Recent Questions - Mathematics Stack Exchange |
- Zero-mean random variable for the number of descents
- Removal of an arbitrary point of the boundary of a closed and connected $A\subseteq\Bbb R^2$ so the new set remains connected
- How can we adapt the Ito's formula if $g \in C^{2}(\mathbb R\setminus \{ x_{1},...,x_{n}\})$ and $g^{''} \leq M$
- Generating pythagorean triples under conditions
- Tubular surface around curve
- Any intuitive answer to the summation of power of 2? [duplicate]
- Markov Chain/Matrix from previous multiple independent states that only one (not both) need to be true.
- Can three quarters of a circle be parametrized by two rational functions on a single interval?
- Finding the permanent of a matrix
- Strongly zero-dimensinal spaces that are homeomorphic to other spaces under certain conditions
- Is there any website/software that let's me compare two mathematical statements and let me know if both are equivalent?
- Most presentations of ZF seem to be incomplete
- Integral of $\frac{1}{2}e^{y^2}$ is it elementary of integral?
- Questions about fixed point iteration
- Let $Y_n,X_n$ be a sequence of r.v. Does $\sup_z \Big|P(Y_n<z\ |\ X_n) - \Phi(z)\Big|\xrightarrow[]{p}0 \implies Y_n|X_n\xrightarrow[]{d}N(0,1)?$
- phones that have a mass greater than 3 standard deviations from the mean are rejected by the quality assurance process. What proportion are rejected?
- $\lim_{n\to\infty} n^2 \int_{0}^{1} \frac{x\sin{x}}{1+(nx)^3} \, \mathrm{d}x$
- Finding $\alpha$ in $Ax=b$
- Cubic expectation of peanuts on squares
- Is the Half Disc Topology normal?
- Why am I getting two different answers while calculating $\int \frac{dx}{\sqrt{2ax-x^2}}$?
- Is this a deterministic proof for no non-trivial cycles? (Collatz Conjecture)
- Why angle form by a secant and tangent outside a circle is half the intercepted arc?
- Simplification of sum expression for inverse of $\gamma(a,z)$ with Lower Incomplete Gamma function and Gamma distribution median
- Intersection point of all tangent planes to a sphere with point of contact on a circle
- Proving a condition of perpendicularity
- What is the set $\{A \,\,\mid \text{ any two eigenvectors of }$A$\text{from different eigenspaces are orthogonal.}\}$?
- If you throw a dice 5 times, what is the expected value of the square of the median?
- How do you show that the three incidence axioms are independent of each other.
- In a metric space $(S,d)$, assume that $x_n \to x$ and $y_n \to y$. Prove that $d(x_n, y_n) \to d (x, y)$.
| Zero-mean random variable for the number of descents Posted: 02 Jan 2022 05:06 AM PST Let $D_n$ be defined as the number of descents. We can create a martingale out of $D_n$ by scaling and subtracting the mean. $Z_n=n(D_n-\frac{n-1}{2})$ is this martingale with filtration $F_n$. Now let $X_{n,i}$ be defined as $$X_{n,i}= \frac{Z_i-Z_{i-1}}{s_n}$$ where $s_n=\sqrt{Var(Z_n)}=n\sqrt{\frac{n+1}{12}}$. With this one can get $$X_{n,i}= \frac{\sqrt{12}}{n\sqrt{n+1}}(iD_i-(i-1)D_{i-1}-(i-1)).$$ Let $Wi:=D_i - \frac{i-1}{2}$ be the zero-mean random variable for the number of descents. How do I show that $$X_{n,i}|F_{i-1}= \frac{\sqrt{12}}{n\sqrt{n+1}}\begin{cases} W_{i-1}-\frac{i}{2},& \text{with prob.} \frac{1}{2}+\frac{W_{i-1}}{i}. \\ W_{i-1}+\frac{i}{2},& \text{with prob.} \frac{1}{2}-\frac{W_{i-1}}{i}. \end{cases}?$$ While it is clear for me that the part before then cases is just from the constant $s_n$, I'm not sure how to get the cases and their respective probabilities |
| Posted: 02 Jan 2022 04:56 AM PST
I think I found some counterexamples: $x$ or $y$ axis or any other line in $\Bbb R^2,$ as well as graphs of unbounded continuous functions defined on an open interval $I\subseteq\Bbb R$ or graphs of continuous functions defined on the whole $\Bbb R.$ Is the unboundedness necessary for the statement not to hold? |
| Posted: 02 Jan 2022 04:54 AM PST Let $B$ be Brownian motion. Consider $g:\mathbb R \to \mathbb R$ that is $C^{2}$ except for some exceptional set $\{ x_{1},...,x_{n}\}\subseteq \mathbb R$. How can we adapt the Ito's formula if $g^{''} \leq M$, for $M > 0$, to show that $$g(B_{t}) = g(B_{0}) + \int_{0}^{t} g^{\prime}(B_{s})dB_{s}+\frac{1}{2}\int_{0}^{t}g^{\prime\prime}(B_{s})ds \; (*)$$ Due to density arguments we can find $(f_{k})_{k \in \mathbb N} \subseteq C ^{2}$ such that: $f_{k} \to g, \; (f_{k})^{\prime} \to g^{'},\; \text{ and }(f_{k})^{\prime\prime} \to g^{''} $ uniformly such that $(f_{k})^{\prime\prime}\leq M$ My attempt: Note that by Ito's formula: $$f_{k}(B_{t}) = f_{k}(B_{0}) + \int_{0}^{t} f_{k}^{\prime}(B_{s})dB_{s}+\frac{1}{2}\int_{0}^{t}f_{k}^{\prime\prime}(B_{s})ds \; $$ Clearly a.s. for all $t \geq 0$, $\;f_{k}(B_{t})\to g(B_{t})$ and $f_{k}(B_{0})\to g(B_{0})$ Now we need to show that a.s. $\int_{0}^{t} f_{k}^{\prime}(B_{s})dB_{s}\to \int_{0}^{t}g^{\prime}(B_{s})dB_{s}$ in $L^{2}$ ( I thought it would be a.s. be the stochastic integral lacks pathwise interpretability) $$ \mathbb E \left[\left(\int_{0}^{t} f_{k}^{\prime}(B_{s})dB_{s}- \int_{0}^{t} g^{\prime}(B_{s})dB_{s}\right)^{2}\right]= \mathbb E \left[\int_{0}^{t} \left(f_{k}^{\prime}(B_{s})- g^{\prime}(B_{s})\right)^{2}ds\right]$$ and then use dominated convergence to let it go to zero. Now onto proving $\int_{0}^{t}f_{k}^{\prime\prime}(B_{s})ds\to\int_{0}^{t}g^{\prime\prime}(B_{s})ds$. Here I am not sure in what convergence sense we should prove (convergence in $L^{2}$ or convergence almost surely, since the Lebesgue-Stieltjes integral does have pathwise interpretation.) I suggest a.s. convergence: $\rvert\int_{0}^{t}f_{k}^{\prime\prime}(B_{s})ds-\int_{0}^{t}g^{\prime\prime}(B_{s})ds\lvert \leq \int_{0}^{t} \lvert f_{k}^{\prime\prime}(B_{s})-g^{\prime\prime}(B_{s})\rvert ds $ using the fact that $\lvert f_{k}^{\prime\prime} \lvert ,\; \rvert g^{\prime\prime}\rvert\leq M$, we use dominated convergence again to get the result. Questions: If the above is correct, we have one term in $(*)$ as a limit in $L^{2}$ while the other terms are a.s. limits. How can we include all of these "different" limits in the equality? |
| Generating pythagorean triples under conditions Posted: 02 Jan 2022 05:04 AM PST I'm trying to generate a pythagorean triple, that is, $a,b,c \in \mathbb N$ that $$a^2 + b^2 = c^2$$ Under the condition that $b > a + n$, $n \in \mathbb N$ and $a+b$ is minimum. I tried expanding the forumula or trying to use the Euclid's formula but I don't know how to add the constraint. |
| Posted: 02 Jan 2022 04:47 AM PST I need help with drawing tubular surface around curve. It is for my college project but we never used program like Sage so I don't know how to use it properly. We have Curve c(t)=(2cos(t),2sin(t),0.5*t) and I need to draw tubular surface around it. I need to calculate frenet trihedron and use it in some formula for tubular surface. Formula is on this link: imgur.com/CUCu0da. My question is how do I make 3d projection out of this formula? |
| Any intuitive answer to the summation of power of 2? [duplicate] Posted: 02 Jan 2022 04:48 AM PST $$2^0+2^1+2^2+2^3+\dots+2^n = 2^{(n+1)} - 1$$ I want to know is there a simple (visual/intuitive/elementary) explanation for this. How can I explain the reason behind this result to a layman? |
| Posted: 02 Jan 2022 04:44 AM PST I am trying to solve this problem generally; let's say you roll a D6 a 9 times, you record these results and put them into a 3x3 matrix like so: { {r1, r2, r3}, {r4, r5, r6}, {r7, r8, r9} } What is the probability (number of true outcomes/total number of outcomes) that when reading the matrix, that any verticals (column) or horizontal (row) has all 5's. So a true some true cases are: { {5,3,6}, {5,5,5}, {1,2,4} } because the second row has all 5's. Also true is: { {5,5,5}, {5,5,1}, {5,5,5} } because the first and third rows hit the criteria, and the first and second columns hit the criteria, only one row or column needs to meet the criteria. I feel like there should be a way to do this using Markov Chains, but I am unable to come or find a method of doing so. Even though I feel like there is a way, I suspect there is also no way to do it. I have tried to unwrap the matrix in its untransposed form and transposed form, but I get the issue of something like this {4,3,5,5,5,1,6,6,2} being a false positive, because the three 5's are not in the same row or column, but do appear next to each other when unwrapped. What I obviously do know, is that for this example, there 6^9 or 10,077,696 possible matrices (not taking into account symmetries). A fraction of these are then true. |
| Can three quarters of a circle be parametrized by two rational functions on a single interval? Posted: 02 Jan 2022 04:47 AM PST Let $u(\theta)=(\cos(\theta),\sin(\theta))$ and $C=u((0,\frac{3\pi}{2}))$ ; thus $C$ consists of three quarters of a circle. If we use the classical transformation $t=\tan(\frac{\theta}{2})$, then $C$ can also be parametrized by $v(t)=(\frac{1-t^2}{1+t^2},\frac{2t}{1+t^2})$ : we have $C=v([\infty,-1) \cup (0,+\infty])$. Question : Can we find another parametrization $w(x)=(f(x),g(x))$ of $C$ where $f$ and $g$ are rational functions of $x$, and $C=w(I)$ where $I$ is a single interval of $\bar{\mathbb R}$ ? All the examples I could find so far of such parametrizations only cover half a circle. |
| Finding the permanent of a matrix Posted: 02 Jan 2022 04:41 AM PST I am trying to find the permanent of a $n\times n $ ($n\geq 4$) matrix $B_n$ of the form $$ \begin{bmatrix} 1 & 1 & 1 & 0 & 0 & 0\cdots& 0 & 0\\ 0 & 1 & 1 & 1 & 0 & 0 \cdots& 0 &0 \\ 0 & 0 & 1 & 1 & 1 &0 \cdots& 0 &0 \\ \vdots &\vdots &\vdots &\vdots &\vdots &\ddots& \vdots &\vdots \\ 1 & 0 & 0 & 0&0& 0 \cdots &1 & 1 \\ 1 & 1 & 0 & 0&0& 0 \cdots& 0 & 1 \end{bmatrix} $$ But I haven't had much luck. There is a hint for this problem though:
I wasn't able to prove $B_n = B_{n−1}+ B_{n−2} − 2$; the rest is clear. |
| Strongly zero-dimensinal spaces that are homeomorphic to other spaces under certain conditions Posted: 02 Jan 2022 04:36 AM PST According to a theorem from van Engelen: Homogeneous zero-dimensional absolute Borel sets (1986): Theorem: If $X$ is a separable metrizable zero-dimensional absolute $F_{\sigma\delta}$ that is nowhere $G_{\delta\sigma}$ and of the first category (in itself), then $X \simeq \mathbb{Q}^\omega$ . My question is: does the Theorem apply for any other spaces that are strongly zero-dimensional? If not with the exact same assumptions, the conditions can even differ from those in the theorem. But I am interested whether for any other strongly zero-dimensional space $Y$, there exists $X$ such that $X \simeq Y$ under some assumptions. I am thinking that maybe Cantor set or irrationals could have the property from the Theorem, but not very sure. Also I am not 100% sure if Cantor set and irrationals are strongly zero-dimensinal or just zero-dimensional. |
| Posted: 02 Jan 2022 04:58 AM PST Sometimes when I am doing my maths homework I sometimes get confused and then decide to make two mathematical statements and then plug some numbers in them to see if the results match. For example : $$\frac{\sqrt{x}}x = \frac{1}{\sqrt{x}}$$ Is there any website/software that lets me type the left hand side and the right hand side of the above equation and then tells me if these two are equivalent statements or not(like telling me in the form of true and false) I searched but was unable to find any, so if you know it then the answer would be appreciated. Thnx in advance. Note: Since I am an Idiot and Newbie I was unable to find the relevant tags for the question. |
| Most presentations of ZF seem to be incomplete Posted: 02 Jan 2022 04:46 AM PST A minor technical issue with ZF (and other set theories such as Morse-Kelley) is that if one isn't careful, the axioms will admit degenerate model, in which there are no sets at all. The axiom of pairing is typical here:
If there aren't any sets, then the axiom is vacuous. Some presentations get around this with an explicit axiom of the empty set: $$\exists x.\forall y. y\notin x$$ From this, we know there's at least one empty set; from extensionality we can can prove that it's unique, and this justifies assigning a symbol $\varnothing$ to it. (Kelley's original presentation of MK has instead an axiom "there exists a set", and this, plus specification, is enough to prove the existence of the empty set.) Then pairing and union and specification get us a universe of other sets. Fine. But many presentations omit the axiom of the empty set. Instead, they claim, the axiom of infinity asserts the existence of a set $\omega$, and with specification we can get $\varnothing$ as a subset of $\omega$. But it seems to me that this approach doesn't actually work. The axiom of infinity states: $$\exists S. (\varnothing\in S) \land (\forall y\in S. y\cup\{y\}\in S)$$ If we're using the axiom of infinity to prove the existence of the empty set, then at this point the symbol $\varnothing$ doesn't refer to anything, and is meaningless, and we have no business using it in the axiom. The symbol "$\varnothing$" also appears in the axiom of regularity, but there it isn't so problematic, because it only appears in the context "$x=\varnothing$". By this we actually mean "$\lnot\exists y. y\in x$", so the issue is merely sloppy notation. But for the axiom of infinity the problem is deeper and it seems to me it can't be fixed notationally, because the axiom of infinity demands that "$\varnothing$" denote an actual object. Both the Wikipedia presentation of ZF (which claims to follow Kunen) and the Mathworld presentation (claims to follow Jech) have this defect. I'm not trying to suggest that set theory itself is flawed; the issue is a minor technical one and is easily resolved by including an axiom of the empty set or even an axiom that asserts the existence of some set. My question is whether the typical presentation of set theory is defective. Or perhaps there is no defect and my understanding is deficient? |
| Integral of $\frac{1}{2}e^{y^2}$ is it elementary of integral? Posted: 02 Jan 2022 04:32 AM PST As $\int \:e^{y^2}dy$ is not a elementary of integral(antiderivative). I am confused is $\frac{1}{2}e^{y^2}$ elementary of integral. I thought it has to be used imaginary error function... I emailed the teacher "Why the answer isn't $\frac{\sqrt{\pi }}{2}erfi\left(1\right)$?", but the teacher reply "Do not know what you are talking about.". So am I wrong? Is that only $\int \:e^{y^2}dy $ not elementary of integral? Here is the answer...
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| Questions about fixed point iteration Posted: 02 Jan 2022 04:42 AM PST I am learning fixed-point iteration and am confused about the convergence rate, which is defined as follows: $$\lim_{k \rightarrow\infty}\frac{x_{k+1}-x^*}{(x_k-x^*)^p}=C,\quad C\neq 0$$ Then we call the iteration process is $p$-order convergent. My question is when the iteration process is $p+1$ order convergent, is it $p, p-1, \cdots, 1$ order convergent? My thoughts, suppose we have $$\lim_{k \rightarrow\infty}\frac{x_{k+1}-x^*}{(x_k-x^*)^{p+1}}=C,\quad C\neq 0$$ then, $$\lim_{k \rightarrow\infty}\frac{x_{k+1}-x^*}{(x_k-x^*)^{p}}*\frac{1}{(x_k-x^*)}=C,\quad C\neq 0$$ If the first term converges to some constant, then contradicts coz the second term goes to infinity. |
| Posted: 02 Jan 2022 04:29 AM PST Let $Y_n,X_n$ be a sequence of random variables. Does $\sup_z \Big|P(Y_n<z\ |\ X_n) - \Phi(z)\Big|\xrightarrow[]{p}0$ implies $Y_n|X_n\xrightarrow[]{d}N(0,1)?$ Note that $P(Y_n<z\ |\ X_n)$ is a random variable. By asymptotic equivalence, the above implies for all value of $z$, $P(Y_n<z\ |\ X_n)\xrightarrow[]{d}\Phi(z)$ which implies $P(Y_n<z\ |\ X_n)\xrightarrow[]{p}\Phi(z)$ for all $z$. This is as far as I could go. This question is closely related to this other question. |
| Posted: 02 Jan 2022 04:22 AM PST Assume the mass of a particular model of mobile phone is normally distributed with a mean of 137 g and a standard deviation of 1,2 q. a) Find the proportion of the mobile phones produced that have a mass below 138.2 g. b) Mobile phones that have a mass greater than three standard deviations from the mean are rejected by the quality assurance process. I understand part A -'s worked oh that 84.134% have a mass below but completely stuck on question B, I have no idea where to start. |
| $\lim_{n\to\infty} n^2 \int_{0}^{1} \frac{x\sin{x}}{1+(nx)^3} \, \mathrm{d}x$ Posted: 02 Jan 2022 04:29 AM PST I have to calculate $$ \lim_{n\to\infty} n^2 \int_{0}^{1} \frac{x\sin{x}}{1+(nx)^3} \, \mathrm{d}x $$ I'm thinking of using the dominated convergence theorem. So I define $$f_n(x)=\frac{n^2 x \sin{x}}{1+(nx)^3}.$$ $f_n$ are measurable owing to the fact that they are continuous. Now, I have to calculate its limit. If I'm not wrong: $$\lim_{n\to\infty} \frac{n^2 x \sin{x}}{1+(nx)^3} = (x\sin{x}) \lim_{n\to\infty} \frac{n^2}{1+(nx)^3} = (x\sin{x}) \cdot 0 =0.$$ So $|f_n(x)|\le0$ and using the dominated convergence theorem: \begin{align*} \lim_{n\to\infty} n^2 \int_{0}^{1} \frac{x\sin{x}}{1+(nx)^3} \, \mathrm{d}x &=\lim_{n\to\infty} \int_{0}^{1} n^2 \frac{x\sin{x}}{1+(nx)^3} \, \mathrm{d}x \\ &= \int_{0}^{1} \lim_{n\to\infty} n^2 \frac{x\sin{x}}{1+(nx)^3} \, \mathrm{d}x \\ &= \int_{0}^{1} 0 \, \mathrm{d}x =0 \end{align*} Is it ok? |
| Posted: 02 Jan 2022 04:50 AM PST I've tried finding $\alpha$ in the following example $$A=\left[ {\begin{array}{*r} 2 & { - 1} & 2 & 1 \\ 4 & { - 3} & 0 & 7 \\ { - 1} & 0 & { - 3} & 2 \\ \end{array}} \right] ,\quad b=\begin{bmatrix} 0 \\ 6 \\ \alpha\\ \end{bmatrix}.$$ When using the row reduced echleon form I get the following augmented matrix: $$A|b= \left[ {\begin{array}{*r} 1 & 0 & 3 & { - 2} \\ 0 & 1 & 4 & { - 5} \\ 0 & 0 & 0 & 0 \\ \end{array}\left| {\begin{array}{*c} 0 \\ 0 \\ 1 \\ \end{array}} \right.} \right]. $$ But I would like something like the following, I don't understand why $\alpha$ disapears for me? |
| Cubic expectation of peanuts on squares Posted: 02 Jan 2022 04:28 AM PST There are 9 squares on the table, and we drop 9 peanuts to the table one at a time. Each peanut fall into every square with equal probability 1/9. After dropping all peanuts, we denote $x_{i}$ as the number of peanuts on the i-th square, i=1,2,3...9, and define Y = $x_{1}^{3}+x_{2}^{3}+...+x_{9}^{3}$. Calculate expectation of Y. In fact this can be calculated by computer simulation in a brute-force approach, but I am wondering if there is any quicker solution since brute-force method can be tedious and definitely not desired mathematical answer. |
| Is the Half Disc Topology normal? Posted: 02 Jan 2022 04:27 AM PST Recall: A metric space is Hausdorff, normal and regular. Here normal implies regular since the singletons are closed. Now consider the Half disc topology in $\mathbb R^2$: https://en.m.wikipedia.org/wiki/Half-disk_topology In the upper half plane P the standard metric on $\mathbb{R^2}$ induces the topology. The line $L=\{(x,0)|x \in \mathbb{R}\}$ is added with open balls defined differently. This space is possible not a metrizable (i.e. metric) space anymore. 1)The Half Disc topology is Hausdorff. Would like to show: 2)The Half Disc topology is normal 3)The Half Disc topology is regular. If I can show that it is normal, since singletons are closed in a top. Hausdorff space, then $2)\rightarrow 3)$. Any hints how to tackle $2)$? Thank you. |
| Why am I getting two different answers while calculating $\int \frac{dx}{\sqrt{2ax-x^2}}$? Posted: 02 Jan 2022 04:26 AM PST I was calculating the antiderivative of the function $\frac{1}{\sqrt{2ax-x^2}}$. I got two different answers in two different ways. First way: $$\int \frac{dx}{\sqrt{2ax-x^2}}=\int \frac{dx}{\sqrt x\cdot\sqrt{2a-x}}=2\int \frac{\frac{1}{2\sqrt x}dx}{\sqrt{2a-(\sqrt x)^2}}=\int \frac{d(\sqrt x)}{\sqrt{(\sqrt {2a})^2-(\sqrt x)^2}}=2\sin^{-1}\sqrt{\frac{x}{2a}}+c$$ Second way: $$\int \frac{dx}{\sqrt{2ax-x^2}}=\int \frac{dx}{\sqrt{a^2-(x-a)^2}}=\int \frac{d(x-a)}{\sqrt{a^2-(x-a)^2}}=\sin^{-1}\frac{x-a}{a}+c$$ It seems that $2\sin^{-1}\sqrt{\frac{x}{2a}}$ and $\sin^{-1}\frac{x-a}{a}$ are unequal. So I think one of my methods is wrong. But I've checked and found nothing wrong. So what is happening here? And if both of my methods are correct, how to show that these two expressions are equal? |
| Is this a deterministic proof for no non-trivial cycles? (Collatz Conjecture) Posted: 02 Jan 2022 04:39 AM PST By generalizing the problem toward infinite cycles and studying limits... my question is, is this a deterministic proof that no non-trivial cycles exist vis-à-vis the Collatz Conjecture: https://www.youtube.com/watch?v=pAEqCwhRmas Curious if it's correct or if there's a glaring error. |
| Why angle form by a secant and tangent outside a circle is half the intercepted arc? Posted: 02 Jan 2022 04:59 AM PST I know that according to Inscribed Angle Theorem that the measure of an angle will be half the intercepted arc's measure. I know that the measure of the angle formed from tangent and secant is equal to one-half the difference of the measures of intersected arcs i.e Angle = $\frac 12(\text{long - short})$ But I don't understand why a secant and tangent outside a circle angle is also half of the intercepted arc? According this Tangent-Secant Proof I saw, it says $\angle CAE = \dfrac12 AC.$ Why?
Image source: http://jwilson.coe.uga.edu/emt668/EMAT6680.2003.fall/Nichols/6690/Webpage/Day%209.htm |
| Posted: 02 Jan 2022 04:50 AM PST This post will generalize: Where no explicit sum representation was found despite other good answers. An answer avoiding Incomplete/Partial Exponential Bell Polynomials was specified since they were too general, but out of no other way, here is an explicit series for the Inverse of the Regularized Gamma function using the formula Wikipedia Lagrange Inversion formula. Note the $(x)_y$ Pochhammer Symbol:
The inverse of the Regularized Gamma function then is derived from the sum representation: $$Q(a,z)=1-\frac{1}{\Gamma(a)}\sum_{k=0}^\infty \frac{(-1)^k z^{k+a}}{(k+a)k!},Q(a,Q^{-1}(a,z))=z$$ And Lower Gamma function: $$y(z)=\gamma(a,z)=1-\Gamma(a)Q(a,z)= \sum_{k=0}^\infty \frac{(-1)^k z^{k+a}}{(k+a)k!}\implies y^{-1}(z)=\boxed{Q^{-1}\left(a,\frac{1-z}{\Gamma(a)}\right)=\sum_{n=0}^\infty\left(-\frac{a+1}{z^a}\right)^n \frac{z^n}{n!} \sum_{m=1}^{n-1}(-1)^m(n)_m \text B_{n-1,m}\left(\frac12,-\frac{a+1}{3(a+2)},\frac{a+1}{4(a+3)},…,\frac{(-1)^{n-m-2}(a+1)}{(a-m+n)(m-n-1)}\right)}\implies \hat f_n=\frac{(-1)^{k+1}(a+1)}{(k+a)(k+1)}$$ Unfortunately, there is not a direct way to simplify to the Complete Bell Polynomials, but maybe we can simplify using the definition of the Incomplete Bell Polynomials: $$\text B_{n,k}(x_1,…,x_{n-k+1})=n! \sum_{m_1=0}^n\cdots\sum_{m_n=0}^n\prod_{j=1}^n\frac{x_j^{m_j}}{{j!}^{m_j}m_j!}\text{ such that }\sum_{j=1}^n j m_j=n\text{ and }\sum_{j=1}^n m_j=k$$ The question how to simplify and transform the Incomplete Bell Polynomial: $$\sum_{m=1}^{n-1}(-1)^m(n)_m \text B_{n-1,m}\left(\frac12,…,\frac{(-1)^{n-m-2}(a+1)}{(a-m+n)(m-n-1)}\right) $$ The kth entry in the Incomplete Bell Polynomial is $\frac{(-1)^{k+1}(a+1)}{(k+a)(k+1)} $ Note that the Incomplete Bell Polynomials are implemented into Wolfram using BellY, so you can try to type the expression into Mathematica. We also now have a complex, but new expansion for the Median of a Gamma distribution. Please correct me and give me feedback! |
| Intersection point of all tangent planes to a sphere with point of contact on a circle Posted: 02 Jan 2022 05:04 AM PST I have solved the following problem, but I am not sure about the way I have solved the third question, so I would be grateful if someone would check it out. "Consider the plane $\pi: 2x+y+2z+6=0$ and the sphere $S$ with centre $C(-2,1,-1)$ and radius $3$. (1) Find the centre and the radius of the circle $K$ intersection of the sphere and the plane; (2) Let $J$ be the point $(0,0,-3)$. Show that $J$ lies on $S$ and find the equation of the plane tangent to $S$ at $J$. (3) All the tangent planes to the sphere $S$ whose point of contact lie on the circle $K$ intersect at a point $Q$. Find the coordinates of $Q$. My solution: (1) $$d=\text{distance center of sphere-plane}=\frac{|ax_c+by_c+cz_c+d|}{\sqrt{a^2+b^2+c^2}}=\frac{|2(-2)+1\cdot 1+2\cdot (-1)+6|}{\sqrt{2^2+1^2+2^2}}=\frac{1}{3}.$$ Radius $$r=\sqrt{R^2-d^2}=\sqrt{9-\frac{1}{9}}=\fbox{$\sqrt{\frac{80}{9}}$}$$ Center $$C=(-2,1,-1)+d\frac{\vec{n}}{|\vec{n}|}=(-2,1,-1)+\frac{1}{3}\cdot\frac{(2,1,2)}{3}=\fbox{$(-\frac{16}{9},\frac{10}{9},-\frac{7}{9})$}.$$ (2) The equation of the sphere $S$ is $(x+2)^2+(y-1)^2+(z+1)^2=9$. $J\in S$ since it satisfies this equation. The equation of the tangent plane at $J$ is $\big( 2(x+2),2(y-1),2(z+1) \big)_{|(0,0,-3)}\cdot \big( (x-0),(y-0),(z+3) \big)=0\Leftrightarrow \fbox{$4x-2y-4z=12$}$ (3) Intuitively I would expect that these tangent planes form a kind of cone and they all meet at the point where each of them intersect the line passing through the center of the sphere $S$ and the circle $K$; such a line has equation $\vec{r}(t)=(-2,1,-1)+t(2,1,2)$ and its intersection with the plane found in (2) is $\fbox{$(-20,-8,-19)$}$, which I would assume belongs to each of the planes tangent to $S$ at a point in $K$ and is thus $Q$. Is this correct? Is there a way to do this rigorously? Thanks. |
| Proving a condition of perpendicularity Posted: 02 Jan 2022 04:51 AM PST
Today I've got an insteresting question about geometry. Let's get into it.
Is there any idea to go along this type of perpendicularity problems? |
| Posted: 02 Jan 2022 04:32 AM PST I am reading "Masahiko Saito's Linear Algebra" (in Japanese) by Masahiko Saito. The following three propositions are in this book.
What is the set $\{A \,\,\mid \text{ any two eigenvectors of } A \text{ from different eigenspaces are orthogonal }\}$? Is there a matrix $A\in\{A \,\mid \text{ any two eigenvectors of } A \text{ from different eigenspaces are orthogonal }\}$ which is not a normal matrix? |
| If you throw a dice 5 times, what is the expected value of the square of the median? Posted: 02 Jan 2022 04:56 AM PST My question: If you throw a dice 5 times, what is the expected value of the square of the median of the 5 results? A slightly modified question would be: If you throw a dice 5 times, what is the expected value of the median? The answer would be 3.5 by symmetry. For the square, it seems to be that symmetry does not hold anymore. Is there a "smart" way to solve this problem? If there isn't a smart way to solve the problem, if there a smart way to estimate the answer? |
| How do you show that the three incidence axioms are independent of each other. Posted: 02 Jan 2022 05:02 AM PST How do you show that the three incidence axioms: Incidence Axiom 1. For every pair of distinct points P and Q there is exactly one line l such that P and Q lie on l. Incidence Axiom 2. For every line l there exist at least two distinct points P and Q such that both P and Q lie on l. Incidence Axiom 3. There exist three points that do not all lie on any one line. are independent of each other (i.e it is impossible to prove any one of them from the other two) by inventing a nontrivial interpretation for each pair of incidence axioms, in which those axioms are satisfied but the third axiom is not. |
| Posted: 02 Jan 2022 04:31 AM PST Let $(X,d)$ be a metric space and $(x_n)$,$(y_n)$ be sequences in $X$. (a) If $x_n \to x$ and $y_n \to y$, prove that $d(x_n,y_n) \to d(x,y)$. (b) If $x_n$ and $y_n$ are Cauchy sequences in $X$, prove that the real sequence $d(x_n,y_n)$ is convergent. Proof. (a) Is this correct?: Let $x,y,w,z∈X$. The triangle inequality implies that $|d(x,z)-d(w,y)| \leq d(x,y)+d(z,w)$, so $|d(x_n,y_n)-d(x,y)| \leq d(x_n,x)+d(y_n,y)$ which implies that $d(x_n,y_n) \to d(x, y)$ as $n \to ∞$ if $d(x_n,x) \to 0$ and $d (y_n, y) \to 0$. (b) I have no ideas, you help me please? |
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