Recent Questions - Mathematics Stack Exchange |
- Find the joint distribution function
- Show that $G$ is group under matrix multiplication.
- Find the rate of change of area of triangle when 2 sides and a angle are increasing.
- How do I convert the argument of a dirac delta function from cartesian coordinates to spherical coordinates?
- Stats probability colours
- Combinatorial Proof With a Story
- Is the statement: "∃m ∈ℤ, ∀n ∈ℤ,m=n+5" true or false?
- Counting Problem (4 Digit Code) - Equality
- Deducing Cayley's theorem from the Yoneda lemma using Wikipedia's recipe
- How to estimate population by using capture-recapture method
- Finding ratio of areas of two triangles
- A nonconstant holomorphic map from a compact connected complex surface to a compact connected complex curve
- Help me to figure it out their maximum possible sum. - Any kind of solutions or hints I need.
- Interesting result for the integral of the $cos(x)^n$
- in the figure, the two squares have the same center. They have side lengths equal to 14 and 12 . What is the area of the shaded region? [closed]
- Just using $+$ ,$-$, $\times$ using any given n-1 integers, can we make a number dividing n? (NO bracket allowed!!)
- a GATE question on radius of convergence
- Z-test with Alpha Level
- Engel's Theorem example
- How to prove the solution of an equality is lower bounded?
- Sums of Möbius between $x$ and $y$
- recurrence relation, $a_{n+1}= 10a_n+n+1, ∀ n ≥ 0$ with $a_0= 0$
- Is there a function that maps each element of $\mathbb{Q}$ between 0 and 1 to $\mathbb{N}$?
- Can one construct a homeomorphism with noncompact periodic point set?
- How can I prove that every word has unique representation?
- number of valuation ring of $\Bbb Q_p((T))$
- Number of valuation ring of a given field
- Players $P_1,P_2...P_m$ of equal skill play a game consecutively in pairs
- Necessary and sufficient conditions for the sum of two numbers to divide their product
- An example of neither open nor closed set
| Find the joint distribution function Posted: 26 Mar 2021 08:52 PM PDT Let $X$ ~ U[0,1], Y=1-X, and X = (X,Y).
$Fx$(X,Y).
$$\frac{\partial^{2}}{\partial y\partial x}Fx(X,Y) = 0$$ for every par $(x,y)$ in the plane $R^{2}$ . Except on some lines or line segments.  |
| Show that $G$ is group under matrix multiplication. Posted: 26 Mar 2021 08:48 PM PDT Following question is from one of the previous year papers. I am feeling that something is misprinted here.
Is this question correct?  |
| Find the rate of change of area of triangle when 2 sides and a angle are increasing. Posted: 26 Mar 2021 08:47 PM PDT In a triangle ABC we have AB = x metres, AC = y metres and angle CAB = z radians. If x,y,z are increasing at a rate of 0.08ms-1, 0.06ms-1 and 0.0001 radian per second respectively, find the rate of change of the area of triangle ABC in square meter per second when x =1521 metre, y = 2021 metres and z =3${\pi}$/5 radians.  |
| Posted: 26 Mar 2021 08:46 PM PDT I have V(x)=Aδ(x) where I needed to convert the argument from cartesian to spherical, namely from δ(x) to δ(r) so I can compute an integral involving V in dr. I don't understand the dirac delta quite well so I have little idea how to do this. Any help would be greatly appreciated. Thanks!  |
| Posted: 26 Mar 2021 08:53 PM PDT A bowl contains $2$ red balls and $4$ blue balls. Consider the following game. You pick a ball at random without looking. If it is red, you win \$ $18$. Otherwise, you pick again from the remaining balls. If the picked ball is red, you win \$ $6$. If not, you must give away \$ $12$ (so you "win" minus $12$ dollars).
I dont understand how to do this.  |
| Combinatorial Proof With a Story Posted: 26 Mar 2021 08:53 PM PDT I have $$\sum_{k=0}^n \binom{n}{k} = 2^n$$ I am proving this using a story, but I was wondering if my story is correct. Our story is going to be we choose $k$ people out of $n$ to be in a club. LHS: Considering all calses, we choose $k$ people to form to club from $n$ people. RHS: Out of $n$ people, they choose either to be a part of the club or not. By this we show that the LHS and RHS count the same thing. Anything I should clarify, or anything I have wrong. I am trying to work out the kinks for these types of proofs.  |
| Is the statement: "∃m ∈ℤ, ∀n ∈ℤ,m=n+5" true or false? Posted: 26 Mar 2021 08:44 PM PDT Is the statement: "∃m ∈ ℤ, ∀n ∈ ℤ,m=n+5" true or false? This was recently asked on a test and according to the solutions, the correct answer is true. Solutions also say that the worded translation is: "there exists an integer m such that for all intergers n, m=n+5" My friend argues that it the statement is asking if one integer could statisfy the equation for all integers (which is false), whereas my teacher, myself, and peers cant really explain why his understanding isn't correct. Thanks for any help.  |
| Counting Problem (4 Digit Code) - Equality Posted: 26 Mar 2021 08:39 PM PDT I have that the number of 4 digit codes (each can be 0-9) is $10^4$. I know that the number of 4 digit codes with no repeats is $P(10,4)$. I figured that (the total number of 4 digit codes) - (4 digit codes with no repeats) should equal (4 digit codes with 3 repeats) : 10 $+$ (4 digit codes with 2 repeats) : ${4\choose 3} \cdot 10 \cdot 9 = 360$ $+$ (4 digit codes with 1 repeats) : ${4\choose 2} \cdot 10 \cdot 9 \cdot 8= 4320$ but $4690 \neq 4960$ My brain is going to explode. Please help!  |
| Deducing Cayley's theorem from the Yoneda lemma using Wikipedia's recipe Posted: 26 Mar 2021 08:37 PM PDT I'm following Wikipedia in trying to prove that Cayley's theorem emerges as a particular case of the Yoneda lemma. In case that article gets edited, here's the screenshot:
A couple of aspects in the proof are unclear:
 |
| How to estimate population by using capture-recapture method Posted: 26 Mar 2021 08:54 PM PDT The scenario as follows. Suppose I want to do a population study for wild ducks in a certain area. I caught 360 ducks on the first day, and 19 of them had markers from last year. Then I gave new markers to the rest of the ducks. I caught 189 ducks on the second day, and 8 of them had markers from last year and 92 of them had markers from the day before. Here is my way to estimate the population. $n_{1}$ = 360 - 19 = 341 (Sample size from the first day exclude the ducks with markers from last year) $n_{2}$ = 189 - 8 = 181 (Sample size from the second day exclude the ducks with markers from last year) $m_{2}$ = 92 (There are 92 ducks that have bands from the day before) $\hat{N}$ = (341*181)/92 = 670.88 I am not sure if this is correct since I don't know if I need to exclude the ducks that have markers from last year in my calculation. I will be grateful if somebody can help me. Also, do I need to exclude the ducks that have markers from last year if I want to calculate $x_{11}$ (Number of ducks present on both days), $x_{12}$ (Ducks present on the first day and absent on the second day), and $x_{21}$ (Ducks absent on the first day and present on the second day)?  |
| Finding ratio of areas of two triangles Posted: 26 Mar 2021 08:38 PM PDT
So what I need to find is just $\frac{BD}{DC}$ or $\frac{AB \cdot BD}{AD \cdot AC}$ but either way I cannot see how I should proceed. Any hints? Thanks in advance.  |
| Posted: 26 Mar 2021 08:22 PM PDT According to this question (holomorphic map between compact Riemann surfaces), every nonconstant holomorphic map between compact Riemann surfaces is surjective (essentially by the open mapping theorem). I've heard that open mapping theorem does not hold in general for a holomorphic map $\Bbb C^n\to \Bbb C$. Let $S$ be a connected compact complex surface and $C$ a connected compact complex curve, and $\pi:S\to C$ a nonconstant holomorphic map. Then is it true that $\pi$ must be surjective? Assuming $\pi$ is surjective, is it true that the set of critical values of $\pi$ is a finite subset of $C$? (I know that Sard's theorem implies that the set of regular values of $\pi$ is a dense subset of $C$, but I can't see that this implies that there are only finitely many critical values.) Now if there are only finitely many critical values, then the set of regular values $R$ of $\pi$ is a connected subset of $C$. In this case must $\pi^{-1}(R)\to R$ must be a fiber bundle map?  |
| Help me to figure it out their maximum possible sum. - Any kind of solutions or hints I need. Posted: 26 Mar 2021 08:49 PM PDT The Squares of two positive integers differ by 2020. Find their maximum possible sum. - Any kind of solutions or hints I need.  |
| Interesting result for the integral of the $cos(x)^n$ Posted: 26 Mar 2021 08:23 PM PDT I was finding the recurrence relation for the following integral: $$I_n = \int_{0}^{\frac{\pi}{2}}\cos(x)^n\,dx .$$ I derived the following relation: $$I_{n} = \frac{n-1}{n}\cdot I_{n-2} .$$ What i than did was to plot the $I_n$ as function of n as the function of n. At first I didn't restrict $n$ to be natural number, which is not what is assumed in those types recurrence relations, as far as I was taught. This resulted it the following link and a function $I(n)$ that looked like an some kind of irrational function, at least from $x\in (-1,+\infty)$. Immediately, this reminded me of the Fourier series, but that's as far as I went with my knowledge. I am wondering is my assumption that this has to do something with the Fourier series right, and if so, is there any method that could be used to get a more closed-form expression of the function that was transformed. I wouldn't expect that to be true for every function, but having even a few examples of something like that could be interesting to me. To recap, my question is does this have anything to do with the Fourier transform, and if so, could the transform me reversed, in such a way that we get the original function back? Thanks to everyone on their answers in advance!  |
| Posted: 26 Mar 2021 08:18 PM PDT enter image description herein the figure, the two squares have the same center. They have side lengths equal to 14 and 12. What is the area of the shaded region?  |
| Posted: 26 Mar 2021 08:27 PM PDT Probably title is slightly ambiguous but I could not see any way of shortening the problem. I am sure many of you have seen the problem like $2$ $\square$ $2$ $\square$ $2$ $\square$ $2$ $\square$ $2$ $=$ $20$ something like this where between the numbers we can use some sort of operation depending on the question. Well I think the question is motivated by this. So given any set of $n-1$ numbers (we can swap the order) and we can only use addition, subtraction and multiplication, can we make a multiple of $n$ always? Another way of putting it is: Given $n-1$ elements of non-integral domain $\mathbb{Z}/n\mathbb{Z}$, can we make zero only using $+,-,\times$? Oh and by the way bracket is not allowed!!!! Well the question I am asking seems to be based on this (I was asked this in an interview randomly and I could not figure it out and he did not know the answer.)  |
| a GATE question on radius of convergence Posted: 26 Mar 2021 08:24 PM PDT Find the radius of convergence of the power series $$\sum_{n=0}^{\infty} 4^{(-1)^{n} {n}}z^{2n}.$$ My attempt: Since the radius of convergence $r$ is given by $$r = \dfrac{1}{\lim\limits_{n\rightarrow \infty}|c_{n}|^\frac{1}{n}}.$$ Please help me further.  |
| Posted: 26 Mar 2021 08:13 PM PDT A medical company has performed and experiment using to treat me to groups: cancer patients taking a new drug (treatment group) and cancer patients taking a placebo (control group). They want to compare the proportion of patient showing improvement in the drug treatment group to the proportions of patients showing improvement in the control group. Because the new drug has severe side effects, they only want to begin production if it can be shown it has a significant improvement in patients. Given these concerns, which of the following inferntial procedures would be the best choice? A. One proportion Z test with alpha level of 5% B. One proportion Z test with alpha level of 10% C. One proportion Z test with alpha level of 20% D. Two proportion Z test with alpha level of 5% E. Two proportion Z test with alpha level of 10% My Answer I Strongly believe the answer to be D, a Two-Proportion z-test with alpha level of 5% My Reasoning It's should be a two-proportion z-test since we are looking at two different groups & the alpha level should be this lowest possible, which would be 5% since we are rear hung about a live saving medicine. The margin of error needs to be low as possible. Thoughts on this  |
| Posted: 26 Mar 2021 08:51 PM PDT On the wikipedia page of Engel's theorem it says that "if we merely have a Lie algebra of matrices which is nilpotent as a Lie algebra, then this conclusion does not follow", meaning that it might not be possible to choose a basis in which all elements are upper triangular. Can someone give me an example and explain why is this true for the adjoint representation? Is it because its kernel is commutative?  |
| How to prove the solution of an equality is lower bounded? Posted: 26 Mar 2021 08:55 PM PDT Let $x_{n,k}\in (0,1]$ be any solution of the following inequality \begin{align} n(1-x)^2+2kx^2(1-x)^2 -x^2-k(1-x)^4 \leq 0, \end{align} where $n, k$ are parameters such that $n\geq k\geq 1$. I want to prove that there exists a strictly positive constant $c$ (i.e. $c>0$) such that \begin{align} x_{n,k} \geq c, \text{ for all } n\geq k \geq 1. \end{align} If you want to have a look at their graphs, they are available here thanks to Wolfram Alpha. I have tried a lot of estimations, but did't success. I guess that we should need some special trick here. Could any one give me a hint? Thanks!  |
| Sums of Möbius between $x$ and $y$ Posted: 26 Mar 2021 08:33 PM PDT For $z_1 > z_2 \geq 0$ define $$M(z_1,z_2) = \sum_{z_2 < a \leq z_1 } \mu(a),$$ where $\mu$ is the Möbius' function. Prove that $$\sum_{k=1}^{\infty} M\left(\frac{n}{k}, 0\right) = 1\,\text{ and that }\, \sum_{k=1}^{\infty} M\left(\frac{n}{2k-1}, \frac{n}{2k}\right) = -1.$$ The only relevant useful identity I could think of using here is $${\displaystyle \mu (n)=\sum _{\stackrel {1\leq k\leq n}{\gcd(k,\,n)=1}}e^{2\pi i{\frac {k}{n}}}}$$ - would it help? Any help appreciated!  |
| recurrence relation, $a_{n+1}= 10a_n+n+1, ∀ n ≥ 0$ with $a_0= 0$ Posted: 26 Mar 2021 08:25 PM PDT Im not sure where to go with only the $a_0 = 0$ value, and the extra $+1$.  |
| Is there a function that maps each element of $\mathbb{Q}$ between 0 and 1 to $\mathbb{N}$? Posted: 26 Mar 2021 08:14 PM PDT Kepler devised a method to enumerate the rational numbers between 0 and 1. This clip from his Harmonices Mundi illustrates the method.
It works like this: Let $m=n=1$. Then $\tfrac{m}{n}$ is the first rational number. The next two are $\tfrac{m}{n}$ and $\tfrac{n}{m}$. These last two rationals become $m$ and $n$ for the next iteration. Wash, rinse, repeat for eternity. Done.
If we read Kepler's diagram left to right; down to up, the first 10 rationals are:
Suppose we want to know where, say, $\tfrac{6067}{9000}$ appears on that list. We could systematically run through Kepler's tree. Eventually the process will terminate. But is there a better way to do this? Is there some function that maps each element of $\mathbb{Q}$ between 0 and 1 to $\mathbb{N}$? EDIT: I'd like to specify that I'd like to know what that function is. I'd like to be able to plug in, say, $\tfrac{3}{8}$, and get the number $10$.  |
| Can one construct a homeomorphism with noncompact periodic point set? Posted: 26 Mar 2021 08:28 PM PDT Can one construct a compact metric space $X$, and a homeomorphism $f\in \text{Homeo}(X)$, such that $f$ has noncompact periodic point set? I am trying to find a homeomorphism on a cylinder $S^1\times [0,1]$, but not succeeded yet. Looking forward to your brilliant answer. Thanks Here are some of my attemptations Let $f\in\text{Homeo}(S^1\times [0,1])$, which sends $(x,y,h)$ to $(-x,-y, g(h))$, i.e the antipodal map on the circle, then lift $g(h)$, so it suffices to find a $g:[0,1]\to[0,1]$ which is surjective,continuous and has dense periodic points (or just fixed points is Okay, I guess). Because no matter whether $g$ is a homeomorphism, $f$ is a bijective continuous map, thus in this case, a homeomorphism, if the periodic points of $g$ is dense in $[0,1]$, but not the whole interval, we can deduce that the periodic points set of $f$ is not even closed, thus a noncompact one, so now the problem is to construct that peculiar $g$.  |
| How can I prove that every word has unique representation? Posted: 26 Mar 2021 08:28 PM PDT Let $\Sigma = \{0,1,*\}$ be a finite alphabet and let $\mathcal{L}\subseteq \Sigma^*$ be a formal language defined as follows
I want to prove that
I was trying to use the fact that every element of $\mathcal{L}$ has an even number of $*$ but maybe I am missing something.  |
| number of valuation ring of $\Bbb Q_p((T))$ Posted: 26 Mar 2021 08:36 PM PDT I would like to know the number of valuation rings of $\Bbb Q_p((T))$. I know $\Bbb Q_p$ has $2$ valuation rings, that is,$\Bbb Q_p$ and $\Bbb Z_p$. Every algebraic extension of $\Bbb Q_p$ has more than $2$ valuation rings because of extension theorem on valuation.But $\Bbb Q_p((T))$ is not algebraic over $\Bbb Q_p$,I am at a loss. How many valuation rings of $\Bbb Q_p((T))$ are there? Thank you in advance.  |
| Number of valuation ring of a given field Posted: 26 Mar 2021 08:34 PM PDT Let $K$ be a field. I'd like to know the number of valuation ring of $K$.
Let $K$ be $\mathbb{Q}$, then $\mathbb{Z}_{(p)}$ is a valuation ring of $\mathbb{Q}$, so there are infinitely many valuation ring according to the number of prime $p$. Let $K$ be a local field, then valuation ring is just $K$ and its integer ring, so in this case the number is $2$. Let $K$ be a finite field,I believe the number is $1$ because $K$ itself is the only valuation ring of finite field. I couldn't find a field $K$ whose number of valuation ring is natural number more than $3$. I know there is a $1$ to $1$ correspondence between the places and valuation rings of a field. So I can confirm above claims from the view point of $places$, but I still cannot find field $K$ which contains more than $3$ valuation rings. Thank you in advance.  |
| Players $P_1,P_2...P_m$ of equal skill play a game consecutively in pairs Posted: 26 Mar 2021 08:41 PM PDT
I haven't really understood the question or what it wants to do, so I don't know how to begin. Can anyone explain the question and give a starter hint?  |
| Necessary and sufficient conditions for the sum of two numbers to divide their product Posted: 26 Mar 2021 08:54 PM PDT How to find necessary and sufficient conditions for the sum of two numbers to divide their product. Thanks in advance.  |
| An example of neither open nor closed set Posted: 26 Mar 2021 08:47 PM PDT I need a very simple example of a set of real numbers (if there is any) that is neither closed nor open, along with an explanation of why it is so.  |
| You are subscribed to email updates from Recent Questions - Mathematics Stack Exchange. To stop receiving these emails, you may unsubscribe now. | Email delivery powered by Google |
| Google, 1600 Amphitheatre Parkway, Mountain View, CA 94043, United States | |
No comments:
Post a Comment