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H: True or False problem on sum principle
If A $\cap$ B $\cap$ C = $\emptyset$, then the sum principle applies so |A $\cup$ B $\cup$ C| = |A|+|B|+|C|.
I think it would be true since there is nothing in common among A,B and C, but just wondering if there is any exceptions to this problem so it would be false?
AI: Let... |
H: How does knowing a ratio help me determine a total?
I am working with this question:
A received 1/3 more votes than B. Which of the following could have been the total number of votes cast for the two candidates? Answer options are 12, 13, 14, 15, 16.
I can see that however many votes B got, A got the same number p... |
H: Determine the value for which a sequence is an arithmetic progression.
We have the following sequence $$ -a, -\dfrac{a}{b}, \dfrac{a}{b}, a$$
Determine the value of $b$ for which this is an arithmetic progression ($a \neq 0$)
I don't know how to do this. I've tried adding a variable and making a couple of equations... |
H: Homework - NFA and its regular expression.
So for (a) I got this:
I THINK that is right, I'm not too sure. But I am also having a lot of trouble doing (b). Any pointers to get me started would be appreciated.
AI: There are four parts, $u,c,v,a$. $u \in \{a,b\}^*$, so write $(a|b)^*$ for that, $c$ is easy, $v$ has... |
H: Connection on complex vector bundle
Let $M$ be a $m$-dimensional
Riemannian manifold. I will follow the notation of the book "From calculus to
cohomology - Madsen and Tornehave"
If $\xi $ is a $k$-dimensional complex vector bundle, what is the connection ?
If $\{e_i\}$ is a basis of $\Omega^0(\xi)$ then any sec... |
H: If $f = u + iv$ is a complex function, is $|f| = (u^2 + v^2)^{1/2}$?
If $f = u + iv$ is a complex function, is $|f| = (u^2 + v^2)^{1/2}$? Where, $|f|$ is the modulus or absolute of $f$.
I thought this should be correct because for any complex number $z = a + ib$, $|z| = (a^2 + b^2)^{1/2}$. At any point $w = x + iy$... |
H: How to differentiate CDF of Gamma Distribution to get back PDF?
CDF of a gamma distribution ($X \sim \mathcal{G}(n, \lambda)$) looks like
$$F(x) = \frac{\Gamma_x(n)}{\Gamma(n)}$$
Where $\Gamma_x(n) = \int_0^x t^{n-1} e^{-t} \, dt$ the incomplete gamma function. Ok so far?
But how do I differentiate such an express... |
H: Finding derivative of three variables
Consider a box with dimensions x, y, and z. x is changing at a rate of 1 m/s, y at -2 m/s and z at 1 m/s. Find the rate that the volume, surface area and diagonal length ($s = \sqrt{x^2+y^2+z^2}$) are changing at the instant when $x = 4$, $y = 3$ and $z = 2$.
I know I need to u... |
H: If $A$ has a positive Lebesgue measure then there exist subsets which are not measurable
I was thinking if there is a solution to this problem without trying to explicitly create Vitali sets in $A$. Does anyone have any ideas?
AI: I don't think the Vitali construction is useful here. Instead, I'd use the other (Ber... |
H: If $f \ge 0$ is zero a.e., then $\int_{\Bbb R}f \,\mathrm d\lambda= 0$
Suppose $f \geq 0$ is measurable, then $f = 0 $ almost everywhere implies $\int\limits_{\mathbb{R}} f \,\mathrm d\lambda= 0 $.
My try
Pick Simple $\phi \leq f $. Since $f = 0 $ ae, then how can we show $\phi = 0 ?$. IF we can show this, then $... |
H: Big Oh and Big Omega clarification
Can I get an explanation of:
Can g(n) be Big O of $n^{2}$ and also the Big O of $n^{3}$? (at the same time)
Can g(n) be Big Omega of $\Omega (n)$ and also be the Big O of $n$?
AI: For your first question: if $g(n)=O(n^2)$, then $g(n)=O(n^{2+k})$ for all $k>0$. Indeed, $g(n)=O(n^2... |
H: Let n∈ℕ. Suppose that p is an odd prime number that divides n^2+1 Show that if p=4k+3 for some integer k≥0, then n^(p−1)≡−1(modp).
Let $n \in \mathbb N$. Suppose that $p$ is an odd prime number that divides $n^2+1$. Show that if $p=4k+3$ for some integer $k \ge 0$, then $n^{p−1}\equiv−1 \pmod p$.
The question says... |
H: Prove that $f$ is bounded if it converges as $x \rightarrow \infty$ and $x \rightarrow -\infty$
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that $f(x) \rightarrow 7 $ as $x \rightarrow \infty $ and $f(x) \rightarrow -25 $ as $x \rightarrow -\infty $ and
Prove that $f$ is a bounded fun... |
H: Inner Product of Real Polynomials
Updated improved question:
Let $V$ be the space of real polynomials in one variable $t$ of degree less than or equal to three. Define
$$
\langle p,q\rangle = p(1)q(1)+p'(1)q'(1)+p''(1)q''(1)+p'''(1)q'''(1).
$$
(i) Prove that $\langle\cdot,\cdot\rangle$ defines an inner product.
C... |
H: Can $\Bbb N$ be topologized to be a compact Hausdorff space?
Can $\Bbb N$ be topologized to be a compact Hausdorff space?
I guess it might be a topology that is strictly finer than cofinite topology and strictly coarser than the topology consists of all infinite subsets of $\Bbb N$. But is there such a topology?
AI... |
H: Rotating and Scaling about centroid.
rotating
$x' = x\cos(\text{angle}) - y\sin(\text{angle})$
$y' = x\sin(\text{angle}) + y\cos(\text{angle})$
Scaling
$x' = x\cdot sx$
$y' = y\cdot sy$
but all formulas will doing about origin point. If i want to do about Centroid point.
(I have $(Cx,Cy)$ ). What the formulas wil... |
H: Calculating residue in exponential fraction
I want to calculate the residue of $$\dfrac{e^{it}}{e^t+e^{-t}}$$ at $t=\pi i/2$.
To calculate the residue at $\pi i/2$, I write
$$\frac{e^{it}}{e^t+e^{-t}}=\frac{e^{it}e^t}{e^{2t}+1}=\frac{e^{it}e^t}{(e^t+i)(e^t-i)}$$so the residue is $$\frac{e^{i\pi i/2}e^{\pi i/2}}{(e... |
H: Are there non commutative rings with no zero divisors?
If there are,
Are there unity (but not division) rings of this kind?
Are there non-unity rings of this kind?
Sorry, I forgot writting the non division condition.
AI: Take the "polynomials" with integer coefficients in two non-commuting variables $x$ and $y$. If... |
H: What is wrong with my reasoning about this conditional probability problem?
There is a box containing 6 red balls and 5 white balls. If the first drawn ball is red and not returned to the box, find the probability that a white ball is taken in the second drawn.
I have 3 solutions.The first two solutions produce the... |
H: Adjoining two primitive n-th roots
Let $\omega_n$ denote a primitive $n^{th}$ root of unity. If $m$ and $n$ are positive integers with $lcm(m,n)=k$, show that $\mathbb{Q}(\omega_n,\omega_m)=\mathbb{Q}(\omega_k)$.
To start, I am aware that $(\omega_n\omega_m)^k=1$, and so $o(\omega_n\omega_m)|k$, I am working towar... |
H: Is there an axiomatic definition of the concept "field equipped with a conjugation operator"?
In some sense, $\mathbb{C}$ is more than just a field, since aside from the usual field operations, it is also equipped with a conjugation operator $\mathbb{C} \rightarrow \mathbb{C}$.
This means a couple of things.
We ca... |
H: Sum of Powers of Coprime Ideals
Let $R$ be a commutative unital ring and let $I$, $J$ be ideals in $R$ such that $I+J=R$.
Show that $II+JJ=R$.
What I've tried:
Clearly $II+JJ \subseteq I+J =R$ as $I$ and $J$ are closed under addition and multiplication. So it suffices to show that $R \subseteq II+JJ$. Since $1\in ... |
H: Help with determining trigonometric limit
Use the relation $\lim_{\theta \to 0}\frac{\textrm{sin}\theta}{\theta}=1$ to determine the limit of $f(x)=\frac{\textrm{tan}(2x)}{x}$
I understand the identity $\textrm{tan}(2x)=\frac{\textrm{sin}(2x)}{\textrm{cos}(2x)}$.
So, $$\frac{\textrm{tan}(2x)}{x} = \frac{\frac{... |
H: $\left( \frac{1 \cdot 2}{p} \right) + \left( \frac{2 \cdot 3}{p} \right) + \cdots + \left( \frac{(p-2)(p-1)}{p} \right) = -1$
Let $p$ be an odd prime number. Prove that $$\left( \frac{1 \cdot 2}{p} \right) + \left( \frac{2 \cdot 3}{p} \right) + \left( \frac{3 \cdot 4}{p} \right) + \cdots + \left( \frac{(p-2)(p-1)}{... |
H: Show that $\lim_{x\to a^{+}} g(x) =g(a)$
Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a bounded function and suppose that $g(x)=\sup_{t>x}f(t)$. Show that $\lim_{x\to a^{+}}g(x) = g(a)$ for all real $a$.
I have a hard time with these kind of proofs because in high school, I would simply replace $x$ by $a$ in $g(x)... |
H: Is this equation to prove that $aRb \iff a^2 - b^2 = 1$ is antisymmetric correct?
Over $\mathbb{R}$, $aRb \iff a^2 - b^2 = 1$.
I tried determining if it was antisymmetric. I seem to have done it, but while doing the equation, I stumbled upon a scenario that always made me doubt my decisions:
Have
$$aRb \land bRa$... |
H: Residue of $f(z)=\frac{\cot(z)\coth(z)}{z^3}$
How can I find the residue of the following function at the point $z=0$
$$f(z)=\frac{\cot(z)\coth(z)}{z^3}$$
AI: Keep in mind that each of $\cot{z}$ and $\text{coth}{z}$ has a simple pole at $z=0$. Thus $z=0$ is a pole of order $5$, and the residue is equal to
$$\frac... |
H: limit of sequence of functions
Suppose
$$ f_n = \frac{1}{(1 + \frac{x}{n})^n x^{\frac{1}{n}}} $$
What is $\lim_{n \to \infty } f_n $ ??
I am having hard time with this sequence which seems like it is going to be something like the exponential, but I cannot see how to simplify it and make it look like the exponenti... |
H: Given $f_{X,Y}(x,y)$, what is the pdf of $Z=XY$?
I approached the problem as following:
$f_{Z}(z) = \int_{0}^{\infty} f_{X,Y}(X=\frac{z}{y},Y=y)dy$
However, according to the textbook, the problem should be solved as below:
$f_{Z}(z) = \int_{0}^{\infty} f_{X}(X=\frac{z}{y}) f_{Y}(Y=y) dy$
Why do we use the marginal... |
H: Extending an embedding $:S^1\rightarrow \mathbb R^{n}$
Assume we have an embedding $f:S^1\rightarrow \mathbb R^n$. I want to extend $f$ to an embedding $\tilde{f}:B\rightarrow \mathbb R^n$, where $B$ is the closed unit ball of $\mathbb R^2$. In fact, I want to see $f(S^1)$ as the boundary of a smooth manifold. I wo... |
H: How to find all surjective functions $f:M_n(\Bbb R)\to\{0,1,2,\cdots,n\}$ satisfying $f(XY)\le\min{(f(X),f(Y))}$
Let $M_n(\Bbb R)$ be the set of all real $n\times n$ matrices. Find all surjective functions $f:M_n(\Bbb R)\to\{0,1,2,\cdots,n\}$ such that
$$f(XY)\le\min{(f(X),f(Y))}$$
for all $X,Y\in M_n(\Bbb R)$.
My ... |
H: Help with numerical analysis proof
Let $u$ be a nonzero vector in $\mathbb{R}^n$, and define $\gamma=\frac{2}{||u||_2^2}$ and $Q=I-\gamma uu^T$.
Prove Q is a reflector satisfying
A) $ Qu=-u$
B) $Qv=v$ if $<u,v>=0$
My approach: I'm letting some $$\hat u=\frac{u}{||u||_2}$$ so that $||\hat u||=1$.
I don't know wh... |
H: Prove $h(x)=\sqrt{x^2-1}$ continuous by $\epsilon,\delta$
Proof:
Let $h\colon (1, \infty)\to \Bbb R$ be a function. Let $h(x)= \sqrt{x^2-1}$.
Let $\epsilon>0$ be arbitrary.
Let $x_0\geq 1$ be arbitrary.
Suppose $x_0 > 1$.
Let $$\delta=\min\left\{1, \frac{\epsilon\sqrt{x_0^2-1}}{2|x_0|+1}\right\}$$
Let $x\geq1$ ... |
H: Finding a vector that is in two subspaces
The following question has been posed to me:
You are given two subspaces $U$, $V$, both in $\mathbb{R}^n$, each with a basis of column vectors forming the columns of the respective matrices $U$, $V$. Find a vector $\vec{x}$ in both subspaces $U$, $V$. That is, find a vector... |
H: Prove $\left\{\frac{n}{2n+3}\right\}$ and $\left\{\frac{n}{2n-3}\right\}$ converge?
Question: prove that the sequences $\left\{\frac{n}{2n+3}\right\}$ and $\left\{\frac{n}{2n-3}\right\}$ converge using the definition.
What I have: I know both of them have limit $1/2$.
The definition says: a sequence $\{x_n\}$ is ... |
H: Find $x$ for inequality of $1+x+x^{2}+x^{3}+...+x^{99}\le0$
Finding the range of $x$ for inequality of $1+x+x^{2}+x^{3}+...+x^{99}\le0$
AI: We observe that $x=1$ is not a solution. Now note that $1+x+...+x^{99}=\frac{1-x^{100}}{1-x}$ (show this!), so that the inequality becomes $\frac{1-x^{100}}{1-x}\ge 0$. From he... |
H: Calculating integral with standard normal distribution.
I have a problem to solving this,
Because I think that for solving this problem, I need to calculate cdf of standard normal distribution and plug Y value and calculate.
However, at the bottom I found that Integral from zero to infinity of 1 goes to infinity ... |
H: Proof that two norms $\lVert\cdot\rVert_1$ and $\lVert\cdot\rVert_2$ are equivalent
Two norms $\def\norm#1{\lVert#1\rVert}\norm\cdot_1$ and $\norm\cdot_2$ are equivalent iff $\;\exists\;c_1,c_2>0$ such that $c_1\norm x_1\le \norm x_2\le c_2\norm x_1$
Show that $\norm x_1=\sum_{i=1}^n \lvert x_i \rvert$ and $\n... |
H: Is $A$ s.t $A_{i, j} = x^T_i x_j$ semi-positive definite?
Let $x_1, x_2, \ldots, x_k \in \mathbb{R}^n$ and set define a $k$ by $k$ matrix $A$ by setting $A_{i, j} = x^T_i x_j$. Is $A$ semi-positive definite? If so, how can I show it?
AI: Let $t:=(t_1,\dots,t_k)\in\mathbb R^k$. Then
$$t^TAt=\sum_{i,j=1}^kx_i^Tx_jt_... |
H: How many different ways can a number be expressed as a sum of any number of integers when order matters?
How many different ways can a number $n \in \mathbb{N} $ be expressed as a sum of any number of positive numbers when order matters?
My solution:
Since I know, that $n$ can be represented as a sum of $k$ positiv... |
H: Existence of $x \in E$ such that $\sup(E)-\epsilon
If $\alpha=\sup\left(E\right)$ exists, I often encounter the argument
that there exists a point $x\in E$ such that
$\sup\left(E\right)-\epsilon<x\leq\sup\left(E\right)$
I realize that if $\sup\left(E\right)\in E$, then I can just chose
$x=\sup\left(E\right)$ and t... |
H: Evaluating $\lim\limits_{n\to \infty}\left\{(1+\frac{1}{n})(1+\frac{2}{n})\dots(1+\frac{n}{n})\right\}^{\frac{1}{n}}$
Question is to evaluate :
$$\lim_{n\to \infty}\left\{ \left(1+\frac{1}{n}\right)\left(1+\frac{2}{n}\right)\dots\left(1+\frac{n}{n}\right)\right\}^{\frac{1}{n}}$$
I tried to do something like this bu... |
H: $\lambda_k \to 0$ implies $T$ is compact?
I am doing an exercise which asks to show that if $\{\varphi_k\}$ is an orthonormal basis in a Hilbert space with $T$ a bounded operator satisfying $T\varphi_k = \lambda_k \varphi_k$, then $\lambda_k \to 0$ implies that $T$ is compact. Now I will be done if I can show that ... |
H: Solving $\int_{-\infty}^{\infty}\frac{x^2e^x}{(1+e^x)^2}dx$
I am attempting to use residues to solve $\int_{-\infty}^{\infty}\frac{x^2e^x}{(1+e^x)^2}dx$; the answer is $\frac{\pi^2}{3}$. I have tried to split $\frac{x^2e^x}{(1+e^x)^2}$ into two parts
$$\frac{x^2}{e^x+1}-\frac{x^2}{\left(e^x+1\right)^2},$$
however n... |
H: Bijection map from a set of subgroup to another set of subgroup under some condition.
Let $G$ be a finite group. Let $N \trianglelefteq G$ and $U \leq G$ such that $G = NU$. Then there exists a bijection, preserving inclusion, from the set of subgroups $X$ satisfying $U ≤ X ≤ G$ to the set of $U-$invariant subgroup... |
H: prove if $(A_n)$ limit is $L$ then $(A_n)^2$ limit is $L^2$
Hello,
What I need to prove is: if $(A_n)$ limit is $L$ then $(A_n)^2$ limit is $L^2$.
I've added my attempt to prove it. I got stuck so I'm guessing I'm missing something here.
help will be appreciated :)
AI: $f(x)=x^2$ is continuous, which by definiti... |
H: Injective $\alpha:A \to B$ has surjective $\beta: B \to A$ such that $\alpha\beta = {\rm id}_A$
Let $\alpha : A \to B$ be an injective function. Show that there is a surjective function
$\beta: B\to A$ such that $\alpha; \beta = {\rm id}_A$.
AI: For every $y$ that belongs to the image of $\alpha$ there is exactly
... |
H: Calculate a total percentage based on individual percentages but without original values.
(I hope the question/title made sense.)
Let's say I have the following list:
+----------+--------------+------------+-----------+
| Item | Expected | Actual | % |
+----------+--------------+------------+---... |
H: About vector bundles on algebraic varieties
If $X$ is an irreducible algebraic variety (over $\mathbb C$), an algebraic vector bundle of rank $r$ over $X$ is a couple $(E,\pi)$ where $E$ is an algebraic variety and $\pi: E\longrightarrow X$ is a surjective morphism, with the following properties:
1) $\pi^{-1}(x)$ ... |
H: Find all positive integers m, n, p such that $(m+n)(mn+1)=2^p$
Find all positive integers m, n, p such that
$$(m+n)(mn+1)=2^p$$
Please give me some hints
Thanks
AI: We have $m+n=2^a$ and $mn+1=2^b$ with $a+b=p$. First note that both $m$ and $n$ need to be odd.
First case: Suppose $m=1$ or $n=1$, then the other is... |
H: Contraction mapping- Help needed
Let $F(x)$ be a continuously differentiable function defined on the interval $[a,b]$ such that $F(a)<0$ and $F(b)>0$ and
$$
0<K_1\leq F'(x)\leq K_2\quad (a\leq x\leq b)
$$
Find the unique root of equation $F(x)=0$.
The given hint is to use the contraction mapping theorem i.e., if $f... |
H: Prove that $\mathbb{Z}\left[\sqrt{-3}\right]$ is not a Dedekind domain.
Prove that $\mathbb{Z}\left[\sqrt{-3}\right]$
is not a Dedekind domain.
AI: Consider $(1+\sqrt{-3})(1-\sqrt{-3})=2\cdot2$ |
H: Does $A \cap (E_1\cup E_2) = [A\cap E_1]\cup [A\cap E_2\cap E_1^c]$?
$$A \cap (E_1\cup E_2) = [A\cap E_1]\cup [A\cap E_2\cap E_1^c]$$
Is this a correct relationship?
I tried not returned substantiated.
Thanks for all.
AI: Yes it is.
Use:
$E_1 \cup E_2 = E_1 \cup (E_2\cap E_1^c)$;
$A \cap (B \cup C) = (A \cap B)\cu... |
H: Show that: $\sum_{n=1}^{\infty} n(n-1)s^{n-2} = \frac{2}{(1-s)^3}$
How can I show that:
$\sum_{n=1}^{\infty} n(n-1)s^{n-2} = \frac{2}{(1-s)^3}$
I'm struggling to figure out how to start on this question. Should I sum the series and then differentiate it and also differentiate the sum term by term and then equate th... |
H: Avoid evaluation of a very large matrix in non-negative matrix factorization
This is somewhere in between a math and a programming question, so please
send me back to SO if you think it's off-topic.
I'm implementing non-negative sparse coding, a regularized variant of
non-negative matrix factorization. This entail... |
H: $\operatorname{Ran} \lambda I - T$ is closed for compact operator $T$ and $\lambda \neq 0$
Let $T$ be a compact operator on a Hilbert space $\mathcal{H}$ and $\lambda \in \Bbb{C} - \{0\}$. I want to show that $\operatorname{ran} \lambda I - T$ is closed. So suppose we have $g_j = (\lambda I - T)f_j \in \operatorna... |
H: Question about polynomials of odd degree with no zeros in formally real fields which are maximal to the property of being ordered
I have encountered this argument while reading Tent and Ziegler's "Course in model theory", and I don't know why it is justified.
It arises during the proof that every ordered field has ... |
H: Does there exist a commutative magma such that $\mid$ is transitive, which is not a semigroup?
Let $M$ denote a commutative magma, and write $x \mid y$ iff $xa=y$ for some $a \in M$. If $M$ is a semigroup, then $\mid$ is transitive. Does there exist a commutative magma such that $\mid$ is transitive, but which is n... |
H: Euclidean algorithm to find integers $s$ and $t$ such that $sa+tb=1$
$a=19845$, $b=218$
I got this far but am now stuck and don't know what to do.
19845 % 218 = 7
218 % 7 = 1
1 = 218 - (7*31)
7 = 19845 - (218*91)
1 = 218 - ((19845-218*91)*31)
Then from there I do not know how to simplify it to get the values for $... |
H: Is there a name for sequences like these?
Starting from an integer value (say $0$ in these cases), I need a sequence of integers to add in a cycle that progress through the integers visiting each exactly once.
For example, the most obvious and simplest sequence would be $(+1)$ which would obviously generate the seq... |
H: What is the maximum value of $4(\sin x)^2 + 3(\cos x)^2$
The question is: What is the maximum value of: $4\sin^2\theta + 3\cos^2\theta$
This is the way I did it:
$4\sin^2\theta + 3\cos^2\theta = \sin^2\theta + 3\sin^2\theta + 3\cos^2\theta = \sin^2\theta + 3$
The max value of $\sin^2\theta$ is $1$, so the answer mu... |
H: Spectral theorem for $n$-tuples of selfadjoint operators
I need a 'good' reference to the following version of the Spectral Theorem:
Given $n$ commuting selfadjoint operators on an infinite-dimensional Hilbert space, there exist a Borel measure $\mu$ on $\mathbb R^n$ and auxiliary Hilbert spaces $h(x)$ such that th... |
H: Proof of an elementary property of Projection Operators
I'm asked to show the following:
Let $X$ be a linear space, and let $P : X \rightarrow X$ be a projection operator. Restricted to the linear space $range(P)$, the projection $P$ is the identity operator, that is, $Px = x$ for all
$x \in range(P)$.
If anyone co... |
H: Anti symmetrical relation
Currently learning about symmetrical and anti symmetrical relations. Been working on assignments but I cant seem to bend my head around this one, especially because I can't understand the solution.
What I thought an anti symmetrical relation is: a pair like $(x,y)$ that also exists as $(y... |
H: Combinations' Problem
A person has six friends and during a certain vacation, he met them during several dinners. He found that he dined:-
with all the six exactly on one day,
with every five of them on $2$ days,
with every four of them on $3$ days,
with every three of them on $4$ days, and,
with every two of them ... |
H: Prove that when $f(A) \subseteq f(B)$ doesn't always mean that $A \subseteq B$
How to prove, when $f(A) \subseteq f(B)$ doesn't "always" mean that $$A \subseteq B$$
when $ f\colon X \to Y $ is total function (not partial)
AI: Consider $f: \mathbb R \to \mathbb R^+, x\mapsto x^2$
And then see
$$f([0, 2]) = f([-2, 0]... |
H: Finding $x+y+z$
If $x+1/x = y$, $y+1/y=z$, $z+1/z=x$, then find $x+y+z$. Is there any way to do so without taking out the values of $x$, $y$, $z$?
Please help,
AI: Let us first write down the equations:
$$x+\frac{1}{x}=y, \; y + \frac{1}{y}=z,\; z+\frac{1}{z}=x$$
We now add these equations together:
$$x+\frac{1}... |
H: Finding real part of fourier series
I have encountered the following problem in one of my textbooks but I'm not really getting anywhere:
Let $f$ be complex-valued and piecewise continuous on the interval $[-\pi,\pi]$. Find the complex fourier series of $Re(f)$ on the basis of the complex Fourier series of $f$.
i.e ... |
H: Reverse of number (numerical)
You can find reverse number $R$ using $x_{n+1} = x_n(2 - x_n \cdot R) \ \ $ where $ n = 0,1,..$
Prove it using Newton method for finding $0's$ of some function $f$
Anyone have idea what that function $f$ might be?
AI: The Newton iteration is
$$
x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}
$$... |
H: Graph Theory : Job Assignment Problem
Problem is to assign 5 jobs to five people. How many non-planar graphs can be drawn such that no vertex is isolated?
AI: The mothers of all non-planar graphs are $K_5$ and $K_{3,3}$ as you know. The second of those is the one you want to avoid. That restricts how you can partit... |
H: How to factorize the quadratic $a(b-c)x^2 + b(c-a)x + c(b-a) = 0$?
How do i factorize this equation:
$a(b-c)x^2 + b(c-a)x + c(b-a) = 0$
I tried the quadratic formula, but the discriminant is not factorising into a perfect square.
Please help!
AI: HINT:
Observe that $$a(b-c)-c(b-a)=-b(c-a)$$
Put the value of $b(c-... |
H: Question about criteria of connectedness
I have to prove the following :
A) No proper non empty subset of $X$ is both open and closed in $X$ $implies$ that $X$ is not the union of two disjoint open subsets of itself.
Attempt at the proof:
I assumed that, No proper non empty subset of $X$ is both open and closed in ... |
H: $a_1=k,a_{n}=2a_{n-1}+1(n\geq 2).$ Does there exist $k\in\mathbb N$ such that $a_n,n=1,2,3,\cdots$ are all composite numbers?
Let $a_1=k,a_{n}=2a_{n-1}+1(n\geq 2).$
If $k=1$ then $a_n=1,3,7,15,31,63,\cdots$ here $3,7,31$ are prime numbers. I'm interested in this problem:
Does there exist $k\in\mathbb N$ such that ... |
H: Example to $\lim f(x)g(x)$ may not exist
Let $A\subset\mathbb{R}$, $c$ a cluster point of $A$ and $f,g:A\rightarrow \mathbb{R}$. Suppose that $f$ is bounded on some neighbourhood of $c$ show by example that if $\lim_{x\rightarrow c}g(x)$ exists, then $\lim_{x\rightarrow c}g(x)f(x)$ may not exist.
By hypothesis, $... |
H: Problem book,typing error?
I have the following problem :
Let $f:(0,\infty)\rightarrow \mathbb R$ be an arbitrary function satisfying the hypothesis, $\lim_{x \rightarrow 0} x(f(x)-1)=0$. Show that $\lim_{x \rightarrow 0 } f(x)$ exists.
The problem has also other parts and 2 other hypothses on $f$. But the soluti... |
H: What is the probability of this event?
The question is: A card is drawn from an ordinary pack(52 cards) and a gambler bets that either a spade or an ace is going to appear. The probability of his winning are?
I think the answer is $\frac{16}{52} = \frac{4}{13}$. Did I go "probably" go wrong somewhere?
AI: There are... |
H: Probability that $j$ persons to get off on the same floor, and $k-j$ persons to get off on separate floors
There are $k$ persons and $n$ floors. Assuming that the probability of any person to get off on any floor is $\frac{1}{n}$, and the decisions taken by the persons are independent, what is the probability that ... |
H: Understanding proofs from paper on Game Theory (Price of Anarchy)
I'm trying to distill the arguments in the paper "Worst-Case Equilibria" (http://cgi.di.uoa.gr/~elias/publications/paper-kp09.pdf). But there are some things I do not understand and would appreciate some help.
Theorem 1: The price of anarchy for $m$ ... |
H: "Comfort" function with tunable parameter
I'm trying to create a "comfort" function with the following characteristics:
its domain is $(-\infty, +\infty)$;
its range is $[0,1]$;
it is at or near its maximum value ($1$) in some interval $[x_{c}-\delta, x_{c}+\delta]$;
it is at or near its minimum value ($0$) in th... |
H: Show a set is in Borel sigma algebra
First of all, sorry for asking again a question.
For all functions, $f:[a,b] \to \mathbb{R}_{+} $ define
$S(f)=\{(x,y)\in \mathbb{R}^{2}: 0\leq y \leq f(x)\}$.
Show if $f$ is measurable, then $S(f) \in \mathbb{B} (\mathbb {R}^{2})$. Compute that two-dimensional Lebesgue meas... |
H: Find $f$ using equation involving real and imaginary part
Suppose $f=u+iv$ is differentiable in the entire complex plane. The real and imaginary parts of $f$ are related by $au(x,y)+bv(x,y)+c=0$ where $a,b,c \in \mathbb{R}$ and $a$ and $b$ are not simultaneously zero. Find $f$.
My attempt: Differentiate the equatio... |
H: Convex hull of the union of two nonempty sets
I was reading about convex hulls on Wikipedia (Convex hull) and I read :
$ Conv(A \cup B)= Conv(Conv(A) \cup Conv (B))$ where $A$ and $B$ are nonempty sets.
I can see intuitively that this equality is true, but I do not know how to write it formally down.
AI: For simpli... |
H: How find this limit $\lim_{n\to\infty}\frac{(2n+1)!}{(n!)^2}\int_{0}^{1}(x(1-x))^nf(x)dx$
Let $f:[0,1]\longrightarrow R$ be a continuous function,Calculate the limit
$$\lim_{n\to\infty}\dfrac{(2n+1)!}{(n!)^2}\int_{0}^{1}(x(1-x))^nf(x)dx$$
My try:use this
$$n!\approx\left(\dfrac{n}{e}\right)^n\sqrt{2n\pi}$$
s... |
H: Limit of metric of sequences
I'm not sure if I'm overcomplicating this, but I'm trying to prove that if $x_n \to x$ and $y_n \to y$, then $\lim_{n\to \infty} \rho(x_n, y_n) = \rho(x,y)$.
So far I have that I want to show that $\rho(\rho(x_n,y_n), \rho(x,y)) \to 0$, and I have tried a tricky triangle inequality:
$... |
H: Proving Riemann integral does not change when finite values of a function is changed.
I know how to prove that the Riemann integral of a function does not change if one point of the function is changed. However, extending that result to a finite set by use of induction is something I have struggled to prove. I just... |
H: Combinatorics - Find the coefficient of $x^{12}$ in...
Would someone be able to help me figure out these two binomial coefficient problems using generating functions? Its a rough concept for me to understand, so a good explanation would be very much appreciated!
$a$) $(1-x)^8$
$b$) $(1-4x)^{-5}$
Thank you in advanc... |
H: Example of 3-regular graph with chromatic index > 3
folks.
For a homework assignment I've been asked to prove that a 3-regular Hamiltonian graph has a chromatic index of 3. I really would like to work through the proof myself, but am having trouble thinking of a 3-regular graph that's NOT Hamiltonian as a negative ... |
H: Riemann Integral (Rudin)
I was reading Rudin's, "Principles of Mathematical Analysis", specifically the section about the Riemann Integral and I've ran into some "shaky" notation. Can someone just explain to me geometrically what is going on here (by here I mean what is def $6.2$ saying)?
$6.2$. Definition. Let $\a... |
H: Prove that $P_{S_N}(t) = P_N(P_X(t))$ for $S_N = X_1 + \cdots + X_N$.
Let $N$ and $(X_i)_{(i \ge 1)}$ be independent random variables ($X_i$ have the same density). Let $S_N = X_1 + \cdots + X_N$. Prove that $P_{S_N}(t) = P_N(P_X(t))$ where $$P_X(t) = E(t^X) = \sum_{k=0}^\infty t^x \Pr(X=k) $$
is probability-genera... |
H: find $\int _\gamma \frac{1}{z+\frac {1}{2}}dz$
I'm asked to find $$\int _\gamma \frac{1}{z+\frac {1}{2}}$$ where $\gamma (t)=e^{it}, 0\leq t\leq 2\pi$.
To do this I deal with two different logarithms, one without the negative imaginary axis which I call $\log _1$ and another without the positive imaginary axis, $\... |
H: Matrix norm inequality involving max and stacked matrices
In a paper I found the following inequality for matrices $A$ and $B$:
$\max\left\{||A||, ||B||\right\} \le \left\| \begin{align}A \\ B\end{align} \right\|_2 $
I suspect that this is a well-known inequality, which I just did not happen to find. Can somebody p... |
H: Is $g(x,y) = f(\frac{x}{2},\frac{y}{2})$ correct notation?
I was a bit confused when I saw this statement $g(x,y) = 2f(\frac{x}{2},\frac{y}{2})$, and seeing it used in a double integral $\int \int g(x,y) = 2 \int \int f(\frac{x}{2},\frac{x}{2}) \, dx dy$.
I understand that idea is for us to use the change of variab... |
H: Prove that $\left( \frac{p-1}{2} \right)! \equiv (-1)^n \mod p$, $n$ is quad. nonres. of $p$ $< p/2$.
Let $p$be a prime number with $p \equiv 3 \mod 4$. Prove that $\left( \frac{p-1}{2} \right)! \equiv (-1)^n \mod p$ where $n$ is the number of positive integers less than $p/2$ that are quadratic nonresidues of $p$.... |
H: Compact but not Hausdorff space
I think this space might be a space which is compact and non-Hausdorff, but I don't how to prove this.
Let $x,y\in\mathbb{R}^n$, define $x\sim y\leftrightarrow\exists t\neq 0(x=ty)$. Then $\sim$ is an equivalence relation and $\mathbb{R}^n/\sim$ is a compact and non-Hausdorff space.
... |
H: About norm and duality in $\mathcal S(\mathbb R^n)$
Reading the book "Classical and multilinear harmonic analysis, Vol. 1" by Muscalu, Schlag, 2013; I have a problem understanding the first step of the proof of Lemma 11.3.
The relevant parts are:
Let $\mu$ be a finite measure on $\mathbb R^n$ with $n\geq 2$ and $g... |
H: Show that prime $p=4n+1$ is a divisor of $n^{n}-1$
Show that the prime number $p=4n+1$ is a divisor of $n^{n}-1$
Ok, the question itself is simple as hell, but I couldn't think of a simple way to solve this question. I tried to solve the question by using $p\equiv 1 \pmod n$ but only to fail miserably... I could... |
H: Problem with change of variables
I have this integral: $$\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}xyz\,dx\,dy\,dz=\frac{1}{8}$$
But when I make this change of variables:$$x=t$$$$y=t$$$$z=t$$ I have $$\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}t^3\,dt\,dt\,dt=\frac{1}{4} ?!$$
What am I doing wrong?
AI: If $D$ is the region you ... |
H: estimate for $\int |f|d\mu$
Let $(\Omega,\mathcal A,\mu)$ be a measure-space with $\mu$ a finite measure (i.e. $\mu(A)<\infty$ for all $A\in\mathcal A$) and $f:\Omega\to\mathbb R$ a measurable function. Prove that:
1) $$\sum_{i=1}^\infty \mu(\{|f|\geq n\})\leq\int |f|d\mu\leq\mu(\Omega)+\sum_{n=1}^\infty \mu(\{|f|\... |
H: Homework - set theory infinite union
A question from my homework I'm having trouble understanding.
We are given:
$A(1) = \{\varnothing\}$, $A(n+1) = A(n)\cup (A(n)\times A(n))$
$A=A(1)\cup A(2)\cup A(3)\cup \cdots \cup A(n)\cup A(n+1) \cup \cdots$ to infinity
The questions are:
1) show that $A\times A \subseteq A$
... |
H: Finding The Order of Elements
This is a homework problem from my Group Theory class.
What is the order of $6$ in $\mathbb Z_{16}$?
I know $\mathbb Z_{16} = \{0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15\}$.
I know that in order to find the $\operatorname{ord}(6)$ I need to find $n,$ such that $6^n=e,$ where $e$ is the ide... |
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