. Can we conclude that: No of complex roots = No of total roots - No of real roots . D......Wed, 12 Aug 2026 22:47:08 -0400https://liberty.io/proving-that-the-reciprocal-of-an-irrational-is-irrational.htmlhttps://liberty.io/proving-that-the-reciprocal-of-an-irrational-is-irrational.htmlThe question I am working on is: Prove that if x is irrational, then 1/x is irrational. My proof differs from the one given in the answer key; but I still feel that mine is valid. Could someone ...Wed, 12 Aug 2026 22:45:22 -0400https://liberty.io/finding-mode-in-binomial-distribution.htmlhttps://liberty.io/finding-mode-in-binomial-distribution.htmlSuppose that $X$ has the Binomial distribution with parameters $n,p$ . How can I show that if $(n+1)p$ is integer then $X$ has two mode that is $(n+1)p$ or $(n+1)p-1?$...Wed, 12 Aug 2026 22:44:26 -0400https://liberty.io/is-there-a-solution-for-parker-039-s-square.htmlhttps://liberty.io/is-there-a-solution-for-parker-039-s-square.htmlFrom wiki: Mathematician Matt Parker attempted to create a 3x3 magic square using square numbers in a YouTube video on the Numberphile channel. His failed attempt is known as the Parker Square. I ......Wed, 12 Aug 2026 22:40:39 -0400https://liberty.io/definition-of-local-ring.htmlhttps://liberty.io/definition-of-local-ring.htmlI was reading some article on local rings, and it gave the following (equivalent) definitions: A ring $R$ is $local$ if it satisfies one of the following equivalent properties (so this needs provi......Wed, 12 Aug 2026 22:37:14 -0400