Buzz Spill Daily

Buzz Spill Dailyhttps://liberty.ioCurated star highlights with fresh energy.en-usWed, 26 Aug 2026 06:26:20 -0400Determine the expected value and variance of $\bar{X}$https://liberty.io/determine-the-expected-value-and-variance-of-bar-x.htmlhttps://liberty.io/determine-the-expected-value-and-variance-of-bar-x.htmlLet $X_1,..,X_n$ be independent, identically distributed (continuous) random variables. Let $$\bar{X} = \frac{1}{n} \cdot \left(X_1+...+X_n\right)$$ Determine the expected value and variance......Wed, 26 Aug 2026 10:14:15 -0400Nullity of T vs. nullity of $T^2$https://liberty.io/nullity-of-t-vs-nullity-of-t-2.htmlhttps://liberty.io/nullity-of-t-vs-nullity-of-t-2.htmlFor a finite dimensional vector space, $V$, and linear transformation, $T$, what is the relationship between the nullity of $T$ and the nullity of $T^2$? I read somwhere that if $nullity(T)=k$, th......Wed, 26 Aug 2026 10:07:28 -0400Good video lectures for studying Calculus?https://liberty.io/good-video-lectures-for-studying-calculus.htmlhttps://liberty.io/good-video-lectures-for-studying-calculus.htmlI am looking for a good online video resource to start studying Calculus. I am studying it alone, not part of any school or university. Trying to learn and enhance my mathematical skills. Thanks!...Wed, 26 Aug 2026 10:02:12 -0400How to find what a given series/sequence converges tohttps://liberty.io/how-to-find-what-a-given-series-sequence-converges-to.htmlhttps://liberty.io/how-to-find-what-a-given-series-sequence-converges-to.htmlGiven a function series, what are the ways often used to find the function that the series converges to? For example, how do you construct the function on the right hand side from the series on th......Wed, 26 Aug 2026 09:43:39 -0400Basic tensor derivationhttps://liberty.io/basic-tensor-derivation.htmlhttps://liberty.io/basic-tensor-derivation.htmlLet $D$ be a tensor derivation on a mnaifold $M$. I have to show that if $D(\partial_i)=\sum F_i^j \partial_j$, then $D(dx^j)=-\sum F_i^j dx^i$. Any help on how to do this? Thanks in advance....Wed, 26 Aug 2026 09:43:09 -0400Proof of vector calculus identitieshttps://liberty.io/proof-of-vector-calculus-identities.htmlhttps://liberty.io/proof-of-vector-calculus-identities.htmlHere is the all identities : http://en.wikipedia.org/wiki/Vector_calculus_identities I need help concerning vector functions and indexing notations. Let $\overrightarrow{a}$ be a (smooth) vector ...Wed, 26 Aug 2026 09:35:32 -0400What is convex combination of two points?https://liberty.io/what-is-convex-combination-of-two-points.htmlhttps://liberty.io/what-is-convex-combination-of-two-points.htmlI am studying algorithms and i saw a definition like the following: Given $3$ points $p_1 = (x_1, y_1)$, $p_2 = (x_2, y_2)$ and $p_3 = (x_3, y_3)$, $p_3$ is a convex combination of $p_1$ and $p_2$......Wed, 26 Aug 2026 09:33:26 -0400Eigenvectors of similar matriceshttps://liberty.io/eigenvectors-of-similar-matrices.htmlhttps://liberty.io/eigenvectors-of-similar-matrices.htmlIf $A$ and $B$ are similar matrices then every eigenvector of $A$ is an eigenvector of $B$. Is the above statement is true? I know that similar matrices have same eigenvalue, but I'm not sure about......Wed, 26 Aug 2026 09:32:14 -0400Is a positively homogeneous convex function lower semi-continuous?https://liberty.io/is-a-positively-homogeneous-convex-function-lower-semi-continuous.htmlhttps://liberty.io/is-a-positively-homogeneous-convex-function-lower-semi-continuous.htmlHere the definition of positively homogeneous function $f$ is \begin{equation} f(\lambda x) = \lambda f(x), \quad \lambda \ge 0, \end{equation} and $f$ is defined on linear space $\mathbb{R}^n$. I ...Wed, 26 Aug 2026 09:24:42 -0400Square Roots of Complex Number $3-4i$https://liberty.io/square-roots-of-complex-number-3-4i.htmlhttps://liberty.io/square-roots-of-complex-number-3-4i.htmlWhat I did $z^2=3-4i$ $(a+bi)^2 = 3-4i$ $a^2-b^2+2abi = 3-4i$ Then got 2 simultaneous equations $a^2-b^2=3$ and $2ab=-4$ Solve for $a^2$ in 1st equation: $a^2=3+b^2$ Subbed into 2nd equation......Wed, 26 Aug 2026 09:19:32 -0400equivalence relation and setshttps://liberty.io/equivalence-relation-and-sets.htmlhttps://liberty.io/equivalence-relation-and-sets.htmlThe question: Let $X$ and $Y$ be two sets, and let $S$ be an equivalence relation on set $X$ and $T$ be an equivalence relation on set $Y$. Define a relation $R$ on $X ×Y$ by $(a,b)R(c,d)$ if and ......Wed, 26 Aug 2026 09:18:27 -0400What comes after seconds?https://liberty.io/what-comes-after-seconds.htmlhttps://liberty.io/what-comes-after-seconds.htmlAngles can be measured in different ways. For example, one can measure angles in degrees/minutes/seconds. So $1^\circ$ is divded into $60$ min. and $1$ min is divided into $60$ sec. That way a tent......Wed, 26 Aug 2026 09:17:31 -0400Prove that the square root of any irrational number is irrational.https://liberty.io/prove-that-the-square-root-of-any-irrational-number-is-irrational.htmlhttps://liberty.io/prove-that-the-square-root-of-any-irrational-number-is-irrational.htmlThe problem I'm having with this proof is that I'm not sure if my proof actually proves the theorem correct or if I'm using circular reasoning. Theorem: Prove that the square root of any irratio......Wed, 26 Aug 2026 09:00:17 -0400Why does an integral change signs when flipping the boundaries?https://liberty.io/why-does-an-integral-change-signs-when-flipping-the-boundaries.htmlhttps://liberty.io/why-does-an-integral-change-signs-when-flipping-the-boundaries.htmlLet us define a very simple integral: $f(x) = \int_{a}^{b}{x}$ for $a,b\ge 0$. Why do we have the identity $\int_{a}^{b}{x} = -\int_{b}^{a}{x}$? I drew the graphs and thought about it but to me ...Wed, 26 Aug 2026 08:51:07 -0400Intuitive and convincing argument that functions are vectorshttps://liberty.io/intuitive-and-convincing-argument-that-functions-are-vectors.htmlhttps://liberty.io/intuitive-and-convincing-argument-that-functions-are-vectors.htmlBack to school time again. As I'm discussing all the mathy stuff and insights gained over the summer, I cannot help but notice that many of my peers in second or third year undergrad cannot bridge ......Wed, 26 Aug 2026 08:48:24 -0400Define a binary composition ( operation ) $\star$ such that $\langle G,\star\rangle$ is a group with $a$ as its identity.https://liberty.io/define-a-binary-composition-operation-star-such-that-langle-g-star-ran.htmlhttps://liberty.io/define-a-binary-composition-operation-star-such-that-langle-g-star-ran.htmlLet $G$ be a group and $a\in G $ be any non identity element i.e. $a\neq e$. Define a binary composition ( operation ) $\star$ such that $\langle G,\star \rangle$ is a group with $a$ as its identit......Wed, 26 Aug 2026 08:32:26 -0400How to prove that the matrix is not invertible?https://liberty.io/how-to-prove-that-the-matrix-is-not-invertible.htmlhttps://liberty.io/how-to-prove-that-the-matrix-is-not-invertible.html$A$ and $B$ are $3\times 3$ matrices ($A,B \in \mathcal{M}_{3 \times 3}(\mathbb{R})$). There are two equations: $$A^2+3BA=I$$ $$A^2=AB$$ I want to prove that $A$ does not have an inverse. I tri......Wed, 26 Aug 2026 08:24:24 -0400Lower Limit Topology Propertieshttps://liberty.io/lower-limit-topology-properties.htmlhttps://liberty.io/lower-limit-topology-properties.htmlI am reading topology from Munkres book. While reading the countability and Separation axioms, I came across several references to Lower limit topology ($\mathbb{R}_l$) which essentially comprises of ...Wed, 26 Aug 2026 08:15:07 -0400How many solutions exists for this equation? [duplicate]https://liberty.io/how-many-solutions-exists-for-this-equation-duplicate.htmlhttps://liberty.io/how-many-solutions-exists-for-this-equation-duplicate.html$$x_1 + x_2 + x_3 + x_4 = 28$$ I tried to solve it with generating functions. Is it correct to get to the form of $${(1 + x + {x^2} + {x^3} + ....)^4}$$ and this equals to: $${(1 - x)^{ - 4}}$$......Wed, 26 Aug 2026 08:08:12 -0400Find center of rotation after object rotated by known angle (2D)https://liberty.io/find-center-of-rotation-after-object-rotated-by-known-angle-2d.htmlhttps://liberty.io/find-center-of-rotation-after-object-rotated-by-known-angle-2d.htmlI need to be able to calculate and find the true center of rotation (x,y) of an object after it has been rotated by a known angle. Previously I would simply find the center of the object, rotate 180 ...Wed, 26 Aug 2026 08:07:07 -0400Negation of biconditional statements?https://liberty.io/negation-of-biconditional-statements.htmlhttps://liberty.io/negation-of-biconditional-statements.htmlLet $p$ and $q$ be two sub statements of the compound biconditional statement given as $p$⇔$q$. The negation of this biconditional statement is given as ($p$^~$q$)∨($q$^~$p$) In the above statem......Wed, 26 Aug 2026 07:58:31 -0400Camera rotation draws twoards Z (Animation script built from mathformula in Excel)https://liberty.io/camera-rotation-draws-twoards-z-animation-script-built-from-mathformul.htmlhttps://liberty.io/camera-rotation-draws-twoards-z-animation-script-built-from-mathformul.htmlI have constructed a script in FME which calculates the position and rotation of the camera along a track "railroad track". I use math when I write my script so no standard functions. The software ......Wed, 26 Aug 2026 07:56:41 -0400Proving $x^n - y^n = (x-y)(x^{n-1} + x^{n-2} y + ... + x y^{n-2} + y^{n-1})$https://liberty.io/proving-x-n-y-n-x-y-x-n-1-x-n-2-y-x-y-n-2-y-n-1.htmlhttps://liberty.io/proving-x-n-y-n-x-y-x-n-1-x-n-2-y-x-y-n-2-y-n-1.htmlIn Spivak's Calculus 3rd Edition, there is an exercise to prove the following: $$x^n - y^n = (x-y)(x^{n-1} + x^{n-2} y + ... + x y^{n-2} + y^{n-1})$$ I can't seem to get the answer. Either I've g......Wed, 26 Aug 2026 07:48:00 -0400How to perform wedge producthttps://liberty.io/how-to-perform-wedge-product.htmlhttps://liberty.io/how-to-perform-wedge-product.htmlI have heard all kinds of great things about Clifford/Geometric algebra, but I can't find any good resources. I have been looking EVERYWHERE for just one actual example of a wedge product being ...Wed, 26 Aug 2026 07:46:31 -0400How is the subgradient method different from gradient descent in practice?https://liberty.io/how-is-the-subgradient-method-different-from-gradient-descent-in-pract.htmlhttps://liberty.io/how-is-the-subgradient-method-different-from-gradient-descent-in-pract.htmlI'm trying to learn about the subgradient descent optimization method. I'm having trouble understanding how it differs from basic gradient descent in a practical sense. According to this lecture,......Wed, 26 Aug 2026 07:42:50 -0400Solving Inequalities with variables in the denominatorhttps://liberty.io/solving-inequalities-with-variables-in-the-denominator.htmlhttps://liberty.io/solving-inequalities-with-variables-in-the-denominator.htmlOk so I've got an exam in 4 hours and I can't ever figure out these problems. Here's the problem I'm struggling with now: $$\frac{x}{3x - 5} \leq \frac{2}{x - 1}.$$ I've learned this so many tim......Wed, 26 Aug 2026 07:37:34 -0400Why are Natural Numbers called Natural Numbers?https://liberty.io/why-are-natural-numbers-called-natural-numbers.htmlhttps://liberty.io/why-are-natural-numbers-called-natural-numbers.htmlWhen we say $1,2,3...$ are natural numbers, why don't we include rational and irrational numbers? Isn't $\pi$ something natural? Shouldn't we say all real numbers the Natural numbers? Shouldn't......Wed, 26 Aug 2026 07:18:53 -0400Determine whether the series $\frac{e^\frac{1}{n}}{n^2}$ converges or divergeshttps://liberty.io/determine-whether-the-series-frac-e-frac-1-n-n-2-converges-or-diverges.htmlhttps://liberty.io/determine-whether-the-series-frac-e-frac-1-n-n-2-converges-or-diverges.htmlThe problem states Determine whether the series converges or diverges: $$\sum_{n=1}^\infty\frac{e^{1/n}}{n^2}$$ Immediately I always like to try the test for divergence: $$\lim_{n\to\infty} \s......Wed, 26 Aug 2026 07:17:55 -0400Find the six trigonometric functions given a point with $y=2x$https://liberty.io/find-the-six-trigonometric-functions-given-a-point-with-y-2x.htmlhttps://liberty.io/find-the-six-trigonometric-functions-given-a-point-with-y-2x.htmlThe terminal side of the angle θ is standard position lies on the given line and satisfies the given condition. Determine the 6 trigonometric functions of said angle. $$y = 2x, \quad \sec θ &gt......Wed, 26 Aug 2026 07:08:44 -0400What is the symbol for imaginary numbers?https://liberty.io/what-is-the-symbol-for-imaginary-numbers.htmlhttps://liberty.io/what-is-the-symbol-for-imaginary-numbers.html$\mathbb Q$ is used to represent rational numbers. $\mathbb R$ is used to represent real numbers. Is there an accepted symbol for imaginary numbers?...Wed, 26 Aug 2026 07:07:16 -0400How to integrate when integration by parts never ends?https://liberty.io/how-to-integrate-when-integration-by-parts-never-ends.htmlhttps://liberty.io/how-to-integrate-when-integration-by-parts-never-ends.htmlSo I have the following integration: $\int{e^{-x}\sin{x}}$ I let u be the $e^{-x}$ and $dv = \sin{x} dx$ After doing the integration by parts I get:$(e^{-x})(-\cos{x})-\int{-e^{-x}*(-\cos{x})}$ ......Wed, 26 Aug 2026 07:04:53 -0400What does the mathematical symbol with an underline below the variable name mean?https://liberty.io/what-does-the-mathematical-symbol-with-an-underline-below-the-variable.htmlhttps://liberty.io/what-does-the-mathematical-symbol-with-an-underline-below-the-variable.htmlI have been reading a research paper. The author has used a variable '$x$' with an underline beneath '$x$'. I know that a variable with a line on top of it implies it's arithmetic mean. But I have ......Wed, 26 Aug 2026 07:03:55 -0400Two different formulas for standard error of difference between two meanshttps://liberty.io/two-different-formulas-for-standard-error-of-difference-between-two-me.htmlhttps://liberty.io/two-different-formulas-for-standard-error-of-difference-between-two-me.htmlI mostly see this formula when searching for a formula for the estimate of the standard error in difference between two means, and it is also used in this video. $$\Delta=\sqrt{s_1^2/N_1+s_2^2/N_2}......Wed, 26 Aug 2026 07:00:52 -0400$\Delta x$ in limit problem?https://liberty.io/delta-x-in-limit-problem.htmlhttps://liberty.io/delta-x-in-limit-problem.htmlI was working on some limit homework and everything was going fine until I reached this problem: $$\lim_{\Delta x \to 0} \frac{2(x + \Delta x) - 2x}{\Delta x}.$$ I am understanding limits but the ...Wed, 26 Aug 2026 06:58:10 -0400Calculate X Y Z from two specific degrees on a spherehttps://liberty.io/calculate-x-y-z-from-two-specific-degrees-on-a-sphere.htmlhttps://liberty.io/calculate-x-y-z-from-two-specific-degrees-on-a-sphere.htmlI am a programmer, don't know much about advanced math. I would need the exact formula(s) that could achieve this, so I can translate it to my programming language. I am having a headache trying to ...Wed, 26 Aug 2026 06:56:00 -0400Why does "$x^2 - 5x + 6 = 0$", which is the same as "$(x-3)(x-2) = 0$", represent a parabola?https://liberty.io/why-does-x-2-5x-6-0-which-is-the-same-as-x-3-x-2-0-represent-a-parabol.htmlhttps://liberty.io/why-does-x-2-5x-6-0-which-is-the-same-as-x-3-x-2-0-represent-a-parabol.htmlConsider the equation $x^2 - 5x + 6 = 0$. By factorising I get $(x-3)(x-2) = 0$. Which means it represents a pair of straight lines, namely $x-2 =0 $ and $x- 3 = 0$, but when I plot $x^2 - 5x + 6 ......Wed, 26 Aug 2026 06:54:40 -0400The sum of series with natural logarithm: $\sum_{n=1}^\infty \ln\left(\frac{n(n+2)}{(n+1)^2}\right)$ [duplicate]https://liberty.io/the-sum-of-series-with-natural-logarithm-sum-n-1-infty-ln-left-frac-n-.htmlhttps://liberty.io/the-sum-of-series-with-natural-logarithm-sum-n-1-infty-ln-left-frac-n-.htmlCalculate the sum of series: $$\sum_{n=1}^\infty \ln\left(\frac{n(n+2)}{(n+1)^2}\right)$$ I tried to spread this logarithm, but I'm not seeing any method for this exercise....Wed, 26 Aug 2026 06:51:24 -0400Integral of $x^2 e^{-x^2}$https://liberty.io/integral-of-x-2-e-x-2.htmlhttps://liberty.io/integral-of-x-2-e-x-2.htmlLike the title says, I'm trying to find $$\int_0^r x^2 e^{-x^2}\,dx$$ Where $r$ is some finite value. I've done one step using integration by parts with $u=x^2$ and $dv=e^{-x^2}dx$, which has le......Wed, 26 Aug 2026 06:45:12 -0400Which algorithm does MATLAB eig() use to diagonalize a complex symmetric matrix?https://liberty.io/which-algorithm-does-matlab-eig-use-to-diagonalize-a-complex-symmetric.htmlhttps://liberty.io/which-algorithm-does-matlab-eig-use-to-diagonalize-a-complex-symmetric.htmlI used MATLAB eig() to find eigenvectors and eigenvalues of a complex symmetric matrix. I searched through MATLAB online documentation to find a link to some algorithm, but failed. My curiosity is ......Wed, 26 Aug 2026 06:23:49 -0400Convert the integral from rectangular to cylindrical coordinates and solvehttps://liberty.io/convert-the-integral-from-rectangular-to-cylindrical-coordinates-and-s.htmlhttps://liberty.io/convert-the-integral-from-rectangular-to-cylindrical-coordinates-and-s.htmlConvert the integral from rectangular to cylindrical coordinates and solve I think I know how to do this, but I just want to double check my method. So assuming I have the below problem: $$\int^......Wed, 26 Aug 2026 06:21:33 -0400How can Cyclic groups be infinitehttps://liberty.io/how-can-cyclic-groups-be-infinite.htmlhttps://liberty.io/how-can-cyclic-groups-be-infinite.htmlI am a little confused about how a cyclic group can be infinite. To provide an example, look at $\langle 1\rangle$ under the binary operation of addition. You can never make any negative numbers with ...Wed, 26 Aug 2026 06:19:58 -0400How to find curves of intersection between two surfaceshttps://liberty.io/how-to-find-curves-of-intersection-between-two-surfaces.htmlhttps://liberty.io/how-to-find-curves-of-intersection-between-two-surfaces.htmlThe curve of the intersection of the surface $x^2+y^2-2z^2=1$ and the plane $y=2z+1$ is given by $\varphi(t)=(\sqrt2\cos t,2\sin t-1,\sin t-1), t\in [0,2\pi)$. I would like to know how to find it ......Wed, 26 Aug 2026 06:17:40 -0400What is the difference between a forest and a spanning forest?https://liberty.io/what-is-the-difference-between-a-forest-and-a-spanning-forest.htmlhttps://liberty.io/what-is-the-difference-between-a-forest-and-a-spanning-forest.htmlIf a graph is labelled as a forest it does not contain any cycles, meaning it consists of all trees, which I realize can even be a single node (since that is technically a tree). If a graph is lab......Wed, 26 Aug 2026 06:07:33 -0400Reduction formula for $\int \cosh^n x dx$https://liberty.io/reduction-formula-for-int-cosh-n-x-dx.htmlhttps://liberty.io/reduction-formula-for-int-cosh-n-x-dx.htmlGiven $n\in\Bbb N$, $n > 2$, find the reduction formula for: $$ \int \cosh^n x dx $$ I've derived an answer, but the signs do not match, so my question would be: where is the mistake in my ...Wed, 26 Aug 2026 06:05:22 -0400Integral: $\int_{-\infty}^{\infty} x^2 e^{-x^2}\mathrm dx$https://liberty.io/integral-int-infty-infty-x-2-e-x-2-mathrm-dx.htmlhttps://liberty.io/integral-int-infty-infty-x-2-e-x-2-mathrm-dx.htmlI don't know how to evaluate it. I know there is one method using the gamma function. BUT I want to know the solution using a calculus method like polar coordinates. $$\int_{-\infty}^\infty x^2 e^......Wed, 26 Aug 2026 06:01:41 -0400Generalization: $x=\text{sup}\{q\in \mathbb{Q}:q<x\}$https://liberty.io/generalization-x-text-sup-q-in-mathbb-q-q-x.htmlhttps://liberty.io/generalization-x-text-sup-q-in-mathbb-q-q-x.htmlHow do I prove that $x=\text{sup}\{q\in \mathbb{Q}:q<x\}$? Provided that $x\in\mathbb{R}$......Wed, 26 Aug 2026 06:00:32 -0400Random math questions (modular arithmetic & notation)https://liberty.io/random-math-questions-modular-arithmetic-notation.htmlhttps://liberty.io/random-math-questions-modular-arithmetic-notation.htmlI found this amazing wall clock picture on the internet but I really don't know a few things. I don't know what's $B'_L$, &#x33;, why $2^{-1}\equiv 4[7]$ and the one with the black and white ci......Wed, 26 Aug 2026 06:00:08 -0400How does a branch cut define a branch?https://liberty.io/how-does-a-branch-cut-define-a-branch.htmlhttps://liberty.io/how-does-a-branch-cut-define-a-branch.htmlI am studying complex analysis and I have problem understanding the concept of branch cut. The lecturer draw this as some curve that starts from a point and goes on and on in some direction (for e......Wed, 26 Aug 2026 05:44:53 -0400What is a "convex" polyhedron?https://liberty.io/what-is-a-convex-polyhedron.htmlhttps://liberty.io/what-is-a-convex-polyhedron.htmlI was doing a spot of light reading (crystallography), when the term "convex" polyhedron came up in a a section (very prominently) in conjunction with something else called the "Euler ...Wed, 26 Aug 2026 05:38:15 -0400Is a hypersurface really defined by an arbitrary polynomial?https://liberty.io/is-a-hypersurface-really-defined-by-an-arbitrary-polynomial.htmlhttps://liberty.io/is-a-hypersurface-really-defined-by-an-arbitrary-polynomial.htmlIn An Invitation to Algebraic Geometry Karen Smith writes at the beginning of the book: The zero set of a single polynomial in arbitrary dimension is called a hypersurface in $\mathbb C^n$. The ...Wed, 26 Aug 2026 05:32:27 -0400Why is Euler's number $2.71828$ and not anything else? [closed]https://liberty.io/why-is-euler-039-s-number-2-71828-and-not-anything-else-closed.htmlhttps://liberty.io/why-is-euler-039-s-number-2-71828-and-not-anything-else-closed.htmlWhy is Euler's number $\mathtt 2.71828$ and not for example $\mathtt 3.7589$? I know that $e$ is the base of natural logarithms. I know about areas on hyperbola xy=1 and I know its formula: $$e =\......Wed, 26 Aug 2026 05:23:36 -0400Why the slope of a line is change in y divided by change in x? [duplicate]https://liberty.io/why-the-slope-of-a-line-is-change-in-y-divided-by-change-in-x-duplicat.htmlhttps://liberty.io/why-the-slope-of-a-line-is-change-in-y-divided-by-change-in-x-duplicat.htmlWhy the slope of a line isn't the change in x divided by the change in y?...Wed, 26 Aug 2026 05:17:42 -0400Product of two independent poisson variableshttps://liberty.io/product-of-two-independent-poisson-variables.htmlhttps://liberty.io/product-of-two-independent-poisson-variables.htmlSuppose we have two independent poisson variables $X_1$ and $X_2$ such that $X_1 ∼ \operatorname{Poisson}(\lambda_1)$ and $X_2 ∼ \operatorname{Poisson}(\lambda_2)$. What will be the probability ...Wed, 26 Aug 2026 05:14:53 -0400If my average speed is 50 mph, is it better to go 10% faster or 5 mph faster? (These are not the same)https://liberty.io/if-my-average-speed-is-50-mph-is-it-better-to-go-10-faster-or-5-mph-fa.htmlhttps://liberty.io/if-my-average-speed-is-50-mph-is-it-better-to-go-10-faster-or-5-mph-fa.htmlBACKGROUND Suppose we have a journey between $A$ and $B$ where different sections of roads have different speed limits and our average speed is $50$ mph. If when I repeat the journey I travel $10......Wed, 26 Aug 2026 05:10:26 -0400Show Marsaglia polar method produce the standard normal distributionhttps://liberty.io/show-marsaglia-polar-method-produce-the-standard-normal-distribution.htmlhttps://liberty.io/show-marsaglia-polar-method-produce-the-standard-normal-distribution.htmlSuppose $X, Y$ are two independent variables that uniformly distributed on the interval $[-1,1]$. If $s = x^2 + y^2 < 1$, show the result of $\frac{x}{\sqrt{s}} \sqrt{-2\ln(s)}$ comes from a sta......Wed, 26 Aug 2026 05:03:07 -0400How is the graph coloring problem NP-Complete?https://liberty.io/how-is-the-graph-coloring-problem-np-complete.htmlhttps://liberty.io/how-is-the-graph-coloring-problem-np-complete.htmlAn NP-Complete problem can be checked efficiently, but has no known way of being solved in polynomial time. Then, how is the graph coloring problem (http://en.wikipedia.org/wiki/Graph_coloring_pro......Wed, 26 Aug 2026 05:01:31 -0400Show that the perimeter of an octagon is $8(\sqrt{2} -1)$https://liberty.io/show-that-the-perimeter-of-an-octagon-is-8-sqrt-2-1.htmlhttps://liberty.io/show-that-the-perimeter-of-an-octagon-is-8-sqrt-2-1.htmlMy question is as follows: The top of a table is made in the shape of a regular octagon by cutting the congruent isosceles triangles from the corners of a $1$ m square piece of wood. Show that the ...Wed, 26 Aug 2026 04:47:44 -0400Finding the limit when denominator = 0https://liberty.io/finding-the-limit-when-denominator-0.htmlhttps://liberty.io/finding-the-limit-when-denominator-0.htmlI'm having trouble solving this limit: $$\lim_{x \to -2^-} \frac{1}{(x + 2)^2}$$ I can't find a way to rationalize the denominator. Also, is there a way to do it without plugging in -2.001 and st......Wed, 26 Aug 2026 04:47:01 -0400Prove that if A and B are mutually exclusive then $P(A)\leq P(B^c)$.https://liberty.io/prove-that-if-a-and-b-are-mutually-exclusive-then-p-a-leq-p-b-c.htmlhttps://liberty.io/prove-that-if-a-and-b-are-mutually-exclusive-then-p-a-leq-p-b-c.htmlSuppose $\textbf{P}(A\cap B) = 0$. Prove $\textbf{P}(A) \leq \textbf{P}(B^{c})$ using the axioms of probability. I know the axioms are: $\textbf{P}(A)\geq 0$. $\textbf{P}(\Omega) = 1$. If A and......Wed, 26 Aug 2026 04:41:00 -0400Find the composition of a piecewise functionhttps://liberty.io/find-the-composition-of-a-piecewise-function.htmlhttps://liberty.io/find-the-composition-of-a-piecewise-function.htmlLet $$f(x)= \begin{cases} 5 &\text{, x $\geq$ 0}\\ x^2 &\text{, x < 0} \end{cases}$$ and $$g(x)= \begin{cases} \sqrt{x} &\text{, x > 3}\\ 4 &\text{, x $<$ 3} \end{cases}$$ ......Wed, 26 Aug 2026 04:37:40 -0400Show that the dimention of the intersection of projective linear sub-spaces of dimentions $d_1$ and $d_2$ of $\mathbb{P}^n$ is bigger than $d_1+d_2-n$https://liberty.io/show-that-the-dimention-of-the-intersection-of-projective-linear-sub-s.htmlhttps://liberty.io/show-that-the-dimention-of-the-intersection-of-projective-linear-sub-s.htmlProposition: Let $L$ and $M$ linear projective subspaces of $\mathbb{P}^n$ of dimention $d_1$ and $d_2$ respectively. Prove that $\operatorname{dim}(L\cap M)\geq d_1+d_2-n$ (we consider the dimenti......Wed, 26 Aug 2026 04:36:48 -0400Difference between imaginary part and imaginary numberhttps://liberty.io/difference-between-imaginary-part-and-imaginary-number.htmlhttps://liberty.io/difference-between-imaginary-part-and-imaginary-number.htmlIs there a difference between imaginary part and imaginary number? Because our teacher says there is. I mean in $5+3i$ , is it right to say the imaginary part is $3i$ and imaginary number is $3$?......Wed, 26 Aug 2026 04:34:58 -0400Find a circle perpendicular to two other circles.https://liberty.io/find-a-circle-perpendicular-to-two-other-circles.htmlhttps://liberty.io/find-a-circle-perpendicular-to-two-other-circles.htmlHere are two circles $C_1 = [ x^2 + y^2 = 1]$ and $C_2 = [ (x-2)^2 + y^2 = 3 ]$. The radii are $1$ and $3$ and the two circles are orthogonal to each other $C_1 \cdot C_2 = 0$. What is the equati......Wed, 26 Aug 2026 04:34:09 -0400Finding the local trivilization of canonical line bundlehttps://liberty.io/finding-the-local-trivilization-of-canonical-line-bundle.htmlhttps://liberty.io/finding-the-local-trivilization-of-canonical-line-bundle.htmlI am going through Nakahara's "Geometry, topology and physics" by myself and I got stuck on a calculation. I have the total space $$L=\{(p,v)\in\mathbb{C}P^n\times\mathbb{C}^{n+1}|v=ap,a\in\math......Wed, 26 Aug 2026 04:32:13 -0400Express $\cos(\tan^{-1}(x))$ in terms of $x$.https://liberty.io/express-cos-tan-1-x-in-terms-of-x.htmlhttps://liberty.io/express-cos-tan-1-x-in-terms-of-x.htmlExpress $\cos(\tan^{-1}(x))$ in terms of $x$. My Attempt: Let $\tan^{-1}(x)=A$. $$x=\tan (A)$$ Then, $$\cos(\tan^{-1}(x))=\cos (A)$$...Wed, 26 Aug 2026 04:30:10 -0400Composition of function with itselfhttps://liberty.io/composition-of-function-with-itself.htmlhttps://liberty.io/composition-of-function-with-itself.htmlFor the set of all functions $f:\{0,1, \dots,2014\} \to \{0,1,\dots, 2014\}$ such that $f\left(f\left(i\right)\right)=i$, for all $0 \leq i \leq 2014$. Consider the following statements: $A$ : For ...Wed, 26 Aug 2026 04:18:25 -0400Boolean algebra - sum of products formhttps://liberty.io/boolean-algebra-sum-of-products-form.htmlhttps://liberty.io/boolean-algebra-sum-of-products-form.htmlI have a logic circuit where the output can be represented with the following boolean expression (1)$\overline {xy}$ + x $\bar y z$ + $\overline {\bar x + z} $ + y Using truth tables I found the ...Wed, 26 Aug 2026 04:05:49 -0400Simultaneous diagonalization of commuting linear transformationshttps://liberty.io/simultaneous-diagonalization-of-commuting-linear-transformations.htmlhttps://liberty.io/simultaneous-diagonalization-of-commuting-linear-transformations.htmlLet $V$ be a vector space of finite dimension and let $T,S$ linear diagonalizable transformations from $V$ to itself. I need to prove that if $TS=ST$ every eigenspace $V_\lambda$ of $S$ is $T$-inv......Wed, 26 Aug 2026 04:04:31 -0400Derived categories of filtered moduleshttps://liberty.io/derived-categories-of-filtered-modules.htmlhttps://liberty.io/derived-categories-of-filtered-modules.htmlFor a ${\mathbb Z}$-filtered ring ${\mathbb k}$ one can consider the category ${\mathbb k}\text{-filt}$ of ${\mathbb Z}$-filtered ${\mathbb k}$-modules, equipped with the exact structure which decl......Wed, 26 Aug 2026 04:02:36 -0400Sum of sets of measure zerohttps://liberty.io/sum-of-sets-of-measure-zero.htmlhttps://liberty.io/sum-of-sets-of-measure-zero.htmlLet $A$ and $B$ be two subsets of $\Bbb R$ of measure zero. Is it true that the Minkowski sum $A+B = \{ a + b \mid a \in A, b \in B \}$ has measure zero as well? I think so but I can't prove it. The ...Wed, 26 Aug 2026 03:59:58 -0400How to evaluate the integral $\int e^{2x} \sin^2x\ dx$ [closed]https://liberty.io/how-to-evaluate-the-integral-int-e-2x-sin-2x-dx-closed.htmlhttps://liberty.io/how-to-evaluate-the-integral-int-e-2x-sin-2x-dx-closed.htmlHow can I evaluate the following integral? $$\int e^{2x} \sin^2x\ dx$$...Wed, 26 Aug 2026 03:58:37 -0400Universal hashing familyhttps://liberty.io/universal-hashing-family.htmlhttps://liberty.io/universal-hashing-family.htmlI'm not really sure this question is proper for MathExchange but I have found it more suitable than StackOverFlow. So, I'm taking the course of Data Structures and Algorithms and we got some examp......Wed, 26 Aug 2026 03:53:12 -0400Proof of intersection and union of Set A with Empty Sethttps://liberty.io/proof-of-intersection-and-union-of-set-a-with-empty-set.htmlhttps://liberty.io/proof-of-intersection-and-union-of-set-a-with-empty-set.htmlI need to prove the following: Prove that $A\cup \!\, \varnothing \!\,=A$ and $A\cap \!\, \varnothing \!\,=\varnothing \!\,$ It's my understanding that to prove equality, I must prove that both are ...Wed, 26 Aug 2026 03:45:43 -0400Calculating the square root of 2https://liberty.io/calculating-the-square-root-of-2.htmlhttps://liberty.io/calculating-the-square-root-of-2.htmlSince $\sqrt{2}$ is irrational, is there a way to compute the first 20 digits of it? What I have done so far I started the first digit decimal of the $\sqrt{2}$ by calculating iteratively so th......Wed, 26 Aug 2026 03:32:14 -0400Convergent or Divergent Integralhttps://liberty.io/convergent-or-divergent-integral.htmlhttps://liberty.io/convergent-or-divergent-integral.htmlConvergent or Divergent? $$\int_0^1 \frac {dx}{(x+x^{5})^{1/2}} $$ I have problem with the fact that if we have integration from 0 to a say and a to infinity. How does this change the way we do......Wed, 26 Aug 2026 03:29:23 -0400Differentiation, using d or deltahttps://liberty.io/differentiation-using-d-or-delta.htmlhttps://liberty.io/differentiation-using-d-or-delta.htmlAre the symbols $d$ and $\delta$ equivalent in expressions like $dy/dx$? Or do they mean something different? Thanks...Wed, 26 Aug 2026 03:27:24 -0400Limacon/cardioid orientation confusionhttps://liberty.io/limacon-cardioid-orientation-confusion.htmlhttps://liberty.io/limacon-cardioid-orientation-confusion.htmlI'm learning about polar equations and am trying to graph the curve r = 3 + 3 sin(theta), with theta between 0 and 2pi. Based on what I've learned and the image below, since a = b = 3, it would see......Wed, 26 Aug 2026 03:26:52 -0400Can you predict the remainder when 111...1 (100 ones) is divided by 1111111https://liberty.io/can-you-predict-the-remainder-when-111-1-100-ones-is-divided-by-111111.htmlhttps://liberty.io/can-you-predict-the-remainder-when-111-1-100-ones-is-divided-by-111111.htmlWe have 111...11 a hundred times. We begin with 1 in our quotient. We have 93 ones remaining. We add six zeros according to the division algorithm to get 1000000 in our quotient. We add another on......Wed, 26 Aug 2026 03:20:48 -0400Limits at infinity involving ehttps://liberty.io/limits-at-infinity-involving-e.htmlhttps://liberty.io/limits-at-infinity-involving-e.htmlI am working with vector functions and one of the problems has $e$ in it. I always thought that $$\lim_{x\to\infty}e^{ax} = \infty$$ but the book says otherwise and when I went all the way back to ...Wed, 26 Aug 2026 03:12:20 -0400For u-substitution of a definite integral, is it acceptable to have the limits (bounds) of the new integral equal to one another?https://liberty.io/for-u-substitution-of-a-definite-integral-is-it-acceptable-to-have-the.htmlhttps://liberty.io/for-u-substitution-of-a-definite-integral-is-it-acceptable-to-have-the.htmlFor example, I have the problem: $$ \int_{-1}^1x^4\left(1-x^2\right)^2dx $$ I set $\,u=1-x^2\,$, which gives $\,x^2=1-u\,$, $\,x=\pm\sqrt{1-u}\,$ and $\,-\frac{1}{2}du=xdx$. For the upper and lower ...Wed, 26 Aug 2026 03:05:14 -0400Inverse of random variablehttps://liberty.io/inverse-of-random-variable.htmlhttps://liberty.io/inverse-of-random-variable.htmlLet's say we have a random variable $X$ of which the distribution is unknown. Now there are these general rules like $E[X + Y] = E[X] + E[Y]$ etc. But what if we would define $ \quad Y = \dfrac{1}......Wed, 26 Aug 2026 03:04:21 -0400Stardew Valley Farm -- Recreational Math Problemhttps://liberty.io/stardew-valley-farm-recreational-math-problem.htmlhttps://liberty.io/stardew-valley-farm-recreational-math-problem.htmlIn the game Stardew Valley, farming is essential. In order to plant seeds, one must till the soil using a hoe. A golden hoe is an upgraded hoe that allows the player to till 1, 3, or 5 squares of d......Wed, 26 Aug 2026 02:57:03 -0400What is the smallest $20$ digit number with at least $13$ distinct prime factors?https://liberty.io/what-is-the-smallest-20-digit-number-with-at-least-13-distinct-prime-f.htmlhttps://liberty.io/what-is-the-smallest-20-digit-number-with-at-least-13-distinct-prime-f.htmlThe number $10^{19}+1035830$ is a $20$-digit number with $12$ distinct prime factors. I am not sure whether it is the smallest example because to save time I only considered numbers with at least $3$ ...Wed, 26 Aug 2026 02:52:05 -0400Why shift the result of subtracting vectors?https://liberty.io/why-shift-the-result-of-subtracting-vectors.htmlhttps://liberty.io/why-shift-the-result-of-subtracting-vectors.htmlWhy do we shift the resulting vector after subtracting any two vectors a and b? I learned that when you add any two vectors a and b, the sum vector looks like the one that can be seen in the picture ...Wed, 26 Aug 2026 02:42:12 -0400Newton method to solve nonlinear systemhttps://liberty.io/newton-method-to-solve-nonlinear-system.htmlhttps://liberty.io/newton-method-to-solve-nonlinear-system.htmlHow can I solve this problem: ${e^x}^y +{x}^2+y-1.2=0 \\ {x}^2 +{y}^2+x-0.55=0$ for various loaded inputs with start points $(x_0 , y_0) = 0.6, 0.5$ Could anybody help me. Thank you in advance...Wed, 26 Aug 2026 02:40:34 -0400What is orthogonal transformation?https://liberty.io/what-is-orthogonal-transformation.htmlhttps://liberty.io/what-is-orthogonal-transformation.htmlWhen $A^{T}A = AA^{T} = I$, I am told it is an orthogonal transformation $A$. But don't really get what it means. Hope to hear some explanations. $\begin{bmatrix}cos\theta & sin\theta \\ -sin\......Wed, 26 Aug 2026 02:28:19 -0400$n$th derivative of $e^x \sin x$https://liberty.io/n-th-derivative-of-e-x-sin-x.htmlhttps://liberty.io/n-th-derivative-of-e-x-sin-x.htmlCan someone check this for me, please? The exercise is just to find a expression to the nth derivative of $f(x) = e^x \cdot \sin x$. I have done the following: Write $\sin x = \dfrac{e^{ix} - e^{-......Wed, 26 Aug 2026 02:27:19 -0400Inverse of percentage?https://liberty.io/inverse-of-percentage.htmlhttps://liberty.io/inverse-of-percentage.htmlWhat is the best way to cacluate the inverse of a percentage? For example: 100 * .85 = 85 85 * x = 100 x = 100 / 85 x = 1.17647 Also, 279 * .85 = 237.15 237.15 * x = 279 x = 279 / 237.15 x = 1.......Wed, 26 Aug 2026 02:24:50 -0400What is the difference between a Definition and a Theorem?https://liberty.io/what-is-the-difference-between-a-definition-and-a-theorem.htmlhttps://liberty.io/what-is-the-difference-between-a-definition-and-a-theorem.htmlThis may get into a discussion, but I have a homework problem and it tells me there is a difference between a definition and a theorem. I don't know how to differentiate the two in this question: ...Wed, 26 Aug 2026 02:24:19 -0400A family has two children. One child is a girl. What is the probability that the other child is a boy? [duplicate]https://liberty.io/a-family-has-two-children-one-child-is-a-girl-what-is-the-probability-.htmlhttps://liberty.io/a-family-has-two-children-one-child-is-a-girl-what-is-the-probability-.htmlMy initial thought process: Sample space: GG, GB, BG, BB. I then crossed out BG because it's the same as GB because order doesn't matter here. And because we know one is a girl, there leaves two ...Wed, 26 Aug 2026 02:18:32 -0400Uniqueness of solution of differential equationhttps://liberty.io/uniqueness-of-solution-of-differential-equation.htmlhttps://liberty.io/uniqueness-of-solution-of-differential-equation.htmlConsider the differential equation $ y'(x)=f(x,y(x)) $ with initial condition $y(x_0)=y_0$. According to some theorem, if $f$ is continuous, then there exists at least one solution that satisfies the ...Wed, 26 Aug 2026 02:14:02 -0400Find the area of the region that is enclosed by the cardioid $r=2+2\sin(\theta)$https://liberty.io/find-the-area-of-the-region-that-is-enclosed-by-the-cardioid-r-2-2-sin.htmlhttps://liberty.io/find-the-area-of-the-region-that-is-enclosed-by-the-cardioid-r-2-2-sin.htmlWe just learned polar integration, so I know that's how we're supposed to do it. I have a problem though: I'm getting a negative answer. What I did: Using the graph, which is: I figured out that......Wed, 26 Aug 2026 02:10:17 -0400set of natural numbers subset of the set of real numbers [closed]https://liberty.io/set-of-natural-numbers-subset-of-the-set-of-real-numbers-closed.htmlhttps://liberty.io/set-of-natural-numbers-subset-of-the-set-of-real-numbers-closed.htmlThe natural numbers are said to be a subset of the real numbers but how is this possible since in the set of natural numbers division is not allowed....Wed, 26 Aug 2026 01:52:49 -0400Limit of product of sequences is the product of the limits of the sequenceshttps://liberty.io/limit-of-product-of-sequences-is-the-product-of-the-limits-of-the-sequ.htmlhttps://liberty.io/limit-of-product-of-sequences-is-the-product-of-the-limits-of-the-sequ.htmlI want to prove that if: $$\lim_{n \to \infty}s_n = L_1, \lim_{n \to \infty}t_n = L_2$$ then $$\lim_{n \to \infty}(s_n t_n) = L_1L_2$$ where $s_n, t_n$ are complex sequences (I'm working through......Wed, 26 Aug 2026 01:49:31 -0400Find a solution for 23x=5 (mod 60)https://liberty.io/find-a-solution-for-23x-5-mod-60.htmlhttps://liberty.io/find-a-solution-for-23x-5-mod-60.htmlThe topic is euclidean algorithm and GCD. This is a simple question, but I'm just stuck finding an inverse for 23 in $Z_{60}$. I performed the algorithm and proved that 23 and 60 are coprime, so th......Wed, 26 Aug 2026 01:46:16 -0400Does $S$ spans $V$ mean every $v \in V$ is a finite linear combination of elements in $S$?https://liberty.io/does-s-spans-v-mean-every-v-in-v-is-a-finite-linear-combination-of-ele.htmlhttps://liberty.io/does-s-spans-v-mean-every-v-in-v-is-a-finite-linear-combination-of-ele.htmlDefine $\mathbb{R}^{\infty}:=\lbrace (a_1,a_2,a_3,\ldots) \mid a_i \in \mathbb{R} \rbrace$ and let $\phi_i((a_1,a_2,a_3,\ldots))=a_i$. Show that $S=\lbrace \phi_i \mid i \geq 1 \rbrace$ does not sp......Wed, 26 Aug 2026 01:24:55 -0400What do we mean by 'defining a function'?https://liberty.io/what-do-we-mean-by-039-defining-a-function-039.htmlhttps://liberty.io/what-do-we-mean-by-039-defining-a-function-039.htmlFirst I will start by quote from Wikipedia about function's defining methods : There are many other ways of defining functions. Examples include piecewise definitions, induction or recursion, ...Wed, 26 Aug 2026 01:16:44 -0400Linearizing equations of motionhttps://liberty.io/linearizing-equations-of-motion.htmlhttps://liberty.io/linearizing-equations-of-motion.htmlI have been looking at the operations of a quadrotor drone. I am reading the maths behind it and in one section it mentions: "These equations of motion are linearized with respect to an equili......Wed, 26 Aug 2026 01:04:31 -0400Discriminant Of A Higher Degree Polynomialshttps://liberty.io/discriminant-of-a-higher-degree-polynomials.htmlhttps://liberty.io/discriminant-of-a-higher-degree-polynomials.htmlI searched a lot in Google and even tried myself to determine the nature of roots of 4th and 3rd degree standard equations.Is their any way to know nature of roots of an equation such as $(b^2-4ac)......Wed, 26 Aug 2026 01:01:33 -0400How can I calculate the centroid of polygon?https://liberty.io/how-can-i-calculate-the-centroid-of-polygon.htmlhttps://liberty.io/how-can-i-calculate-the-centroid-of-polygon.htmlWhat is the way to calculate the centroid of polygon? I have a concave polygon of 16 points, and I want know the centroid of that. thanks...Wed, 26 Aug 2026 01:01:01 -0400