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H: $f$ measurable, $g$ either 1 or 0 if $f$ rational or irrational
I have trouble advancing with this problem, though it seemed quite straight forward at glance.
$f$ is a measurable function, and let
$$g(x)=
\left\{
\begin{array}{ll}
0 & \mbox{if } f(x)&\mbox{is rational} \\
1 & \mbox{if } f(x)&\mbox{is irratio... |
H: Open sets in $\mathbb{R}^2$ as countable union of disjoint open rectangles
From this question I realize that there exists an open set in $\mathbb{R}^2$ that is not a disjoint union of open rectangles. The example given is the set of points lying below the line $y=-x$.
However, I can't quite see how one would prove ... |
H: Finding the limit of $\left( 1-\frac{1}{n} \right)^{n}$
How would one compute the following limit?
$$\lim_{n \to \infty} \left( 1 - \frac{1}{n} \right)^{n}$$
I know
$$\lim_{n \to \infty} \left( 1 + \frac{1}{n} \right)^{n} = e$$
but right there is a minus keeping that limit from being used.
Another problem I am que... |
H: stuck in finding $\frac{d(\sin^{-1}(x))}{dx}$
what is wrong with the following reasoning?
$$\sin(\sin^{-1}(x))=x$$
This is of the form $f(g(x))=h(x)$. Then, by implicit differentiation:
$$(f(g(x)))'=f'(g(x))\cdot g'(x)=h'(x) $$
Or in this case:
$$\cos(\sin^{-1}(x))\frac{d(\sin^{-1}(x))}{dx} = 1$$
Then... |
H: Tate's Thesis: in what sense is Tate's Theorem 4.2.1 the Riemann-Roch theorem for curves?
I am reading Tate's Thesis. Tate derives a theorem which he calls "the number-theoretic analogue of the Riemann-Roch theorem" from an abstract Poisson summation formula. I am accustomed to thinking of the Riemann-Roch theorem ... |
H: Complex Numbers....
Suppose a is a complex number such that:
$$a^2+a+\frac{1}{a}+\frac{1}{a^2}+1=0$$
If m is a positive integer, find the value of:
$$(a^2)^m+a^m+\frac{1}{a^m}+\frac{1}{(a^2)^m}$$
My Approach:
After I could not solve it using the usual methods I tried a bit crazier approach. I thought that as the ... |
H: Diagonal sequence trick: Uniform bound necessary?
There is a treatment of the "diagonal sequence trick" in Reed and Simon (Functional Analysis Vol.1) stated there as follows:
Let $f_n(m)$ be a sequence of functions on the positive integers which is uniformly bounded, i.e. $|f_n(m)| \le C$ for all $n,m$. Then there... |
H: Let $\mu$ be the measure constructed in the Riesz representation theorem. Is $\mu(\partial A)=0$?
I am currently self-studying Rudin's real complex analysis. The Riesz representation theorem in the book states:
Let $X$ be a locally compact Hausdorff topological space. Let $T:C_c(C)\rightarrow \mathbb{C}$ be a posit... |
H: Equation of a line parallel to $5x-3y=7$ That goes through the point (3,-1)
This is a study question in preparation for my midterm. It's multiple choice. The answers are:
A) $y=(5/3)x-(7/3)$
B) $y=(3/5)x-(14/5)$
C) $y=(5/3)x-6$
D) $y=-(3/5)x+(4/5)$
Here is my process:
$5x-3y=7$
Move the 3x over and change the sign.... |
H: How to prove algorithm for solving a square congruence when p ≡ 5 (mod 8)
I'm having trouble understanding why this algorithm works and where it comes from:
"Suppose p ≡ 5 (mod 8) is a prime and y is a square (mod p); that is, for some $ x, x^2
≡ y\ (mod\ p)$. This can be solved by the following algorithm
Compute... |
H: Interesting Medieval Mathematics Lecture/Activity Ideas?
Recently I have been invited to give a talk about Medieval Mathematics or mathematics in the 500 AD - 1500 AD time frame. I have been researching the time frame for the past week and have found interesting snippets of Arabian mathematicians and Fibonacci. I k... |
H: Nonlinear difference equation
Maybe this is a trivial question, but how to find the general solution to the following first order difference equation?
$$
y_{t+1}=a+\frac{b}{y_{t}}
$$
Also, could someone recommend a reference textbook on difference (and possibly differential) equations? I would need a quite comprehe... |
H: Uncountable open cover of $\mathbb{R}$
The question posed to me is to find an uncountable open subcover of $\mathbb{R}$ such that it has no finite subcover, but I can't even think of a way to define an uncountable open cover.
AI: Consider the collection $\{(a,b)\mid a<b\in\Bbb R\}$, the collection of non-trivial fi... |
H: The Decomposition VS. The Partition of a set
The book that I am reading says that the decomposition of a set $A$ is any representation of $A$ as the union of a disjoint family of set, $A=\sum_{i\in I}A_i$ (if a family of set is pairwise disjoint the author writes $\bigcup_{i\in I}A_i$ as $\sum_{i\in I}A_i$).
The fa... |
H: A strange characterization of connected spaces
Let $X$ be a topological space. Recall that an open cover $\mathcal U$ of $X$ is point finite if every point of $X$ is contained in finitely many elements of $\mathcal U$.
Let us say that a point finite open cover $\mathcal U$ is fair if any two points of $X$ are conta... |
H: How many possible deals in straight poker.
Each player is dealt 5 cards.
5 players how many deals are possible?
I know that for one player there is 2598960 possible outcomes i.e. 52 C 5, I need to know how I can do this for the next four players and why if possible
AI: Hint: having chosen five cards for the first ... |
H: The convergence of a sequence of sets
A sequence $\{A_n : n=0,1,2,...\}$ is said to be monotone nondecreasing if we have
$$A_0\subseteq A_1\subseteq \cdot \cdot \cdot \subseteq A_n \subseteq \cdot \cdot \cdot $$
The same sequence is said to be monotone nonincreasing if we have
$$A_0\supseteq A_1\supseteq \cdot \cd... |
H: Counting the number of integer sequences
Count the number of sequences of integers, a(1), a(2), .... a(n), containing n positive integers such that1<=a(i)<=m, for all 1<=i<=n and max_value - min_value = q, where *max_value* means the highest integer in the above sequence and similarly for *min_value*. Note that rep... |
H: Show $P(A \subset B) = (\frac{3}{4})^n$
Let $S = \{1, 2, \dots, n \}$ and suppose that $A$ and $B$ are,
independently, equally likely to be any of the $2^n$ subsets of $S$.
Show that $P(A \subset B) = (\frac{3}{4})^n$ and show that $P(A \cap B = \emptyset) = (\frac{3}{4})^n$.
Applying the law of total probab... |
H: CFG - whose words contain exactly twice as many b's as a's.
I am trying to built a CFG for the language that accepts all words that have twice as many b's as a's. The only idea I could come up with is:
Start -> S
S-> SaSbSbS | SbSaSbS | SbSbSaS | $\epsilon$
But obviously it will not be able to parse the word aaabbb... |
H: Can a directed hamiltonian path be found in polynomial time?
I was discussing a programming competition problem with one of my math professors in Linear Algebra that reads as follows:
A matrix is an $r\times c$ array of numbers, where $r$ is the number of rows and $c$ is the number of columns. Given two matrices $... |
H: A question about a sequence.
I'm preparing for the subject exam in November. This is a question that I thought I had the correct answer to.
Let $\left \{a_n \right \}_{n=1}^\infty$ be defined recursively by $a_1=1$ and $a_{n+1}=\left( \frac{n+2}{n} \right) a_n$ for $n\geq 1$. Then what is $a_{30}$ equal to?
So I ... |
H: A question about Riesz - Fischer theorem's proof
In Riesz-Fischer theorem's proof, when we put
$$
g_k =|f_{n_1}|+|f_{n_2}-f_{n_1}|+ \cdots + |f_{n_k}-f_{n_{k-1}}|
$$
it is easy to get (by Minkowski's inequality)
$$
\left \| g_k \right \|_p \leq \left\|f_{n_1}\right\|_p+\left\|f_{n_2}-f_{n_1}\right\|_p+ \cdots + \... |
H: answer check for a calculus 3 variable chain rule problem
yo.
find $\frac{\partial{V}}{\partial{t}} = ?$
$V =\frac{1}{3}{x^2}h$
$x = \frac{t}{t+1}$
$h = \frac{1}{t+1}$
$$\frac{\partial{V}}{\partial{t}} = \frac{2t-t^2}{3(t+1)^4}$$
Does this look correct?
AI: The way I solve this problem is to plug $x, h$ into $V$. ... |
H: Prove that $x_n$ converges, and find $\lim_{n\rightarrow \infty} x_n$
I need some critique and suggestions on how to go over this question:
Prove that $x_n$ converges, and find $\lim_{n\rightarrow \infty} x_n$:
$x_1 > 0 $, $x_{n+1} = \frac{1}{2}(x_n + \frac{5}{x_n}) \ \forall n \geq 1 $.
Here's how I solved it:
L... |
H: Are the only $b$ such that $b^2 + 4c$ and $b^2 - 4c$ are both perfect squares primes of the form $4k + 1$?
And of course, multiples of primes of the form $4k + 1$
$b, c, k$ are positive integers.
Ran into a math problem essentially involving this, just curious. I observed it to hold for the first $50$ integers, but... |
H: If a commutative ring with identity is the sum of two ideals, then their product is equal to their intersection.
My problem is to prove exactly as the title says; particularly if I+J=R for some commutative ring R with identity and ideals I and J of the ring R, then IJ = I ∩ J.
I know already that IJ is an ideal in ... |
H: $n$ points on a circle
Choose $n$ points on a circle so that no three of the $\binom{n}{2}$ chords have a common point inside the circle. Let $a_{n}$ be the number of regions formed inside the circle by drawing the cords.
Obtain the recurrence relatin $a_{n}=a_{n-1}+f(n)$ for $n\geq 1$ where $f(n)=n-1+\sum_{i=1}^{... |
H: How can I find the maximum velocity if I've already found when it occurs?
I've been working on this problem for awhile and got the second half figured out, but I can't seem to get what the maximum actually is.
Here's the question:
A bullet is fired in the air vertically from ground level with an initial velocity 27... |
H: A function of $f\circ g$
This is studying for my midterm.
Let $f(x)=x^2/(x+1)$ and $g(x)=2x-3$ A function of $f\circ g$ is:
So I begin with the equation:
$$x^2/(x+1) \cdot 2x-3$$
Add one to the denominator of the second equation.
$$x^2/(x+1)\cdot(2x-3)/1$$
Multiplying i get:
$$(2x^2-3x^2)/(x+1)$$
Although this isn'... |
H: inf A of $n^22^{-n}$
Let $A=\{n^22^{-n}, n \in \mathbb{N}\}$. Find $\inf A$, $\sup A$.
I tried starting by proving that $\frac{n^2}{2^n} \leq 1/n$ by induction. After, I showed that $\frac{n^2}{2^n} \geq 0$. By the squeeze theorem,
$$0 \geq \frac{n^2}{2^n} \geq \frac{1}{n}.$$
But my solution is awfully complicated ... |
H: Is the set consisting of $0$ and all polynomials with coefficients in $\mathbb{F}$ and with degree $m$ a subspace of $P(\mathbb{F})$?
Is the set consisting of $0$ and all polynomials with coefficients in $\mathbb{F}$ and with degree $m$ a subspace of $P(\mathbb{F})$?
I want to say no, since it seems that it is not ... |
H: Flipping a coing probability
Given that we have already tossed a balanced coin ten times and obtained zero heads, what is
the probability that we must toss it at least two more times to obtain the first head?
One thing I know is that tossing it initially 10 times is of no use for the answer. However, how can I give... |
H: How to prove this assertion?
If $a<b$, then there exists a positive integer $n$ such that $a \leq b - 1/n$.
I was thinking I could use the Archimedean Property, but I don't know how since I need to get a non-strict inequality $\leq$.
I'd appreciate any hints or ideas. Thank you.
AI: This is really equivalent to sho... |
H: Formally prove: $\lim_{n\to\infty}x_n=L_1\Longrightarrow\lim_{n\to\infty}x_{n+k}=L_1,\forall k\in\mathbb{N}$
OK, so I'm given the following:
$$\lim_{n\to\infty}x_n=L_1\iff\forall\epsilon>0,\exists N(\epsilon)\in\mathbb{N}\ni\forall n>N(\epsilon),\ \left|x_n-L_1\right|<\epsilon$$
I just have no idea how to use tha... |
H: Why must we account for the domain of an original function when we reduce?
Why are reduced rational functions not always "equal to" the original function? Why must we account for the domain of an original function when we reduce? For example,
$$g(x) = \dfrac{x^2 + 3x - 4}{x - 1}$$
has a limited domain, because $x =... |
H: Analytic function bounded by polynomial
Prove that a function which is analytic in the whole plane and satisfies the inequality $|f(z)|<|z|^n$ for some $n$ and all sufficiently large $|z|$ reduces to a polynomial.
The function is analytic, so $f^{n}(z)$ exists for all $n$, all $z$. We have the Cauchy's integral f... |
H: Is the function $ f(x,y)=\frac{xy}{x^{2}+y^{2}}$ where $f(0,0)$ is defined to be $0$ continuous?
Is the function $ f(x,y)=\frac{xy}{x^{2}+y^{2}}$ where $f(0,0)$ is defined to be $0$ continuous? I don't think it is and I am trying to either show this by the definition or by showing that maybe a close set in $\mathbb... |
H: In which spaces,$F$ is irreducible iff $F=\overline{\{x\}}$, for all closed $F$?
I'm wondering which spaces $X$ meet the following condition:
For any closed $F\subseteq X$, $F$ is irreducible if and only if $F=\overline{\{x\}}$ for some $x\in X$.
Where $F$ is irreducible if and only if for all closed sets $F_1,F_... |
H: Calculate $\lim_{n\to\infty}\binom{2n}{n}2^{-n}$
I would like to show that:
$$\lim_{n\to\infty}\binom{2n}{n}2^{-n} = \infty$$
I have gotten as far as:
$$
\binom{2n}{n}={(2n)!\over (n!)^2}=\left({n\over1}+1\right)\left({n\over2}+1\right)(\dots)\left({n\over n}+1\right)\ge2^n
$$
But the $2^{⁻n}$ factor defeats that a... |
H: How many ways are there to distribute 8 teachers to 4 schools where each school must get at least 1 teacher?
Additional details: the teachers are considered distinct from one another.
So here is what I thought:
1) Choose four teachers to go to each one of the schools: $\binom{8}{4}\cdot4!$
2) For each of those situ... |
H: A linear transformation defined by a system of equations carries $\mathbb{R}^n$ onto $\mathbb{R}^m$ iff the rank of the coefficient matrix is $m$.
How do I show that a linear transformation defined by a system of equation carries $\mathbb{R}^n$ onto $\mathbb{R}^m$ if and only if the rank of the coefficient matrix o... |
H: Want to check analyticity of a series on a open disk.
How do we check the analyticity of a any power series? For example:
How will we show that $$f(z):= \sum_{n=1}^\infty z^{n!}= z^1+z^2+z^6+z^{24}......+z^{n!}......$$ is anaytic on disk {$z : |z|<1$}
Thanks.
AI: The coefficients $a_n$ of your power series are $0$ ... |
H: Radius of Convergence of $\sum ( \sin n) x^n$.
Thank you very much in advance for any assistance/advice on solving this problem. I am fairly new to power series and determining the radius of convergence.
Determine, with proof, the radius of convergence of $\sum ( \sin n) x^n$.
My understanding of this problem is ... |
H: Uniform continuity on $\mathbb{C}$
If a continuous function $f: \mathbb{C} \rightarrow \mathbb{C}$ satisfies $f(z) \rightarrow 0$ as $|z| \rightarrow \infty$, then $f$ is uniformly continuous on $\mathbb{C}$.
Should I be thinking about the Riemann sphere here? I have no clue what my intuition should I have be. Any ... |
H: Infinitely many moons, or one ring to bring them all, a limit to bind it?
The Kanagy clan makes its home on a distant planet of mass $M_p$ with $k$ moons. Suppose the moons are identical with mass $m$. Furthermore, these moons share a common circular orbit on an orbital plane. The circular orbits are at distance $L... |
H: Sequence of Ordinals and Ordinal Definability in Levy Collapse Extensions
Let $\kappa$ be an inaccessible cardinal. Let $G$ be generic for $Col(\omega, < \kappa)$, the Levy Collapse.
If $f\in \text{ }^\omega \text{Ord}^{V[G]}$, is $f \in OD_{\text{ }^\omega\omega}^{V[G]}$?
A possibly useful fact is that if $f\in ... |
H: Limit of $n$-th derivative over factorial and exponential function
Suppose $f(z)$ is analytic on the disk $|z|<1$. Prove that $\lim_{n\rightarrow\infty}\dfrac{f^{(n)}(0)}{n!n^n}=0$.
When I see the $n$-th derivative, I think of the Cauchy's formula:
$$\dfrac{f^{n}(0)}{n!n^n}=\dfrac{1}{2\pi in^n}\int_C\dfrac{f(z)}{... |
H: Question on Discrete metric space
Let $X = \{1,1/2,1/4,...,1/2^n,...\} \cup \{0\}$ and $Y = \{X\} - \{0\}$.
Is $Y$ dense in $X$? The metric is the usual. If yes, why a separable discrete metric space is then countable? In this setting $X$ is not discrete? We cannot have accumulation points in discrete spaces?
AI: ... |
H: Matrices as Functions
A friend of mine was criticized in undergrad by a Professor for saying that a matrix is a function.
Now, a matrix can be represented by a linear transformation, and linear transformations by definition are functions.
Is there any theoretical reason as to why a matrix can't be dubbed a functio... |
H: How prove this $\lim_{x\to+\infty}(f'(x)+f(x))=l$
let $f(x)$ is continous and $f'(x)$ is continous on $[0,\infty)$,show that
$$\lim_{x\to+\infty}(f'(x)+f(x))=l$$
if and only if: $\displaystyle\lim_{x\to+\infty}f(x)=l$ and $f'(x)$ is uniformly continuous on $[0,+\infty)$.
How prove this it? Thank you.
I can pr... |
H: Left and Right Cosets of a Subgroup of Index 2
Show that if $H$ is a subgroup of index $2$ in a finite group $G$, then every left coset of $H$ is also a right coset of $H$.
AI: $H$ is a coset, and the only other possible coset is $G \setminus H$ by order considerations. |
H: What is $O(\sqrt{2^n}n^2)$?
What is $O(\sqrt{2^n}n^2)$?
Is it $O(2^n)$, or does the square root cause it to be reduced? I'm trying to analyze an algorithm that I came up with, and if it still has exponential time cost, I'm going to have to try harder.
Thanks!
AI: $\sqrt {2^n}=2^{\frac n2}$, still exponential in the... |
H: showing that there is no simple group of order 48
I tried to solve this prob. by using Sylow theorems.
But i can't solve it because it is not seen in the text as a example or exercise.
I even use a diagram of sets and consider all possible cases, but it is not effective....
AI: Suppose G is simple. Since G is simp... |
H: Complex integral of product of two polynomial powers
Compute $$\int_{|z|=1}z^n\left(z-\dfrac{1}{2}\right)^mdz$$ where $m,n$ are integers.
If $m,n\geq 0$, the function is entire, and so the integral is $0$.
If $n<0$ and $m\geq 0$, the function becomes $\dfrac{\left(z-\dfrac12\right)^m}{z^{|n|}}$. This can be handl... |
H: Problem with $\text{Tor}$ functor
Please explain to me about small $\text{Tor}$ functor problem.
I use $\text{Tor(A,B)}$ define at http://en.wikipedia.org/wiki/Tor_functor.
we take a projective resolution:
$\cdots\rightarrow P_2 \rightarrow P_1 \rightarrow P_0 \rightarrow A\rightarrow 0$ (1)
then remove the A te... |
H: Limit at infinity $\lim_{x\rightarrow \infty}\left(x-\lfloor x\rfloor\right) = $
$\displaystyle \lim_{x\rightarrow \infty}\left(x-\lfloor x\rfloor\right) = $
I have Tried like this way:: Let $x = k\rightarrow \infty$. Then Limit Convert into $\lim_{x\rightarrow k}(x-\lfloor x \rfloor)$
Now When $x = k\rightarrow \i... |
H: Why is the modulus of a complex number $a^2+b^2$?
Why is the modulus not $\sqrt{a^2-b^2}$? Carrying out standard multiplication this would be the result-why is this not the case? I know viewing the complex plane you can easily define the sum as being the distance to the points, but what meaning does $\sqrt{a^2-b^2... |
H: Help Solving First Order nonlinear ODE
$e^y + [t e^y -sin(y)] \frac{dy}{dt}=0 \quad y(2)=1.5$
Is it possible to find an exact solution to this ODE using some method?
Following Adriano's hint: (second edit)
$d( e^y t + cos(y) ) =0$
or
$e^y t + cos(y)=c_1 $ for some constant $c_1$
and then using the initial value con... |
H: How can we prove that a Fourier Series exists?
How does one show that an arbitrary periodic function, so long as it is reasonably well behaved, can always be represented as a sum of sine and cosine functions? It sounds like the first thing you would learn when it comes to Fourier Series but my text simply claims th... |
H: Can Prims and Kruskals algorithm yield different min spanning tree?
In this problem I am trying to find the min weight using the Prims and Kruskals and list the edges in the order they are chosen.
For Prims I am getting order (A,E), (E,F), (F,C), (C,D), (C,B) with a weight of 21
With Kruskals I get (C,F), (A,E), ... |
H: Invertible polynomials and power series
Consider the polynomial ring $A[x]$ and $f(x)=\sum_{i=0}^{n}a_ix^i\in A[x]$, where $A$ is a commutative ring with unit. Show that $f$ is a unit in $A[x]$ if and only if $a_0$ is a unit in $A$ and $a_1,\dots, a_n$ are nilpotent.
Consider the formal power series ring $A[[x]]$ ... |
H: Condition on spectrum of T
Let $T$ $\in \mathfrak{B}(\mathbb{H})$ be normal. Let $A$ be the closed subalgebra generated by $T$, $T^{*}$ and $I$. Suppose $T$ can be approximated in norm by finite linear combinations of projections in $A$. What is the necessary and sufficient condition on $\sigma(T)$ for this to happ... |
H: ways to fill n places with fixed values in a given range and avoid diplicates formed.
what are the number of ways in which we can fill n places with 2 fixed values and rest places with values between the 2 selected ones such that we get no duplicates?
example : n=4 , fixed values 1 and 3
case 1 : fix 1 on position ... |
H: General question on parameter functions
I have always wondered why considering paths, $\gamma(t)$, there is inevitably a condition that $\gamma'(t) \neq 0$ associated within the same sentence.
Can someone please give me a motivational reasoning behind this; whether by examples or goals etc.
I have intentionally lef... |
H: How do we validate the equality of function while reducing the indeterminate form to simpler function?
To evaluate the limit of an indeterminate form, we need to reduce the function to its simplest form in order to get a meaningful answer.
For example: $f(x_1)$=${x^3-1\over x-1}$, evaluate $lim_{x->1} {x^3-1\over x... |
H: Calculate the Factor Group $(\mathbb{Z}_4 \times \mathbb{Z}_6)/\langle(0,2)\rangle$
I am attempting to understand and compute: $(\mathbb{Z}_4 \times \mathbb{Z}_6)/\langle(0,2)\rangle$
I know $(0,2)$ generates $H = \{(0,0),(0,2),(0,4)\}$, which has order 3 because there are 3 elements. Now, I must find all the co... |
H: Probability of 4 or fewer errors in 100,000 messages
The probability of an error occurring in a message is 10^-5. The probability is independent for different messages. There are 100,000 messages sent. What is the probability that 4 or fewer errors occur?
AI: In principle, the number $X$ of errors in $100000$ messa... |
H: Complete separable metric space
Let $K$ be an algebraically closed field of cardinality continuum containing the rationals.
Is there always a (canonical if possible) way to equipp $K$ with a topology which makes it into a separable complete metric space?
AI: If $K$ is an algebraically closed field of cardinality $... |
H: Real random variable $X$ such that $ \lim_{n\to \infty}{nP(|X|>n)}=0$ what about $ E|X|<\infty$?
Let $X\colon (\Omega,\mathcal F,\mathbb P)\to (\mathbb R,\mathcal B(\mathbb R))$ be a random variable from a probability space to the real numbers with the Borel sets.
I proved that if $\mathbb E|X|<\infty$ then $ \lim... |
H: Question About Dedekind Cuts
Rudin gives the definition of a Dedekind Cut to be:
A set of rational numbers is said to be a cut if
(I) $\alpha$ contains at least one rational, but not every rational;
(II) if $p\in\alpha$ and $q<p$ (q rational), then $q\in\alpha$;
(III) $\alpha$ contains no largest rational.
I'm conf... |
H: Why is orthogonal basis important?
Lets take the $\mathbb{R}^3$ space as example. Any point in the $\mathbb{R}^3$ space can be represented by 3 linearly independent vectors that need not be orthogonal to each other. What is that special quality of orthogonal basis (extending to orthonormal) that we choose them over... |
H: In topological group left and right multiplication are homeomorphic
From Wikipedia (http://en.wikipedia.org/wiki/Topological_group): The inversion operation on a topological group $G$ is a homeomorphism from $G$ to itself. Likewise, if $a$ is any element of $G$, then left or right multiplication by $a$ yields a hom... |
H: When does the limit of a sum become an integral?
In many maths and physics texts and courses, I've been told in many cases that the limit of a sum becomes an integral, i.e. (very roughly):
$$\lim_{n\to\infty} \sum_{x=0}^n f(x) = \int_0^\infty f(x) dx$$
However I know that this equation can't be true in every case, ... |
H: A quick question on scalars and commutators.
I just had a quick question about the factoring of scalars within the commutator.
Say we have operators $\hat{A}$ and $\hat{B}$, and scalars $a$ and $b$. If we take the commutator: $$[a\hat{A}, b\hat{B}]$$That is just equivalent to $ab(\hat{A}\hat{B} - \hat{B}\hat{A})$, ... |
H: Prove that flow is a linear combination of flow cycles and flow paths
Let $D=(N,A)$ be a directed graph, and for an arc $e=xy$ define $h(e)=x$ and $t(e)=y$. A flow is $\mathbf{x}=(x(e_1),\dots,x(e_k))$ with $\sum_{e:t(e)=v}x(e)=\sum_
{e:h(e)=v}x(e)$ for all $v\in A\backslash \{s,t\}$. A flow cycle is a flow $\mathb... |
H: Can someone help me find the sum of the following series?
I am working on one of the fractals and finding its convergent area.
$$\begin{align}
S & = 1+3\left(\frac{1}{9}+4(\frac{1}{9^2})+4^2(\frac{1}{9^3})+...\right)\\
& = 1+3*\sum_{i=0}^{\infty} \left[\frac{4^i}{9^{i+1}}\right]
\end{align}$$
I want to say that... |
H: How does the law of cosines help with this problem?
I've gotten stuck on this problem from Stewart Calculus, 7th edition:
A runner sprints around a circular track of radius 100 m at a constant speed of 7 m/s. The runner's friend is standing at a distance 200 m from the center of the track. How fast is the distance... |
H: Prove that $|\cos(\sin(x_1)) - \cos(\sin(x_2))| \leq |x_1 - x_2|, \forall x_1, x_2 \in \mathbb R$.
I asked this question without any limitation on methods that might be used. I believe it's turned out to be interesting to see a variety of different approaches. It turns out that the aim of the exercise that my stude... |
H: $z$-transform of $1/n$
How can one calculate the $z$-transform of:
$x(n) = \frac{1}{n}$ , where $n \geq 1$? I have searched for table entries, then got stuck while trying to do it with the definition of $z$-transform (summation).
AI: From what I can gather, by definition, the $z$-transform is the sum
$$
\sum_{n = ... |
H: if $f(x) = x-\frac{1}{x}.$ Then no. of solution of the equation $f(f(f(x))) = 1$
If $\displaystyle f(x) = x-\frac{1}{x}.$ Then no. of solution of the equation $f(f(f(x))) = 1$
$\underline{\bf{My\;\; Try}}::$ Given $\displaystyle f(x) = x-\frac{1}{x} = \frac{x^2-1}{x}.$ Now Replace $\displaystyle x\rightarrow \frac{... |
H: remainder on division by 9 for a tricky number
What will be the remainder when $ 32^{32^{32}} $ is divided by 9?
I was able to solve this by using the cyclicity of remainders when $2^{2^n}$ is divided by 9. For $n$ =even it gave remainder 7 and for $n$ =odd it gave remainder 4. So using this I worked out the ab... |
H: Invariant Subrepresentation of Induced Representation
In what follows we let $G$ be a compact Lie group and $H\lneqq G$ a
closed subgroup. We denote by $\mu_{G},\mu_{H}$ the respective Haar
measures.
Let $(\mathcal{H},\langle,\cdot\,,\cdot\rangle_{\mathcal{H}})$ be a
Hilbert space. A unitary representation $\Phi$ o... |
H: Bounding the integral of the tails of a random variable.
I found an argument like this in a book, but I couldn't understand how we got this bound.
Suppose $X_n$ is a sequence of random variables. For some $\delta > 0$ and all
$n \geq 1$,
$$
\int_{|X_n| \geq M} |X_n| \, dP \leq \frac{1}{M^\delta} E[|X_n|^{1 + \delta... |
H: Derivation of weak form of Euler Lagrange Equation
In Giaquinta's and Giusti's 1982 paper entitled "On the regularity of the minima of variational integrals", they look at the following quadratic functional:
\begin{equation}
F(u)=\int_{\Omega}\sum_{i=1}^{N}\sum_{\alpha, \beta=1}^{n}A^{\alpha\beta}(x, u)D_{\alpha}u^... |
H: Does $i^T=-i$ or $i^T = i$ ?(T is transpose)
Assume $A$ is a skew-symmetric matrix(its eigenvalues is zero or purely imaginary). If $x$ is its eigenvector, we have $$x^TAx=\lambda|x|^2$$, take transpose on both sides. we have $$-x^TAx=(\lambda|x|^2)^T$$
If $\lambda \in \Bbb{R}$,then $\lambda = 0$. If $\lambda$ is ... |
H: Solve the limits by alternative method
We know, that $\displaystyle\lim_{n\to ∞}$ $(1+2+3+...+n)\over n^2$
= $\displaystyle\lim_{n\to ∞}$ $1\over n^{2}$.${n(n+1)\over 2}$
= $\displaystyle\lim_{n\to ∞}$ $1\over 2$$({1}+{1\over n})$=${1\over 2}$$({1}+{1\over ∞})$=$1\over 2$
But, doing it in alternate way by the ... |
H: Asymptotic behaviour of real sequences
Let's say we have two real sequences $(a_n)_{n\in\mathbb{N}}$ and $(c_n)_{n\in\mathbb{N}}$ with $c_n\in o(\frac1n)$ (i.e. $c_n(\frac1n)^{-1}\xrightarrow{n\rightarrow\infty}0$). And for all $\epsilon>0$ holds that
$$\frac{1-\epsilon}n-\frac1\epsilon |c_n|\leq a_n\leq \frac{1+\e... |
H: Given $n \in \mathbb{N}$ prove that a polynomial result gives a natural number.
A friend asked me this question:
Prove that for every $n\in \Bbb N$ the next equation result: $\dfrac{n^3}{6}+\dfrac{n^2} {2}+\dfrac{n}{3}$ would be a natural number.
My instincts were that i need to use induction, Though i'm not sure h... |
H: How do I prove that $\lim_{x\to0^+} x^{1/x}=0$
How do I prove that $\lim_{x\to 0^+} x^{1/x}=0$
It looks like on of those situations in which l'hospital will come in handy but that doesn't seem to work as well.
AI: Hint: Try to find the limit of logarithm of the function and use $\ln (x^{1/x})=\frac{\ln(x)}{x}$. Th... |
H: Video lectures on Group Theory
The web is full of video lectures these days but, try as I might, I can find very little for Introduction to Group Theory. The closest I found was http://www.extension.harvard.edu/open-learning-initiative/abstract-algebra . Are they any online introductory group theory lectures people... |
H: Closed form for $\int \frac{1}{x^7 -1} dx$?
I want to calculate:
$$\int \frac{1}{x^7 -1} dx$$
Since $\displaystyle \frac{1}{x^7 -1} = - \sum_{i=0}^\infty x^{7i} $, we have $\displaystyle(-x)\sum_{i=0}^\infty \frac{x^{7i}}{7i +1} $.
Is there another solution? That is, can this integral be written in terms of eleme... |
H: How to prove that $x/y$ is continuous in R
$f:R^2$ \{y=0} $\Rightarrow R$ , $f:(x,y)\Rightarrow x/y$.
Prove (formally) that $f$ is continuous.
I think what I should show is that any point that belongs to an open ball of radius $\epsilon$ of image, has a pre-image that belongs to an open ball around (x,y), and sinc... |
H: Is $\Gamma L(r,q^t)$ a subgroup of $\Gamma L(rt,q)$?
I know that $GL(r,q^t)$ can be seen as a subgroup of $GL(rt,q)$ since every linear transformation on the vector space $V(r,q^t)$ extends in a unique way to a linear transformation on the vector space $V(rt,q)$.
For $\Gamma L(r,q^t)$ however, I run into trouble if... |
H: Map from $\mathbb {R}$ to $\mathbb {R^2}$
Is there a way to construct an injective function that map from $\mathbb{R}$ to $\mathbb{R^2}$? If yes, please give me an example. Thank you!
AI: Injective means that for $(x,y) \in \mathbb R^2$ and $a,b\in\mathbb R$ we have $f(a)=(x,y) = f(b) \Rightarrow a=b$ thus all we n... |
H: Singular values in SVD
I have recently started reading about SVD. If factorization of a matrix $A$ is required, we calculate the eigenvectors of $AA^T$ and $A^TA$ and they become the column vectors of $U$ and $V$ matrices correspondingly. The $\Sigma$ matrix is filled with square roots of the eigenvalues. Now we fi... |
H: Analytical solution to Poisson’s equation 1D
I need some help in finding u(x) analytically where equation and the boundary conditions are satisfied
AI: Anything wrong with integrating twice?
$$u(x) = -\frac{x^2}{2} + A x + B$$
$$u(0) = 1 \implies B = 1$$
$$u(1) = 2 \implies -\frac12 + A + B = 2 \implies A=\frac{3... |
H: Why “Syracuse” in “Syracuse problem”
Is “Syracuse” in “Syracuse problem” (a variant name of Collatz conjecture) a reference to the city of Syracuse in Sicily, to one of several Syracuses in USA or something else (a person's name, for instance)?
AI: "The name Syracuse problem was proposed by Hasse during a visit to ... |
H: prove uniqueness of a measure
If $(X,\mathcal{M},\mu)$ is a measure space and $\mathcal{\overline{M}}:=\{E\cup F:E\in\mathcal{M}\text{ and }F\subset N\text{ for some }N\in \mathcal{N}\}$ is a completion of $\mathcal{M}$ with respect to $\mu$ where $\mathcal{N}:=\{N\in\mathcal{M}:\mu(N)=0\},$ then $\mathcal{\overlin... |
H: Irrationality of a unique positive root of $\sin{x} = x^2$
The equation $\sin{x} = x^2$ has a unique positive real root. I wonder if there is any standard technique how to show that this number is irrational (rational), preferably a technique which works also in other similar scenarios.
I tried an inverse symbolic ... |
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