instruction stringlengths 22 27.2k |
|---|
Abe and Bernard are dealt five cards each from the same $52$ card deck. Let $A$ be the event that Abe gets a flush (five cards of the same suit) and $B$ be the event that Bernard’s five cards are of pairwise different kinds (i.e. pairwise independent). Are $A$ and $B$ independent?
Thoughts. Is $P(A) = P(A|B)$? In other... |
Three people have been exposed to a certain illness. Once exposed, a person has a 50-50 chance of actually becoming ill.
a) What is the probability that exactly one of the people becomes ill?
I am a bit unsure, how to solve this question.
|
Find $x$ such that
\begin{equation}
x\tanh(x\sqrt{2\alpha})=\frac{2}{\sqrt{2\alpha}}
\end{equation}
|
An operation . is defined on the set $Z×Z$, ie. the set containing all pairs of integers by:
$(u,v).(x,y)=(u+v,v.y)$
How can i show that the pair ($Z×Z$, . ) forms a semigroup?
|
We have $11$ bins with $10$ objects each. Every object is either black or white, and the $i$th bin ($1 \le i \le 11$) has precisely $(i -1)$ black objects in it. Someone selects, uniformly at random, one of those bins and then selects, also uniformly at random, two objects from it. What is the probability that these tw... |
What are the values of b such that the matrix [(1,1)(b,1)] is invertible mod 26.
I figured that the matrix is only invertible if its determinant and the n value 26 's gcd is 1, meaning they are relatively prime.
So, det(matrix) = (1-b), and then for all values gcd((1-b),n) = 1 would conclude that that's the values of ... |
Let $f$ be an entire function and $\xi=e^{\frac{2\pi i}{n}}$ for some $n\in \mathbb{N}$. Suppose that $f\left(\xi z\right)=f(z)$ for all $z\in \mathbb{C}$. Show that there is a entire function $g$ such that $f(z)=g\left(z^{n}\right)$ for all $z \in \mathbb{C}$.
Remark: I have tried to express $g$ in terms of $f\left(\... |
Is $\{(x,y,z) \in \mathbb{R}^3 :x^2+3y^2+12z^2 = 0\}$ a vector space?
My inclination is that the only real solution to $x^2+3y^2+12z^2=0$ is $(0,0,0)$, which is the trivial subspace of $\mathbb{R}^3$. However my true/false assignment is telling me that this is incorrect, that this set is in fact NOT a vector space. Why... |
Let $\{e_1,\ldots,e_n\}$ be an arbitrary basis in a finite dimensional inner product space $V$. Prove there exists vectors $\{f_1,\ldots,f_n\}$ such that $(e_i,f_j)=\delta_{ij}$.
I tried using the the Gram-Schmidt process to obtain the $f_i$'s but the resulting $f_i$'s should apparently be uniquely determined and $(e... |
An inversion in a permutation $\sigma = \sigma_1...\sigma_n$ is a pair of indices $i < j$ such that $\sigma_i > \sigma_j$.
Let inv($\sigma$) be the number of inversions of $\sigma$.
If $\sigma$ is a product of $k$ transpositions, prove that $k \equiv$ inv $(\sigma)$ (mod $2$).
attempt: Let inv($\sigma$) be the number ... |
I recently had a problem:
Show that if $X$ is contractible, and $Y$ is path-connected, show that the homotopy class $[X,Y]$ has a single element.
I have been able to prove this (I think) in a fairly "standard way":
Take $f,g:X \to Y$ and homotopy $F(x,t)$ between $i_x$ and constant function at $x_0$. Then consider the ... |
I was working on the motorway problem:
At time t=0, cars are launched from the same entrance of highway following a Poisson process with parameter $\lambda$, and the speed of cars ($v$) follows a known distribution of $S$.
The observer is on one of the cars with a constant speed of $v_0$, and the amount of observation ... |
Show that $K := \{x \in C[0,1] : x(0) \in [-3,4], |x(t)-x(s)| \leq d
|t^2-s^2|, \forall t,s \in C[0,1]\}$ is compact.
I already know that $C[0,1]$ is a compact set. So it is only necessairy to show that $K$ is closed.
Is anyone is able to prove it or give me a good hint?
|
The chances of being born with a certain disease are estimated as $1$ in $1200$. What is a good estimate of the chance that an island with $10000$ inhabitants has precisely $8$ people born with that particular disease? We assume that all $10000$ events of being born with that particular disease are mutually independent... |
I'm looking for a standard way I can approach problems where I am tasked to find the branch points and branch cuts of a complex function.
For instance,
$$ f(z) = e^{(z^2+1)^{1/2}}$$
or
$$ f(z) = \log(z^2-3)$$
My current understanding is that you relate a more complicated function to what you know.
For example,
$$ (z-... |
is there a short proof of the fact that there is a finite amount of consecutive smooth numbers (meaning Given a finite set B, there is a finite amount of pairs $n,n+1$ so that both can be expressed as the product of elements from B only.
I was told there is a proof via the Thue–Siegel–Roth theorem but the proof of the ... |
I have a set of objects of different sizes (measured in square metres). I know the price of some of them. I want to use the known prices to find the missing prices.
Here is the data I have:
\begin{array}{c c}
0.13 m^2 & 7500\\
0.13 m^2 & 7500\\
0.15 m^2 & ?\\
0.20 m^2 & ?\\
0.20 m^2 & ?\\
0.40 m^2 & ?\\
0.65 m^2 & ?\... |
Please, how do I write the following as a combination of a sum and a product:
$$ (c-a_1)(c-a_2)(c-a_3)b_1 + (c-a_2)(c-a_3)b_2+(c-a_3)b_3 ?$$
Also, how can I generalize it?
|
$\newcommand{\R}{\mathbb{R}}$
I am trying to follow a derivation of the first variation formula for the energy functional. (In "Selected Topics in Harmonic maps").
Here is the context:
$M,N$ are Riemannian manifolds, $\phi:M \to N$ is a smooth map which is a critical point of the energy functional $\int_M |d\phi|^2$ wh... |
I posted this same question on space.stackexchange but never received any answer. So I am posting here hoping to get an answer as this is a quite mathematical topic.
I am trying to fully understand the TRIAD algorithm for attitude determination of a satellite.
https://en.wikipedia.org/wiki/Triad_method
So so far I kno... |
It seems that in every paper mentioning universal side divisors, they are defined very succinctly and with a bunch of symbols, so that I remain completely confused as to what they are and how to find a concrete specific example of such a divisor.
An answer to another question talks about polynomial rings, which I don't... |
Let $a>0$ be a real number, such that for all integers $n\geq 1$: $n^a \in \mathbb N$
Show that $a$ must be an integer.
It's not difficult to show this when $a$ is a rational number: $2^\frac{p}{q}$ is irrational when the fraction is in lowest terms and $q \neq 1$.
When $a$ is irrational, then for all $n\geq 1$, there... |
If $a$ is an integer, prove that $gcd(14a + 3, 21a + 4) = 1$
-We have just started the section on greatest common divisor, one thing I know is that $gcd(a,b) = ax + by$
-My initial thought is that we would plug our values into that equation to get the following:
$$(14a + 3)x + (21a + 4)y = 1$$
-I am a bit confused as t... |
Let $\mathcal{M}$ be a surface and $\gamma_1, \gamma_2$ two smooth curves contained in $\mathcal{M}$ in natural parameterization s.t.:
$\gamma_1(0)=\gamma_2(0) = p$ , $\ddot{\gamma}_1(0),\ddot{\gamma}_2(0)\in T_p\mathcal{M}$.
Let $\alpha\in [0,\pi/2]$ be the angle between the two curves. Given that $\alpha \neq 0$. Le... |
In need to prove with boolean algebra that XOR complement (negado) is equal to XNOR but i cant do it, can you help me?
!(!xy+x!y)=xy+!x!y
how to prove it?
|
My textbook does not provide much about counting measures and integration. So I decided to set up integration on space $(N , P(N) , \mu_c ,R)$ myself, imitating the construction of the Lebesgue integral. (All functions are measurable since we are considering the power set.) I started with simple functions, which are ba... |
I ended up in the wrong math class (trigonometry) for my level but am trying to survive by catching up on some more basic principles. I'm wondering if the same principle (and if so, what is it) is applying in the second and third part of each equation. In the first equation, how does it get reduced to $_y^x$ if, in th... |
As I'm not even sure what type of problem this is, so I can't research it.
$$3+5x^{1/2}=2x$$
My Question:
Could I get an explanation (with example) of how to solve the above equation? Or direct me to a resource with the explanation (preferably Khan Academy)? I am not looking for the answer to this problem, just the way... |
I just learned about Bisection Method, Fixed-Point Iteration Method, Newton- Raphson Method, and Secant Method. My prof wants us to be able to write new Algorithm of root finding by coming 2 or 3 standard algorithms to solve a particular equation numerically.
I have no idea what to do.
Could you guys help me out this p... |
In reviewing complex analysis, I stumbled upon the following problem:
Find a function $g(x,y)$ harmonic on $\{ 1<x^2+y^2<16\}$ such that $g(x,y)=3$ when $x^2+y^2=1$ and $g(x,y)=8$ when $x^2+y^2=16$.
In my notebook, I saw some notes on the side that $g(x,y)$ should have the form $a \cdot \ln(x^2+y^2)+b$, but not only ... |
If $\gcd(a, c) = 1$ and $b \mid c$, prove that $(a, b) = 1$
-Not sure how to approach this problem.
-We have just started the greatest common divisor section, and looking at my notes I see that $(a \mid c = 1) \implies ax+by = 1$, would this be useful?
-My train of thought is $b \mid c$ then this proof would be relat... |
I am a high school student who is deeply passionate about mathematics and I have written many different mathematical proofs. I was wondering what sort of discovery warrants writing a mathematical paper for publication. Does the amount of content matter, or the significance of the discovery? Any advice you can give woul... |
Doing some reviewing and I'm not 100% sure if my thought-process is correct.
I have the following two vectors and need to prove they're a basis for $R^3$:
$$B=
\begin{bmatrix}
1 \\
0 \\
-1 \\
\end{bmatrix},
\begin{bmatrix}
0 \\
1 \\
-1 \\
\end{bm... |
I am trying to solve a couple integrals of the form:
\begin{equation}
I(g)=\int_{0}^{\infty} x \, e^{-a(gx-b)^{2}}\,e^{-\beta_{1}x}\, {_{1}}F_{1}(-\alpha_{1};-\alpha_{3};\beta_{3} x) \ \mathrm{d}x
\end{equation}
where, $a>0$, $g\in\mathbb{R}$, $b\in\mathbb{R}$, $0<\beta_{1}<\beta_{3}$, and $0<\alpha_{1}<\alpha_{3}$.
$\... |
For example: (5,6,7)8,9,10(11,12,13)
(227,228,229)230,231,232,233,234,236,237,238(239,240,241)
|
This question is related to my trouble in abstract algebra in general so some general advice is really appreciated!! (I'm really new to this.)
Let A be an infinite subset of G with $n = \#A$.
\begin{equation}
H = \{g∈G: gag^{-1}∈A, \forall a∈A\}\\
N = \{g∈G: ga=ag, \forall a∈A \}
\end{equation}
First I'm asked to pro... |
Roundness is the measure of how closely the shape of an object approaches that of a circle.
I am trying to find a similar measure which shows how closely is something to an ellipse. Is there any similar measure?
|
Let $V := R^n$ be a vector space and let $I \in O(n)$ be an operator satisfying $I^2 = -Id$. I want to show that the $span\{x,Ix\}$ is an invarient subspace of $I$.
Let $W = span\{x,Ix\}$. I need to show that $IW \subseteq W$. That is, if we transform $W$ by $I$, then we are still within $W$. But, my linear algebra is ... |
I have been given the following problem to solve:
In a right pyramid whose base is an equilateral triangle, the angle between 2 side-faces is 70 degrees. Compute the base angle of a side-face.
I have little idea how to tackle this problem. I understand the definitions, but don't feel like any of my attempts is even wor... |
If $m | (8n + 7)$, $m | (6n + 5)$,prove that $m = ± 1$
-We have just starting going over the "divides" notation, and I am aware of a few properties and theorems from my notes. I am; although, a bit lost as to how to properly set up this proof. Any help is appreciated.
So far I have:
$$3(8n + 7) - 4(6n + 5) = (24n + 21... |
use the formula $P_n(x) = \dfrac{1}{2^nn!}\dfrac{d^n}{dx^n}((x^2-1)^n)$ to show that $P_{2n}(0) = \dfrac{(-1)^n(2n)!}{4^n(n!)^2}$ and odd terms are 0.
I first subbed in 2n to the formula and got
$P_{2n}(x) = \dfrac{1}{4^n(2n)!} \dfrac{d^{2n}}{dx^{2n}}((x^2-1)^{2n})$ but I am not sure how to deal with differentiating th... |
So, my lecturer says that $-\cos(\frac{\pi}{8}) = \cos(\frac{9\pi}{8})$.
What did he do to get that?
Please recommend a source where I can brush up on my knowledge of angles.
|
I read the answer to this question:
https://stackoverflow.com/questions/34794664/how-should-i-calculate-the-average-speed-by-road-segment-for-multiple-segments/34795821#34795821
Can anyone please explain why the harmonic mean is in use when averaging speed and not the "regular average" ?
|
Language $L$ is specyfied by grammar : $(\{S,A,B\},\{c,d\},S,\{S \rightarrow SA, A \rightarrow Bc | \epsilon, B \rightarrow d\})$.
My task is to construct LR(1) parsing table for language $L^R$ (with, as far as I understand, is a reverse of language L). I know very well how to construct LR(1) parsing table, what I requ... |
Given is the following series
I want to calculate the limit for $n \to \infty$.
I already recognised the geometric series $\sum_{i=2}^n a^{i} = \frac{1}{1-a}$ for $a=e^\rho$ (since rho is supposed to be negative and $e^\rho$ muss be <1), but I am confused with $i-1$ in the exponent.
I changed the expression to:
$lim_... |
Let $f : \mathbb{R}^n \to \mathbb{R}^m$ be a continuous function. We
suppose that there exist $c \in \mathbb{R}^m$ such that $$\lim_{\|x\|
\to \infty} f(x) = c.$$ Show that $f$ is uniformly continuous.
We know that if $\forall \epsilon_1 > 0$, $\exists M > 0$ such that $\forall x \in \mathbb{R}^n\setminus B(0,M) ... |
Using the infinite sum series, an infinite sum of (1/5)to the nth power, where n goes from zero to infinity, the general summation equation tells us that the answer is 5/4. However, how is this possible, considering that each additional sum is smaller than the previous sum? If you begin to do the math by hand, calcul... |
Prove that $\lim_{n \to \infty } \left | z_{n} - z \right | = 0$ if and only if $\lim_{n \to \infty } Re(z_{n}) = Re(z)$ and $\lim_{n \to \infty } Im(z_{n}) = Im(z)$.
I understand the epsilon delta definition for limits of complex functions, but I'm getting confused since there is no clear function of z.
|
I was trying to solve this limit:
$\lim_\limits{n\to \infty} \binom {3n}{n}^{1/n} $
I solved it with Cesaro theorem:
$\lim_\limits{n\to \infty} \binom {3n}{n}^{1/n} $= $\lim_\limits{n\to \infty} \frac{((3(n+1))!}{(2(n+1))(n+1)!}\frac{2n!n!}{3n!}= \frac{27}{4}$
But when I have tried to use Stirling I arrived to the form... |
I was computing some calculations, when I got stuck about a possible closed form for this series:
$$S = \sum_{k = 2}^{N}\ \frac{k!}{k^k - k!}$$
I proved by hands that it's absolutely convergent by using the ratio test, but now I have to find something with which to approximate that series, or a closed form if it does e... |
Just today, I was making tortilla chips, and I began to wonder, what is the most efficient way to pack circular quarters onto the plane?
This sort of circle packing is most efficient for circles, but what if quarter circles are used instead of whole circles, is it still this method?
Does anyone have an answer for this... |
I'm doing a calculus project where we have to make a model of some graph rotated about the y axis. I am doing a fish bowl and I have most of it understood and ready. The only thing I'm not sure of is what the equation of the semicircle is if its center is at (0,1.5). I added a picture to help illustrate it. Thank you f... |
Let $G = S_4$ be a group, $N = \{1, (1, 2)(3, 4), (1, 3)(2, 4), (1, 4)(2, 3)\}$ a normal subgroup of G.
It's easy to see that $G/N$, the set of cosets is $G/N = \{a, b, c\}$, where
$$a = \{(1), (1, 2)(3, 4), (1, 3)(2, 4), (1, 4)(2, 3)\},$$
$$b = \{(1, 3)(1, 2), (2, 3)(2, 4), (1, 2)(1, 4), (1, 4)(1, 3)\},$$
$$c = \{(1, ... |
Can I get a reference for the statement that Cartier duality gives a contravariant equivalence from the category of finite etale group schemes to itself?
|
If I take sphere and remove two open disks from it and on the boundary of that space I make identification like on the picture, what do I get? Are both of those objects Klein's bottles?
This is what I want: ]1
|
There's a theorem in Linear Algebra which says that if ${\bf A}$ is an $m \times n$ matrix and $m < n$, then the homogeneous system of linear equations ${\bf A}{\bf X}=0$ has a non trivial solution.
I read a proof but what happens if I have a matrix like ${\bf B}$, whereas
B = \begin{pmatrix}
1 & 0 & 0 &... |
A functional $\phi$ on a $C^*$-algebra $A$ with unit element, i.e. $\phi: A \rightarrow \mathbb{C}$ is called a state if $\phi(T^*T) \ge 0$ for all $T \in A$ and $\phi( \operatorname{id}) = 1.$ Now, I started wondering whether such a functional $\phi$ is automatically bounded or not?
|
Say we design a simple Turing machine that adds two numbers together. Is there any way to formally prove that the machine actually computes the function we 'know' it does?
Is there a general method for all functions? Induction seems the most relevant, but will it work for any function? Any proof for this?
|
A bit string is a finite sequence of $0$’s and $1$’s. How many bit
strings of length $8$ begin and end with a $1$?
My answer would be: $2^6$. Because we know, that the bit starts with $1$ and ends with $1$: Which means, that $2$ of the length $8$ is used, also there are $6$ positions remaining, which is $2^6 = 64$.... |
How do I prove that there is no isomorphism between $\Bbb Z$ under addition and $\Bbb Q$ under addition?
They both are infinite order. I thought they might be isomorphic.
Help would be appreciated!
Thanks in advance!
|
If I have that $\limsup_{n}E|X_n|^{r} \leq E|X|^{r}$, is that enough to show that $\{|X_n|^{r}:n\geq 1\}$ is uniformly integrable? I am not sure here if the limsup condition here is as strong as if I had a uniformly bound. Does anyone have any hints as to how I can work with such a definition? Thanks.
|
I understand how to find the image($A$). The basis of Im($A$) would be the first two columns of the matrix $A$ (given the two leading 1's in ref are in the first and second columns).
So the $\text{Ker}(B) = [1,1,1,1],[1,2,3,4]$
But I do not get how to find $B$ based on its kernel? Any ideas?
|
A hint is provided with the proof prompt:
$(abc) = (1ca)(1ab)$, $(1ab) = (1b2)(12a)(12b)$, and $(1b2) = (12b)^2$.
My idea:
$(1ab) = (12b)(12b)(12a)(12b)$.
To solve for the other half of $(abc)$, I'm guessing $(1a2) = (12a)(12a)$. Thus, $(1ca) = (1a2)(12c)(12a)$
Therefore, $(abc) = (1a2)(12c)(12a)(1b2)(12a)(12b)$
Thus,... |
I have a problem where I need to know what J is.
I do x^J and get y.
For example, if I do 5^J, I would want to get 55 as y. Same with
4^J = 30.
When J is 2.455, it works up to 4 only! I need for decimals of J! Is there an equation or something so I can make a program to calculate J?
I want to see if It is possible to g... |
I need to try to solve, by some numerical method, a variant of the Navier-Cauchy equation of motion for an elastic, linear, isotrophic, not homogeneus body:
$$ \rho_0 (\ddot{\textbf{u}} - \textbf{g}_0) = \nabla ( \lambda \nabla \cdot \textbf{u}) + \nabla \cdot ( \mu ( \nabla \textbf{u} + \nabla \textbf{u}^{\textsf{T}} ... |
If $a$ and $b$ are positive integers, and $\frac{1}{a} + \frac{1}{b}$ is an integer, prove that $a=b$. And show that $a = 1$ or $2$
-I played around with numbers and the conditions and it seems that it's common sense that if $a = b$ then $a = b = 1$ or $2$, For example:
If $a = b = 4$, then $\frac{1}{a} + \frac{1}{b}$ ... |
Problem Statement: Let $A=\begin{bmatrix} 2 && 1 \\ 1 && 2 \end{bmatrix}$. Find an orthonormal basis for $\mathbb{C}^2$ with respect to the Hermitian form $\bar{x}^TAy$.
I am trying to figure out this proof, but not sure if what I am currently trying to do will help me. I was goin to start out by taking an arbitrary ... |
Let $z_{1}, \dots, z_{n} \in \mathbb{C}$. Then, there exists a subset $S \subset \{1,\dots,n\}$ such that:
$
\left| \displaystyle\sum_{j \in S}z_{j} \right| \geq \displaystyle\frac{1}{6}\sum_{j=1}^{n}|z_{j}|
$
I'm not sure on how to start for the proof of this result (I need it as a lemma for some proof involving comp... |
Showing the set $A = \{ x \in l_1 : |x_n| \leq 1/n^2 ,\forall n \}$ is closed. I had to show it is compact, and I am done showing it is relatively compact, but now I am stuck showing it is closed.
$l_1$ is the space of finite sequences $x = (x_1,x_2,...)$ with the norm $\|x\|_1 = \sum_k^\infty |x_k| < \infty$ i.e. the... |
Given a sequence of sequence processes $(X_N(\cdot))_{N \geq 0}$, I want to show this sequence converges to another process $X(\cdot)$ by considering that the sequence of generators $(A_N)_{N \geq 0}$ converge to some generator $A$ which corresponds to the process $X$. Could someone please point towards a proposition f... |
An incompressible inviscid fluid, under the influence of gravity, has the velocity field
$$\textbf u = (− \cos(x)\sin(y), \, \, \sin(x)\cos(y), \, \, 0)$$
with the $z$-axis vertically upwards, where $g$ is the acceleration due to gravity.
Use Euler’s equation to find the pressure $p$.
In my notes, the equation is given as ... |
If $|z-i|<1$ what can we deduce about how large or small $|z-1|$ and $|z+i|$
could be?
I tried drawing a diagram to get a feel but I don't know how to do anything more.
I feel like $ 1< |z+i|<3$ based on diagram but how can I go about the $|z-1|$
|
I am working with an inequality and I need to prove something of the shape $$c\cdot a+d\cdot b \leq a\cdot b$$
The numbers $a$ and $b$ have a specific form, but for the $c$ and $d$ I only know that $c<b$ and $d<a$. I also know that in some cases that $c>b/2$ or $d>a/2$, so I wanted to know if there is any criteria or a... |
Even before attempting the problem, I immediately defaulted to an answer: $\frac{1}{2}$.
I thought that this was a possible answer since the probability of flipping a head on one flip is definitely $\frac{1}{2}$.
I then worked through the problem:
Let E be the event in the problem statement.
The total number of possibl... |
Is there an algorithm for finding all of the subgroups of $D_{15}$? Also, is there a formula for finding the size of that subgroup?
Not sure where to start with finding all the subgroups of $D_{15}$ so any tips would be helpful! Thanks.
|
I have a question about the ODE (weak formulation) given by $$u''-u=f$$ where $u\in H^1(\mathbb{R})$ and $f\in L^2(\mathbb{R})$.
I want to see if there is an explicit formula for the weak solution. If $f\in C^{\infty}_0(\mathbb{R})$, then clearly we can find the strong solution, which have the form $$u(x)=c_1e^{x}+c_2... |
Question:
Referring to the Wikipedia article on Adjoint Functors in Section 2 (Motivation), they talk about "turning rngs into rings" (can be rephrased as "constructing rings from rngs"). I do not quite understand how this "turning" of algebraic structures into new algebraic structures is done. Which brings up my mai... |
I am trying to find
$$\lim\limits_{x \to \infty} n-\sqrt{n^2-4n} $$ using the definition of a limit.
I have tried to multiply top and bottom by $n+\sqrt{n^2-4n} $ giving $\frac{4n}{n+\sqrt{n^2-4n}} $ but this does not seem to help too much. Any hints would be much appreciated. Thank you.
|
I want to find numerically (the functional expression might become too complicated) the derivative of a complex function (to use it in a Newton-algorithm). Can I simply do something like
$$ \frac {df}{dz} = \lim_{h \to 0} {f(z+h)-f(z))\over h} + {f(z+ih)-f(z))\over h}
$$
? Or how do I have to do this?
[update] ... |
I couldn't find any suitable substitution for this integral and hence I couldn't solve it.
$$\int_{0}^{1}\left \{ \sqrt{1-x^2}+2 \right \}^2 dx$$
|
Let $G = (V,E)$ be a connected graph. Suppose $e$ is an edge in a circuit of $G$. Show that the new graph $(V,E-\{e\})$ is still connected.
Attempt: Let $v,w \in V$ be vertices. Then inside $G$, $v$ and $w$ are connected. Suppose the circuit starts and ends at vertex $a \in V$. Then the only problem that could occur... |
Can someone tell me why the first choice is the correct answer?
|
According to wolfram a function is monotonic if its derivative never changes sign, but the derivative doesn't have to be continuous. So I feel the answer is Yes, tangent is monotonically increasing. Maybe not?
|
I would like to find the geodesics in the Poincare disk.
I know that the metric is $$\frac{dx^2+dy^2}{(1-x^2-y^2)^2}$$ so $$s=\int \frac{\sqrt{1+y'^2}}{1-x^2-y^2}\, dx$$ Then I try to find y(x) using the Euler-Lagrange equations. The ODE in which I end up is: $$yy'^2+xy'+y=0$$ I can't solve the above ODE.
I also tried... |
The dual to $c_0$ is $l^1$, but if $\{x_n\}_{n\in\mathbb{N}}\in c_0$, than according to Dirichlet's test
$\sum_{n\in\mathbb{N}}(-1)^nx_n$ converges. But $\{(-1)^n\}_{n\in\mathbb{N}}\notin l^1$. So why is the dual space $l^1$ and not space of sequences with bounded partial sums, as $\{(-1)^n\}_{n\in\mathbb{N}}$ is?
I'd ... |
I am trying to understand the proof of a theorem which uses Zorn's lemma. I understand quite well all parts of the proof except for one point:
Let $R$ be a ring and define $K\doteq \{I\subseteq R \;\vert\; I \textrm{ is an ideal} \}$. Let $T$ be a totally ordered (by inclusion) subset of $K$. Then
$$J\doteq\bigcup_{... |
Let $p \in (0,1)$ and $n \in \mathbb{N}$. We consider a sample of $n$ i.i.d. Bernoulli variables $X_1,\dots,X_n$ with parameter p.
Computer $E[e^{\lambda\bar{X_n}}]$ such that $\bar{X_n}= \frac{1}{n} \sum_{i=1}^n X_i$
$E[e^{\lambda\bar{X_n}}]=E[e^{\frac{\lambda}{n} \sum_{i=1}^n X_i}]=e^{\frac{1}{n}}E[e^{\lambda\sum_{i=... |
$(X,d_1)$ and $(X,d_2)$ are two connected metric spaces if and only if $X\times Y$ is a connected metric space with metric $$ D((x_1,y_2), (x_2,y_2)) = \max(d_1(x_1,x_2),d_2(y_1,y_2)).$$
I know that product of two connected space is connected but how I have to use their metrics.
|
I took mathematical probability last semester and now I am taking financial mathematics, but only probability was a pre requisite for financial math (no finance classes were required). These types of questions re confusing me because I don't quite understand financial terminology and I guess my professor thinks that we... |
I always have a problem with the word generic in the literature of mathematics. Let me ask you a specific question about "non-degenerate $\mathbb{Z}$-graded lie algebras''. The definition I'm working with says:
A $\mathbb{Z}$-graded lie algebra (over $\mathbb{C}$) $\mathfrak{g}$, i.e.
$\mathfrak{g}=\bigoplus_{n\in \... |
I am working with the equation
$T(n) = T(n/2) + n^2$,
given $T(1) = 0$.
I started by using backwards substitution arriving at
$T( ( ( n - 1 ) / 2 ) + ( n - 1 ) ^ 2 ) + n ^ 2$
and eventually arrived at
$T( ( n - k ) / 2 )$ + the summation of $(n-i)^2$ from $i = 0...k$.
However, I am not sure where to go from here becaus... |
I am a beginner in calculus and I want to know what is the difference between sum and integral. More specifically I came across this example:
Compare $$\sum^\infty_1\frac1x\space \text{and} \space \int_1^\infty\frac1xdx$$
It would be really helpful if someone explains this to me. I want to know the difference between... |
I have to show, that $\mathbb{R}^3$ is not diffeomorphic to $\mathbb{R}^3\setminus \{0\}$.
That means, I have to show that there are no two smooth maps $f\colon\mathbb{R}^3\rightarrow \mathbb{R}^3\setminus \{0\}$ and $g\colon \mathbb{R}^3\setminus \{0\} \rightarrow \mathbb{R}^3$ such that $f \circ g$ is the identity o... |
I'm trying to solve this equation
$$
\left \lfloor{x +\frac{1}{100}}\right \rfloor + \left \lfloor{x +\frac{2}{100}}\right \rfloor + ... + \left \lfloor{x +\frac{223}{100}}\right \rfloor = 521
$$
I haven't faced until now problems like this one... How can I find $x$?
Rather than a specific solution, I'm looking for th... |
This is a question from a review package that is causing me some trouble.
Let $U,W$ be subspaces of a finite dimensional vector space. Show if $\dim(U+W) = 1+\dim(U \cap W)$, then $\{U+W,U\cap W\}=\{U,W\}$.
I know that for $\{U+W,U\cap W\}=\{U,W\}$ to be true, one of two cases must happen. Either $U\subseteq W$ or $W\s... |
Let $R$ be a commutative ring and $f=a_0+ \cdots +a_nt^n \in R[t]$. Define $f':=a_1+2a_2t+ \cdots + na_{n-1}t^{n-1}$. Show that $f$ is divisible by a square of non-constant polynomial if and only if $f,f'$ are not relatively prime.
The forward direction is easy: suppose $g^2 | f$ and $\deg(g) \neq 0$. Then $f=g^2h$ f... |
I need to show that if $G$ is non-abelian group of order $6$ then it contains an element of order $3$. I don't know how to proceed. Any kind of help/hint is appreciated.
|
The problem is to evaluate the improper integral:
$I = \int_0^{\infty} \frac{\sin(xt)(1-\cos(at))}{t^2} dt$.
This can be written as follows:
$$I = \int_0^{\infty} dt \frac{\sin(xt)}t \int_0^a \sin(yt)dy = \int_0^{\infty} dt \int_0^a \frac{\sin(xt)\sin(yt)}t dy$$
In a previous problem, I determined:
$$J = \int_0^{\inft... |
Let $f:\mathbb{C} \rightarrow \mathbb{C}$ be an entire function and $\xi=e^{\frac{2\pi i}{n}}$ for some $n\in \mathbb{N}$. Suppose that $f(\xi z)=f(z)$ for all $z\in \mathbb{C}$ and consider the function
$$g(z)=\frac{1}{n}\sum_{k=0}^{n-1} f \left(\xi^{k}\sqrt[n]{z}\right)$$
where $\sqrt[n]{z}$ is the $n$th root with a... |
I like the way there a physical meaning tied to the determinant as being related to the geometric volume. Since the determinant can be calculated through Laplace's formula where the cofactor matrix is involved, I'm wondering what would be the physical/geometric meaning of the cofactor matrix? What about the adjugate ma... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.