text stringlengths 83 79.5k |
|---|
H: How to calculate the optimal number of slices to use for a Riemann sum
I wrote some code that calculates the area under a curve using Riemann sum. The way I have it working right now is that a user enters the upper and lower bounds for the integral and then the number of slices to use for the calculation. I want ... |
H: Using $\lim_{x \to \infty}$ to determine whether $f(x) = \Theta(g(x))$?
I'm learning it in the context of Running time complexity.
to determine whether $f(x) = O(g(x))$, you can check whether the folloing limit:$$\lim_{x \to \infty} {f(x) \over g(x)} < \infty$$
if so, then you know that $f(x) = O(g(x))$.
Is there a... |
H: Can uniform local integrability implies uniform local absolute continuity?
Suppose we have $u\in L_{u,}^{loc}$, i.e.,
$$
\sup_{x\in \mathbb{R}^n}\int_{|x-y|<1}|V(y)|dy<\infty
$$
then can we obtain that $\forall \epsilon>0$, there exists a $r>0$, such that
$$
\sup_{x\in \mathbb{R}^n}\int_{|x-y|<r}|V(y)|dy<\epsilon... |
H: Does $\sum_{n=1}^{\infty} \sin\left(\frac{\pi}{n}\right)$ converge?
I'm not really sure how to start here, the basic tests don't work.
AI: $\sin(\pi/n) = \pi/n + O(1/n^2)$ for $n$ large. Now use the fact that the harmonic series diverges. |
H: Evaluating Sums $\sum_{i=1}^{n} \sum_{j=0}^{n-i}$
I'm unsure how to evaluate sums when the second sum has $n-i$ on the top.
$$\sum_{i=1}^{n} \sum_{j=0}^{n-i} (3j^{2} - 2)$$
$$=\sum_{i=1}^{n} (\sum_{j=0}^{n-i} 3j^{2} - \sum_{j=0}^{n-i}2)$$
$$=\sum_{i=1}^{n} (\sum_{j=0}^{n-i} 3(\frac {n(n+1)(n+2)} 6 ) - 2n)$$
From h... |
H: Reference for integration
Does anyone have a good reference for a book that already assumes knowledge of calculus/analysis and whose main focus is computing more difficult integrals? I'm looking for something which will have a lot of worked examples for differentiation under the integral, tricky substitutions, unus... |
H: Solve second order ODE with undetermined coefficients method
Consider the differential equation
$$y''+5y'=-sin(x)-1$$
Find the general solution.
Here's my work:
I found the solution to the homogeneous equation to be:
$y_h(x)=C_1e^{-5x}+C_2$
And for the particular solution, I guessed
$y_p(x)=Acos(x)+Bsin(x)+C$
$y_... |
H: Let H be a subgroup of Z(G). Then show that H is normal in G.
Hi i am kinda stuck here. How do i show that H is normal in G???
Any tips or suggestions would be appreciated.
Thank You!
AI: Any subgroup $H$ is normal in $G$ iff $gHg^{-1} \subset H$ for all $g \in G$. This is equivalent to $gH = Hg$ for all $g \in ... |
H: Find the set of all vectors in R3 that are orthogonal to (-1,0,2) and (3,1,-2). Write the set in the standard form of a plane through the origin
Let $v = (-1,0,2)$ and $u = (3,1,-2)$ so we need to find $x = (x_1,x_2,x_3)$ such that $v \cdot x = 0$ and $u \cdot x = 0$. This gives us the set of linear equations
$$ -... |
H: Jordan measurable for closure and interior
Give an example of a bounded set $S\subseteq\mathbb{R}^n$ such that $\overline{S}$ and $\operatorname{Int}S$ are Jordan-measurable, but $S$ is not.
Jordan-measurability of a set $A$ is equivalent to the condition that $\operatorname{Bd}A$ has measure zero, where $\operator... |
H: Equating coefficients $A-2B\sin x=2-\sin x$
I'm trying to find out how to find $A$ and $B$ for the equation
$A-2B\sin x=2-\sin x$
I know I'm supposed to get $A=2$ and $B=\frac{1}{2}$, and I've looked on Google for help but didn't understand how any examples I found would help me solve the problem. Any help would b... |
H: Augmenting «$\Bbb Z[x]$ f.g. $\Rightarrow x$ integral» for ${\frak p}[x]$
In KCd's blurb on ideal factorization, page 5:
$\hskip 0.3in$
The situation is this: $K$ is a number field, ${\cal O}_K$ its ring of integers, ${\frak p}\triangleleft{\cal O}_K$ a prime ideal, $x\in K$.
We assume $x \frak p\subseteq p$ and w... |
H: absolute and uniform convergence of a Fourier-like series
I am following stein's real analysis book and he claims that if $a_n=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-inx}dx$ where $f\in L^1([-\pi,\pi])$ then $\sum_{n=-\infty}^{\infty} a_n r^{|n|}e^{inx}$ converges absolutely and uniformly for each $r$, $0\leq r <1$... |
H: The Image of the Function: $f$:
I am a bit confused how they arrived at this answer:
The image of the function $f: \mathbb N \to \mathbb R , f(n) := \frac{(-1)^n+1}{3} $
The answer they got was $\{0, \frac{2}{3}\}$
Can someone explain how to arrive at this conclusion?
Thank you!
AI: If $n$ is even, then write $n ... |
H: Show that: $(E^{\circ})^{\circ}=E^{\circ}$ i.e. the interior of the interior of a set equals the interior of the set.
Let $(M,d)$ be a metric space. Show that $(E^{\circ})^{\circ}=E^{\circ}$.
I know that $A=A^{\circ}\Leftrightarrow A \text{ is an open set}$
So I want to prove that $E^{\circ}$ is an open set and th... |
H: Probability of at least 1 out of 4 televisions breaking?
If a television has an 8% chance of breaking in its lifetime, and I buy 4, what are the chances of at least 1 breaking in its lifetime?
If I only had 2 I suppose I would do: P(A) + P(B) - P(AB).
But I have 4.
Thanks
AI: Note that the event "at least $1$ break... |
H: If a linear transformation is onto and one-to-one, is it always invertible?
I know that if $T: \mathbb{R}^m \to \mathbb{R}^n$ is invertible then it is also onto and 1-1.
But is it equivalent? In other words, are all linear transformations that are bijective considered invertible?
AI: By basic set theory, as alluded... |
H: How to approximate this series?
How to approximate this series, non-numerically?
$ S_n = \sum_{n=1}^{50} \sqrt{n}$
AI: For a very simple approximation,
which is the first step
to the Euler-McLaurin formula,
use this
$f'(n)
\sim f(n)-f(n-1)
$
so
$f(n)
\sim \int_{n-1}^{n} f(x) dx
$
so
$\sum_{n=1}^N f(n)
\sim \int_0^{... |
H: Elementary Algebra Inequality question
$$
\frac{1}{x-1} < -\frac{1}{x+2}
$$
(see this page in wolframalpha)
Ok, so I think the main problem is that I don't really know how to do these questions. What I tried to do was move $-1/(x + 2)$ to the LHS and then tried to get a common denominator. I ended up with
$$
\frac... |
H: Help with exact value of: $\tan (\sin^{-1}(-1/2) - \tan^{-1}(3/4))$
Ok, so I used the tan formula of difference of angles, and so far I've got to:
I found that $$\tan \alpha = \frac{-1/2}{\sqrt{3}/2} = \pm \frac{\sqrt3}{3}$$
so
$$\tan \left(\sin^{-1}\left(\frac{-1}{2}\right) - \tan^{-1}\left(\frac{3}{4}\right)\righ... |
H: Basic limit to infinity
Trying to work out a limit from a past exam in Calculus and Linear Algebra $1$, now I know the answer is $0$, and I have a worked solution from Wolframalpha is, but the answer is definitely not done the way they did it(computationally) as it was some 20 steps long.
My thought is, I take it t... |
H: Image of $C^1$ function does not contain open set
Let $f:\mathbb{R}\rightarrow\mathbb{R}^2$ be a $C^1$ function. Prove that the image of $f$ contains no open set of $\mathbb{R}^2$.
So say $f(x)=(g(x),h(x))$. Since $f$ is $C^1$, we have that $g'(x),h'(x)$ both exist and are continuous functions in $x$. To show that ... |
H: Find the rank of a matrix representing $p$ distinct polynomials of maximum degree $n$
I can't find the Reduced Row Echelon Form to find the number of pivots because I don't have numbers to work with. I know an upper bound for the rank is the smaller amongst $p$ and $n+1$. Any tips on how to approach this problem? B... |
H: Find the interval on which $f$ is increasing if the **derivative** of $f$ is $f'(x)=(x+4)^6(x-3)^7(x-4)^6$.
Find the interval on which $f$ is increasing if the derivative of $f$ is $f'(x)=(x+4)^6(x-3)^7(x-4)^6$.
Can anybody please verified that it increases at $(3,\infty)$, or do i have to do product rule with the... |
H: How many non-zero quadratic residues are there for $p^k$, where $p$ is an odd prime and $k$ a positive integer?
How many non-zero quadratic residues are there for $p^k$, where $p$ is an odd prime and $k$ a positive integer?
Hi everyone, just need a bit of help for this practice question, I have proved that for $k=1... |
H: abstract algebra problem..
Let $U = \{1,0,c\}$ be a ring with three elements ($1$ is the unity). Which
statements are true?
\begin{align*}
&I.\,\,\,\,\,\,\,\,1 + 1 + 1 = 0 \\
&II.\,\,\,\,\, 1 + 1 = c \\
&III.\,\,c^2 = 1
\end{align*}
The solution is I II and III are all true. For III, the solution says by multiplic... |
H: use combinatorial reasoning to calculate $ \sum{\binom{100}{a}\binom{200}{b}\binom{300}{c}}$
Given $ a + b + c = 100 $. $a,\ b,\ c $ are non-negative integers. Calculate
$$ \sum {\binom{100}{a} \binom{200}{b} \binom{300}{c} } $$
Can someone help me with this question? I have no idea how to start it.
AI: This sum i... |
H: Why must this function have a critical point inside the sphere?
Suppose we have $f: \mathbf{R}^{3} \to \mathbf{R}$ with the following property: $\langle \nabla f(x), x \rangle > 0$ for every $x \in S^{2}$, that is, it's gradient points outwards the unit sphere. It's asserted that there must a point $p$ inside the s... |
H: How to find the limit of recursive sequence?
Suppose $a_0 = 0$, $a_1 = 1$, and
$$ a_{n+1} = a_n+2 a_{n-1}$$
if $n \ge 1$.
Find
$$\lim_{n\rightarrow\infty}\frac{a_{n+1}}{a_n}$$
Usually, what's the approach we use for recursive sequence like this?
I know another type of recursive question with no fraction i... |
H: Prove $\sum_{k=0}^{n}(-1)^k(k+1)\binom{n}{k}=0$
Prove that $\sum_{k=0}^{n}(-1)^k(k+1)\binom{n}{k}=0$.
I know $\sum_{k=0}^{n}(-1)^k\binom{n}{k}=0$ and even $\sum_{k=0}^{n}(-1)^k\dfrac{1}{k+1}\binom{n}{k}=0$ because I can multiply left side by n+1 and divided by n+1.
Then I have $\dfrac{1}{n+1}\sum_{k=0}^{n}(-1)^k\d... |
H: Can the direction of Contour Integral be affect to the result of integration?
Now i doing the home work about Residue Integration and i doubt that "Can the direction of Contour Integral be affect to the result of integration?" I mean with the same shape of contour but direction of contour changes from clockwise to ... |
H: Difference equation: $y_{k+1} = y_{k} + \frac{c}{2k}$
I'd like to solve this difference equation. Unfortunately, the forcing term is not geometric, so I don't know how to find the solution:
$$
y_{k+1} = y_k + \frac{c}{2k}.
$$
AI: Move $y_k$ over to the left side of the equation and take the partial sums on both sid... |
H: If a derivative of a continuous function has a limit, must it agree with that limit?
Suppose we have a continuous function $f : \mathbb{R} \to \mathbb{R}$. Suppose also that for a certain point $c$, $\lim_{x \to c} f'(x)$ exists. Must $f'(c)$ exist as well, and be equal to this limit?
This isn't quite the same as a... |
H: Find the Radius of Convergence of the Series $\sum a_{n}x^{n^{2}}$ Using $\sum a_{n}x^{n}$?
I want to show that $$\sum a_{n}x^{n^2}$$ has radius of convergence of 1, using the fact that the power series $\sum a_{n}x^{n}$ has radius of convergence $R>1$, where $R$ is a real number (actually the problem says $R=2$).
... |
H: Prove $n^3+3n$ is even in case $n=2a+1$
Prove that $\forall n \in Z$, $n^3+3n$ is even.
Attempt: I am solving this problem using proof by cases. Case 1 is when $n$ is even, i.e. $n=2b$. This one is easy. However, in case 2 when $n$ is odd, i.e. ($n=2a+1$) I am having difficulties with showing that $n^3+3n$ is eve... |
H: How do we say the behavior of the limit of cosine square?
$\lim_{n\rightarrow \infty } (cosx)^{2n}$
Is it correct to say that this sequence diverges?
$\lim_{n\rightarrow \infty } ((cosx)^{2})^{n}$
$ 0\le (cosx)^{2}\le1$, hence the behavior of $(cosx)^{2n}$ would be like oscillate between 0 and 1?
But in some... |
H: Locally Lipschitz implies continuity. Does the converse implication hold?
Let $A$ be open in $\mathbb{R}^m$; let $g:A\rightarrow\mathbb{R}^n$. If $S\subseteq A$, we say that $S$ satisfies the Lipschitz condition on $S$ if the function $\lambda(x,y)=|g(x)-g(y)|/|x-y|$ is bounded for $x\neq y\in S$. We say that $g$ ... |
H: Derivative of ${x^{x^2}}$
Studying past exam problems for my exam in ~$4$ weeks, and I came across this derivative as one of the questions. I actually have no idea how to solve it.
$$\frac{d}{dx} (x^{x^2})$$
Using the chain rule on it letting $x^2 = u$ led to me getting $2x^{x^2-2}$, which isn't right. The function... |
H: number of distinct ways of writing each element of the set $HK$ in the form $hk$
Let $H$ and $K$ be subgroups of the group $G$. The number of distinct
ways of writing each element of the set $HK$ in the form $hk$, for some $h \in H$ and $k \in K$ is $|H \cap K|$.
My thoughts:-
Let $|H \cap K|=n$. Let $h_i=k_i$ w... |
H: what is the limit of $l_p$ at p=0?
The p-norm is defined as:
$$ \ \|x\|_p=\left(|x_1|^p+|x_2|^p+\dotsb+|x_n|^p\right)^{\frac{1}{p}} $$
When $p<1$, this is no longer a "norm" because it violates the triangle inequality (- it is super additive and not subadditive). However, it is still valid to ask, what is its limit... |
H: Analysis True or False Question. (Simple)
First one, I think it's true just by definition.
Second one, I think it's true as well.
Third one, i dont think sinx = cosx in interval [0,pi/2].
Am i missing anything?
If there are counterexamples that I simply ignored, please tell me.
Thank yOu
AI: The third one is also ... |
H: Does the rectangle contain the point?
A rectangle is defined by the 4 points ABCD. How can I tell if a given point, (x,y), is in the interior of the rectangle?
My current guess is the following:
To be inside the rectangle, the point should be between the lines AB and CD, and between the lines AD and BC.
The equati... |
H: Prove for a connected graph $G=(V,E)$, $\kappa(G)=1+\min_{v\in V}\kappa(G-v)$
$\kappa(G)$ here is the connectivity of $G$. I'm just at a loss for where to start. I thought about induction on the number of vertices since I can see that it clearly holds for trivial cases, but I can't make that work. I thought about i... |
H: Does this series converge? Can I use a rearrangemt to prove it doesn't?
$$1-\frac{1}{2}+\frac{2}{3}-\frac{1}{3}+\frac{2}{4}-\frac{1}{4}+\frac{2}{5}-\frac{1}{5}+\frac{2}{6}-\frac{1}{6}+ \cdots$$
I was thinking to use a rearrangment to get the harmonic series, but I don't know if that's valid, because the series shou... |
H: Gamma function in the sight of Lebesgue and Riemann integration.
I am taking a somewhat hard measure theory course and I was asked to prove this:
a) Let $\alpha > 0$ be a real number. Prove that
$$\Gamma(\alpha):=\int_0^\infty e^{-x}x^{\alpha-1}dx$$
exists. (We are studying the relationship between being Lebesgue... |
H: prove $\sum_{i=0}^{n}\binom{2n+1}{i}=2^{2n}$
Can someone help me to prove $\sum_{i=0}^{n}\binom{2n+1}{i}=2^{2n}$.
The right side means the total number of subsets of $[1,2,3,..,2n]$. Then What does
the left side mean? Can someone please help me? Thank you.
AI: You have that $$2^{2n+1}=\sum_{i=0}^{2n+1}\binom{2n+1}{... |
H: Limit goes to infinity, show that the f has a finite minimum.
So limit goes to infinity, and I have to show that there exists a finite infimum.
how do i show this?
AI: Think of a chain you hold by both ends very high. It has to ''drop down'' somewhere. That's roughly the idea.
So let's hold that rope ''very high''... |
H: Integration: easy, but how?
Once again I have to deal with an Integral, which seems to be not that hard:
$ \int x^{n-1} * sin(x^n) dx $ and $(n \in \mathbb{N})$
Do you have any hint for me ? :)
AI: Hint: If $u = x^n$, then $$x^{n - 1} dx = \frac{1}{n} du$$ |
H: Why does the limit of $F_X$ from the left equal $P (X < x)$?
I am trying the understand the definition of the cumulative distribution function:
We use $F_X(x^-)$ to denote the limit of $F_X$ from the left $ lim_{ 0 < \Delta \rightarrow 0} F_X(x - \Delta )$ and $F_X(x^+) $ the limit from the right $ lim_{0<\Delta \... |
H: Logical proof of the statement $xy = 0 \implies x=0\text{ or } y=0$
Claim:
If $xy=0$, then $x=0$ or $y=0$.
My proof is as follows:
case 1: $x=0$, so $0y=0$
case 2: $y=0$, so $x0=0$
Either way, $xy=0$.
I'm very confused by this myself. So if I let $xy=0$ be $P$, and $x=0$ or $y=0$ be $Q$, then the claim "if $xy... |
H: proofing the $ \nabla (FG) $
$\\ F $and$ G$ are vector fields
my problem is that why the answer is not $ (\nabla. F)G + (\nabla .G)F $
AI: Following on from above, if $F, G$ are indeed vector fields in $\mathbb{R}^3$ then write them out accordingly, viz;
$$ F = F_{1}\hat{i} + F_{2}\hat{j} + F_{3}\hat{k}$$
And appl... |
H: Calculus Integration, using the substitution method
Please help solve this problem using the u-substitution method. Thanks in advance.
AI: Make the change : $2 x^2 - 3 = y^2$. The integral becomes very simple. |
H: Co-prime power modulo
Is there $p$ such that $a^p\,$mod$\,r=0$ and $r$ is a prime number and $1\le a<r$.
I am believing that $a^p\,$mod$\,r=0$ will be not equal to $0$ any value of $p$ for all values of $a$ from $1$ to $r-1$.
Is it correct? please provide any counter example if not.
AI: If $a^p\equiv 0\mod r$, it ... |
H: Are there any integers $a,b$ s.t. ${ a }^{ 2 }-{ b }^{ 2 }=8$?
For $a$ and $b$ are integers greater than $1$, ${ a }^{ 2 }-{ b }^{ 2 }=8$ holds?
AI: Hint: We have $8= a^2 - b^2 = (a+b)(a-b)$. As $a,b \ge 1$, we have $a+b > a-b$, that is either $a+b=8$ and $a-b = 1$, or $a+b=4$ and $a-b=2$. |
H: Integrate $\int_0^{\pi/2}\frac{\sin^3t}{\sin^3t+\cos^3t}dt$?
How to integrate
$$
\int_0^{\pi/2}\frac{\sin^3t}{\sin^3t+\cos^3t}dt\,?
$$
I tried to use $\sin^3tdt=-(1-\cos^2t)d\cos t$. But the term $\sin^3t$ in the denominator can not be simplified. Can anyone give me a hand? Thanks.
AI: General Hint:
By setting $x=... |
H: Etymology of the term "filter"
According to this article, filters were introduced in general topology by Henri Cartan in 1937.
I wonder why he called them filters.
AI: It may not have been Cartan who named them. According to Michèle Audin, ‘Henri Cartan & André Weil du vingtième siècle et de la topologie’, p. $8$, ... |
H: How is the circle that fits beneath two adjacent circles related?
This is hard to search and probably easy to solve, but I keep finding articles about intersecting circles, and that is not what I'm after. I don't know what to tag this under, so if you know how to classify this better, please do.
I'd like to know ... |
H: How do I solve this limit? $\lim_{x \to \frac{\pi}{4}}\frac{\sin x-\cos x}{\ln(\tan x)}$
I can't really see the right way to solve this limit. My attempt is:
$$\lim_{x \to \frac{\pi}{4}}\frac{\sin x-\cos x}{\ln(\tan x)}=\left(\lim_{x \to \frac{\pi}{4}}\frac{\sin x-\cos x}{\ln(\tan x)}\right):\cos x = \lim_{x \to \f... |
H: Autocorrelation problem, regression analysis
Bit stuck on my econometrics course (old exam q), not big on mathematical statistics, anyway this is the problem:
Given some model $y_{it}=\beta_0+\beta_1x_{it}+u_{it}$ and suppose that the idiosyncratic errors are serially uncorrelated with constant variance i.e. $var(u... |
H: If $\sqrt[3]{a} + \sqrt[3]{b}$ is rational then prove $\sqrt[3]{a}$ and $\sqrt[3]{b}$ are rational
Assume there exist some rationals $a, b$ such that $\sqrt[3]{a}, \sqrt[3]{b}$ are irrationals, but:
$$\sqrt[3]{a} + \sqrt[3]{b} = \frac{m}{n}$$
for some integers $m, n$
$$\implies \left(\sqrt[3]{a} + \sqrt[3]{b}\right... |
H: Step in proof: Sum of euler phi function over divisors (Group Theory)
Proof: $$\sum_{d|N}\phi(d)=N$$ where the sum is over $d\in div(N)$
Let $G$ be the cyclic group $\mathbb Z/N \mathbb Z$. Then $$N = \sum_{g \in G} 1 = \sum_{d|N}\sum_{g \in G, ord(g)=d} 1=\sum_{d|N}\phi(d)$$
The step from $$\sum_{d|N}\sum_{g \in G... |
H: Showing that any sequence in $[0,1]$ has a convergent subsequence.
One should show that any sequence in $[0,1]$ has a convergent subsequence.
Now before even trying to prove it in general, I take one sample sequence, $x_n = |\sin(n)|$.
I think that for this sequence, the only convergent subsequence could be a const... |
H: Co-prime binomial coefficient
A number 'r' is prime if and only if $\binom{r-1}{k} \equiv(-1)^k \pmod r$
Since 'r' is a prime and it gives non-zero remainder by dividing $\binom{r-1}{k}$ .
So $\binom{r-1}{k}$ and 'r' are co-primes
If a0,a1,a2,a3,..........,ar-1 are coprimes to r .
Then
Is $\binom{r-1}{0}$a0+... |
H: Linear Algebra, reflected linear image
If I have a linear image of the room where v1 and v2 is an image of theirselves and v3 is an image of the null vector.
If that gives me the matrix A=(a, b, c; d, e, f; g, h, i;) then A^n = A because you've already applied a vector to the plane.
However, if v3 is an reflection ... |
H: Can I simplify logarithm of logarithm.
It it possible to simplify Logarithm of logarithm:
log log x
(for example, make it log (base 4) x.
AI: No, it's not possible to simplify (although it might be possible to make it very, very complicated).
As a side note, we have
$$
\log_{4} x = \frac{\log x}{\log 4}
$$
so there... |
H: which is bigger $\log\log x$, $\sqrt x$ , $(\log x)^2$
I need to compare these 3 and rank them based on which is bigger:
$\log \log x$
$\sqrt x $
$(\log x)^2$
As $x\to\infty$
AI: For every $x>1$, you can find a $t$ such that $x:=\mathrm {exp}(\mathrm {exp}(t)))^2$.
Plug this reparameterization into you functions... |
H: Useful relationships that are true for every norm
I am looking for useful identities that are true for every normed vector space $(V,||.||)$ on either $\mathbb{R}$ or $\mathbb{C}$(if your identity is restricted to either one of them, please say so).
I am looking for things like $|||f||-||g|||\le ||f-g||$ and so on.... |
H: Cumulative pdf, integration problems.
Given the cumulative pdf $f(x,y)=\begin{cases}xy/96, & \text{if 0<=x<=4 and 1<=y<=5} \\0, & \text{otherwise} \end{cases}$
Find $P(X+Y<3)$
My attempt:
$$P(X+Y<3)=\int_{1}^{5}\int_{0}^{3-y}xy/96 dxdy=1/12$$
$$P(X+Y<3)=\int_{0}^{4}\int_{1}^{3-x}xy/96 dydx=0$$
Shouldn't those two... |
H: Is the classification of finite simple groups not a bit arbitrary?
I've never been able to find any details on what exactly decides what the classifications ought to be for finite simple groups. We have:
Cyclic groups
Alternating groups
Groups of Lie type
Sporadic groups
But why does the classification stop there... |
H: How to solve $\lim_{x\to 0}\frac{\sqrt{x^2+4x+5}-\sqrt{5}}{x}$?
How to solve the following question?
\begin{eqnarray}
\\\lim_{x\to 0}f(x)&=&\lim_{x\to 0}\frac{\sqrt{x^2+4x+5}-\sqrt{5}}{x}\\
\\&=&\lim_{x\to 0}\frac{\sqrt{\frac{x^2+4x+5}{x^2}}-\sqrt{\frac{5}{x^2}}}{1}\\
\\&=&\frac{\sqrt{\lim_{x\to 0}(1+\frac{4}{x}+\... |
H: Properties of Digit root
Why the digit root of any number calculated in any way remains same...e.g
Let $f(x)$denote the digit root of $x$
$f(1237)=f(12+37)=f(49)=f(123+7)=f(130)=4$
I checked numerically with many numbers but I found that the digit root remains constant irrespective of the way the numbers are summed... |
H: Expected number of sixes tossed before success
Given a fair dice, what is the expected number of sixes that get tossed before throwing 2 sixes in a row? Would I start by finding all possible sets that occur without two sixes in a row and multiply them by the corresponding probability? ie
$$\sum_{k=0}^\infty (A+B)^... |
H: showing that some quotient ring is a field.
To show that $$\mathbb{Z}[x] / \langle 5, x^3+x+1\rangle$$ is a field; I tried to show that $\langle 5, x^3+x+1\rangle$ is a maximal ideal of $\mathbb{Z}[x]$, but I failed.
Because I have not seen the maxmal ideal generated by 2 generateors, the problem is hard.
Please i... |
H: Show that a language is not regular using Myhill-Nerode Theorem
I'd like to show that the language below is not regular using Myhill-Nerode Theorem. How can I do that?
Let Σ = {0, 1}.
Let L = {ww|w ∈ Σ*}
I am not sure where or how to go about this...
AI: HINT: I use the notation and terminology of the Wikipedia a... |
H: Inverse of orthogonal matrix is orthogonal matrix?
Is inverse of an orthogonal matrix an orthogonal matrix? I know its inverse is equal to its transpose, but I don't see where the orthogonality would come from.
AI: If $A^t = A^{-1}$, then taking inverses of both sides, we have $(A^{t})^{-1} = A = (A^t)^t$. |
H: True/False about ring and integral domain
I have some true or false questions and would like to have your help to check on it.
A). in a ring R, if $x^2=x$, $\forall x\in R$, then R is commutative
For (A), when looking at $(x+y)^2$, it has $x+y=(x+y)^2=x^2+xy+yx+y^2$
and then yx+xy=0, and from 2x=4x, therefore 2x=0.... |
H: Number of values of x
$$a\dfrac{(x-b)(x-c)}{(a-b)(a-c)}+b\dfrac{(x-c)(x-a)}{(b-c)(b-a)}+c\dfrac{(x-a)(x-b)}{(c-a)(c-b)}=x$$
How many values of $x$ satisfy this equation? It is clear that x=a, x=b, x=c do satisfy the equation, but are those the only three possible solutions?
AI: This has to be true for all $x$.
Fi... |
H: Showing symmetry of a relation between a commutative identity ring and a multiplicative subset of the ring
R is a commutative ring with identity, not necessarily an integral domain, and S is a multiplicative subset of R\0 containing 1 which is closed under multiplication. A relation ~ is defined on R x S so that $(... |
H: How to solve $\lim_{x\to\infty}\frac{\cos x}{x-1}$?
How to solve the following question?
\begin{eqnarray}
\\\lim_{x\to\infty}f(x)&=&\lim_{x\to\infty}\frac{\cos x}{x-1}\\
\\&=&\lim_{x\to\infty} \frac{\sqrt{1-\sin^2x}}{x-1}\\
\\&=&\lim_{x\to\infty} \frac{ \frac{\sqrt{1-\sin^2x}}{x}}{\frac{x-1}{x}}\\
\\&=&\lim_{x\to... |
H: Subspaces and span?
Let $S$ be the subspace spanned by $(\text{u}_1, \text{u}_2, ... , \text{u}_m)$. Then, $S$ is the smallest subspace containing $(\text{u}_1, \text{u}_2, ... , \text{u}_m)$ in the sense that if $S_2$ is any other subspace containing $(\text{u}_1, \text{u}_2, ... , \text{u}_m)$, then $S \subseteq... |
H: Finding a polynomial $g(x)$ such that $ g(x)g(x-1)=g(x^2)$
Find all polynomials $g(x)$ with real coefficients with the property $$g(x)g(x-1)=g(x^2).$$
My try: I found $$g(x)=(x^2+x+1)^n$$
satisfies the condition; maybe there are other solution? If so, how to prove it (and/or find them)?
Thank you.
AI: Hint: What c... |
H: Condition for subgroup lattice of $\mathbb{Z}_{n}$ to be a straight line
I was asked what was the condition(s) on n for the lattice subgroup of $\mathbb{Z}_{n}$ to be a straight line (ie each subgroup is a subgroup of another).
Then the order of each subgroup divides the order of the one it is included in.
So I fi... |
H: Why is $\tan(x)$ a function?
A function $f:X\rightarrow Y$ maps each $x\in X$ to some $y \in Y$. So consider $\tan{\frac{\pi}{2}}$ for which $\tan(x)$ is undefined, so in this case, $\tan(x)$ does not map to an element of its range. This conflicts with my understanding of what a function is. Why do we still conside... |
H: Invertible functions and their properties
If an n × n matrix A is singular, then the columns of A must
be linearly independent. Is this true?
Invertible functions must be bijective
Invertible functions must have square matrices
Invertible functions must span R^n
Also, am I missing some other must conditions of inve... |
H: Volume of a Special Pyramid
Let $P$ be a plane in $\mathbb{R}^3$ parallel to the $xy$-plane. Let $\Omega$ be a closed, bounded set in the $xy$-plane with $2$-volume $B$. Pick a point $Q$ in $P$ and make a pyramid by joining each point in $\Omega$ to $Q$ with a straight line segment. Find the $3$-volume of this pyr... |
H: Dimension of a subspace of finite-dimensional product space $V$ equals $\dim V - 1$
Suppose $w$ is a nonzero vector in a finite dimensional inner product space $V$. Let $P = \{ v \in V | \langle v,w\rangle = 0\}$. Show that $\dim P = \dim V - 1$ where $P$ is a subspace of $V$.
AI: Let $v=(x_1,x_2,\ldots,x_n), w=(a_... |
H: How many surjective function are there in infinite sets?
Say we have two sets $|A| = a$ and $|B| = b$, where $a \geq b$ and $a$ is infinite. How would you go about proving that the number of surjective functions from A onto B is $b^a$?
AI: HINT: Partition $A$ into sets $A_0$ and $A_1$, where $|A_0|=b$ and $|A_1|=a$... |
H: Question about the roots of an irreducible polynomial
How can we deduce that each field is isomorphic?
$g(x) := x^4 - 10x^2 + 1$ is an irreducible polynomial with $\alpha_{1} := \sqrt{2}+ \sqrt{3}$ as a zero. Show that $\alpha_{2} := \sqrt{2}- \sqrt{3},\alpha_{3} :=- \sqrt{2}+ \sqrt{3}, \alpha_{4} := - \sqrt{2}- \s... |
H: Discrete mathematics Relations Question
if r2 is in the set of N*N ( natural numbers) with (X,y) in the subset of r2, if and only if x+y=0
is it reflexive?
is it symmetric?
is it anti symmetric?
is it Transitive?
i said it is reflective because 0+0 =0
but i'm a little stuck beyond that. the only relation in the s... |
H: Proof with quantifiers
$(\forall x)(\exists y)(x+y=0)$
$x$ and $y$ are real numbers
The statement reads: for all $x$ there exists some $y$ such that $x+y=0$ is true.
My proof is: take $y=-x$
Is this valid? I'm just paranoid that since the proof is so simple that something is wrong.
AI: Yes, you've got the right id... |
H: Finite lebesgue Integral
Hi guys I've been trying to prove this for a very long time, if someone could help me i would appreciated very much! let $(X,S,\mu)$ be a mesurable space, if $\mu(X)$ is finite and $f$ is a mesureble non negative function then:
if $\int f d\mu$ is finite then $\sum_{n=0}^{\infty} 2^n \mu(\... |
H: discrete mathematics relations question 2
I am a little confused by this relation
R3 is a subset of Z×Z defined by (x,y) in the set R3 if and only if x>2y
is it reflexive? Symmetric? antisymmetric? or transitive?
i say its NOT reflexive because (1,2) is not in the set R3
i say it is NOT Symmetric because (1,2) is n... |
H: Convergence of $ x_n = \left( \frac{1}{2} + \frac{3}{n} \right)^n$
I need to show that the sequence $ x_n = \left( \frac{1}{2} + \frac{3}{n} \right)^n$ is convergent.
Using calculus in $ \mathbb{R}$, we could see that $ \lim _{n \to \infty} \left( \frac{1}{2} + \frac{3}{n} \right)^n = \lim_{n\to \infty} e^{n \ln \... |
H: Find all permutations in increasing order
Given a set of distinct numbers, say, {1, 2, 3, 4, 5, 6}, find all permutations containing 3 numbers. All the permutations have to be in ascending order.
For e.g., some correct permutations would be {1, 2, 3}, {2, 4, 6}, etc. {2, 3, 1} would be incorrect because it is not i... |
H: Is $\mathcal O_L$ an $\mathcal O_K$-lattice in $L$?
This is a basic question. Let $L/K$ be a finite extension of algebraic number fields and let $\mathcal O_L$ and $\mathcal O_K$ be their respective rings of integers. Is it true that $$K\otimes_{\mathcal O_K}\mathcal O_L \cong L$$
If so how can we prove it?
Thanks... |
H: Line bundles of the circle
Up to isomorphism, I think there exist only two line bundles of the circle: the trivial bundle (diffeomorphic to a cylinder) and a bundle that looks like to a Möbius band. Although it seems obvious geometrically I did not find a good argument to justify it. Do you have an idea?
AI: Line b... |
H: Limit problems..
http://s23.postimg.org/xedyol4kr/limit.jpg
I got stuck on Q2 , could someone shed light on how to do it? The proof bit. Finding limits in terms of α and β is easy.
As for Q3, I know it is probably not the way I was supposed to do but is the following method not correct?
Lim (A+B)= lim(a)+lim(b) an... |
H: Combinations of two letters from a $124$-element alphabet
The telephone company wants to add an area code composed of $2$ letters to every phone number. In order to do so, the company chose a special sign language containing $124$ different signs. If the company used $122$ of the signs fully and two remained unused... |
H: Choosing a Set of r elements from a set having n elements.
Define a set $X$={$1$,$2$,$...$,$n$} .
Determine the number of ways of selecting a subset of $X$ such that it contains no consecutive integers .
AI: Let $a_n$, be the number of subsets satisfying the condition, including the empty set. By adding one mo... |
H: Can we find an example of non-mesuarable set which their outer measure could be computed?
We know there is non-measuarable set and we know every set has outer measure, so can anyone give me an example of a non-measuarable and there outer measure could be computed ?
AI: Let us consider the Lebesgue (outer) measure o... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.