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H: How to check if matrices form the basis for a subset?
I have worked out a solution to a hw problem, but it felt too easy. That always means I'm wrong.
Here's the question...
Consider the subspace S of 2×2 symmetric matrices. Show that the following three matrices form a basis for S
a = [1 0 0 0] b = [0 1 1 0] c = [... |
H: Ordered pairs with 2 sets with some conditions
$\langle B,C\rangle$ such that $ |B|=|C|=2$ and $ B,C\subseteq A $ and $B \cap C=\emptyset $
where $A=\{1,2,3,4,5,6\}$ when the number of ordered pairs B,C
Is equal to the number of words length $6$ such that the numbers 0,1,2 is repeated each twice exactly
I suppose :... |
H: Does the proof of uniqueness of a solution depend on a proof that the solution is actually a solution?
To prove: The equation $a + x = b$ has the unique solution $x = b - a$
The proof I am looking at has two parts.
There is at first the proof that $x = b - a$ solves the equation by plugging it into the equation and... |
H: Operator norm of a matrix in terms of its coefficients
Let $M:\mathbb{C}\to \mathbb{C}$ be a matrix and equip $\mathbb{C}$ with the norm
$$\|x\|_\infty=\max_{1\le j\le n}|x_j|.$$
If the operator norm is given by
$$\|M\|=\sup_{\|x\|=1}|Mx|,$$ is it possible to compute the operator norm exactly in terms of the matrix... |
H: Show that $ \Phi(x,z) = x^{\delta} \cdot \prod_{p \le z} \bigg(1-\frac{1}{p^{\delta}}\bigg)^{-1}$
I am stuck at the following exercise:
Let $\Phi(x, z)$ be the number of $n \le x$ all of whose prime factors are less than or equal to $z$. Prove that for any $\delta > 0$ holds
$$ \Phi(x,z) \le x^{\delta} \cdot \prod... |
H: How to solve this integral with multiple variables
$\int_{-1}^{-2}\int_{-1}^{-2}\int_{-1}^{-2}\frac{x^2}{x^2+y^2+z^2}dxdydz$
I've tried looking it up and as far as I get is that I probably need to use cylindrical coordinates but I haven't been able to solve it.
I also tried those sites that calculate them for you a... |
H: Using Uniqueness Result for Analytic Functions
I am reviewing for an Analysis qual and stumbled upon this question. In particular, I am having difficulties with part (ii). My attempt is the following:
Using the hint, let $\Omega = \mathbb{C}$, $S=\{1/n : n\in \mathbb{N}\}$, and $g(z)=z^2$. We have that since $S \s... |
H: What is $i^j$ for quaternions?
Given complex numbers, we can calculate e.g. $i^i$.
Given quaternions, how can we calculate something like $i^j$? Wolfram Mathematica choked on that and googling did not produce any useful results. My guess is that this could be something ill defined, similar to quaternion derivative ... |
H: If a function $f$ is $L$-periodic and even, then $f'$ has $2$ zeros in $[0,L)$?
Let $f: \mathbb{R} \longrightarrow \mathbb{R} $ be a differentiable and even function. If $f$ is periodic and the (minimal) period $L>0$, then $f'$ has $2$ zeros in $[0,L)$?
For example, this occurs if we consider $f(x)=\cos(x)$, for a... |
H: Approximation of a function’s derivative
I came across this result when I was tinkering with some summations and integrals. Any ideas if it could be useful?
$\frac{d}{dx}f(x)\approx f(x+\frac{1}{2})-f(x-\frac{1}{2})$
AI: This centered difference is often used to estimate $f'(x)$. More generally, $f'(x)$ can be appr... |
H: Prove that, If $f(0)=f(1)=0,$ and $M:=\max _{[0,1]}\left|f^{\prime}\right|,$ then $\int_{0}^{1}|f| \leq \frac{M}{4}$.
Let $f:[0,1] \rightarrow \mathbb{R}$ is continuously differentiable, $M:=\max _{[0,1]}\left|f^{\prime}\right| .$ Prove the following statement.
(a) If $f(0)=f(1)=0,$ then $\int_{0}^{1}|f| \leq \frac... |
H: Set of random numbers with uniform distribution - justify the distribution of the differences
Make a set of random integers, uniformly distributed between 0 and n=10^4. The size of the set is n^2.
import pandas as pd
import numpy as np
n = 10 ** 4
seq = pd.DataFrame(np.random.randint(0, n, n ** 2))
Plot the histog... |
H: Connection between trigonometric identities and secant/tangent lines
Assuming the relationship I am asking about is obvious to most students, I hope this post is an opportunity for some to have fun exploring a basic question. What I'm wondering about is the relationship to the trigonometric identities I learned abo... |
H: Limitation of eigenvalues and eigenvectors
Consider a simple example of a 2x2 matrix. Let's say we assign two numbers $\lambda_1$ and $\lambda_2$ and for each of these numbers, a corresponding $x_1,x_2$ vectors with two values each and then assume that these are the eigenvalues and corresponding eigenvectors of som... |
H: Sorted extraction
I have numbered balls from $1$ to $N$ in an urn and take out $n$ balls, one at a time, putting it back each time. I want to calculate the probability of getting a strictly growing sequence. I thought about doing it with success cases over total cases.
If I'm not mistaken, total cases would be $N^n... |
H: How does one handle series generating functions with multiple equals signs?
How would somebody walk through this equation? I'm looking for $q(n)$. If I'm given an input of 10, for example, how would this play out? The two equals signs is throwing me off. Does the result of the far right equation serve as input to t... |
H: Basic ODE with Initial Value
old guy here working through an old ODE book (separable equations):
$(\ln y^x)dy = 3x^2ydx$ which rearranges to:
$ln (y)/y dy = 3x dx$
solving this yields:
$(ln y)^2 = (3x^2)/2 + C$
initial boundary: $y(2) = e^3$
My answer: $(ln y)^2 = 3x^2/2 +3$
Book's answer: $(ln y)^2 = 3x^2 + 3$ <- ... |
H: Is it possible to show $(\lnot p \implies p) \implies p \vdash (\lnot \lnot p \implies p)$ in constructive logic?
I was given the task of showing that $(\lnot p \implies p) \vdash p$ cannot be proven in constructive logic (that is, a system with no excluded middle, double negation, or $\lnot$-elimination).
I'm try... |
H: Show that $\lim_{n\rightarrow\infty} \frac{\binom{n}{k}}{2^n} =0$
Show that the following limit holds
$$
\lim_{n\rightarrow\infty} \frac{\binom{n}{k}}{2^n} =0
$$
for a fixed value of $k$
I really am just stuck at the first step here. Normally I would consider tackling this using L'Hopitals rule, however $\binom{n}... |
H: Understanding why the hyperplane is perpendicular to the vector
I am following along Stephen Boyd's lectures on convex optimization and am having trouble understanding the diagram in this screenshot.
I have read through a few answers such as this one and this one.
My question is I am having trouble understanding wh... |
H: Find the minimum perimeter of the triangle.
Consider point $A(5, 2)$ and variable points $B(a, a)$ and $C(b,0),\, a\in R, \, b\in R$. If the perimeter of $\triangle ABC$ is minimum, find $a$ and $b$.
My attempt:
$\begin{align}
P&=AB+BC+CA\\
&=\sqrt{(a-5)^2+(a-2)^2}+\sqrt{(a-b)^2+(a)^2}+\sqrt{(5-b)^2+(2)^2}
\end{al... |
H: Unable to prove an exercise in Continuous functions in Topology
I am self studying Topology from C. Wayne Patty and I am unable to solve the following question in exercise 1.7
Adding image->
I tried by assuming a sequence $x_n$ $ \epsilon $ A which converges to x . I got f($x_n$) = g($x_n$ ) but I am not able to m... |
H: Applying the mean value theorem to sine function
Prove that $ \pi x\cos(\frac{1}{2}\pi x^2) = c$ has a solution in $0 < x< 1$ for $c=1$?
Is this true for all positive values of $c$?
I said define $f:[0,1] \to \mathbb{R}$ by $f(x) =\sin(\frac{1}{2}\pi x^2)-cx$. Then $f$ is (in particular) continuous on $[0,1]$ an... |
H: Asking about number of parellogram in a figure
This question was asked by my younger brother and I couldn't solve it.
So, I am asking it here.
Question is ->
I think directly calculating it is a bit lenthy and could lead to error.
Can someone please tell a method for such questions.
I am a masters of mathematics st... |
H: Consider the function$ f : \mathbb Z \to \mathbb Z$ given by$ f(n) = n^2.$ Write down the set $f ^{−1} (\{9, 10\}).$
How can I solve this?
I think they for $f$ inverse $9$. Answer is $3$. And for $f$ inverse $10$ I am confused. Is this possible to find the solution for $f$ inverse $10$?
AI: By definition $f^{-1} (\... |
H: Prove the therem of function f that is differentiable at only one point
Let $f$ be $f:\mathbb{R}→\mathbb{R}$. I understand and I saw function as $f(x)=x^2g(x)$ that can show that a function can be differentiable at one point. If such $x0$ exists is differentiable at this point then:
$\lim_{x \to x0} \frac{f(x+\Del... |
H: Show that a sequence of PDF of normal distribution with running mean and unit variance is not bounded by an integrable function
I am trying to show that the condition of bounded by an integrable function is crucial in the Dominated Convergence Theorem.
Consider a sequence of functions $(f_n)$ on $\mathbb{R}, $ whic... |
H: Meaning of probability of the intersection of two events
Suppose X is a random variable of a rolling 6-side die. Suppose A is the event that the outcome is even.
The example asks for the conditional probability of X=k, given A.
The answer to this is 1/3. We also know that P(A) = 1/2.
If we write down the formula of... |
H: Prove that $n^2>n+1$ for all $n\geq 2\in\mathbb{Z}$
I'm a bit unsure of my reasoning here:
Clearly the proposition $P_2$ which states that $2^2>2+1$ is true.
Suppose that it holds true for $k^2>k+1$ where $k>2$.
We define $d_k=k^2-(k+1)=k^2-k-1$
Suppose we also have $(k+1)^2>(k+1)+1$.
As before we define $d_{k+1}=(... |
H: Proving a function to be continuous in Topology
I am trying exercises of section 1.7 of C. Wayne Patty and I am unable to think about solution of this question.
Note that I want to ask b part only.
My attempt -> The definition of continuity is inverse of the definition of open sets given here. So, I don't know... |
H: Choice of a vector for supporting hyperplane theorem
I'm having trouble relating the content on these notes to these ones from MIT OCW here.
Specifically, the question I'm having is the first set of notes describes the specific half space where $a^Tx \leq a^T x_0$ and the second concludes that any one of the half... |
H: Prove that $\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{4}}+\frac{1}{\sqrt{5}}+\frac{1}{\sqrt{6}}\cdots>\infty$ .
I need to a fresh solution, but mine
We have that
$$\begin{align*}
\frac{1}{\sqrt{1}} &+ \frac{1}{\sqrt{2}}+ \frac{1}{\sqrt{3}}+ \frac{1}{\sqrt{4}}+ \frac{1}{\sqrt{5}}+ \fr... |
H: the example that CLT holds, but LLN does not
When a sequence of random variables $(X_k)$ have different probability distributions, CLT is not necessarily stronger than LLN. The image above is captured from Feller Vol.1, P.255. This shows one example that CLT holds but LLN does not. The author says that the $\bf{s... |
H: Let $A = \{1, 2, 3\}$. Find distinct functions $f:A\to A$ and $g:A\to A$ such that $g\circ f\neq f\circ g$.
This question appeared in my text book while solving. I am not super sure about the solution to it. Can anyone please check my solution? I can't consult my professor right not (by mail) because it's midnight ... |
H: Domain of $\arccos(x)$
Is there an explanation of why the domain of $\arccos(x)$ is $[-1, 1]$?
AI: Because it is meant to be the inverse function to $\cos(x)$, which has a range of $[-1,1]$. The inverse $f^{-1}$ of a function $f$ has $f$'s codomain as its domain by definition; i.e., if $f: A \to B$ then $f^{-1} : B... |
H: Question about a proof related to the preimage of an measurable function.
I have a question regarding the proof of Thm A. I know that f is measurable if the set {x: f(x)<c} for all real value c, is measurable.
I have a hard time understanding(fill in the detail) of how did the set A transforms into a union of the i... |
H: When can you switch the limites of integration of a line integral?
I was looking into why the property that $\int_a^b f(x) \ dx = -\int_b^a f(x) \ dx$ holds true. I found that 2 common answers were that
It comes from $\int_a^b f(x) dx + \int_b^c f(x) dx = \int_a^c f(x) dx$ for arbitrary $a \le b \le c$ (for exampl... |
H: What can we say about the probability of two events when $A$ implies event $B$?
Suppose we have two events $A$ and $B$ where $A$ implies $B$. What can we say about their probabilities?
My try:
I can come up with two events
$A=\{\text{rainy weather}\}$
and
$B=\{\text{cloudy weather}\}$
where $A \rightarrow B$.
Also,... |
H: If a convex combination of conformal matrices is conformal, are they all proportional?
$\newcommand{\CO}{\text{CO}}$
$\newcommand{\SO}{\text{SO}}$
$\newcommand{\dist}{\text{dist}}$
Let
$\CO(2) =\{\lambda R : R \in \SO(2)\, | \, \lambda > 0\} $ be the set of $2 \times 2$ conformal matrices.
Let $A_i \in \CO(2)$ be a... |
H: How to write multiplication of series of numbers in factorial form?
I want to write
$$1 \times 2 \times 3 \times \dotsb \times 1 \times 5 \times 9 \times 13 \times \dotsb$$
in factorial form, but I don't know how?
AI: You could use multifactorial notation:
$1 \times 2 \times 3 \times \dotsb \times n \times 1 \times... |
H: Passage missing in simple proof of orthogonal decomposition
I am reading this nice book on linear algebra. Specifically, I am reading the proof of the Theorem for Orthogonal decomposition of a vector $x\in \mathbb{R}^n$, given a subspace $W$. I think there is a step missing on the proof of its uniqueness.
Theorem:
... |
H: Is there a precise definition of "arbitrary union"?
Is there a precise formulation of what "arbitrary union" means? For example, for a topology, we require that it be closed under arbitrary unions. Do we mean the union of any subcollection of the topology is also in the topology?
In general, do we define "arbitrary... |
H: List of operations on a set to make it a 2-dimensional vector space
This question might be very silly. I was working on two Examples of Friedberg- Insel-Spence's Linear Algebra. In example $6$, in $\mathbb{R}^2$(not 2-dimensional real vector space, consider it as the set $\mathbb{R} \times \mathbb{R})$ scalar multi... |
H: Lower limit and subsequence
Let $x_n$ be a convergent sequence $x_n \to x$ in a topological space $X$ and $F:X\longrightarrow \mathbb R$ any function (not necessarily continuous). If there exist a subsequence $x_{n_k}$ such that
$$F(x) \leq \liminf_k F(x_{n_k})$$
then can we conclude that
$$F(x) \leq \liminf_n F(x_... |
H: Proof verification for $a > b \implies -a < -b$
I am aware that this question asks for the verification of a proof of (almost) the same problem but my proof is different and in my opinion, a bit simpler and more intuitive. Here's how it goes :
Let us assume that $a > b$.
We can write this inequality in the form o... |
H: Max and Min value of a function on a circle
Find the maximum and minimum values of the function $f(x,y) = 5x^2 + 2xy + 5y^2$ on the circle $x^2 + y^2 = 1$.
After substituting the equation of the circle in that of the function and then equating $f'(x) = 0$, I get the values of $y$ to be $\pm1\sqrt2$. Plugging thes... |
H: Consider the system $\dot{x}=4x^{2}-16$ find an analytical solution.
I am working through a text book by Strogatz Nonlinear dynamics and chaos . In chapter 2 question 2.2.1 , I am looking for an analytical solution. I have the question's answer but would like to ask how a certain step was performed.
Question
Consid... |
H: A simple question about conditional expectation
If I got
$$E\left(\min\left(X,Y\right)\right)$$
Why is it equal to
$$E\left(\min\left(X,Y\right)\right)=E\left(\min\left(X,Y\right)\mid\min\left(X,Y\right)=X\right)P\left(X\le Y\right)+E\left(\min\left(X,Y\right)\mid\min\left(X,Y\right)=Y\right)P\left(X>Y\right)$$
AI:... |
H: $0 \cdot \infty$ object. Does it make sense?
There maybe a mistake in the question but, let that someone asks you to calculate something like this: $$0\cdot \lim_{x\to0}(log(x))$$ with no further information. The assumptions that one makes is just that log is the natural logarithm, $x\epsilon\mathcal{R}$ and genera... |
H: Lie Group has non degenerate two form
Show that
Every even dimensional Lie Group has a non degenerate two form.
How does one answer this question?
I can see it's true for $R^{2n}$ and I was thinking about pulling back the forms on $R^{2n}$ on coordinate patches. But I don't know how to show they will agree on int... |
H: What went wrong in the evaluation of $\int \frac{1}{3-2\sin(x)}dx$?
I tried to evaluate the following integral:
$$\int \frac{1}{3-2\sin(x)}dx\,\, $$ with universal substitution, using the fact that
$t(x):=\tan\left(\frac{x}{2}\right)$
$\sin(x)=\frac{2t(x)}{1+t(x)^2}$ and $t'(x)=\frac{1+t(x)^2}{2}$
$\displaystyle\i... |
H: Algorithm to find orthnormal eigenvectors of a symmetric matrix
I have a symmetric matrix $S$ and I'm trying to implement the following algorithm to find first $k$ orthnormal eigenvectors
Note: The picture is from http://www.wisdom.weizmann.ac.il/~harel/papers/highdimensionalGD.pdf
I use a very simple $2x2$ matrix... |
H: Find residues at the singularities
I have a function $f(z) = \frac{\cos(z)}{z^6}$. I have to find the singularities and the corresponding residues. I think there is a single pole at $z=0$, which has order 6.
For the residue, I did this:
$\text{Res}(f,0) = \frac{1}{5!} \lim_{z\to 0}\frac{d^5}{dz^5}(z^6 \cdot \frac{\... |
H: Contour integration $\frac{e^{iz}}{2\sqrt{z}}$
When $z=u+iv$,
I would like to compute the integral of $\frac{e^{iz}}{2\sqrt{z}}$ along above curve.
The imaginary axis
$$\frac{1}{2}\int_{R}^{0} \frac{e^{-v}}{\sqrt{iv}}d(iv)$$
$R$ goes to $\infty$.
Here because of $\sqrt{iv}$. I confuse to use the change of variab... |
H: Binomial coefficient sum: $\sum_{k=0}^{i-1}{i-1 \choose k}{j-1\choose k} = {i+j-2\choose j-1}$
I'm having problems showing this equation, hope someone here can help me:
$$\sum_{k=0}^{i-1}{i-1 \choose k}{j-1\choose k} = {i+j-2\choose j-1}$$
where $1 \leq i\leq j $.
AI: Note that $\binom{j-1}{k} = \binom{j-1}{j-1-k}.... |
H: Finding a sequence $x_n$ with $\lim \sup(x_n)= 2$ and $\lim \inf(x_n)=-5$
Find a sequence $x_n$ with $\lim \sup(x_n)= 2$ and $\lim \inf(x_n)=-5$.
I am stuck on this one. I am trying to find a sequence $x_n$ with the given information.
AI: What about $$X_n=\begin{cases}-5&n\text{ even}\\2&n\text{ odd} \end{cases}\... |
H: $\sum_{i=1}^{\infty} v_i $ converges?
If I have a series of vectors $(v_i)_{i\in V}$ in a vector space $V$. Then if $\sum_{i=1}^{\infty} v_i$, can I say that if $\sum_{i=1}^{\infty} \mid \mid v_i \mid \mid < \infty$ then the series converges? Or do I need the vector space to be complete, or some other conditions?
A... |
H: Sorting socks after laundry.
Yesterday sorting socks after laundry I came across the following problem. Assume we have a bin with $n$ pairs of objects. We draw an object from the bin. If there is a paired object on the table we put both objects in another bin. Otherwise we lay the unpaired object on the table.
What... |
H: Solving Boundary Value Problem using the Finite Difference Method - What Values to Substitute for $y'$?
I am given a boundary value problem of the form $y'' = f(y', y, x)$ and asked to solve the system subject to boundary conditions using the finite difference method.
I proceed to develop a system of equations with... |
H: What does $x\in (0,1)$ mean?
What does $ x \in (0,1)$ mean? Does it mean $x = 0$ or $x=1$ or does it mean $0<x<1$?
I know $x \in \{0,1\}$ but in this case it has parentheses instead of curly brackets which confuses me. Sorry for the very easy question I am just a bit confused what the different kinds of brackets me... |
H: Two integrating factors: same solution to differential equation
If we have to valid integrating factors to a differential equation we must get the same general solution using each of them although we get two new equations by multiplying the original for each of the integral factors?
So if the original equation is n... |
H: What is the coefficient of a6b6 in (a+b)12
I can't use binomial theorem, how should I solve it?
AI: From the $21$ factors of $(x+y+z)^{21}$, you have to choose $7$ that you pick the $x$ from, and from the remaining $14$ factors, you have to choose $9$ to pick an $y$ from. From the remaining $5$ factors you will pic... |
H: How to find the volume by triple integral?
I'm new to triple integral and this triple integral volume problem seems impossible to solve, and I have no idea where to start and how to solve it, could someone have a look at it please.
Let $G$ be the wedge in the first octant that is cut from the cylindrical solid $y^... |
H: Union and Difference in Set Theory
Let x ∈ A. {x,x,x,x,x} ∪ {x,x} = {x,x,x,x,x,x,x}
This statement is false right? Because the union of two sets is a set of the first and second set's elements with no elements repeating. I guess I am confused because x seems to be an arbitrary value.
Let x ∈ A. {x,x} / {x} = {x}
Wo... |
H: If $a:b:c=d:e:f$, how to show that $\frac{(a+b+c)^2}{(d+e+f)^2}=\frac{(ab+bc+ca)}{(de+ef+df)}$?
If $a:b:c=d:e:f$, how to show that $\frac{(a+b+c)^2}{(d+e+f)^2}=\frac{(ab+bc+ca)}{(de+ef+df)}$?
AI: Let $\dfrac ad=\dfrac be=\dfrac cf=k$(say)
$$\dfrac{ab+bc+ca}{de+ef+fd}=\dfrac{k^2(de+ef+df)}{de+ef+df}=k^2\text{ if } d... |
H: Primitive Wreath Product Example
I'm trying to make sense of the primitive Wreath product action by looking at an example.
I took $S_3$ acting on a triangle $\Delta=\{1,2,3\}$, $C_2$ acting on $\Gamma=\{1,2\}$ and constructed the Wreath product $S_3 wr C_2$. I looked at the action of $S_3 wr C_2$ on $\Delta\times \... |
H: If I take limit, the strict inequality may become equal.
Suppose $$x \lt 1+\frac{1}{n} \;\; \forall n\in \Bbb N$$
If I let $n\to \infty$, I get $$x \le 1$$
But I am wondering the reason why $\lt$ should turned to $\le$ ?
This just an example in my analysis class, and this one is simple that $x$ can be $1$. But I wa... |
H: Limit of a Function Definition (Why restrict domain?)
The definition I use for a limit of a function is the following:
$\lim_{x\rightarrow c}f(x):=L$ if $\forall \epsilon>0, \exists \delta>0, \forall x\in \mathbb{R}^{\neq c}$, [ $|x-c|<\delta\implies|f(x)-L|<\epsilon$]
However, the video here gets rid of the requ... |
H: Show that the sum function $f(x) = \sum_{n=1}^\infty \frac{1}{ \sqrt{n} } (exp(-x^2/n)-1)$ is continous
Consider for $x \in \mathbb{R}$ the sum function defined as
$$
f(x) = \sum_{n=1}^\infty \frac{1}{ \sqrt{n} } (exp(-x^2/n)-1)
$$
I have shown that the series converges point wise by using that
$$
|exp(-x^2/n)| \le... |
H: Complex polarization identity proof getting stuck towards the end w.r.t. the imaginary part
I'm working on a homework problem regarding the proof for the polarization identity for complex scalars. I've taken a look at another question on this community (Polarization Identity for Complex Scalars) and have tried work... |
H: Verify the identity $\frac{\tan(a+b)}{\tan(a-b)}$ = $\frac{\sin(a)\cos(a)+\sin(b)\cos(b)}{\sin(a)\cos(a)-\sin(b)\cos(b)}$
I've been asked to verify the following identity but I don't know how to do it.
$$\frac{\tan(a+b)}{\tan(a-b)} = \frac{\sin(a)\cos(a)+\sin(b)\cos(b)}{\sin(a)\cos(a)-\sin(b)\cos(b)}$$
When I try I... |
H: Consider $\dot{x}=4x^{2}-16$.
I am solving the ODE above, it is a question from Strogatz Nonlinear dynamics and chaos, chapter 2 question 2.2.1.
Question
\begin{equation}
\dot{x}=4x^{2}-16
\end{equation}
Answer
\begin{equation}
\frac{\dot{x}}{4x^{2}-16} = 1\\
\frac{\dot{x}}{x^{2}-4} = 4\\
{{dx\over dt}\over x^2-4}... |
H: Solving $k(k+1)(k-1)=k$
The function $y=e^{kx}$ satisfies the equation
$$\left(\frac{d^{2}y}{dx^{2}}+\frac{dy}{dx}\right)\left(\frac{dy}{dx}-y\right)=y\frac{dy}{dx}$$
I found the derivative and the second derivative of $y$, resulting in
$$\left(k^{2}e^{kx}+ke^{kx}\right)\left(ke^{kx}-e^{kx}\right)=ke^{2kx}$$
Dividi... |
H: Uniform continuity of continuous real-valued function from space of trace-class operators on a Hilbert space
Let $\phi$ be a function from the space of trace-class operators on a separable Hilbert space $\mathcal{H}$ into the reals. Assume that $\phi$ is continuous and consider the restriction of $\phi$ to an open ... |
H: Use the laws of logic to show that $[a\Rightarrow(b\lor c)]\Leftrightarrow[(a\land\lnot b)\Rightarrow c]$
I am trying to prove that $[a\Rightarrow(b\lor c)]\Leftrightarrow[(a\land\lnot b)\Rightarrow c]$.
My proof is the following:
$a\Rightarrow(b\lor c)~$ Premise
$(a\Rightarrow b)\lor c~$ Associative Law
$(\lnot a... |
H: Complex conjugations
I'm stumped on an equation from a coursera course (Intro to DSP) that has to do with complex exponential multiplication:
In line 2, by simple rules of complex multiplication, it should be (h+k). I understand it being (h-k) has to do with complex exponential conjugation but not really sure why/... |
H: An example of a field $F$ such that $F^n$ uses element-wise operations, but $F$ is not a subfield of $F^n$?
I'm going through The Linear Algebra a Beginning Graduate Student Ought to Know, and I came across this idea that I can't seem to understand. Suppose $F$ is a field, then he asks "is it possible to define mul... |
H: Diffrentiability of modulus function
$f(x) = (x-1)|(x-1)(x-2)|$. My teacher explained that since the since the effective power of $(x-1) = 2$ which is greater than $1$, it will be differentiable at $x = 1$. But since the effective power of $(x-2)$ is $1$, it wont be differentiable at $x = 2$.
$f(x) = |x-a|^n$
if $n... |
H: Doubt on strong law of large number theorem
Suppose $\{X_1,X_2,.....\}$ sequence of independent and identical random variable.
Let $\mathbb{E}(X_1^{+})<\infty$, i.e. expectation of positive part of the random variable $X_1$ is finite. Instead of saying $\mathbb{E}(X_1)<\infty$
From here, can I conclude that
$$
\fra... |
H: If $T$ is a bounded linear map and $\sum x_n$ is an absolutely convergent series, then $T(\sum x_n) = \sum T(x_n)$
Is the following true? If so, how to prove it?
If $T:X \to Y$ is a bounded linear map between the Banach space $X$ and the normed vector space $Y$ and $\sum x_n$ is an absolutely convergent series, th... |
H: Connected components of free loop space
Let $X$ be a topological space. And let $\Lambda X=\mathrm{Top}\left[S^1,X\right]$ be the space of continuous loops in $X$. Then how do we calculate $\Pi_0\Lambda X$, the set of connected components of $\Lambda X$. Is it realted to $\pi_1\left(X,x_0\right)$ for some $x_0$?
Fo... |
H: Probability of a Gaussian random variable
Hi have the following problem... I have a tank represented with a Gaussian random variable: $X\sim N(6;0.25)$, the request:
Check that the sum of two independent measures is greater than 10 meters
How can I check my solution? Thank you
AI: Hint: The sum of two independent... |
H: Construct a Reed Solomon code: find the parity check matrix
I am trying to solve the following exercise, but I need a check/opinion on how to solve it.
Construct a Reed-Solomon code with dimensions $[12,7]$ over $\mathbb{F}_{13}$ and find a parity check matrix for the code $C$. Hint: $2$ is a primitive element of ... |
H: Finding the p-value between two numbers
I am trying to solve this example in statistics, but my results are different from the ones in the solution and I do not understand why. The example say:
A study finds a test statistic $t$-value of $1.03$ for a t-test on a single population mean. The sample size is $11$, and... |
H: Value of $\alpha$ for which $x^5+5\lambda x^4-x^3+(\lambda\alpha-4)x^2-(8\lambda+3)x+\lambda\alpha-2=0$ has roots independent of $\lambda$
Consider the equation $$x^5 + 5\lambda x^4 -x^3 + (\lambda \alpha -4)x^2 - (8\lambda +3)x + \lambda\alpha - 2 = 0$$ The value of $\alpha$ for which the roots of the equation ar... |
H: If $H$ and $K$ are normal subgroups of $G$ and $G/H \cong K$, does this imply that $G/K \cong H$?
If $H$ and $K$ are normal subgroups of $G$ and $G/H \cong K$, does this imply that $G/K \cong H$?
I wasn't able to find a counterexample or to prove that the implication is true. I would appreciate any help with this... |
H: Fixed points Cross products
Let $(A,G,\alpha)$ be a $C^*$-dynamical system with $G$ amenable (So that I need to consider just cross-product since reduced and full are the same). Let $\theta$ be an automorphism of $A \rtimes G$. Is it always true that $A$ is in the fixed point set of $\theta$? I have the $C*$ algebr... |
H: Question regarding specific steps of Fatou's lemma proof.
I have a question regarding to the following two steps of the proof.
How can we get $\int g_n \leq \int f_n$, or $g_n \leq f_n$ for all n, Since by definition of liminf, $g_n \leq f_m $ for all $m \geq n$. Where did the $m$ go?
Why did $\int f_n$ become ... |
H: Definition of basis in topology
In the definition of a basis on a set $X$ in topology, one of the properties is that, "for any two basis elements $B_1, B_2$ and any point $x \in B_1 \cap B_2$, there is a third basis element $B_3$ containing $x$ and contained in $B_1 \cap B_2$". I am wondering what happens if we cha... |
H: integrals in terms of gamma function
$\displaystyle I_1 = \int^1_0 \sqrt[3]{\log( x)}\,dx$
I am trying to write this in terms of gamma functions.
After I substitute
$\displaystyle y = \log( x)$ , $x = e^y$ , $dx = e^y dy$
Then,
$\displaystyle I_1 = \int^1_0 e^yy^{\frac{1}{2}}\,dy$
but the upper limit isn't infinit... |
H: Calculating $\cos\frac\pi4$ from the half-angle formula gives $\sqrt{\frac12}$ instead of $\frac{\sqrt{2}}2$. What went wrong?
I am using the formula $$\cos\left(\frac x2\right)=\sqrt{\frac{1+\cos(x)}2}$$ to find $\cos\left(\frac{pi}{4}\right)$ but it does not give me the correct result.
$$
\cos\left(\frac{\pi}{4}\... |
H: Markov chains: showing $P$ has unique eigenvalue $1$
I have a $4\times 4$ matrix and I tried solving for the determinant of $P-\lambda I$. This came out really messy and when I put the matrix into a matrix calculator my solution was $1,0$ and $-1$. Does this still mean $1$ is a unique eigenvalue solution? Is there ... |
H: Vanishing derivative at infinity implies slowly varying
The functions $$f(x)=e^{(\ln x)^{1/3} \cos((\ln x)^{1/3}) } \quad g(x)=e^{\sqrt{\ln x} \cos((\ln x)^{1/3}) }$$
are oscillating but slowly varying at infinity, that is for all $\lambda >0$, we have
$$\lim_{x\to \infty} \frac{f(\lambda x)}{{f(x)}}=\lim_{x\to \in... |
H: What does this definition of a polynomial mean?
In a book I am reading, there is the following definition for a polynomial:
A function $p: \mathbb{F} \rightarrow \mathbb{F}$ is called a
polynomial with coefficients in $\mathbb{F}$ if there exists $a_0, \ldots, a_m \in \mathbb{F}$ such that
$p(z) = a_0 + a_1z + a_2... |
H: Covering $\Bbb RP^\text{odd}\longrightarrow X$, what can be said about $X$?
I am looking for any argument related to the following fact, which may or may not be true.
Let $f:\Bbb RP^n\longrightarrow X$ be a covering space, where $n\geq 2$. Then, $X=\Bbb RP^n$.
Now, for $n=\text{even}$, this is surely true and thi... |
H: Why is it necessary to exclude empty set to accomplish this relation proof?
Working on the book: Daniel J. Velleman. "HOW TO PROVE IT: A Structured Approach, Second Edition" (p. 201)
∗20. Suppose $R$ is a relation on $A$. Let $B = \{X \in P (A) \colon X\neq\emptyset \}$, and define a relation $S$ on $B$ as follows... |
H: Prove that $\int_2^x \frac{dt}{\log(t)^n} = \mathcal{O}\bigg(\frac{x}{\log(x)^n}\bigg)$
I am stuck at the following exercise:
Show that for $n \in \mathbb{N}$ holds
$$\int_2^x \frac{dt}{\log(t)^n} = \mathcal{O}\bigg(\frac{x}{\log(x)^n}\bigg).$$
I do not see how I could prove this. I know that the following identi... |
H: When is the supremum of a function on a subset equal to the supremum on a set?
Let $f$ be a continuous, real function on a set $X$. Suppose the set $D \subset X$ is dense in $X$, and the set $S \subset X$ is closed and has empty interior, i.e. it is nowhere dense. Moreover, $S\cap D$ is nonempty.
Are there any know... |
H: Is there a solution to $ \lim_{x\to1^+} \sin\frac{\sqrt{x+1}}{{x^2-1}} $
Is there a solution to the below /Does it exist?
$
\lim_{x\to1^+} \sin\frac{\sqrt{x+1}}{{x^2-1}}
$
When I use conjugate and factor it, I get:
$$
Sin(\sqrt2)
$$
$$
x \neq 0
$$
I found this question asked before and the answer back was it doesn'... |
H: Property of decreasing functions
I was reading the proof of a theorem, and then I got stuck into this sentence:
"Since $f(k)=e^{-\frac{k^2}{m}}$ is a decreasing function, we have that:
$$\int_k^{k+1}e^{-\frac{x^2}{m}}d x\le e^{-\frac{k^2}{m}}\le\int_{k-1}^ke^{-\frac{x^2}{m}}dx"$$
I cannot understand how can I prove... |
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