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H: Definition of a real-valued random variable
I have trouble understanding the definition of a random variable:
Let $(\Omega, \cal B, P )$ be a probability space. Let $( \mathbb{R}, \cal R)$ be the usual
measurable space of reals and its Borel $\sigma$- algebra. A random variable
is a function $X : \Omega \right... |
H: Double set difference
Let $A,B \subset X$ and $A \cap B = \emptyset$. What is $A \setminus (A \setminus B)$? Since $A,B$ are disjoint, $A \setminus B = A$, whence $A \setminus (A \setminus B) = A \setminus A$. Now what is this set? It's the set of elements of $A$ that are not elements of $A$. Well, there are no su... |
H: Can all functions be restricted to a dense set from which they may be continuously extended?
Given a function $f:\mathbb{R}\rightarrow\mathbb{R}$ ($f$ is not necessarily continuous), will there always be a dense set $S\subseteq\mathbb{R}$ such that, if we restrict $f$ to $S$, we may then extend it to a continuous f... |
H: Regarding basis for a topology
We know that $\{[a,b):a,b\in \mathbb{Q}:a<b\}$ is a basis for a topology which is strictly finer than the usual topology on $\mathbb{R}$. I have also shown that this topology is contained in the lower limit topology on $\mathbb{R}$. But I couldn't find an example to show that this top... |
H: how to prove $f^{-1}(B_1 \cap B_2) = f^{-1}(B_1) \cap f^{-1}(B_2)$
I am given this equation:
$f^{-1}(B_1 \cap B_2) = f^{-1}(B_1) \cap f^{-1}(B_2)$
I want to prove it: what i did is
I take any $a \in f^{-1}(B_1 \cap B_2)$, then there is $b \in (B_1 \cap B_2)$ so that $f(a)=b$. Because of $b \in (B_1 \cap B_2)$, it ... |
H: How to find the general solution of $\sin \left(\theta +\frac{\pi }{6}\right) =\cos 3\theta $
Find the general solution of the equation.
\begin{eqnarray} \sin \left(\theta +\frac{\pi }{6}\right)\space
=\space \cos 3\theta \\ \end{eqnarray}
The answers in my book are $\left(6n+1\right)\frac{\pi }{12},\space \left... |
H: $ \{ x : ( \inf_{n \geq k} f_n)(x) > a \} = \bigcap_{n=k}^{\infty} \{x: f_n(x) > a \} $
Pick $x \in \{ x : ( \inf_{n \geq k} f_n)(x) > a \} \iff ( \inf_{n \geq k} f_n)(x) > a \iff f_n(x) > a \forall n \geq k$. Therefore, $x \in \{y : f_n(y) > a \}$ for all $n \geq k$. And this holds iff $x \in \bigcap_{n=k}^{\inft... |
H: Am I correctly determining the convergence or divergence of these series: $\sum \frac{1}{n(\ln(n))^{5/4})}$?
I'm doing online homework, which immediately tells me if I got it right or wrong, and I got the following question correct, but I just want to make sure I'm getting it correct for the right reasons.
It asks ... |
H: Divisible abelian groups.
From Hungerford's Algebra (with slight changes)
1) Let $p$ be a prime and let $Z(p^{\infty})$ be the following subset of the group $\Bbb{Q}/\Bbb{Z}$ $$Z(p^{\infty}) = \{ \overline{a/b} \in \Bbb{Q}/\Bbb{Z} | a,b \in \Bbb{Z}, b=p^i \operatorname{for some} i \geq 0\}.$$ Show that for each pr... |
H: Find the absolute value of the difference between the area of these triangles.
Let $\triangle ABC$ and $\triangle ABC'$ be two non congruent triangles with side $AB=4$, $AC=AC'=2$$\sqrt{2}$ and $\displaystyle\angle B=30^\circ$. Find the absolute value of the difference between the area of these triangles.
Diagram:-... |
H: Is Dedekind completion of ${}^{\ast}\Bbb R$ a Archimedean field?
Here's Theorem 1.2 on page 6, Martin Andreas Väth's Nonstandard Analysis(See here on googlebooks)
The Dedekind completion $\overline{X}$ of a totally ordered field $X$
is a complete Archimedean field with $\Bbb{Q}_{\overline{X}}$ as the canonical ... |
H: Why, in the Rabin cryptosystem, during decryption, do we get four possibilities instead of two?
The encryption algorithm : c=m^2 modn, should mean that we have two(or one) possibilities for m. Why do we get four squareroots?
AI: For a prime $p$ the congruence $x^2 \equiv a \pmod p$ has either $0$ or $2$ solutions f... |
H: $(123)$ and $(132)$ are not in the same conjugacy class in $A_4$
Could you tell me how to show that $(123)$ and $(132)$ are not in the same conjugacy class in $A_4$?
I know that all 3-cycles can't be in the same class, because the order of each class must divide $|A_4| = 12$ and there are eight 3-cycles in $A_4$.
... |
H: About continued fractions as best rational approximations
I'm reading this notes about continued fractions:
http://www.math.jacobs-university.de/timorin/PM/continued_fractions.pdf
I had no problems understanding everything there, except one thing that has me stuck.
At page 9, the author proves that the "convergent... |
H: Removable singularity theorems for manifolds
All the theorems on removable singularities are for functions defined on open domains $\Omega \in \mathbb{C}$. But what are the corresponding theorems for functions defined on Riemann surfaces? How do they differ and are there any extra issues we need to take into accoun... |
H: Tips for simplifying
I'm working on my homework for Uni. I'm stuck on some problems. I'm not looking for a solution, just some hints on how to do it.
$ \frac{3R^2-6Rv}{2v-R} \Leftrightarrow -3R$
I think here I'm also missing the same idea as in the problem above:
$\frac{R}{v}-\frac{2t}{R} = \frac{R}{v+3t} \Leftrig... |
H: Conditional probability in continuous distribution
I'm struggling understanding how conditional probability for a continuously distributed random variable is to be calculated. The task is as follows:
$f(t) = 1/8 * (4-t)$ for $0 < t <= 4 $ and $0$ elsewhere
We know that in this interval exactly one event $E$ will ... |
H: Tips on solving precalc equation
Need some hints on my uni homework.
I'm sure I made a mistake somewhere, but I don't see it (I'm solving for x):
$$\frac{x+\frac{1}{y}}{\frac{2}{z}-x}=\frac{2-\frac{2z}{y}}{\frac{y}{z}-1}
\\Solution: x=\frac{3}{y+2z} $$
$$\frac{xy+1}{y}*\frac{z}{2-xz} = \frac{2y-2z}{y}*\frac{z}{y-... |
H: Prove that for each prime $p$ there exists a nonabelian group of order $p^3$
Can anyone helo me out with this question: "Prove that for each prime there exists a nonabelian group of order $p^3$".
This is my attempt:
I know how to show that $|Z(G)|= p$.
So, since $C(x)$ contains $Z(G)$ for every $x$ not in $G$ then ... |
H: Strictly convex function: how often can its second derivative be zero?
It's a basic fact that a twice-differentiable function from $\mathbb{R}$ to $\mathbb{R}$ is strictly convex if its derivative is positive everywhere.
The converse is not true: consider, e.g., $f(x) = x^4$, which is strictly convex, with $f ''(0)... |
H: A = {x/x^2+1 : x belongs to reals } Show that A is a subset of [-1/2, 1/2]
I am really stuck with this assignment and have very little idea on how to proceed. I think using derivative is not allowed. How should I proceed, could anyone give any tips?
I've thought something like.
If $x>0$
$$
x^2+1 > x \geq 1
$$
But t... |
H: How to find the general solution of $\tan \left(x+\frac{\pi }{3}\right)+3\tan \left(x-\frac{\pi }{6}\right)=0$
Find the general solution of the equation.
\begin{eqnarray} \tan \left(x+\frac{\pi }{3}\right)+3\tan \left(x-\frac{\pi }{6}\right)=0\\ \end{eqnarray}
The answers in my textbook are $n\pi $ and $n\pi +... |
H: Conditional expectation $E(X\mid XY)$ when $(X,Y)$ is standard normal
I have to calculate
$\mathbb{E}(X|X*Y)$ with X,Y being independent and standard normal distributed. I got at tip in this post (Conditional expectation on components of gaussian vector), that I should use the definition and Bayes Thm to solve the... |
H: if X not finite then O is not a $\sigma$ - algebra
let O (family of sets) consist of those sets which are either finite or have a finite complement. then O is an algebra. I did this part!
my question now is:
if X (space) is not finite, then O is not a $\sigma$ - algebra.
The definitions of algebra and $\sigma$ - ... |
H: If $F$ has characteristic $p$, then $pa$ =0 for all $a \in F$
I have to prove the statement in the title, i.e
If $F$ has characteristic $p$, then $pa = 0$ for all $a \in F$, $p$ prime.
From the definition of a characteristic of a field, we have that
If F is a field of characteristic p then the prime field P of F is... |
H: Does $\det(I+A(I+B)^{-1})=\det(I+A^*(I+B)^{-1})$ hold for $A,B$ positive semi-definite matrices?
Is
$$\det(I+A(I+B)^{-1})=\det(I+A^*(I+B)^{-1})$$
where $I$ is identity matrix, $A,B$ are positive semi-definite complex valued matrices and $A^*$ is the conjugate (Hermitian) transpose of $A$.
Thanks a lot in advance.
... |
H: Help solving this Linear First order ODE
I'm trying to solve this ODE :- $x (dy/dx) + y \log(x) = e^x x^{(1-1/2 \log(x))}$
I divided the equation throughout by $x$, obtaining $(dy/dx) + y\log(x)/x = e^x x^{(\log(x^{-1/2}))}$.
Then, I obtained the Integrating factor as $e^{((\log (x)^2)/2)}$.
Then, $y e^{((\log(x)^2... |
H: Why do mathematicians use only symmetric matrices when they want positive semi-definite matrices?
Why do mathematicians use only symmetric matrices when they want positive semi/definite matrices?
I mean I haven't seen using non-symmetric positive semi/definite matrices. If non-symmetric positive semi/definite matri... |
H: Bijection question
I need to show that $(m,n) \mapsto 2^{m-1}(2n-1)$ is a bijection of $\mathbb{N} \times \mathbb{N}$ on $\mathbb{N}$
I think I need to show that the expression is both injective and surjective, but I am not sure how to do that.
Maybe a kind person can help me in some direction?
Thanks in advance.
A... |
H: Finding inverse of functions[methods of]
I am now trying to understand functions, inverses and composites. I must admit am not getting a thing. But following some leads, I managed to work one as below. Is this a good understanding on hows and whys?
Find the inverse of function $ y=\sqrt{6+x}$
$ y=\sqrt{6+x}$ can b... |
H: Find expected value of $f(x)=\frac{1}{\pi(1+x^2)}$ (Integral does not converge)
I want to find expected value of random variable given by pdf:
$$f(x)=\frac{1}{\pi(1+x^2)}$$
So I have:
$$E(X)=\int_{-\infty}^{\infty}xf(x)dx=\int_{-\infty}^{\infty}\frac{x}{\pi(1+x^2)}dx$$
Integral is equal to: $$\frac{\log(1+x^2)}{2 \... |
H: How does this differentiation come about ?
The question is that: If $f(z)$ is analytic, show that $\frac{\partial f}{\partial \bar z} = 0$
Now, assuming $f(z) = u + iv$
$\frac{\partial f}{\partial \bar z} = \frac{\partial}{\partial \bar z}(u + iv)$
What the book does is this:
$$
(\frac{\partial u}{\partial x}.\... |
H: Interpreting the sign of fourier coefficients
I am studying the Fourier series right now. Hopefully it's going okay. Now I have been playing a little bit with taking the product of a wave function (a sine or cosine with some phase) with a sine and integrating the result (using a convolution sum) over the period of ... |
H: How to find the general solution of $\sin \left(x+\frac{2\pi }{3}\right)=3\sin \left(x+\frac{\pi }{3}\right)$
Find the general solution of the equation.
\begin{eqnarray} \sin \left(x+\frac{2\pi }{3}\right)=3\sin \left(x+\frac{\pi }{3}\right)\\ \end{eqnarray}
The answers in my textbook are $n\pi -\tan ^{-1}\left(\... |
H: Does $ x^p+y^p=kz^p$ have any solutions when $x,y,z,k,p>2, gdc(x,y,z)=1$?
Does the Diophantine equation
$$\displaystyle x^p+y^p=k(z^p)$$
have any solutions when $x,y,z,k,p>2, $ and $ x,y,z$ are co-primes?
AI: Let $a,b$ be two positive integers, and $p$ odd prime.
Then
$$(a^{p}-b)^{p}+(a^{p}+b)^{p} \equiv 0 \pm... |
H: In my calculator why does $\sqrt4 -2=-8.1648465955514287168521180122928e-39?$
I have tried this on my Windows 7 calculator with $\sqrt9 -3 $ it too gives some weird answer- ie$1.1546388020691628168216106791278e-37$. And so for any $n$(positive) $\sqrt n^2-n= wierd_ .answer$
Why does this happen?
AI: Presumable, the... |
H: Restricted permutations revisited!
In how many ways can we arrange $n$ different things at $r$ places (each of $r$ places can have any of the $n$ things)repetition allowed,such that $2$ of the $n$ things are always included?
Foe example if n=3 lets say {2,3,4} and r=3 and(2,4) being compulsory,we have $12$ permut... |
H: The transform of a Hermitian matrix by a unitary matrix is Hermitian
In the following document, p91 (or p4 of the PDF) , section 6.03, it is stated that "The transform of a hermitian matrix by a unitary matrix is hermitian." Apparently the proof is obvious, but not to me... could someone elaborate?
AI: A matrix is... |
H: Multivariable Calculus question, show limit of equation DNE
Show that
$$\lim_{(x,y)\to(0,0)} \frac {3xy} {x^3+2y^2}$$ Does not exist.
This was on an old test. I am not sure what to show since it's intuitively obvious that as $x$ and $y$ approaches $0$, the denominator will become $0$ and thus DNE.
Am I wrong to a... |
H: Simple question that I can't solve
Here is a relatively simple question that I'm unable to solve :/
There are $10000$ closed lockers in a hallway. A man begins by opening all $10000$ lockers. Next, he closes every $2^{nd}$ locker. Then he goes to every $3^{rd}$third locker and closes it if it is open or opens it if... |
H: How to show that $\mu$ is normally distributed
Suppose we have:
$$p(\mu \mid \sigma, \boldsymbol{w}, \boldsymbol{y}) \propto \exp\left[-\frac{1}{2\sigma^2}\sum_{t=1}^T\left(\frac{(y_t-\mu)^2}{w_t^2}\right) \right]$$
where $\boldsymbol{w} = (w_1, w_2, \cdots, w_t)$ and $\boldsymbol{y} = (y_1, y_2, \cdots, y_t)$
How ... |
H: Determine all the extrema of a function subject to a non-linear constraint.
QUESTION
Determine all extrema of the function
$$f(x,y) = x+ 2y $$ subject to $$x^2 + y^2 - 80 = 0$$
ATTEMPT
I don't think I understand what I'm supposed to do. This was in a test and I ended up trying to "graphically"or ïntuitively" find o... |
H: Can we identify the time if we know every angle between three hands of a watch?
Let $M, H, S$ be the minute hand, the hour hand, the second hand of a watch respectively. Also, let $A_{MH}, A_{MS}, A_{HS}$ be the angle between $M$ and $H$, $M$ and $S$, $H$ and $S$ respectively.
Then, here is my question.
Question : ... |
H: Picking Numbers for a Digit
How many three digits numbers are out there with two digits the same and the other different?
If I imagine I have three slots. __ __ __. The first slot is the only one with a restriction. The range of digits can be $[1,9]$ while the 2nd and 3rd can take any digit value from $[0,9]$ Overa... |
H: What is wrong with this reasoning about free modules?
Let $A$ be any set. Let $F(A)$ be the free $R$-module of base $A$ and the map $i: A \to F(A)$ be such that $i(a) = a$. Then for any $R$-module $M$, and a map $g:A \to M$, we can find an unique map $f: F(A) \to M$ by defining it as $f(a) = 0$ for $a \notin A$, an... |
H: A persistent difference
Here's a fun math problem. I wasn't able to get it - am curious what you guys have to say.
Pick a four-digit number whose digits are not all the same. From its digits form the smallest four-digit number $m$ and the largest, $M$. Find $(M-m)$. Keep repeating the procedure. (Treat, say $234$ a... |
H: Factoring of a Pro-$\mathcal{C}$ morphism
Let $\mathcal{C}$ be a category, $X$ an object of $\mathcal{C}$, and $p:I^{\circ}\rightarrow \mathcal{C}$ a projective system in $\mathcal{C}$.
Let $\alpha\in \mathrm{Mor}_{Pro(\mathcal{C})}(p,X)$, where $X$ is identified with the constant projective system of $X$. Can $\al... |
H: $\int_{1}^{2}\frac{1}{(3-5t)^2}dt$
Let $g(t)=(3-5t)^2$, $f(x)=\frac{1}{x^2}$, $g(1)=-2$, $g(2)=-7$.
$$
\begin{align}
& \phantom{={}}\int_1^2 \frac{1}{(3-5t)^2} \, dt \\[6pt]
& =-\frac{1}{5}\int_1^2 \frac{1}{f(g(t))}g'(t) \, dt \\[6pt]
& =-\frac{1}{5}\int_{-2}^{-7}\frac{1}{x^2} \, dx \\[6pt]
& =-\frac{1}{5}\lef... |
H: Solving a second order homogenous ODE with double roots
I'm working on solving the following homogenous equation:
$$ y'' - 8y' + 16y = 0 $$
Seems like a straight forward $y=e^{rx}$ substitution and then solve for r1 and r2:
$$ y=e^{rx}=0 $$
$$ y=re^{rx}=0 $$
$$ y=r^2e^{rx}=0 $$
$$ r^2e^{rx} - 8re^{rx} + 16e^{rx}=0 ... |
H: $\mathbb Q+X\mathbb R[X]$ is not Noetherian
Let $A=\{q+r_1X+ \cdots +r_nX^n: q \in \mathbb{Q}, r_i \in \mathbb{R}\}$ be the polynomial ring with rational costant terms. I have to prove that $A$ isn't a noetherian ring. How can I prove it?
AI: consider the chain of ideals:
$\pi xA \subset \pi xA + \pi^2xA \subset\pi... |
H: Given $f(x+1)=x^2-3x+2$, how can I find $f(x)$?
Given $f(x+1)=x^2-3x+2$, how can I find $f(x)$?
AI: Let $y=x+1$. We then have $x=y-1$. Hence,
\begin{align}
f(y) & = f(x+1) = x^2 -3x+2 = (y-1)^2 - 3(y-1)+2\\
& = y^2-2y+1-3y+3+2 = y^2-5y+6
\end{align} |
H: Let $\,\displaystyle f \colon [0,1] \to [0,1]$ be continuous and $\,f(0)=0,f(1)=1.$
I am stuck on the following problem that says:
Let $\,\displaystyle f \colon [0,1] \to [0,1]$ be continuous and $\,f(0)=0,f(1)=1.$ Then $f$ is necessarily
injective ,but not surjective
surjective,but not injective
bijective
su... |
H: How many n digit numbers can be formed using 1,2,3 which contain at least one 1 and one 3?
How many n digit numbers can be formed using $1,2,3$ which contain at least one $1$ AND one $3 $?
For $n= 5$,
$11133$ is valid.
$12312$ is valid.
$11122$ is not valid.
$33333$ is not valid
Please suggest a formula which works... |
H: NFA from grammar productions
Based on this grammar:
\begin{align} G = (\{S,A,B\}, \{a,b, c\}, S, P) \end{align}
\begin{matrix}
\\P:
\\S → abaS | cA
\\A → bA | cB | aa
\\B → bB | cA | bb
\end{matrix}
I created this NFA:
I'm not sure about $q1 \to q2$ and $q1 \to q3$, if maybe someone can clarify if this is wrong ... |
H: Does this integral make sense in some way?
I have a very simple, possibly silly question...
Can this integral make sense in some way? $$ \int \frac{dx}{dx}$$
And does it actually mean something to write things like $$ \int f(x)$$ without the differential?
I just got an expression of this type and I probably made so... |
H: ii) Show that $u$ is not a real part of the function which analytic on $\mathbb{C} \backslash \lbrace 0 \rbrace$
Suppose $u(x,y)=\ln(x^2+y^2)$
i) Show that $u$ is harmonic on $\mathbb{C} \backslash \lbrace 0 \rbrace$
ii) Show that $u$ is not the real part of a function which analytic on $\mathbb{C} \backslash \lbr... |
H: Get x and y in quadrat equations system
I need help in solving following system of quadratic equations :
$$
2x^2+y^2=4$$
$$2xy-2x=-5$$
I used every known me equations solving methods, but no was helpful for me... Can you help me by giving step-by-step solution (that is homework and I really need it...) ?
Thanks in ... |
H: write $\lvert 10+4i\rvert$ in the form $z=re^{i\theta}$
write $\lvert 10+4i\rvert$ in the form $z=re^{i\theta}$
I am not sure what to do with the absolute value in this case.
AI: Since the absolute value is a real number, $\theta = 0$ and $r = \sqrt{ 10^2 + 4^2 } $ |
H: Is the kernel of this group action the centralizer?
In Dummit and Foote, they state
"... let the group $N_G(A)$ (normalizer) act on the set $A$ by conjugation. It is easy to check that the kernel of this action is the centralizer $C_G(A)$."
From what I understand, the kernel of this action is the set
$$ \{ g \in N... |
H: What is the result of $x^{\top}A\dot{x}+\dot{x}^{\top}Ax$?
What is the result of
$x^{\top}A\dot{x}+\dot{x}^{\top}Ax$, provided that $A=A^{\top}$?
Actually, I wanted to expand $d(x^{\top}Ax)\over{dt}$.
AI: I'll use $y^T$ for the OP's $y^{\top}$.
Observe that $x^TA{\dot x}$, being a scalar quantity, is automatic... |
H: How to find $p$ and $q$ if we have $\operatorname{lcm}(p,q)=b$ and $p+q=a$ where ($a,b \in \mathbb{N}$) and $p>q$.
What is the general formula to find $p$ and $q$ if we have $\def\lcm{\operatorname{lcm}}\lcm (p,q)=b$ or $\gcd(p,q)$ and $p+q=a$ where ($a,b \in \mathbb N$) and $p>q$?
Example: $\lcm(p,q)=84$ and $p+q=... |
H: Proof that $n^3 + 3n^2 + 2n$ is a multiple of $3$.
I'm struggling with this problem:
For any natural number $n$, prove that $n^3 + 3n^2 + 2n$ is a multiple of $3$.
That $n^3 + 3n^2 + 2n$ is a multiple of $3$ means that: $n^3 + 3n^2 + 2n = 3 \times k$ where $k \in \mathbb N$
So I tried to find a the number $k$.
Th... |
H: What is the correct spelling of Paul Erdős's name?
I am not quite sure that this question belongs to the math.stackexchange.com. What I am sure is that definitely there are people here who know the answer to my question.
On the Internet and in the literature I saw two different types of spelling of the last name o... |
H: $\mathbb{R}$ \ $\mathbb{Q}$ and $\mathbb{R}^2\setminus\mathbb{Q}^2$ disconnected?
If I want to prove that $\mathbb{R} \setminus \mathbb{Q}$ is disconnected, does it suffice to say that there are two open disjoint sets that cover $\mathbb{R}\setminus\mathbb{Q}$, namely:
$$(- \infty, 0), (0, \infty)\text{ ?}$$
Along ... |
H: Con ZF implies Con ZFC using set sized models
Can we use forcing to construct models of ZFC and ZFC + GCH starting from c.t.m s of ZF? The usual way to obtain the associated relative consistency results (Con ZF implies Con ZFC and Con ZF implies Con ZF + GCH)is to look at the class sized model L (of ZF) and I was i... |
H: How to solve these series?
Can anyone help me understand how to solve these two series? More than the solution I'm interested in understanding which process I should follow.
Series 1:
$$
\sum_{i = 3}^{\infty} i * a^{i-1}, 0 < a < 1.
$$
Series 2:
$$
\sum_{i = 3}^{\infty} i\sum_{k = 2}^{i-1} a^{i-k} * b^{k-2} ... |
H: The sum of three square roots bounded below by $\sqrt{82}$
Let $a,b,c >0$ and $a+b+c \le 1$. Prove: $$\sqrt{a^2+1/a^2}+\sqrt{b^2+1/b^2}+\sqrt{c^2+1/c^2} \ge \sqrt{82}$$
Progress: I tried to have 3 vectors $(a,1/a)$, $(b,1/b)$ and $(c,1/c)$, played around with the vector inequalities but no success.
AI: Assume fi... |
H: Determine whether the following function is continuous at $(0,0)$
Determine whether the following function is continuous at $(0,0)$:
$$f(x,y)=\begin{cases} \frac{xy}{\sqrt {x^2+y^2}} & \text{if $(x,y)$ $\neq(0,0)$} \\
1 & \text{if $(x,y)=(0,0)$} \\
\end{cases}$$
Here's what I did. I know that $f(0,0)=1$ and I... |
H: Proving that there are n Equivalence Classes Modulo n
For $a,b,n \in \mathbb{Z}$ and $n \geq 2$, I want to prove that there are $n$ equivalence classes mod $n$. I'm not sure how to do it - would I do it inductively? Any help would be appreciated.
AI: There are certainly at least $n$ equivalence classes: Namely, $[0... |
H: Proof of Rudin theorem 2.34 compact subsets of metric spaces are closed
Rudin writes at a certain point in the proof:
$\textbf {Since $K$ is compact}$, there are finitely many points $q_1,...,q_n$ in $K$ such that:
$K \subset W_{q1} \cup W_{q2} \cup ... \cup W_{qn}$ where the $\textbf{$W_{qi}$'s are neighborhoods}$... |
H: Proving linear independence of vectors which are functions of other independent vectors
If the $n$-component vectors $a,b,c$ are linearly independent, show that $a+b, b+c, a+c$ are also linearly independent, Is this true of $a-b,b+c,a+c$?
What I did was write the new vectors as sums and set it equal to zero
$$\beg... |
H: Show $5 \cdot 4^{\log_{2}{n}}$ is $\Theta(n^{2})$.
I'm having trouble working out the algebra for this problem. I know that we need to show $\exists c$ s.t. $5 \cdot 4^{\log_{2}{n}} \leq c \cdot n^{2} \forall n \geq n_{0}$, and also the other direction.
AI: $$
4^{\log_2 n} = 2^{2\log_2 n} = 2^{\log_2 n^2} = n^2.
$$ |
H: How do I use quantifiers to specify a pair of elements?
I'm just learning about quantifiers in class, and my professor has so far only given us examples where he uses quantifiers to specify one element/variable at a time from a set.
So he would write something like: $\forall$ a $\in$ $\mathbb{R}$ ...
Am I limited... |
H: How to find $x$ and $y$ in $(x+c_1)(y+c_2)=c_3$ where $x, y \in \mathbb{N}$ and $c_1,c_2,c_3$ are non-zero constants?
Is there a general formula to find $x$ and $y$ in $(x+c_1)(y+c_2)=c_3$ where $x, y \in \mathbb{N}$ and $c_1,c_2,c_3$ are non-zero constants?
Because I was struggling trying to find solutions to this... |
H: Question about the closed set
Is there a topological space on $\mathbb{R}$ in which the interval $(0,1)$ or intervals like $(a,b)$, $a,b\in\mathbb{R}$ is a closed set?
AI: Yes, the discrete topology, where every subset of $\mathbb R$ is open (hence every subset of $\mathbb R$ is closed as well). |
H: Proof that aleph null is the smallest transfinite number?
The wikipedia page on the cardinal numbers says that $\aleph_0$, the cardinality of the set of natural numbers, is the smallest transfinite number. It doesn't provide a proof. Similarly, this page makes the same assertion, again without a proof.
How does one... |
H: Dervation of $\sum_{k=0}^n(r+1)^k= \frac{(r+1)^{n+1}-1}{r}$
How can one derive the following identity?
$\sum_{k=0}^n(r+1)^k= \frac{(r+1)^{n+1}-1}{r}$
I have playing around with binomial coefficients and index shiftings but wasn't able to get anywhere.
AI: Let's call $q:=r+1$ and look for $x$, then
$$1+q+q^2+\dots+q... |
H: Inequality with argument of quotient of complex number
Show on a complex plane:
$$\left\{ z \in \mathbb{C}: \operatorname{arg} \frac{i}{z} \leq \frac{3 \pi}{4}\right\}.$$
After some easy steps I came to:
$$-\operatorname{arg}(z) =\operatorname{arg} \overline{z} \leq \pi/4.$$
What now?
AI: If $z=r e^{i \theta}$, wit... |
H: Constructing a closed, convex subset of $X^{\ast}$ that is not weakly-* closed
I'm asked to show that if $X$ is a non-reflexive Banach space, there exists (norm) closed and convex subsets of $X^\ast$ that are not $w^{\ast}$-closed. In other words, there's no analogue of Mazur's Lemma for the $w^{\ast}$-topology. As... |
H: Can a subset have $0$ elements?
Let $S = \{1, 2, \dots, n \}$. Must any subset of $S$ have at least $1$ element?
For example, suppose the subset $\{ \emptyset \} \in \mathcal{P(S)}$ . Is the cardinality of $\{ \emptyset \} = 0$ or $1$?
AI: The empty set is a subset of any set, and it has zero elements. That is, w... |
H: Factoring $a^m + 1$, an odd prime
Why is it that if $a^m + 1$, an odd prime, with $m = kl$, and $l$ odd.
We get:
$$a^m + 1 = (a^k + 1)(a^{k(l-1)} - a^{k(l-2)} + \dots + a^k + 1)?$$
What is the name of this property?
AI: If $n$ is odd we have $$\frac{1+x^n}{1+x}=\frac{1-(-x)^n}{1-(-x)}=\frac{1-r^n}{1-r}=\cdots?$$
In... |
H: Integral of function having modulus
Evaluate
$$ \int c e^{-|x-U|} dx$$ Assume $c$ and $U$ to be constant.
I am not used to do integrals since long time. I suddenly have to use this in a computer program.
AI: Let $F$ be a primitive such that $F(0)=0$. Then
$$
F(X)=\int_{U}^X c \, e^{-|x-U|}dx=\left\{\begin{array}{... |
H: Question on Power Sets: if $|A|=n$, what is $|P(P(A))|$?
(Let $P$ be the symbol for Power Sets)
My exercise states, if $|A|=n$, what is $|P(P(A))|$?
The given answer is:
If $|A| = n$, then $|P(A)| = 2^n$ and $|P(P(A))| = 2^{2^n}$.
For $n = 2$, we have $|P(P(A))| = 2^4 = 16.$
Where does the 2 come from in the if st... |
H: Help me this proof! Related to RSA public key cryptosystem
Basically it is similar to the RSA algorithm.
Let p and q be distince primes and let e and d be the integers
satisfying $de≡1$ (mod (p-1)(q-1)).
Suppose further that c is an integer with gcd(c,pq)>1.
Prove that $x≡c^d$ (mod pq) is a solution to the cong... |
H: Using integrals to prove that the mean of the sampling distribution is the population mean
Let the random variables $X_1, X_2, \dots X_n$ denote a random sample from a population.
The sample mean of these random variables is:
$\overline{X}=\frac{1}{n}\sum\limits_{i=1}^{n}X_i$
I would like to show that the mean of t... |
H: Does $f'(0)$ exist?
This is a problem from my introductory calculus homework.
\begin{equation}
f(x) = \begin{cases}
x^2\sin\frac{1}{x}, & x \neq 0,\\
0, & x = 0.
\end{cases}
\end{equation}
Does $f'(0)$ exist?
If I approach the problem algebraically, I obtain the answer \begin{equation}f'(x)=\lim_{x \rightarrow 0}... |
H: $A \cap(A \cup B) = A$ for all sets $A$ and $B$
For all sets $A$ and $B$, $$A \cap(A \cup B) = A.$$
I get that this is true informally, but what would be the notation to formally prove this?
AI: Hint: Frequently, the easiest method for showing some set $X$ and some set $Y$ are equal is this: first, show that any... |
H: Find equation of tangent at given point given x and y equations
The question asks to find an equation of the tangent line to the
curve at the given point
x = t^2 + t
y = t^2 - t^3
Point (0,2)
I've only done questions like this where there is a given t value and not where there is an actual point given, could any... |
H: Cardinality of product of two sets is the product of the cardinalities of both sets
Suppose $|X| = n$ and $|Y| = m$. We want to show $|X \times Y | = mn$.
MY attempt: By hypothesis, we can find bijections $f: X \to \{ 1,...,n\}$ and $g : Y \to \{1,...,m\}$. We want to find a bijection from $X \times Y \to \{ 1,....... |
H: how to calculate the normal vector for a bezier curve
Say we have a cubic Bezier curve (so 4 control points) named Q. I understand how to calculate the tangent at by taking the derivative of Q and substituting but i'm not sure how to calculate the normal vector.
Thank you for the help!
AI: If the curve lies in some... |
H: The Three-Number Problem
I am currently working on a little extra credit for my 9th grade math class and I am stuck on the one of the problems.
The Three-Number Problem
I have chosen three numbers. The second is twice the first, and the third is three times the second. The sum of the first two when multiplied by t... |
H: Isometric identification of $c_0^*$ and $ \ell^1$
Let $\{x_n\}_{n=1}^{\infty}\subset \ell_1$ be a sequence in $\ell_1$ with $x_n = (x_n(1),x_n(2), x_n(3),\ldots )$
I want to show that $$\lim_{n\to\infty}\sum_{j=1}^{\infty} x_n(j)y(j) = 0 $$
for all $y\in c_0$ if and only if $\sup_n \left\|x_n\right\|_1<\infty$ and ... |
H: Post-concatenation of the languages represented by the null set
I have a small question regarding concatenation of regular languages:
Is it true that the concatenation $L\varnothing$, where $L$ is any regular language, result in $\varnothing$?
Namely, does $L\varnothing$ = $\varnothing$?
Thanks a lot!
AI: Yes. If ... |
H: If a sequence of summable sequences converges to a sequence, then that sequence is summable.
Let $(a_i)^n$ be a sequence of complex sequences each of which are summable (they converge). Then if they have a limit, the limit sequence $(b_i)$ is also summable. All under the sup norm for sequences.
Let $(a_i)^n$ sum ... |
H: Looking for guidance on a Fourier integral
Working with a Fourier transform problem, I've encountered the following integral:
$$
\int_{-\infty}^{\infty}\frac{\exp\left(-a^2x^2+ibx\right)}{x^2+c^2}dx
$$
where $a$, $b$, and $c$ are real coefficients. Mathematica claims that this has no closed-form solution, but I su... |
H: Graph Theory Connectivity and Diameter of Graphs
Let G be a graph of order n with κ(G) ≥ 1. Prove that n ≥ κ(G)[diam(G) - 1] + 2.
κ(G) meaning connectivity of G and diam(G) meaning diameter of G.
I know that κ(G) and diam(G) are inversely related, but I still need help approaching this problem.
AI: Have you had a... |
H: Formula for the number of digits in the number $2^x$
I'm wondering if there is a formula for the number of digits in $2^x$.
For example if $x = 3$ then the number of digits is equal to $1$ because $2^3 = 8$
or for example if $x = 4$ then the number of digits is equal to $2$ because $2^4 = 16$.
In an attempt to solv... |
H: Why is the degree of the node pictured 5, not 4?
On the Wikipedia page for degree (graph theory), the bottom-right vertex pictured is said to have a degree of 5.
My understanding is that the degree of a vertex is defined as the number of edges incident to the vertex. In this case, it appears as if the bottom-right... |
H: Proving an inequality:
I have this inequality but I am unsure how to prove it:
0$<\alpha \leq$1
a$^\alpha$+b$^\alpha$ $\geq$ (a+b)$^\alpha$
$\forall a,b \geq 0$
I was given a hint: we can assume b$\geq$0
$(\frac{a}{b})^\alpha$+1 $\geq$ $(\frac{a}{c}+1)^\alpha$
so it suffices to prove f(x)$\geq$ 0:
f(x)=x$^{\alph... |
H: For $a,n \in \mathbb{Z}$, where $n>a>0$, prove that $\nexists x \in \mathbb{Z}$ s.t. $nx=a$
Suppose $a,n \in \mathbb{Z}$, and $n>a>0$. How do I prove that $\nexists x \in \mathbb{Z}$ s.t. $nx = a$ ? I'm really not sure where to start on this one. I'd be happy if someone could give me a hint.
Edit: I've solved this ... |
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