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An urn contains a white balls, b blue balls and c red balls. Balls are drawn one by one without replacement. Find the expected number of draws needed to get all 3 colors.
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If $a$ is the only element with order $2$ in a group $G$, then $a \in Z(G)$.
I'm studying for a test and I can't figure out how to prove it. What kind of methods might I try to solve this problem? I'd prefer suggestions rather than a proof, so that I can figure it out for myself.
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$$\int \frac{x}{(x^2-3x+17)^2}\ dx$$
My attempt:
$$\int \frac{x}{(x^2-3x+17)^2}\ dx=\int \frac{x}{\left((x-\frac{3}{2})^2+\frac{59}{4}\right)^2}\ dx$$
let $u=x-\frac{3}{2}$
$du=dx$
$$\int \frac{u+\frac{3}{2}}{\left((u)^2+\frac{59}{4}\right)^2}\ du$$
How can I continue from here?
|
I have the following task:
Let $A\in \mathcal{M}_n(K)$ be a diagonalizable square matrix. Show
using the spectral decomposition of $A$ that the only nilpotent
matrix in $\langle A\rangle =\{p(A)=c_0A^k+\cdots+c_{k-1}A+c_kI\mid
c_i\in K\}$ is zero matrix $O$. $K$ is a field.
The spectral decomposition of $A$ is... |
I am stumped at my professor's answer to this predicate logic. all x and y are natural numbers.
∃y∃x(x >= y)
I think it is true, since there is a pair $(Y, X)$ like $(1, 5)$ such that $X \geq Y$.
But my professor says it is false!
Why?
Thank you
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I was thinking about the following integral if I could solve it without using trigonometric formulas. If there is no other way to solve it, could you please explain me why do we replace $x$ with $2\sqrt 2 \sin(t)$? I'm really confused about these types of integrals.
$$\int \sqrt{8 - x^2} dx$$
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What I am aware of
(1) A cyclic permutation is a permutation that consists of a single nontrivial cycle (cycle of length $> 1$). Let $k$ be the length of the cyclic permutation $\tau$. Therefore the following applies: $\tau^k = id$.
What I am unsure about
Let's say we have the following non-cyclic permutation $\pi=$$\l... |
I am doing some old exam questions - and I don't know the answer, can some one calculate the result and show how you did it?
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$\pi$ is dependent on properties of geometry, assuming that we define it as $C/d$. Could there be a geometry where $\pi$ is a rational number or an integer?
|
Let $s$ be the number of variables and $n$ be the degree of the monomials we want to count in $R[X_1,\dots,X_s]$. Then show, that the count is
$$\delta(n,s):=\binom{s-1+n}{s-1}.$$
The question is a duplicate, but I have so far not been satisfied with any of the seen answers. Not here, not on google/lecture notes e... |
Is there any solution of the equation other than $x=2$?
Please help me. Thank you in advance.
|
I am trying to prove this but just don't see it. We are talking about openness in the metric sense, yes?
So, my attempt is
Let $x \in \mathbb{R}^n$ and $d$ represent the Euclidean metric, $d(x,y)=\sqrt{\sum(x_i-y_i)^2}$. An open ball around a point $x$ is defined for some $r>0$ to be $B_r(a)=\{y \in \mathbb{R}^n:d(x,y... |
Sometimes you want to find the common tangent line of two functions. The first thing that comes to mind to a person that is learning basic calculus is that you should equal the derivatives of those functions. Nevertheless, this way to resolve a problem like this is inaccurate. I saw some questions in the site that show... |
I've been assigned the following partial differential equation as an introductory exercise:
$$u_{xy}+au_x+bu_y+abu=0$$
Where $a,b$ are constants and $u=u(x,y)$. Having seen barely anything except for a couple easy examples, I have no clue how to even start. We haven't seen any actual solving methods, and so far we've o... |
Let $P_{20}$ be a path of length 20 like so: $x_0$-$x_1$-$~\cdots~$-$x_{20}$ and $G$ a cycle of order 3.
Allegedly there are $3 \cdot 2^{20}$ mappings $P_{20}\rightarrow G$, which I don't quite see.
We defined a mapping $\Psi$ of graphs $(V,E)\rightarrow (V',E')$ as a pair of mappings $\Psi_1:V\rightarrow V'$ and $\Ps... |
I need to find the asymptotic upper bounds in $O$ notation for $T(N)$ in two recurrences. Assuming that $T(N)$ is constant for sufficiently small $N$, I need to make the bounds as tight as possible.
$T(N) = T(N-3) + 3 \log N$
$T(N) = 2T(N/4) + \sqrt{N}$
I know I should use master theorem. I can't implement it correctly... |
I need to compute first derivatives of the following function $S(w)$ with respect to $w$. Then solve it. The reason behind that is to minimize $S(w)$.
$S(w)=\sum_{i=1}^{n} w_i^{1/2} \bigg(y_i - \sum_{j=1}^{n} w_j y_j\bigg)\bigg(\sum_{j=1}^{n} w_j y_j (1-\sum_{j=1}^{n} w_j y_j) \bigg)^{-1/2} E_i + \lambda (\sum_{j=1}^{... |
Prove that for all $n\in\mathbb{N}$ the number $\sqrt{n(n+1)}$ is irrational.
My first move would be:
Let's assume that it's not, that it $\sqrt{n(n+1)} = \frac{a}{b}$, where $a,b\in\mathbb{N}$ and $a,b$ are coprime. Then:
$$
n(n+1)=\frac{a^2}{b^2} \iff a^{2}=n(n+1)b^2
$$
I do not know how to continue this proof (I sup... |
I am having trouble with these compositions.
$$T = \{(a,a), (a,b), (b,c), (b,d), (c,d), (d,a), (d,b)\}$$
$$U = \{(a,a), (a,d), (b,c), (b,d), (c,a), (d,d)\}$$
I need to find $T \circ T$, $U \circ T$, and $T \circ U$.
My problem is when I get down to, for example $U \circ T$ where $(d,a)$ corresponds with both $(a,a)$ an... |
Polygonal numbers are of the form $\cfrac{n^2(s-2)-n(s-4)}{2}$, where $s$ is the number of sides of the polygon and $n$ is to say which one it is (the $n^{th}$ $s$-gonal number)
So my question is, how can you tell if a number is polygonal, and if it is, which $n,p$ plugged into the formula get you the number. Basically... |
Probably I had nonstandart question, but I hope to find some help and valueable advice.
Assume I have an online game with $n$ players (let's say $n$ is about 100.000).
There's also $m$ maps ($m$ is about 10) and every map goes with its probability (some of the maps played much more frequently than others).
So, assume ... |
I have tried to solve this limit : $\lim_\limits{n\to\infty}\sin(2\pi(n^2+n^{1/2})^{1/2})$.
Where n $\in\mathbb{N}$.
I have understood that the limit exists and goes to 0 if the argument becomes like $k2\pi$ with k $\in$ $\mathbb{Z}$, so I have collected $n^2$ and I have taken it out of the root , highlighting that $... |
In the Ehrenfest Chain model:
There are M balls which are divided between urn A and urn B. At each stage, if a ball is chosen, then it would be moved into a different urn.
Let $X_n$ be the # of balls in urn A, then the transistion is
$P_{i,i+1}=1-i/m$,
$P_{i,i-1}=i/m$.
It turns out that the stationary distribution ... |
How would I go about proving the following two regular expressions are equal to one another:
$$
( a + b )^* a ( a + b )^* b( a + b )^* = (a + b)^* ab(a + b)^*
$$
I can "see" why they are equal to one another because the second $( a + b )^*$ is a redundant term as the expression will always have an $ab$ together. Bu... |
By mathematical induction, how would you show $\gcd(n,n+1)= 1$ for every integer $n$?
I'm thinking you would start by knowing that some integer, $d$, would divide both $n$ and $n+1$. That's all I have so far.
|
I have the sets A, B, and C:
$A = \{x\in\mathbb{Z} | 2 < x < 5\}$
$B = \{x\in\mathbb{Z} | 4 ≤ x ≤ 7\}$
$C = \{x\in\mathbb{Z} | 2 ≤x< 6\}$
What is $B ∩ C^c$?
If the complement of C is all numbers ∉ C, then wouldn't $B ∩ C^c$ be equal to all integers, since set $C ⊆ B$, and $C^c$ is all the numbers outside of C?
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What is the specific definition for power $+$ operator in automata theory? For example, when $x$ is a binary what does it mean that $x = 0^+$. Does it mean that x is a string with at least one $0$?
|
I was assigned this exercise by my Stochastic Analysis Professor.
Exercise.
Let $B$ be a one-dimensional Brownian Motion, and consider the following processes:
$X_t=\int_0^tB_sds\quad Y_t=\int_0^ts^2dB_s$.
Compute, for $s,t\geq0$, $Cov(X_s,Y_t)$.
Solution.
This is how i would solve it.
It is known that the stochast... |
Though understanding these diagrams, I do not understand how to visualise the following explanation:
$\color{green}{[P1.]}$ Suppose you were to grab the edges of $A$ and stretch it out so it covers all of $\Omega$. $B$ stretches out with it. Now we ask, what proportion of $\Omega$ is covered by the intersection of $... |
I want to prove a function $f(t)$ is continuous at $ t_o$. By def of continuity, I need to show that for $\epsilon >0 $, there exist $ \delta>0$ s.t
$ |t-t_o| < \delta$ implies $ |f(t)-f(t_o)|< \epsilon $ .
If in order to show this, I need other condition to satisfy along with a bound on $ |t-t_o|$ to control $|f(t... |
Using the invariant formula for the exterior derivative, one gets that for a $3$-form $\omega$ its derivative is evaluated on vector fields according to
$$3 \ {\rm d} \omega (X, Y, Z) = X \ \omega (Y, Z) + Y \ \omega (Z, X) + Z \ \omega (X, Y) - \omega ( [X, Y], Z) - \omega ( [Y, Z], X) - \omega ( [Z, X], Y) .$$
This i... |
Let C be the ternary repetition code of length 4 over the alphabet $\{0,1,2\}$.
If the probability of each symbol being wrongly received is t and each symbol is likely the probability of a word being incorrectly decoded is $t^2 (3-2t)$
I do not understand how to get this answer and this is the working in the solutions ... |
Consider a mapping $\xi:\mathbb{R}^d\rightarrow\mathbb{R}^k$ such that $D\xi \, D\xi^T>\delta\, I_k$. Here $D\xi:\mathbb{R}^d\rightarrow \mathbb{R}^{k\times k}$ is the Jacobian.
Consider a function $f:\mathbb{R}^{d}\rightarrow \mathbb{R}$ and $A:\mathbb{R}^k\rightarrow \mathbb{R}$ defined by
\begin{equation}
A(z)=\int... |
Can a Lebesgue measurable function be modified on a set of first category so as become continuous except on a set of Lebesgue measure zero?
OR
Can a Baire-measurable function be modified on a set of Lebesgue measure zero so as to become continuous except on a set of first category?
|
In Hartshorne Proposition II.5.8, he shows, given a morphism $f \colon X \to Y$
where X and Y are schemes and $\mathcal{G}$ a quasi-coherent sheaf of $\mathcal{O}_{Y}$- modules, that $f^{*}\mathcal{G}$ is a quasi-coherent sheaf of $\mathcal{O}_{X}$-modules. Similarly, he shows that given $f$ quasicompact and separated,... |
I'm pretty sure this is equal to B, but I'm not sure how to go about reducing this step by step. Could I use the double negative law to eliminate the complements? I'm not positive if that would work in the current situation.
|
I am trying to solve the recurrence:
$$
T(n) = 5T\left(\frac{n}{4}\right) + n\lg \lg n.
$$
I tried to apply the Master Theorem but it didn't get me anywhere:
$$
a=5,\; b=4\; \text{ and } f(n) = n\lg \lg n.
$$
We have $\log_b a \simeq 1.16$, but I am not understanding where this is taking me.
|
For example, the piecewise function
$
f(x) =
\begin{cases}
\sqrt{1 - (x + 1)^2} &-2 \leq x \leq 0 \\
-\sqrt{1 - (x - 1)^2} &0 \leq x \leq 2
\end{cases}
$
will, at $f(0)$, give $f'(0) = $ undefined (vertical tangent). Once deriving I can prove this algebraically, there is a zero in the denominator. It looked like a... |
If $x_1 , x_2,......x_n$ are non-zero elements of a field so is $\prod_{k=1}^n x_k$; and
$\left(\prod_{k=1}^n x_k\right)^{-1} = \prod_{k=1}^n x_k^{-1}$.
Assume $n = 2$ true;
How I did it:
First:
$$P(1) \implies \prod_{k=1}^1 x_k = x_1 non zero$$
from condition so $P(1)$ holds true.
For $n = 2$ also holds true from a... |
Let $X$ and $Y$ be Riemannian manifolds and consider a function
\begin{align}
f\colon X\times Y &\to \mathbb{R},\\
(x,y) &\mapsto f(x,y)
\end{align}
Now I have to integrate $f$ with respect to the Riemannian volume element of $X\times Y$. Is the infinitesimal volume element of $X\times Y$ the product of the ... |
I need some help with a program in MATHEMATICA or MAPPLE, that solves the following transcendental equation:
$$\alpha+2x+2\sqrt{\beta+\alpha x+x^{2}}=e^{-t\frac{2}{\alpha}}(\alpha+2\sqrt{\beta})e^{{\frac{2}{\alpha}} (\sqrt{\beta+a(\alpha+a)}-\sqrt{\beta})}$$.
Can someone, please, help me with this issue?
P.S: How can I... |
What's your option for calculating this integral? No full solution is necessary, it's optional as usual.
Calculate
$$\int_0^1 \frac{2 \zeta (3)\log ^3(1-x) \text{Li}_2(1-x) }{x}-\frac{2 \zeta (3) \log ^2(1-x) \text{Li}_3(1-x)}{x}+\frac{ \log (x) \log ^5(1-x)\text{Li}_2(1-x)}{x}+\frac{\log ^4(1-x)(\text{Li}_2(1-x){})^2... |
I am doing some old exam questions - and I don't know the answer, can some one calculate the result and show how you did it?
|
Let $c_i$ be a given non-negative integer for all $i\in\{1,\ldots,n\}$. I would like to find the non-negative integers $a_i$ and $b_i$ for all $i\in\{1,\ldots,n\}$ such that:
\begin{align}
\begin{cases}
c_i = a_i+b_i, & \text{ for all } i\in\{1,\ldots,n\}\\
a_i < a_{i+1}, & \text{ for all } i\in\{1,\ldots,n-1\}\\
b_i >... |
$$\int dx \frac{2x+1}{(x^2-9)^\frac{5}{2}}$$
$x=\frac{3}{\sin\theta}$
$dx=\frac{3\sin\theta}{\cos^2\theta}d\theta$
$$\int d\theta \frac{\left(\frac{6}{\sin\theta}+1\right)\cdot\frac{3\sin\theta}{\cos^2\theta}}{\left(\frac{9}{\sin^2\theta}-9\right)^\frac{5}{2}}=\int d\theta \frac{\left(\frac{6}{\sin\theta}+1\right)\c... |
I know that
$\:\:\frac{1}{k\left(k+1\right)}\:\:\:\:=\:\frac{1}{k}\:-\:\frac{1}{k+1}\:$
And that
$\sum_{k=1}^{n-1}\:k$ $= \frac{n(n-1)}{2}$
But I'm not completely sure how to turn $\sum_{k=1}^{n-1}\:\frac{1}{k\left(k+1\right)}$ into a simple expression using this information.
|
I'm asked to verify if the following set is polyhedral,
$$ X = \{[x_1;x_2]: min(x_1,x_2) \leq 0\}$$
Definition of a polyhedral set,
A set $Y$ is polyhedral if $Y = \{y: Ay \leq b\}$, for finite $A$ and $b$.
I think that it isn't polyhedral but I'm not sure how to go about proving it.
By looking at the feasible space ... |
Given $y_1(x)=\sin(x^2)$ and $y_2(x)=\cos(x^2)$, I constructed a linear, homogenic ODE of order 2 by solving:
$$
\begin{vmatrix}
y & y_1 & y_2 \\
y' & y_1' & y_2' \\
y'' & y_1'' & y_2'' \\
\end{vmatrix}=0
$$
Now, I noticed that the Wronskian of $y_1$ and $y_2$ at $x=0$ equals $0$... |
A https://en.wikipedia.org/wiki/Random_variate is a particular outcome of a random variable. I was wondering what is meant in the Wikipedia article with "a random variate is corresponding to a probability distribution". Does this mean a random variate is in principle a functional $\Phi[f(x)]$ with $f(x)$ being some pro... |
The rate at which customer arrive to the bookstore is $\lambda (t)=-(t-4)^2+16 $ where $t$
measured in hours. The customers can buy a book with probability $0.5$ and they can also buy a coffee with probability $0.5$ given they bought a book or $0.8$ given they bought no book. Find the probability of $k$ people buyin... |
Prove, that for any $a>0$, $$a>\ln(1-a)>\frac{a}{1+a}.$$
Prove, that for any $0<a<1$, $$-\frac{a}{1+a}<\ln(1-a)<-a.$$
Proof of 1: We will prove (1) by doing a proof by converse. Assume that $$a>\ln(1-a)>\frac{a}{1+a}.$$ We need to show that this is true for $a>0$. Well, when $a\neq 0$
\begin{equation*}
\begin{aligne... |
I am trying to solve the following problem, please help me to complete the proof:
I need to find the relation between the number of comas in a term $p_c$ of language L = {f,g} and the number $p_f$ of f functional symbols and the number $p_g$ of g functional symbols in term $t$ .
$f$ is an unary func symbol, $g$ is tern... |
I want to prove the following statements:
(i) $X \setminus Y ^ \circ $ = $cl ( X \setminus Y) $
I wrote down that $Y ^\circ$ is open so $Y ^\circ = Y $. Therefore $X \setminus Y ^ \circ $ is closed so is containing $cl ( X \setminus Y) $. Is that the correct way? How can I show the other way that $X \setminus Y ^ \c... |
I was given
$$x^4 + 1$$
and was told to find its real factors. I found the $((x^2 + i)((x^2 - i))$ complex factors but am lost as to how the problem should be approached.
My teacher first found 4 complex roots ( different than mine)
$$( x - \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} i)( x - \frac{\sqrt{2}}{2} + \frac{\sq... |
Background:
I am trying to optimize what we call AJAX request polling frequency in the domain of web design, and I wanted to check if I could use some help from math guys to explore a better alternative.
Here's my illustration of the problem statement:
Problem Statement:
Suppose there are 3 persons - consumer (C), pro... |
So with this question, I came to this math:
I have a J, Q, K, and A in four suits, and after having one of those face cards in a hand, now we are left to choose 12 more cards. so then I figure we get 4$4 \choose 1$ $48 \choose 12$.
However, I believe I am overcounting, I just don't know how. Can someone help me?
|
Check if matrices $A= \begin{bmatrix}
1 & 1 & 5 \\
0 & 2 & 0 \\
0 & 0 & 2 \\
\end{bmatrix}$ and $B=\begin{bmatrix}
1 & 7 & 0 \\
0 & 2 & 7 \\
0 & 0 & 2 \\
\end{bmatrix}$ are similar.
Is it sufficient to show that $A$ and $B$ have the same characteris... |
Given that $N$ is Poisson distributed with a random intensity $Y$, the conditional distribution of $(X|Y)$ is defined as, for $n=0,1,\dots$
$$P[N=n|Y=\lambda]=e^{-\lambda}\lambda^n\frac{1}{n!}$$
$Y$ is a random variable exponentially distributed with parameter $\theta$.
Find:
$P[N=n]$
$E[N]$
$E[N^2]$
I'm struggling ... |
Let $SO_2(\mathbb{R})$ be the group of rotations of the circle under the operation of composition. And consider $\mathbb{R}$ as an additive group.
Prove that $SO_2(\mathbb{R}) \cong \begin{Bmatrix} A\in GL_2(\mathbb{R}) : A^tA = I, detA = 1 \end{Bmatrix}$
Hint: $SO_2(\mathbb{R}) \cong \mathbb{R}/2\pi\mathbb{Z}$
Attempt... |
Suppose we have two diagonal matrices
$$
A_{\mu \nu}=\left(\begin{array}{cccc}
\rho(t) & 0& 0& 0\\
0& 0& 0& 0\\
0& 0& 0& 0\\
0& 0& 0& 0
\end{array}\right)
$$
and
$$
B_{\mu\nu}=\left(\begin{array}{cccc}
1 & 0& 0& 0\\
0& a(t)& 0& 0\\
0& 0& b(t)& 0\\
0& 0& 0& 0
\end{array}\right)
$$
Now if we vary $B_{\mu\nu}$ by a smal... |
I know that the closed form of $\sum\limits^n_{k=1}\:k=\frac{n(n+1)}{2}$
But I'm not sure what the closed form for $\sum\limits^5_{i=2}\:\left(3i\:-\:5\right)$ would be.
Any push in the right direction would be helpful.
|
Let $V=\{a_2x^2+a_1x+a_0|a_1, a_2, a_3\in \mathbb{R}, a_2\ne 0\}$ with operation defined by $$(a_2x^2+a_1x+a_0)+(b_2x^2+b_1x+b_0)=(a_2+b_2)x^2+(a_1+b_1)x+(a_0+b_0)$$
$$c(a_2x^2+a_1x+a_0)=ca_2x^2+ca_1x+ca_0$$
Here I am wondering if the identity is $(0x^2+0x+0)$ and the additive inverse is $-(a_2x^2+a_1x+a_0)$.
For other... |
I am reading a text on non-standard analysis. I need to prove the following: Suppose that $f$ is non-decreasing on the real interval $[a,b]$ and that $f$ satisfies the intermediate value property. Then $f$ is continuous on $[a,b]$.
I need to show this by showing if $y-x$ is infinitesimal, then so is $f(y)-f(x)$.
M... |
Imagine the following situation:
A new virus is discovered that is believed to have infected 20% of the population.
Anyone infected with the virus has a chance of 50% of dying in their sleep every day.
The virus has absolutely no symptoms to let anyone know whether they've been infected or not, before its too late.
Bo... |
I think it's easiest if I start with my actual use-case:
In a video game (XCOM), soldiers shoot at aliens. When they do, they have a % chance to hit. Hitting deals damage.
I want to look at each shot fired in a match and produce a "percentile" measurement of how lucky the player was in the match.
I have done a bunch of... |
If have $T$="The virus test is positive." and $V$="There really is a virus."
and I know that $Pr(V)+Pr(\bar{V})=1$, can I then say that
$Pr(T)=Pr(T|V)+Pr(T|\bar{V})$
and how do I show that formally, based on calculation rules?
|
I've noticed that when going through the steps of Euclidean GCD algorithms, very often the sum of the remainders in the steps $s_{i+1}$ and $s_{i+2}$ will be equal to the remainder in step $s_i$. However, this isn't always the case.
Example
To illustrate what I'm talking about consider this:
$$
gcd(612,93):\\
\begin{al... |
Quaternion has components X, Y, Z, and W.
If you created a Quaternion with input being a 3D Vector representing the axis (X,Y,Z) and a floating point number representing the amount to rotate around that axis(W). Can you extract the X,Y, and Z components used to construct the Quaternion from the Quaternion itself.
I'm ... |
I'm trying to derive the dual problem of a very simple example of a Lagrange multiplier (note: please correct my terminology if it's off).
This isn't homework, I've just picked the example off some slides somewhere and am trying to solve it to understand the process. I haven't found many simple examples that work throu... |
Here I am thinking of using $-(x+y)$ and show that it equals $-(y+x)$.
$-(x+y)=-x-y$ by distributivity
=$-x-y+0=...$ Here I don't know how to continue, could someone suggest?
|
Let $p$ be a prime number. I've to figure out for which $p$ $$G_p =
\{\sigma \in S_{12}: \text{ord}(\sigma) \text{ divides } p \}$$ is a
subgroup of $S_{12}$.
I realize that we have to consider $p \le 11$ and I can find an example to show that it is not the case for $p = 2$: in fact $(1,2)\circ(2,3) \not\in G_... |
I have trouble figuring out the asymptotic behavior of two functions.
If $f(n) = n \log n$, then what do we know about $f^{-1}(n)$? I.e. what is the asymptotic behavior of $g(n)$ such that $g(n) \log(g(n)) = \Theta(n)$, perhaps in terms of some familiar functions?
Behavior of $n^{\frac{1}{\log \log n}}$. I know that ... |
I had to show that if $p: L \to L$ is a projector, then $im ~p \oplus \ker p = L$.
This was easy. Now I have to show that the matrix of $p$ is divided on four blocks where one of them is $r$ dimensional (The dimension of image of p) and the others are just composed by zeros.
(Sorry, I don't know ho to insert matrix her... |
Which mathematical questions can be answered with an n-body computer? An n-body computer is an arrangement of n-bodies moving under mutual gravitation that is prepared in an initial state to answer a specific question. If the solution remains bounded, the answer is "no". If at least one ball shoots out to infinity, the... |
I have an ordinary differential equation:
$$ \frac{\mathrm{d}^2u}{\mathrm{d}t^2} + u = \mathrm{e}^{-t}\cos(t)$$
with $u(0) = u_0$ and $\dot{u}(0) = v_0$, when using the method of undetermined coefficients, I obtain the solution
$$u(t) = u_0\cos(t) + v_0\sin(t) + \frac{\cos(t) -2\sin(t)}{5e^t}$$
however using the laplac... |
Let $$\sigma(n)=\sum_{d\mid n}d$$ the sum of divisor function. I've deduced, but I don't know how prove directly
$$\sigma(n)=\sum_{m=1}^{n}\sum_{k=1}^{m}(-1)^{nk}\cos\left(\frac{\pi n(2-m)k}{m}\right).$$
Question. Can you prove it? Thanks in advance.
|
Let $\mathbb{P}$ denote the set of prime numbers. How would one evaluate
$$\prod_{p\in \mathbb{P}}\frac{p-1}{p}$$ I do not think that the fact that $$\prod_{n=2}^{\infty}\frac{n-1}{n}=\lim_{n\to\infty}\frac{1}{n}=0$$ can be applied, but it is worth noting. Additionally, if we take the log, we obtain
$$\log\left(\prod_{... |
how do you prove that when the limit of n approaches towards positive infinity while n^2/(log n)! We tried to used Stirling theorem but this may not work due to the fact that it may or may not exist on all intervals.
|
I am implementing the quadratic sieve algorithm. And I got run in unexpected problem.
Take a look at those two final steps of the algorithm as described in wiki.
Use linear algebra to find a subset of these vectors which add to the zero vector. Multiply the corresponding $a_i$ together naming the
result mod $n$: a ... |
Let $ \space n,k \in \mathbb Z \space $ such that $1 \le k \le n \space$. Let $\space A=\{1,2,...,n\}$. Find the number of ways to choose $k$ subsets $\space B_1,B_2,...,B_k\space $ of $A$ such that $ B_1 \cap B_2 \cap \cdot \cdot \cdot \cap B_k = \emptyset$.
Well, I did it using inclusion exclusion, but I'm wondering ... |
I'll try to format my question in a manner such that you can skip (irrelevant) parts.
Exercise:
Find all natural $n$ such that $3^{2n+1}-4^{n+1}+6^n$ is prime.
Motivation:
I'm trying to prepare talented elementary school children for mathematics competition and the question from the title popped up somewhere as sugge... |
Prove the reverse triangle inequality :$|z\pm w|\ge||z|-|w||$ for all $z, w \in \mathbb C$, with equality holds if and only if either $z$ or $w$ is a real multiple of another.
I have proved the inequality but not able to solve for equality.
It will be great if someone can help me in this.
What I have tried is this:
let... |
Prove that $(E \land G) \lor ( G \rightarrow F )$ is formally deducible from $(E \lor F)$
where:
"$\land$" means AND
"$\lor$" means OR
"$\rightarrow$" denotes an implication
This is what I have so far:
$E \lor F \vdash E \lor F$
$E \lor F , E, F, G \vdash F$
$E \lor F , E, F \vdash (G\rightarrow F)$
But I have no i... |
Let $G$ be a finite group such that $x^2 = e$ for each $x \in G$. I know already that $G$ is abelian and that the order of $G$ is $|G| = 2^n$ for some $n \geq 0$. Now I wish to show that
$$G \cong \mathbb{Z}_2 \times ... \times \mathbb{Z}_2$$
where there are $n$ factors (we define the direct product of zero groups as t... |
Define the inner product $\langle\cdot,\cdot\rangle$ for continuous functions defined on $[0,1]$ as:
$$\langle\,f\mid g\rangle=\int_{0}^{1}f(x)g(x)e^{\rho x}dx,$$
where $\rho$ is a real number.
I wish to show that:
\begin{multline*}
\langle 1\mid (1-t)^{a-b-1}\rangle \cdot \langle t^{b}\mid (1-t)^{a-2}\rangle \cdot \la... |
I saw on facebook some image on which these identities that I am going to write below are labeled as "amazing math fact" and on the image there are these identities:
$1^3+5^3+3^3=153$
$16^3+50^3+33^3=165033$
$166^3+500^3+333^3=166500333$
$1666^3+5000^3+3333^3=166650003333$
and then it is written under these identitie... |
Can anyone show an example of a function $f$ of a real variabile such that
$f$ is differentiable on a neighborhood of a point $x_0 \in \mathbb{R}$, except at $x_0$ itself;
$f$ is continuous at $x_0$;
$\displaystyle \lim_{x \to x_0} f'(x)$ does not exist;
$\displaystyle \lim_{x \to x_0} \frac{f(x) - f(x_0)}{x - x_0} = ... |
In my opinion, let $v\in$ Span($S_1\cap S_2$) and therefore $v\in$ Span$(S_1)$ and $v\in$ Span$(S_2)$. Write $v=c_1z_1+...+ c_nz_n$ where $z_k\in S_1\cap S_2$ and $c_k\in R$. Here I am feeling I have the wrong direction. Could someone suggest how to approach the question?
And how do you show that $W_1\cap W_2\ne$ Span$... |
I have a square matrix $C$, whose entries I will denote by $c_{ij}$, and I would like to bound the magnitude of its eigenvalues.
Each $c_{ij}$ is defined in terms of $s_{ij}$ and $S_j$ as follows:
$$ c_{ij} = s_{ij}/S_i $$
The $s_{ij}$ and $S_i$ are real and satisfy the following properties:
(1) For any $j$, $\sum\limi... |
Conjecture :
Let $p$ be a prime number , $f\in \mathbb Z[X]$ an irreducible polynomial
with degree $p$ and coefficients in the range $[-1,1]$. Then the galois
group of $f$ over $\mathbb Q$ is $S_p$
Can anyone prove or disprove this conjecture ?
The conjecture is true for the primes upto $p=11$. To prove the c... |
I am finishing up my bachelor's degree in mathematics at the University of North Florida, and I plan on going to graduate school, but I feel very behind. One of my professor's gave us this problem:
$\frac 1a + \frac 1b + \frac 1c \ge \frac 1 { \sqrt{bc}} + \frac 1 { \sqrt{ac}} + \frac 1 { \sqrt{ab}}$
I had no idea how... |
Give a sequence of functions $\varphi_n\in \mathcal{C}^\infty(-1,1)$, Cauchy with respect to the Sobolev space $H^1_0$ norm
$$\| \varphi\|_1=\sqrt{\int_{-1}^1 (\varphi')^2+\int_{-1}^1 \varphi^2}$$
which does not converge to a function in $\mathcal{C}^1\left[-1,1\right]$.
I was told an useful approach would be taking a ... |
Given the ODE $(1-x^2)y''-xy'+\alpha^2 y=0$, I need to show that if $y(x)$ is a solution, then $y(-x)$ is also a solution. From what I understand, because $y(0)=y(-0)$, it means that all solutions are even functions.
I solved the ODE using series around $x=0$ which is a regular point.
I got:
$$y(x)=\sum_{k=0}^\infty a_... |
Hi, for these problems I generally get the gist of it. If you have some linearly dependent vectors $v_1, \ldots, v_m$ in $\mathbb{R}^n$ then when you transform those vectors $T(v_1), \dots, T(v_m)$ they will still be dependent, no matter what space they are in.
And for the second problem, if $n < p$ and $m < p$ then ... |
Suppose $\{f_n(x)\}$ is a sequence in $C[a,b]$ such that for every $x \in [a,b]$ there exists a $M_x$ such that $|f_n(x)|<M_x, n\in \{1,2,3,\cdots\}$
Prove that there exist a $c,d \in [a,b]$ with $c<d$ and $M \geq 0$ such that $|f_n(x)| \leq M$ for all $x \in [c,d]$.
I think i need to use Baire's category theorem but i... |
Let $f\in \mathbb Z[X]$ be an irreducible polynomial. Suppose, the discriminant of $f$ is a perfect square.
Can the galois group of $f$ over $\mathbb Q$ be $S_d$, where $d$ denotes the degree of $f$ ?
Additional question : What can we conclude if the discrimiant is the negative of a perfect square, lets say $-81$ ?
H... |
Just to be upfront, this is a homework question, but I'm just stuck on one particular part and I want to see what I'm doing wrong.
The PDE in question is the following:
$$(1+t)u_t + xu_x = e^{t}\\
u(x,0) = x$$
The way my professor has discussed solving this equation in class has been by setting a function:
$$w(t) = u(x... |
An incoming freshman Mark believes that he has a 25% chance of earning a GPA in the 3.5 to 4 range, a 35% chance of graduating with a 3.0 to 3.5 GPA and 40% chance of finishing with a GPA less than 3.0. Also, Mark has an 8/10 chance of getting into med school if his GPA is above a 3.5, a 5/10 chance if his GPA is betwe... |
The following is an exercise from D. Robinson: A Course in the Theory of Groups.
Let $G$ be a $k$-transitive permutation group of degree $n$ which is neither alternating nor symmetric. Assume $k > 5$. Prove that $(n-k)! \ge 2n$. Deduce that $k \le n-4$.
If $G$ is $k$-transitive, then $n(n-1)\cdots (n-k+1)$ divides $|... |
$$P((A \cap B^c) \cup (A^c \cap B))=P(A) + P(B) -2P(A \cap B).$$
I need to show this holds.
I see it with Venn diagrams but I need to show it using only the axiom, for the union of two disjoint sets:
$$P(A \cup B) = P(A) + P(B).$$
I know the first step is:
$$P((A \cap B^c) \cup (A^c \cap B)) = P((A \cap B^c))+ P((A^c \... |
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