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H: Prove that $R(P) = N (I − P) = X$ and $R(I − P) = N (P) = Y$.
Suppose that $V = X ⊕Y$, and let $P$ be the projector onto $X$ along
$Y$. Prove that
$R(P) = N (I − P) = X$ and $R(I − P) = N (P) = Y$.
I know that from $V = X ⊕Y$ I got $v=x+y$ for $v,x,y$ are element of $V,X,Y$ and the intersect of $X$ and $Y$ is zero... |
H: Show that $\{1, \sqrt{2}, \sqrt{3}\}$ is linearly independent over $\mathbb{Q}$.
My apologies if this question has been asked before, but a quick search gave no results. This is not homework, but I would just like a hint please. The question asks
Show that $\{1, \sqrt{2}, \sqrt{3}\}$ is linearly independent over... |
H: Existence of isomorphism between tensor products.
In multilinear algebra many maps are usually proven to exist rather than simply defined. For example, commutativity is one such example. In the book I'm studying the author says: let $V_1,\dots,V_k$ be a collection of vector spaces over $K$, then if $\sigma \in S_k$... |
H: Zeros of complex function sequence (Application of Rouche's Theorem).
For a given sequence of complex functions: $\phi_n(z)= 1+\frac1n-z-e^{-z}$; here $z\in${$z| Rez>0$}.
I want to prove that :
(1).
$\phi_n $ has a unique zero $z_n$ in the half plane. (i.e. there exists a unique $z_n$ in half plane {$z| Rez>0$} s.... |
H: Probability - $a$ white balls, $b$ black balls
From a urn containing $a$ - white balls and $b$ - black balls are pulled out $k$ balls ($k < a+b$) which are putted aside without knowing their color. Then, it is pulled out another ball. Which is the probability this last ball to be white?
I don't know very much abou... |
H: Continuity and openness proof
I need some help proving the following theorem. My professor said that it was a "local" version of an important theorem:
Suppose f:X→Y and a ∈X. The function f is continuous at a if
and only if for every open set U containing b = f(a), there is an open set V containing a so that V ⊂ f−... |
H: Homework - combinatorics
How many solutions does this inequality have in non negative integers $x_i$:
$n \leq x_1+x_2+x_3+ \dots +x_n \le 2n$ ? Im stumped
I know I should add another variable but...I don't know.
AI: Suppose $s$ is positive integer. Now create a row of $n + s - 1$ blanks, and imagine you have $s-1... |
H: How to show that $\frac{\partial}{\partial y}\left(\int_{0}^{y}\frac{1}{x+it-2}dt\right)=\frac{1}{x+iy-2}$
I'm trying to show that $$\frac{\partial}{\partial y}\left(\int_{0}^{y}\frac{1}{x+it-2}dt\right)=\frac{1}{x+iy-2}$$ In an area that doesn't contain the point $2+0i$. If the function under the integral was a re... |
H: Prove by induction that a^n+a^-n is an integer.
I am to prove by induction that given that $a+1/a$ is an integer (i.e. belongs to Z ) then $a^n+1/a^n$ is an integer too.
I'm pretty much clueless here. Thanks in advance.
AI: HINT:
We need Strong induction here
Use
$$\left(a^n+\frac1{a^n}\right)\left(a+\frac1a\righ... |
H: Simple Divisor Summation Inequality (with Moebius function)
Show that
$$\left| \sum_{k=1}^{n} \frac {\mu(k)}{k} \right| \le 1 $$
where $\mu$ is Moebius function and n is a positive integer.
The hard thing here is that the sum is not directly divisor sum; it's just a normal summation. What I know is that when $F(n... |
H: convergence of a function serie to norm 2
need to show if the following function serie converge on $||.||_2$ on [1,5]:
$$\sum _1^\infty {sin^2(nx) \over n^2} $$
I have no idea how to approach that one, would like
for some directions...
AI: In fact it converges uniformly. Let $f_n(x)=a_nsin^2(nx)$ with $a_n=1/n^2$... |
H: Question related to calculation of probality?
question given in my text book
There are three events A, B and C out of which only one and one can happen. The odds are 8 to 3 against A and 5 to 2 against B. Find odd against C.
Solution in my textbook
Let the total no. of cases = $m + n + p$
$m$ are in favor of $A$,... |
H: Proving that there exists a local minimum between two local maximums of a continuous function
Suppose f is a continuous function that has local maximums at points $x_{1}$ and $x_{2}$. How do I prove that there is a third point between these two points which is a local minimum of f? Need some help thanks.
AI: Hint: ... |
H: How to solve the equation $x^2 + 4 |x| - 4 = 0$
How do I get the value of x from the equation that is provided
AI: Putting $x=a+ib$ where $a,b$ are real
we get $\sqrt{a^2+b^2}=4-(a+ib)^2=4+b^2-a^2-2abi$
Equating the imaginary parts $ b=0$
Now for real $x,$
$$|x|= \begin{cases} +x &\mbox{if } x\ge0 \\
-x & \mbox{if... |
H: Can $\|f\| = \|a\|_{q}$ to arbitrary values of $p$ and $q$ satisfying ${1 \over p} + {1 \over q} = 1$
We all know that:
Suppose $a = (a_{1}, a_{2}, ..., a_{n})$ is a point in Euclide space $R^{n}$. Consider the mapping $f: R^{n} \rightarrow R$, $f(x) = \sum_{i=1}^{n}a_{i}x_{i}$. Then $\|f\| = \|a\|$.
So I wond... |
H: Maximal graph that does not contain Hamiltonian cycle
My lecture notes in Graph Theory states that a graph of order $n$ and with size (= number of edges) $\binom n 2-(n-2)$ is the maximal graph that does not contain a Hamiltonian cycle.
My question now is how to find such a graph? How can I construct a graph of or... |
H: Question about an element at $\mathbb{Z}_{n}^{*}$
Assume that $n,q>1$ and $n=\frac{q^r-1}{q-1}$.
How do I prove that $q\in \mathbb{Z}_{n}^{*}$?
$$\mathbb{Z}_{n}^{*}= \left\{a\mid1\le a<n;\ \gcd(a,n)=1 \right\}$$
Thank you!
AI: You have $\frac{q^r-1}{q-1}=1+q+q^2+\ldots +q^{r-1} = n$.
So $q < n$ and $n + q(-1-q-\ld... |
H: Limit at infinity -
I don't even know where to start on this one. It's obvious that it's infinity, but how to prove it? $$\lim_{n\to\infty}\frac{3^n+2n^n+n!}{(n+1)^4+\sin n+(3n)!}$$
AI: Hint: start rewriting it as
$$
\lim_{n\to\infty}
\frac{n^n
\left(
\dfrac{3^n}{n^n}+2+\dfrac{n!}{n^n}
\right)
}
{(3n)!
\left(
\dfra... |
H: Show that $\frac{(2n)!}{(n)!}=2^n(2n-1)!!$
Show that $\frac{(2n)!}{(n)!}=2^n(2n-1)!!$ is the question I am struggling with.
I started by saying: $(2n)!=2n(2n-1)(2n-2)(2n-3)...3*2*1$
But then I'm stuck.
AI: $$(2n)!=\prod_{k=1}^{2n}k=\prod_{k=1}^{n}(2k)\cdot\prod_{k=1}^{n}(2k-1)=2^n\cdot\prod_{k=1}^{n}k\cdot\prod_{k=... |
H: Prove that $n(n^2 - 1) = \frac{(n+1)!}{(n-2)!}$
Prove that for all $n \in \mathbb{N}$,
$$n(n^2 - 1) = \frac{(n+1)!}{(n-2)!}.$$
Thanks in advance.
AI: $\require{cancel}$
If you really want to express $n(n^2 - 1)$ using factorials, note that $$n(n^2 - 1) = n(n+1)(n-1) = (n+1)(n)(n-1) = \frac{(n+1)n(n-1)\cancel{(n-2)}... |
H: 0-1 Law in a sigma algebra Conditional Expectation
I get stuck on a simple question.
Let T be a sigma algebra with for all its elements A in T P(A)=0 or P(A)=1.
Assuming X is L1 find E[X/T]
I was thinking that T had to be a partition or not necessary?
Thx in advance
AI: $\mathcal T$ is not necessarily generated by ... |
H: function as a net?
One can see net as a generalization of a sequence.
As done in http://en.wikipedia.org/wiki/Net_(mathematics), in the special case where $f: M\backslash \{a\} \rightarrow X$ where M is a metric space and X is a topological space, one can see $f$ as a net from the directed set $M\backslash \{a\}$ ... |
H: Prove that $\binom{n}{r} + \binom{n}{r+1} = \binom{n+1}{r+1} $
Prove that $\binom{n}{r} + \binom{n}{r+1} = \binom{n+1}{r+1} $
Thanks in advance, my professor asked us to this a couple weeks ago, but I was enable to get to the right answer.
Good luck!
Here is what I got up to;
$\frac{(n+1)!}{(n-r)!(r+1)!} = \frac{... |
H: System of linear equations with parameters, using a matrix
Let there be the following system of linear equations:
$$x+z+bw=a \\
ax+y+az+(a+ab)w=1+a^2 \\
bx+(a+b)z+(1+b^2)w=4+a\\
bx+bz+(a-ab+b^2)w=a+1+ab$$
a,b parameters. The question is, for which a,b there is no solution to the system, for which there are infinit... |
H: is the set of all sequences that converge to zero first category
the metric space is: the set of all convergent sequences with metric supremum.
the subset is: all sequences that converge to zero.
is the subset first category in this metric space?
we have figured out that the subset is closed and it is rare (nowhere... |
H: Integral from $0$ to $\infty$ of $\frac{1}{3}\ln\left(\frac{x+1}{\sqrt{x^2-x+1}}\right)+\frac{1}{\sqrt 3}\arctan \left(\frac{2x-1}{\sqrt 3} \right)$
Evaluate the integral $$ \int_0^\infty \left( \frac{1}{3}\ln\left(\frac{x+1}{\sqrt{x^2-x+1}}\right)+\frac{1}{\sqrt 3}\arctan \left(\frac{2x-1}{\sqrt 3} \right) \right)... |
H: How do you prove that $\lim f(x) = 0$, when $f$ is rapidly decreasing?
Let $f: \Bbb{R} \to \Bbb{R}$ be rapidly decreasing in the sense than $\sup_{x \in \Bbb{R}} |x|^k |f^{(\ell)}(x)| \lt \infty$ for all $k, \ell \geq 0$, where $f^{(\ell)}$ is the $\ell$th derivative. It seems like $\lim_{x \to \infty} f(x) = 0$, ... |
H: How to solve this system of equations.
Solve the system of equations: $$\begin{cases}\dfrac{x^2+xy+y^2}{x^2+y^2+1}=\dfrac{1}{xy} \\\left(\sqrt{3}+xy\right)^{\log_2x}+\dfrac{x}{\left(\sqrt{3}-xy\right)^{\log_2y}}= 1+\dfrac{x}{y}\end{cases}$$
My try:
$\dfrac{x^2+xy+y^2}{x^2+y^2+1}=\dfrac{1}{xy}\\\Leftrightarrow xy\l... |
H: combinatorics implementation in real life problems
How many ways there are to organize $7$ men in a row, if two insist on not standing next to each other? How do I approach this?
AI: There are a total of $7!$ ways to organize the $7$ men in a row with no restrictions. Now we must count all the ways in which $2$ spe... |
H: show that 210 is a triangular number
Show that 210 is a triangular number. Would it suffice to solve the equation 210=((n)(n+1))/2 ? Then n is equal to 20 and -21but n in this case must be positive, so 210 would be the 20th triangular number.
AI: Yes, indeed, your solution suffices. Well done! |
H: "closure preserves homeomorphism"
Let me explain the title of the problem and the problem very clearly :
If $X$ and $Y$ are subsets of a topological spaces $A$ and $B$ respectively, which are homeomorphic in the respective subspace topology, does it imply that their closure $\bar{X}$ and $\bar{Y}$ are homeomorphic ... |
H: How many different 8-letter words can be made with three $a$s, two $b$s, two $c$s and a $d$?
How many words, without making any reference to their meaning can be written from the letters: $ a,a,a,b,b,c,c,d$ ?
what is the best approach to solve this kind of problem ?
AI: Assuming you need to find the number of dist... |
H: Why this two spaces do not homeomorphic?
Consider $\Bbb Q$ with subspace topology and $\Bbb Q\times \Bbb Q$ with product topology. Why this two spaces are not homeomorphic?($\Bbb Q$ is the rational numbers)
AI: It is a theorem of Sierpinski that every countable metric space without isolated points is homeomorphic t... |
H: Find the sum of the series
For any integer $n$ define $k(n) = \frac{n^7}{7} + \frac{n^3}{3} + \frac{11n}{21} + 1$ and $$f(n) = 0 \text{if $k(n)$ is an integer ; $\frac{1}{n^2}$ if $k(n)$ is not an integer } $$
Find $\sum_{n = - \infty}^{\infty} f(n)$.
I do not know how to solve such problem of series. So I could no... |
H: Solving the differential equation $y'' + 2y' + 2y = 0$ given constraints
How can I solve this initial value problem?
$$ y'' + 2y' + 2y = 0,$$ given $y\,(\pi/4)=2$ and $y'(\pi/4)=0$.
I've found $y(t)=e^{-t} \left(C_1\cos t + C_2\sin t \right)$ but I wasn't able to find $C_1$ and $C_2$. How can I find them?
AI: Using... |
H: What is the meaning of "integral point"?
While reading this paper (http://cowles.econ.yale.edu/P/cd/d04b/d0473.pdf) I encountered the concept of "integral point", used first in definition 5.1, on page 34. Does anybody know more details about this?
AI: In this context it simply means a point in $\mathbb{R}^n$ with i... |
H: Finding $\lim_{x \to 0} x^x$ without l'Hôpital
I have to find the limit of $x^x$ as $x$ approaches $0$ without derivatives.
AI: We wish to find $\lim_{x\to0^{+}}x^{x}$. Notice
$$x^{x}=e^{x\ln(x)}=e^{(-1)\frac{\ln(\frac{1}{x})}{(\frac{1}{x})}}$$
so it suffices to find $\lim_{y\to\infty}\frac{\ln(y)}{y}$.
$$\frac{\ln... |
H: Test for convergence/divergence of $\sum_{n=1}^{\infty}(-1)^n\sin\left(\frac{n}{\pi}\right)$
Given the series
$$\sum_{n=1}^{\infty}(-1)^n\sin\left(\frac{n}{\pi}\right)$$
I need to test for convergence/divergence. I think the divergent test might work here. I could see that the $\lim_{n\rightarrow\infty}(-1)^n\sin... |
H: Poles of abelian differentials
Let $X$ be a smooth projective curve of genus $g$ over an algebraically closed field $k$. As a corollary of the Riemann-Roch theorem we know that for every abelian differential $\omega$ on $X$ we have
$$ \deg(\omega) = 2g - 2. $$
Now assume that $g\geq 2$, then $\deg(\omega)\geq 0$. ... |
H: Proof of the properties of limits of CDFs
The cumulative distribution function is defined as $F(a) = \mu((-\infty,a])$ where $\mu$ is a probability measure on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$. Given this definition, it is easy to prove right-continuity (I think).
We have also: $$\lim_{a\to -\infty} F(a) = 0$$... |
H: Validity of a Limit Proof
I am trying to show that if $\displaystyle{\lim_{s\to\infty}} s_n = s$, $\displaystyle{\lim_{s\to\infty}} \sqrt{s_n} = \sqrt{s}$ for some sequence $s_n$.
We must note that $s_n > 0$ for all $n$ for the limit to be defined in $\mathbb{R}$.
I've put together this proof, but I'm not sure if ... |
H: Fourier transform of a function over finite group
Let $G$ be finite abelian group and $\hat G$ be its character group.
The Fourier transform of a function $f:G \to \mathbb C$, is the function $\hat{f}:\hat{G}\to \mathbb C$ defined by $\hat{f}(\chi)=\sum_{a\in G}f(a)\chi(-a)$.
I need hint proving that $\hat{\hat{f}... |
H: Integral curve for vector field tangent to sphere
Let $S^1$ be the unit sphere $x_1^2+x_2^2=1$ in $\mathbb{R}^2$ and let $X=S^1\times S^1\in\mathbb{R}^4$ with defining equations $f_1=x_1^2+x_2^2-1=0, f_2=x_3^2+x_4^2-1=0$. The vector field $$w=x_1\frac\partial{\partial x_2}-x_2\frac\partial{\partial x_1}+\lambda\le... |
H: If a Banach space $X$ is isometric to its first dual $X^*$, must $X$ be reflexive?
Suppose that $X$ is a Banach space such that there exists a linear isometry $X \rightarrow X^*$. Must $X$ be reflexive?
Of course, this implies that $X$ is isometric with its second dual $X^{**}$. But with this alone it is not possib... |
H: Properties of Lie derivative
Let's have Lie derivative:
$$
L_{V}\varphi = V^{\mu}\partial_{\nu}\varphi , \quad L_{V}A_{\mu} = V^{\nu}\partial_{\nu}A_{\mu} + (\partial_{\mu}V^{\nu})A_{\nu}.
$$
How to show that for scalar and vector fields
$$
L_{V}L_{U} - L_{U}L_{V} = L_{[U, V]}, [U, V]^{\nu} = U^{\mu}\partial_{\mu}... |
H: Constructing a cochain complex out of a chain complex
Let $(C,\partial)$ be a chain complex where $C_i$ is an $R$-module ($R$ is a given ring) , we can always construct a cochain complex out of the chain complex $(C,\partial)$ in the following way: We construct the $i$th $R$-module of the cochain complex as $C^i=... |
H: Norm in $L^2$ goes to zero for function away from zero
Let $f\in L^2(\mathbb{R})$ and $M>0$. Define $f_M(x)=f(x)$ for $|x|\leq M$ and $f_M(x)=0$ for $|x|>M$. Show that $\|f_M-f\|_2\rightarrow 0$ as $M\rightarrow \infty$.
Well, we have $$\|f_M-f\|_2^2=\int_{|x|>M}|f|^2dx$$
Since $f\in L^2(\mathbb{R})$, we know that ... |
H: $Z$ score probability
I was given a question where I was supposed to find the probability of obtaining $y$ between two scores, however when I input my answer it tells me that I'm wrong, the question is given below along with my answer to the question:
Question
For a normal distribution with sample mean $= -19$ and ... |
H: Prove that $\sqrt{x}$ is continuous on its domain $[0, \infty).$
Prove that the function $\sqrt{x}$ is continuous on its domain $[0,\infty)$.
Proof.
Since $\sqrt{0} = 0, $ we consider the function $\sqrt{x}$ on $[a,\infty)$ where $a$ is real number and $a \neq 0.$ Let $\delta=2\sqrt{a}\varepsilon.$ Then, $\forall x... |
H: Elementary Properties of cyclic groups
Homework Problem from Group Theory:
Prove the following:
For any cyclic group of order n, there are elements of order k, for every integer, k, which divides n.
What I have so far..
Take G as a cyclic group generated by a. >>>> G=, a^(n)=e, where e is the indentity.
I know that... |
H: Using Plancherel formula to compute exponential integral
Let $c>0$. Use the Plancherel formula to compute the integral $$\int_{-\infty}^\infty \dfrac{1}{c^2+y^2}dy$$
We note that the Fourier transform of the function $f(x)=e^{-cx}$ for $x\geq 0$ and $0$ for $x<0$ is $\dfrac{1}{c+iy}$. (This follows from direct co... |
H: Hemitian operator inequality
I am trying to find two Hermitian operators $A$ and $B$ (whose representations are $2 \times 2$ complex matrices) for which neither $A \leq B$ nor $A \geq B $ holds.
Note that $A \geq B$ iff $\langle(A-B)x, x \rangle\geq 0$ where $\langle~{,}~\rangle$ is an inner product.
Any help would... |
H: 15 distinguishable balls into five distinguishable boxes, Assume there is no restriction on which box gets one ball, which box gets two balls, etc.
I am preparing for an exam and I came across this problem. I am a little confused.
Give the expression of ways to distribute 15 distinguishable balls into five
distingu... |
H: Ring of continuous functions on $\mathbb{R}$, maximal ideal, quotient
Let $I(S) = \{f \in \mathcal{C}(\mathbb{R}) \ | \ \ \forall x \in S: f(x)=0\}$
I've already proven that it is an ideal in the ring $\mathcal{C}(\mathbb{R})$.
However, I have troubles proving that if $I(S)$ is maximal, then it must have the form... |
H: Is there ever a requirement to change the limits of integration?
I don't have issues with doing integration problems, but occasionally I see the solution changing the limits of integration whenever a $u$-substitution is done.
I obviously don't have a problem doing this, and I just recently noticed my book doing thi... |
H: Is there a compact complex manifold with trivial $H_2$?
I don't believe that every complex manifold should have nontrivial $H_2$, otherwise we would easily prove the Chern's conjecture... But the problem is I don't have any counterexample. The Kähler manifold will have nontrivial $H_2$ and so do Riemann surface. He... |
H: Showing that $x^me^{-ax}$ is bounded
Let $a>0$, $m\geq 0$ and let $f:[0,\infty)\rightarrow\mathbb{R}$ be defined as $f(x)=x^me^{-ax}$. It should be true that this function is bounded, because near $0$ both terms are bounded, and far from $0$ the term $e^{-ax}$ decreases more rapidly than the term $x^m$ increases. W... |
H: understanding the proof of even-odd handshake problem
I´d like to know if I understand correctly the argument behind the even-odd handshake problem. Basically the theorem says that the number of persons who have shaken an odd number of hands is even.
My reasoning goes like this. Suppose that the number of people wh... |
H: Question on one theorem for uniform continuity.
In my text book, one theorem states this. A real-valued function $f$ on $(a,b)$, is uniformly continuous on $(a,b)$ if and only if it can be extended to a continuous function $g$ on $[a,b].$ And the book gives two examples. The first example says that the function $f(... |
H: homework combinatorics carousel
I was asked this question in my homework:
How many different combinations are there to paint a carousel with $n$ seats, in $r$ different colors, such that any combination that you can get via rotating is considered the same combination?
Example: $red -> green -> blue$ is the same arr... |
H: Show that if $a \sim b$, then $C(a) = C(b)$ where $C(x)$ is the equivalence class containing $x$.
I'm working through a textbook on my own, so I don't want the full answer. I'm only looking for a hint on this problem.
Show that if $a \sim b$, then $C(a) = C(b)$ where $C(x)$ is the equivalence class containing $x$.... |
H: What's the proper proof to show that H is a subspace of V?
Let $V$ be the vector space, which is defined as $$V=\left\{f \colon \mathbb{R} \to\mathbb{R} \right\}.$$ and let H be the set of functions which satisfy $f{(0)}=0$, i.e. $$H=\left\{ f\colon \mathbb{R} \to \mathbb{R}\colon f{(0)}=0 \right\}.$$ How would one... |
H: Ensuring positive definite matrix defined by variables in Matlab
I am working with a piece of Matlab where I have:
n = 4;
Mbar = zeros(n,n);
Mbar(1,1) = M;
Mbar(2,2) = Ixx;
Mbar(3,3) = Iyy;
Mbar(4,4) = Izz;
L = zeros(n,n);
L = chol(Mbar)';
An error message said that it didn't like M, Ixx, Iyy, and Izz. So I chang... |
H: Is my proof correct for: $\sqrt[7]{7!} < \sqrt[8]{8!}$
I have to show that
$$\sqrt[7]{7!} < \sqrt[8]{8!}$$
and I did the following steps
\begin{align}
\sqrt[7]{7!} &< \sqrt[8]{8!} \\
(7!)^{(1/7)} &< (8!)^{(1/8)} \\
(7!)^{(1/7)} - (8!)^{(1/8)} &< 0 \\
(7!)^{(8/56)} - (8!)^{7/56} &< 0 \\
(8!)^{7/56} \left(\left... |
H: Prove that the improper integrals are equal
Prove that
$$\int_0^{\infty} \frac{\cos{x}}{1+x} dx = \int_0^{\infty} \frac{\sin{x}}{(1+x)^2} dx$$
Things that I tried so far: I tried to create integral (0, infinity) cos x/1+x - sin x/(1+x)^2 and prove that it converges to 0 but it did not work. Another thing that I t... |
H: Question on geometrically reduced, geometrically connected.
I have a question from a book which I am trying to attempt.
Let $k$ be a field not of characteristic 2 and let $a\in k$ be not a square (i.e. for all $b\in k$, $b^{2}\neq a$). I want to show that $X=\mbox{Spec}(k[U,T]/(T^{2}-aU^{2}))$ is geometrically redu... |
H: Show $|1-e^{ix}|^2=2(1-\cos x)$
Show $|1-e^{ix}|^2=2(1-\cos x)$
$$|1-e^{ix}||1-e^{ix}|=1-2e^{ix}+e^{2ix}=e^{ix}(e^{-ix}-2+e^{ix})=e^{ix}(2\cos x-2)=-2e^{ix}(1-\cos x)$$
Not sure how they got rid of the $-e^{ix}$ factor. Did I expand the absolute values wrong? thank you
AI: $$|1-e^{i x}|^2 = (1-e^{i x}) (1-e^{-i x})... |
H: multiplying a non-square number to get a square
I have a theory that the only way how can I get a square from a non-square is to multiply it by some power of itself. For example 3 multiplied by 27 gives 81.
Is this always true? If yes, how would one go about proving that?
AI: No, it is not true: $$3 \cdot 12 = 36 =... |
H: Integral and Summation Exchange
Why is it possible to do the following?
$\int \space[\space \sum_{n=0}^\infty(-1)^n x^n \space]\space dx = \sum_{n=0}^\infty (-1)^n\int x^n dx$
I know that this is legal:
$\int \sum_{n=0}^\infty x^n dx = \sum_{n=0}^\infty\int x^n dx$
But I am not sure why the $(-1)^n$ can be take... |
H: Unitary invariance
Why is it that for any non-negative matrix $M$ and unitary matrix $U$, we have
$$\sqrt{UMU^\dagger}=U\sqrt{M}U^\dagger$$?
This question has to do with Problem 2c from this sheet. I think I am allowed to assume the "fact" but I'd like to know why.
AI: Suppose $\sqrt{ U M U^\dagger } = B$. Then
$$... |
H: Inverse Function Theorem - Challenging question
I found the following question on the internet, and I think it would be useful to solve it as part as studying to my midterm:
Let $f:\mathbb{R}^n \to \mathbb{R}^n $ having continuous partial derivatives up to first order , such that:
$ \| f(x)- f(y) \| > \frac{1}{10} ... |
H: Relations in Discrete Math/ tables
Does anyone know how to make this table? I can do a table with normal values but the $x^2$ throws me off.
'Write the relation as a table, the relation $\mathbb{Z}$ on $\{1,2,3,4\}$ by $(x,y) \in \mathbb{Z}$ if $x^2\geq y$.'
Thanks!
AI: $$\begin{matrix}
xRy & 1 & 2 & 3 & 4 \\
1 & T... |
H: Is the set of rational numbers a vector space?
Is the set of all rational numbers $\mathbb{Q}$ a vectorspace?
I assumed no, because if
$$\vec{x} \in \mathbb{Q}$$
$$\pi \vec{x} \notin \mathbb{Q}$$ failing scalar mult closure
This was a question out of my linear algebra book, looking at the solutions says that it is... |
H: Conjugacy classes of rotational symmetry tetrahedron
I am struggling with this question, as I don't know how to give a "short proof" as requested by our professor, as a non-graded exercise. Especially, since we have to do this without using any shapes, and just using our head!!
Problem: Let T be the tetrahedral rot... |
H: Expected value of max(x, y)
We are given a square of unit length which has its centre at $(0,0)$ and its edges are parallel to the axes. How to find the expected value of $\max(x, y)$ where $(x,y)$ is a point in the square.
Edit:
Firstly I thought about just $x$ axis. $x$ axis varies in $[-0.5, 0.5]$. And $y$ will ... |
H: Riemann Integrable Functions Sequence
I cannot use the definition of limsup for this problem so I'm kind of stuck.
Let $f$ be continuous on $[a,b]$, let $f(x)\geq 0$ for $x\in [a,b]$, and let $M_n := \left(\int_a^b f^n\right)^{1/n}$. Show that $\mathrm{lim}(M_n)=\mathrm{sup}\{f(X): x\in [a,b]\}$.
My attempt: I tho... |
H: If $A^2+A=0$,then $\lambda=1$ cannot be an eigenvalue of A.
Prove the following statement:
If $A^2+A=0$,then $\lambda=1$ cannot be an eigenvalue of A.
I've been struggling on this question for a couple of hours and don't know how to approach it.
AI: If there were an eigenvector $v$ of $A$ with eigenvalue $1$, we'... |
H: Integration of $c(y^2)(1-y)^4$
Could anyone please help with integrating $f(y)=cy^2(1-y)^4$? Where $c$ is a constant.
AI: Hint: Substitute $u = 1 - y$, giving
$$-c\int (1 - u)^2 u^4 du$$
Now expand $(1 - u)^2$ and integrate term-by-term. |
H: Limit of $\text{ex} (n;P)/ \binom n2$ for the Petersen graph
This question is linked to
For a graph $G$, why should one expect the ratio $\text{ex} (n;G)/ \binom n2$ to converge?
where an argument was given that this specific ratio converges for $n\rightarrow\infty$. I would like to find the specific limit in the c... |
H: Number of ways of sorting distinct elements into 4 sets
This was on a test I just had. The first part says:
"A person donates nine antique clocks to four different museums. Supposing all clocks are identical and he can distribute them in any way he chooses, how many ways are there of donating the clocks"?
This is c... |
H: An application of Fubini-Tonelli
Let $f$ be a nonnegative measurable function which is finite $\mu$ almost everywhere. Suppose that $\mu(E_t) < \infty$ for all $t>0$, where $E_t = \{x: f(x)>t\}$. Let $\lambda$ be another measure, where $\lambda((a,b]) = \mu(E_a) - \mu(E_b)$ for all $a<b<\infty$. It is true that $... |
H: Prove anti-symmetric-ness of partial ordered set in lattice.
Prove anti-symmetric-ness of partial ordered set in lattice.
Definition:
If $(A, \le_{A})$ is a lattice and $C$ is a set, $([C \rightarrow A], \le)$ is also a lattice.
And $\rightarrow$ is defined as follows:
$f \le g$ if and only if for any $c \in C$, ... |
H: Find the Least Prime Divisor of $2^{17}-1$
Show that the least prime divisor of $2^{17}-1$ is $2^{17}-1$ itself.
This question is really anoying. Let $N=2^{17}-1$. What I know is that if $q\mid N$, then $q=34k+1$ for some $k \in \Bbb{N}$ and $q \equiv \pm1 \pmod 8$. Therefore,
$$q \equiv 2k+1 \equiv \pm1 \pmod 8... |
H: Which step is wrong in this proof
Proof: Consider the quadratic equation $x^2+x+1=0$. Then, we can see that $x^2=−x−1$. Assuming that $x$ is not zero (which it clearly isn't, from the equation) we can divide by $x$ to give
$$x=−1−\frac{1}{x}$$
Substitute this back into the $x$ term in the middle of the origina... |
H: Writing the identity permutation as a product of transpositions
I am reading Introduction to Abstract Algebra by Keith Nicholson and ran into a lemma that states:
If the identity permutation $\varepsilon$ can be written as a product of $n \geq 3$ transpositions, then it can be written as a product of $n - 2$ trans... |
H: A normed space of continuous functions with norm $\int_{0}^{1}|f(t)|dt$ is not complete
Suppose $E$ is a normed space of all continuous functions on $[0,1]$ with norm $\int_{0}^{1}|f(t)|dt$. Prove that $E$ is not complete
I know that we must do is to find a Cauchy sequence of continous functions that doesn't conv... |
H: Relationship between group actions and homomorphisms
I know that there exist no nontrivial homomorphism from $S_3$ into $Z_5$ as they are groups of co-prime order. I am not looking for an explanation of this but for an explanation concerning the obvious misunderstanding I have between group actions and homomorphism... |
H: Decide the dimension of maximal ideals
Let $A=\mathbb C[x,y]/(y^2-x^3,y^5-x^3)$. I want to know the dimension of each maximal ideal over $\mathbb C$. Actually I can't decide it's maximal ideal. And how to decide its dimension?
AI: To find the maximal ideals, use the Nullstellensatz. If $I$ is an ideal in a polynom... |
H: Find a polynomial $p$ of degree $3$ if its value in $4$ points is given
Find a polynomial $p$ of degree $3$ such that
\begin{align*}
p(−4) &= −142, \\
p(1) &= −2, \\
p(−5) &= −242, \\
p(4) &= 10.
\end{align*}
Then use your polynomial to approximate $p(2)$.
\begin{align*}
p(x) &= ?, \\
p(2) &= ?.
\end{align*}
I can'... |
H: Solve for an Undetermined Coefficent by using annihilator method.
The problem is
I know that we need to find and
There are three roots and at least two of them are complex.
Edit: Sorry I forgot the D = 1
The is supposed to be . The + right next to the sinx is just a typo.
Now for the . Since there is a on... |
H: Find a plane parallel to a line and perpendicular to $5x-2y+z=3$.
Find a plane parallel to the following line: $$x=3t+2, y=-t+1, z=t-1$$ and perpendicular to:
$$5x-2y+z=3.$$
I've tried the following:
The normal vector to the plane above is <5,-2,1>.
The normal/direction vector to line is <3, -1, 1>.
I did the dot p... |
H: Help with a proof of Analysis (subsequences)
I have a bad time with the next problem. I'd appreciate any help.
Problem: Let $(a_n)_{n=0}^{\infty}$ be a sequence which is not bounded. Show that there exists a subsequence $(b_n)_{n=0}^{\infty}$ of $(a_n)$ such that $(1/b_n)\rightarrow0$.
Scratchwork:
Claim 1: The ... |
H: Is there a named theorem for the fact that two points define a line, and three points define a quadratic function?
In particular, is there a theorem stating the fact that a polynomial function of degree d is defined by d+1 points?
I'm asking because I want to use this fact in a different proof but I want to be able... |
H: Normalization of Orthogonal Polynomials?
The generalized Rodrigues formula (Hassani, Mathematical Physics, p. 174) is of the form
$$p_n=K_n\frac{1}{w}\left(\frac{d}{dx}\right)^n(wp^n)$$
The constant $K_n$ is seemingly chosen completely arbitrarily, and I really need to be able to figure out a quick way to derive wh... |
H: Prove there exists a natural number $n$ such that $p_1+n(p_2-p_1)$ is prime but $p_2+n(p_2-p_1)$ is not, where $p_1 < p_2$ are primes
I am told $n$ should be taken to be the smallest possible n such that $p_2 + n(p_2 - p_1)$ is composite without actually picking a specific value for n and then somehow reach a contr... |
H: How to tell if an integral can be integrated (has an elementary anti-derivative)?
When dealing with improper integrals I sometimes have to figure out whether or not to use the comparison test. Everything I read says something along the lines of "Also, there will be some integrals that we simply won’t be able to in... |
H: Find the maximum and minimum values of $A \cos t + B \sin t$
Let $A$ and $B$ be constants. Find the maximum and minimum values of $A \cos t + B \sin t$.
I differentiated the function and found the solution to it as follows:
$f'(x)= B \cos t - A \sin t$
$B \cos t - A \sin t = 0 $
$t = \cot^{-1}(\frac{A}{B})+\pi n$
... |
H: A question regarding $\,3 \times 4$ matrices
Good day, I'm currently studying for an exam and need to learn about matrices too. Well, since I'm not good at English I'll just write what I've done so far. Below is a photo showing the full sheet of paper with the steps I did so far.
The thing I'm wondering about is, t... |
H: Change of variables to derive Fourier series
Let $f\in C^{\infty}(\mathbb{R})$ be a periodic function of period $2L$. Define $$a_n=\dfrac{1}{2L}\int_{-L}^Lf(x)e^{-in\pi x/L}dx$$ Show by change of variables that $$f(x)=\sum_{n=-\infty}^\infty a_ne^{i\pi nx/L}$$
I'm quite confused about what "change of variables" r... |
H: Clean characterization of the matrix of a linear transformation
Does anyone have a clean characterization of the matrix of a linear transformation? I would like one that is concise and clean. This is what I have, although I am not sure if it is correct:
Matrix of the transformation T with respect to B,C:
The matrix... |
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