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H: FInd conditional extrema $z = x+2y$
I need to find conditional extrema of a function.
Here is function:
$z = x + 2y$ when $x^2 + y^2 = 5.$
AI: The gradient of $z$ is $(1,2)$. So the extrema are attained on the intersection between $x^2+y^2=5$ and $y=2x$ giving us $(x, y) = (1,2)$ and $(x, y) = (-1,-2)$. Thus, the... |
H: For a complex number $\alpha $ which is algebraic over $\Bbb Q$, determining whether $\bar{\alpha}\in \Bbb Q(\alpha)$ or not
Let $\alpha =3^{1/3}+3^{5/4}i$, which is clearly algebraic over $\Bbb Q$. How can we determine whether $\Bbb Q(\alpha)$ contains $\bar{\alpha}$ or not?
This would be certainly true if $\Bbb Q... |
H: Question about ratio word problem
Here is the question and solutions:
It is possible that there are 10 cups of peanuts, 6 cups of walnuts, 4 cups of cashews. So total number of cups in this mix is 20. So 10:6:4 can be reduced to 5:3:2.
Another combination of the mix could be 20 cups of peanuts, 12 cups of walnuts,... |
H: Question about proof of Riemann removable singularity theorem.
Theorem:
Let $f:D^*(z_0,r)=D(z_0,r)-\{z_0\}\to \Bbb C$ be holomorphic and bounded. Then $\lim_{z\to z_0}f(z)$ exists and the function $\hat{f}:D(z_0,r)\to \Bbb C$ defined by $$\hat{f}(z) =
\begin{cases}
f(z) & \text{if }z\in D^*(z_0,r) \\
\lim_{z\to z_... |
H: Topology on Reals generated by Predicates
What is the topology that the Real number has that is generated by the sets of the form $\{x\in \mathbb{R}:P(x)=T\}$ where P() is a predicate of the language of ordered fields (which is a predicate calculus with the constants $1,0$ the unary functions $-, $ and $()^{-1}$ de... |
H: Question about approximating an area based on a graph
Here is the question and image:
Hi,
here is my attempt:
Each block has an area of $(1/2)^2 = 1/4$ miles$^2$. This grey region covers $14$ blocks. So $14*1/4 = 3.5$. So the approximate area is $3.5$. However, the answer is $2.75$ miles. I do not know why the an... |
H: Can any pattern be sequence?
Can any pattern be sequence ?
like $2$, $e$, $\pi$, $13$, $12$, $67$ ... can be considered as sequence?
I've read that we consider only those pattern as sequence in which we can predict the next term.
Please elaborate this thing in details .
AI: A sequence is simply a function from $\Bb... |
H: Why is a 3-D sphere/cube centered at the origin not a subspace?
My class notes states that: a 3-D cube or sphere centered at the origin is not a subspace as it is not closed under addition and scalar multiplication. Why is this the case? I am having trouble understanding how it is not closed under addition.
AI: Let... |
H: Proof Verification: $M^t\in SO_3$
By the Euler's theorem, the set $SO_3$ of all 3x3 orthogonal matrices $M$ such that $det(M)=1$, which is called the special orthogonal group of 3x3 matrices, is the set of all 3x3 rotation matrices. The rotation $\rho$ of $R^3$ can also be represented by spin $(\mu, \theta)$ where ... |
H: Evaluation of Limit involved in the proof of Asymptotic Unbiasedness of S
We know that $S^{2}$ is an unbiased estimator of $\sigma^{2}$ and $S$ is a biased estimator of $\sigma$. But if $n\rightarrow\infty$, then $S$ is an asymptotically unbiased estimator of $\sigma$. I found a proof here(first answer). But in the... |
H: Find real numbers r and s so that $a_{n+2}+ra_{n+1}+sa_n = 0$ and $b_{n+2}+rb_{n+1}+sb_n = 0$
I already know that $b_{n+1} = a_n +3b_n$ and $a_{n+1} = 3a_n - b_n$. So
$a_{n+2} = 3(3a_n-b_n)-(3b_n+a_n) = 9a_n-3b_n-3b_n-a_n = 8a_n-6b_n$ and
$b_{n+2} = 8b_n+6a_n$.
So we can rewrite the whole thing as
$8a_n-6b_n+r(3a_n... |
H: $E[1/(1+e^X)] = 1/2$ for standard normal $X$
I have normally distributed $X\sim\mathcal{N}(0, 1)$, and I want to compute
\begin{equation*}
\mathbb{E}[1/(1+e^X)] = \int_{-\infty}^\infty \frac{e^{-x^2/2}/\sqrt{2\pi}}{1+e^x} dx
\end{equation*}
I found numerically (and confirmed with Mathematica) that $\mathbb{E}[1/(1... |
H: Analytical way to find the third root of $x^{2}=2^{x}$ other than 2 and 4
How to find analytically the third root (-ve) of $x^{2}=2^{x}$ other than 2 and 4?
Does differentiation of the equation make a sensible way?
I tried with $\log_a$ for different $a$' s. But I couldn't find the root.
AI: Looking for the negativ... |
H: Well-definedness of $\mu$-integral
The paragraph above is from: Foundations of Modern Probability (1st edition) - Kallenberg (Page 11)
I am struggling to show that the extended integral as described in the attached paragraph is independent of the choice of representation. I can see that if $f = g-h$ as written, th... |
H: What's the sum of all possible values of a number which is the sum of the digits of another number in this question
I came across a question in an exercise booklet for Mathematic Olympiad for primary school students in Australia. The question is described as follows:
Let A be a 2018-digit number which is divisible... |
H: Why does that fact that $a_n \equiv 3 \pmod 5$ and $b_n\equiv 1 \pmod 5$ imply $1/\pi(\arctan(1/3))$ is irrational
So since I've started this multi-part question I've learned:
$(3+i)^n = a_n+ib_n$
$a_{n+1} = 3a_n-b_n$
$b_{n+1} = 3b_n+a_n$
$a_n \equiv 3 \pmod 5$
$b_n \equiv 1 \pmod 5$
Now I am asked why the fact tha... |
H: If $A^m = 0$, then $\mbox{rank}(A) \leq \frac{m-1}{m}{n}$
Let $A$ be a $n \times n$ real matrix. Show that if $A^m = 0$, then
$\mbox{rank}(A) \leq \frac{m-1}{m}{n}$
My attempt:
If $m=1$, then $A=0$ so $\mbox{rank}(A)=0$.
If $m=2$, we have $\mbox{im}(A) \subset \ker(A)$ so $2\operatorname{rank}(A) \leq \dim \mbox{... |
H: Why does the formula to get $(e^{x})'s$ slope differ from itself?
right now, and I've noticed that when the professor shows the formula to get the slope of $e^x$, it's different from the formula to get $e^x$. Why does this happen when the slope of $e^x$ is $e^x$? There's no reason it should be different.
The formul... |
H: How many 6 digit numbers can be formed from two sets of digit?
There are two sets of digit :
$ \text{set 1 :} \{~0,1,2,3,4~\}$
$ \text{set 2 :} \{~5,6,7,8,9~\}$
Now how many 6 digit number can we make by taking numbers from these two sets ? From $\text{set 1}$ repetition is permitted but from $\text{set 2}$ repetit... |
H: Sequence without average density
How to (if it's possible) build infinite binary sequence in that 0-elements density will not converge to any value?For example, in $30\%_{one}\over70\%_{zero}$ random sequence (average) density (of 0-elements) converges to 70%,or in 11010010001000010000010000001000... (I don't know ... |
H: Show that $\sum_{n=1}^{\infty}\left(1+\frac{1}{n}-\frac{1}{n^2}\right)^{-n^2}x^n$ does not converge for $x=\pm e$
Show that $$\sum_{n=1}^{\infty}\left(1+\frac{1}{n}-\frac{1}{n^2}\right)^{-n^2}x^n$$ does not converge for $x=\pm e$. Mathematica says that $$a_n:=\left(1+\frac{1}{n}-\frac{1}{n^2}\right)^{-n^2}e^n\xrigh... |
H: Write down an expression in the form $ax^n$ for: $\lim_{h\to 0} \frac{\sqrt{x+h}-\sqrt{x}}{h}$
Write down an expression in the form $ax^n$ for
$$\lim_{h\to 0} \frac{\sqrt{x+h}-\sqrt{x}}{h}$$
What I have tried so far:
multiplying by the conjugate to give:
$$\lim_{h\to 0} \frac{\sqrt{x+h}-\sqrt{x}}{h} \cdot \frac{h... |
H: Equivalence of two elements in a quotient $C^*$-algebra
Let $A$ be a unital $C^*$-algebra, and $I_1$ and $I_2$ closed (two-sided) $C^*$-ideals of $A$ such that $A=I_1+I_2$. Suppose we have two positive elements $a_1\in I_1$ and $a_2\in I_2$ such that
$$a_1^2+a_2^2=1.$$
Question: Then is it true that $a_1$ and $1$ d... |
H: Average speed of train
Distance between two stations $A$ and $B$ is $778$km. A train covers the journey from $A$ to $B$ at a uniform speed of $84$km per hour and returns back to $A$ with a uniform speed of $56$km per hour. Find the average speed of the train during the whole journey?
The correct answer is:
Let dis... |
H: Geometric Interpretation of a question related to complex numbers..
QUESTION: I just encountered that $$\sum_{n=0}^8e^{in\theta}=0$$where $\theta=\frac{2π}9$
First I prove it, and then ask my question :)..
($i=\sqrt{-1}$)
MY ANSWER: We know, $e^{i\theta}=\cos(\theta)+i\sin(\theta)$ and that $e^{in\theta}=\cos(n\th... |
H: Give an example of distribution $u \in \mathcal{D}'((0,+\infty)) $ that is not extendible to $\mathbb{R}$
i.e.
find a $u \in \mathcal{D}'((0,+\infty))$, such that for any $v \in \mathcal{D}'(\mathbb{R})$,
$v|_{(0, +\infty)} \neq u$. In order to find such an example,
my question:
I tried to prove $e^{1/x^2}$ is an e... |
H: Caratheodory's theorem for vectors in a cone
I am studying the book "matching theory" by Lovasz and Plummer, and I found the following statement (page 257):
Comparing it with Caratheodory's theorem in Wikipedia reveals two differences:
The book speaks about vectors in a cone, particularly, in the conic hull of so... |
H: How does the indicator probability stays stable throughout the experiment
I've been dealing with this question-
An urn has 12 blue balls and 8 red balls. You extract balls one after the other. What is the expected value of the number of blue ball instances with a following red ball
This could be solved by indicator... |
H: Fundamental group of the torus minus a point (Van Kampen thm)
i had the exercise to compute the fundamental group of the torus minus one point p.
I know that the fundamental group of the torus is $\pi_1(T^2) = \pi_1(S^1) \times \pi_1(S^1) = \Bbb Z \times \Bbb Z$. So:
$U :=$ open neighborhood of p
$V := T^2 \backsl... |
H: Prove that for every real number $x$, if $|x-3|>3$ then $x^2>6x$.
Not a duplicate of
Prove that for every real number $x$, if $|x − 3| > 3$ then $x^2 > 6x$.
This is exercise $3.5.10$ from the book How to Prove it by Velleman $($$2^{nd}$ edition$)$:
Prove that for every real number $x$, if $|x-3|>3$ then $x^2>6x$.
H... |
H: If every subsequence of a sequence has a convergent subsequence then sequence is bounded or not?
In this question, for statement 1 i've proved that the later sequence is bounded and hence has a convergent subsequence.
For statement 2, i tried to construct a counter-example but i couldn't. So i suspect it is true an... |
H: Integrate $\operatorname{PV}\int_0^{\infty}\frac{x\tan(\pi x)}{(1+x^2)^2}dx$
A friend of mine send me the problem to integrate
$$\operatorname{PV}\int_0^{\infty}\frac{x\operatorname{tan}(\pi x)}{(1+x^2)^2}dx$$ where $\operatorname{PV}$ is Cauchy principle value.
I'm getting $\frac{1}{2}\psi^{(1)}\left(\frac{1}{2... |
H: The algebraic structure of ring $\mathbb{C}[x^2,xy,y^2]$
$1.$ What are maximal ideals of this ring?
$2.$ Is this ring local?
$3.$ Is this ring regular?
EDIT:
I tried to construct some ring that is isomorphic to $\mathbb{C}[x^2,xy,y^2]$
, but I cannot find it.
In addition, I think ideal $(x^2,xy,y^2)$ is a maximal i... |
H: Meromorphic continuation of $1+z+z^2+z^3+\ldots$
Consider the function $f(z)=1+z+z^2+z^3+\ldots$. This series is absolutely convergent on the disc $|z|<1$ and is equal to $1/(1-z)$ in this region. Now, $1/(1-z)$ is a meromorphic function on $\mathbb{C}$ with a simple pole at $z=1$. Is this sufficient to show that $... |
H: Determining whether the set of lines in $\Bbb R^2$ with at least two points whose coordinates are rational is a countable set
Let $A$ denote the subset $\Bbb Q\times\Bbb Q$ of $\Bbb R^2$ and $U$ denote the set of all lines in $\Bbb R^2$ that intersect with $A$ in at least two points.
Now I know $\Bbb Q\times\Bbb ... |
H: Let $X$ denote the closed unit interval, and let $R$ be the ring of continuous functions $X \to \mathbb{R}$.
$\textbf{Problem}$
$\bullet~$ Let $X$ denote the closed unit interval, and let $R$ be the ring of continuous functions $X \to \mathbb{R}$.
$\textbf{(a)}~$ Let $f_{1}, f_{2}, \dots, f_{n}$ be functions with n... |
H: Complex logarithm and the residue
The integral expression for the complex logarithm is defined by:
$$\int_{\gamma} \frac{1}{z}\,dz$$
where $\gamma$ represents a rectifiable path in $\mathbb{C}\setminus\{0\}$. The above integral defines $\text{log}(z)$, which has a branch cut emanating from $0$. And this is the sou... |
H: How to prove divisibility by $7$?
I am currently doing some preparatory maths for which I have an oral examination at the end of August, and am currently completely stuck trying understand how to solve a problem.
The problem is as follows:
Two three digit numbers, $\overline {abc}$ and $\overline {def}$, are such ... |
H: Certain open subsets of $L^1$ for $\sigma$-finite measure
Let $\mu$ be a $\sigma$-finite Borel measure on a metric space $X$, let $B$ be a Borel subset of $X$ of positive $\mu$-measure. Then when does the set
$$
\left\{
I_K g:\, g \in L^1_{\mu}(X)
\right\}\subseteq L^1_{\mu}(X)
$$
define a subset of $L^1_{\mu}(X)$... |
H: Differentiation + integration: how to solve for acceleration and displacement given a specific velocity time graph?
The velocity-time graph shown below is for a particle moving in a straight line, from rest at A, through B to C and then back to rest at B.
I have a few questions below regarding this velocity-time gr... |
H: Are these two products of random variables independent?
Assume $a,b_1$ and $b_2$ are independent random variables. I am wondering whether the $x=ab_1$ and $y=ab_2$ are independent.
Under the condition $a$ is known, these two random variables are obviously independent. However, when $a$ is also a random variable, ho... |
H: How many roots does $(x+1)\cos x = x\sin x$ have in $(-2\pi,2\pi)$?
So the nonlinear equation that I need to find the number of its roots is
$$(x+1)\cos x = x\sin x \qquad \text{with } x\in (-2\pi,2\pi)$$
Using the intermediate value theorem I know that the equation has at least one root on this interval, and if I ... |
H: Proof by Induction: Prove that $2^n > n^2$, for all natural numbers greater than or equal to $5$
Problem: $2^n > n^2, \forall n \in \mathbb{N} , n \geq 5$
Base: $2^5 > 5^2$
Induction Hypothesis: Assume for $n = k \geq 5$ that $2^k>k^2$
Inductive Step:
$$2^k > k^2$$
$$2^k \times 2 > k^2 \times 2$$
$$2^{k+1} > 2k^2$... |
H: If $V_{1} \subset V \subset V_1 + V_2\subset \mathbb{R}^{n}$. Is it true $V = V \cap V_{1} + V \cap V_2$?
Given subspaces $V$, $V_{1}$, $V_2$ of $\mathbb{R}^{n}$such that $V_1 \subset V \subset V_1 + V_2$. Is it true $V = V \cap V_{1} + V \cap V_2$?
Could you please verify my proof or propose other ways to solve ... |
H: A wrong law of large numbers for dependent variables
Suppose we are given $Y, X_1, X_2,\ldots$ i.i.d. standard normal random variables and define
$$Z_i = \sqrt{\rho}Y + \sqrt{1-\rho}X_i$$
for some given $\rho\in[0, 1)$. The random variables $Z_i$ are not independent if $\rho > 0$. Fix some threshold $T\in\mathbb{R}... |
H: Can anyone solve this Pell equation?
I have solved the Pell equation $ p^2 - 95 q^2 =1$ . By looking at the convergents corresponding to the simple continued fraction of $\sqrt{95}$ I was able to find the fundamental solution $p=39$ and $q=4$ .
I found the five smallest pairs of positive integers $p,q$ that satis... |
H: Theorem 2.9 Rudin functional analysis - Inferring exists $n$ such that $K \cap nE \neq \emptyset$
Follow up to this question.
I realized that question, which I've asked, explains "why" we can apply Baire's Theorem to $K$. It doesn't address however why $\exists n$ such that $K \cap nE \neq \emptyset$, so this quest... |
H: Show $U_1 \cup U_2=V \implies U_1=V$ or $U_2=V$
Let $V$ be a vectorspace over the field $K$ and $U_1, U_2$ subspaces of $V$.
Show $U_1 \cup U_2 = V \implies U_1=V$ or $U_2=V$
my thoughts:
Let $x_1 \in U_1$ and $x_2 \in U_2$, then $x_1+x_2 \in U_1 \cup U_2$. But this would mean
$x_1+x_2 \in U_2$ or $x_1+x_2 \in U_1$... |
H: If $L_M:(\mathbb{R}^m, \|\cdot\|_p) \to (\mathbb{R}^m, \|\cdot\|_q)$ is an isometry where $p\neq q$, must $M$ be orthogonal?
Given any $m\times m$ square matrix $M$, let $L_M:(\mathbb{R}^m, \|\cdot\|_p) \to (\mathbb{R}^m, \|\cdot\|_q)$ be defined by $L_M(x) = Mx$ where $1\leq p,q< \infty$ and
$$\|(x_1,...,x_m)\|_p=... |
H: How to show that $\sum_{n=1}^{N} \cos(2n-1)x = \frac {\sin(2Nx)}{2\sin(x)} $
I am studying Fourier analysis and have been given the following question:
Show that $$\sum_{n=1}^{N} \cos(2n-1)x = \frac {\sin(2Nx)}{2\sin(x)} $$
I used the formula for a finite geometric sum and Euler's formula to get to the following:
$... |
H: Prove that for all real numbers $a$ and $b$, $|a|\leq b$ iff $-b\leq a\leq b$.
Not a duplicate of
If $a\leq b$ and $-a\leq b$, then $|a|\leq b$.
if $-a\leq b\leq a$, then $|b|\leq a$
Is my proof of $|a| \leq b \iff -b \leq a \leq b$ correct?
Prove that for all real numbers $a$ and $b$, $|a| \leq b$ iff $-b \leq a \... |
H: Diffeomorphism theorem for Lie Groups?
The integral lattice $\Bbb Z^n$ is a discrete subgroup of the Lie group $\Bbb R^n$. Therefore, it acts freely and properly discontinuously on $\Bbb R^n$ and the orbit space $\Bbb R^n/\Bbb Z^n$ has a smooth manifold structure. I wanted to show that $\Bbb R^n/\Bbb Z^n$ is diffeo... |
H: Length of line segment at intersection of three spheres
For laying out a grid of spheres, I need to calculate the the length of the line segment (highlighted in red) at the intersection of three spheres:
Each sphere has an equal radius and is centred on the corners of an equilateral triangle; the $y$-coordinate fo... |
H: Integratethe following function: $\int \frac{1}{x(x+1)(x+2)\cdot\cdot\cdot(x+n)}dx, n \in \mathbb{N}$
Integrate the following function:
$$\int \frac{1}{x(x+1)(x+2)\cdot\cdot\cdot(x+n)}dx, n \in \mathbb{N}$$
I saw this question as an exercise on a University of Colorado website, and I'm not quite sure how to solve... |
H: Do properties in linear algebra proved by using matrix transformations hold true irrespective of the choice of the bases for the vector spaces?
Let us say I am required to prove that V (dimension $= n$) and $\Bbb{R} ^ n$ are isomorphic and have chosen the matrix representation way of doing this.
Assume a linear tra... |
H: Find $g(x)$ from the following condition: ${g(x)}=(\int_{0}^{1}{e}^{x+t}{g(t)}dt)+x$
Find $g(x)$ from the following condition: $${g(x)}=\left(\int_{0}^{1}{e}^{x+t}{g(t)}dt\right)+x$$ I have tried to solve it by applying Newton-Leibnitz formula and solving the linear differential equation with the help of integratin... |
H: Find the PDF of $U = {XY \over \sqrt{X^2+Y^2}}$ where $X, Y \sim N(0, 1)$ where X,Y are iid RV. Hence find the mean and variance of $U$.
The task is to find the PDF of $U = {XY \over \sqrt{X^2+Y^2}}$ where $X, Y \sim N(0, 1)$ where $X,Y$ are iid RV.
I approached this question as first finding the joint distribution... |
H: Invertbility of an element in a subalgebra.
Let $A$ be a unital algebra over the complex numbers and $B$ be a subalgebra of $A$ with $A=B + \Bbb{C}1_A$. Suppose that $B$ has a unit $1_B \neq 1_A$ and that $\lambda \in \Bbb{C}\setminus \{0\}$. The book I'm reading claims:
$$b + \lambda 1_A \mathrm{\ invertible \ i... |
H: 2 questions in text of Lesson : Inner Product spaces of Hoffman Kunze Linear Algebra( Related to Orthogonality)
I am self studying linear Algebra from Hoffman Kunze and I have 2 questions in text given just after Corollary of Theorem 3 whose image I am adding below.
It's image:
Question (1): Why in last line of... |
H: How to express the tail bound of this series as a function of $N$
I encountered a problem which asks to show how the tail bound of a series converges to zero, where the tail bound has to be expressed as a function of $N$. For example, given the series:
$$
\sum_{k=0}^{\infty} \frac{1}{2^k}
$$
I can derive an estimat... |
H: alternating series where $0
$0<x<1$ , then $1-x+{x}^2-\dots=$ $L$ $-(i)$
This is as much as I can say about this series:
Rewriting the series as $$\sum_{n=0}^{\infty} (-1)^{n}x^{n} $$
Ignoring the sign, I know that (i) is a geometric series and it would converge to $\dfrac1{1-x}$. Also, the terms are nonincreasing ... |
H: Proper punctuation of cases in statement of Lemma
I found a very similar question asked on the TeX StackExchange here but the answer was not as definitive as I hoped for.
My question is in regard to how one properly punctuates the following Lemma. Should a comma be used at the end of each case? If not, what would ... |
H: Integration: find as an exact value the enclosed area between $y=\frac{3x}{5π}$ and the curve $y=\sin x$ for $0≤x≤π$ shown shaded in the diagram.
The diagram shows the line $y=\frac{3x}{5\pi}$ and the curve $y=\sin$
$x$ for $0\le x\le \pi$.
Find (as an exact value) the enclosed area shown shaded in the diagram.
... |
H: Algebra structure of $\mathbb{k}^S$
I'm trying to write explicitly how the algebra structure of $\mathbb{k}^S$ works on its basis elements, where $\mathbb{k}$ is a field and $S$ any finite set.
Let's call its basis $B:=\{e_s:s \in S\}$, where the elements are defined as $e_s: t \mapsto \delta_{s,t}$. Now, I would l... |
H: How to find $k$ from $f(x)\;=\;\frac c{1+a\cdot b^x}$?
The population of a culture of bacteria is modeled by the logistic equation:
$P(t)\;=\;\frac{14,250}{1+29\cdot e^{-0.62t}}$
To the nearest tenth, how many days will it take the culture to reach 75% of its carrying capacity?
What is the carrying capacity?
What i... |
H: Let $ABC$ be a triangle and $M$ be the midpoint of $BC$. Squares $ABQP$ and $ACYX$ are erected. Show that $PX = 2AM$.
$\textbf{Question:}$ Let $ABC$ be a triangle and $M$ be the midpoint of $BC$. Squares $ABQP$ and $ACYX$ are erected. Show that $PX = 2AM$.
I could solve this problem using computational techniques... |
H: Order of multiplication of matrices $A$ and $A^n$
I've come across a problem where I need to find a matrix $A^{n+1}$, where I was given matrices $A$ and $A^n$. I multiplied them like this: $A^n\cdot A$, but I was obviously wrong since the result is adequate for $A\cdot A^n$.
Now, I know that $A\cdot B\neq B\cdot A$... |
H: $\frac 1 {1 + \epsilon} \le 1 - \frac \epsilon 2$ for $\epsilon \in (0, \frac 1 2)$
How can we show that the following holds for $\epsilon \in (0, \frac 1 2)$?
$$
\frac 1 {1 + \epsilon} \le 1 - \frac \epsilon 2
$$
I thought, maybe it would be more convenient to try to show somehow that $\frac 1 {1 + \epsilon} + \fr... |
H: A set of $n$ distinct items divided into $r$ distinct groups
A set of $n$ distinct items is to be divided into $r$ distinct groups of respective sizes $n_1, n_2, n_3$, where $\sum_{i=1}^{r}n_i=n$.
How many different division are possible ?
Because every permutation yields a division of the items and every possible ... |
H: How to calculate the triple integral $\iiint_{\Omega_t}\frac{1}{(x^2+y^2+z^2)^\frac32}$?
$\iiint_{\Omega_t}\frac{1}{(x^2+y^2+z^2)^\frac32}$, where $\Omega_t$ is the ellipsoid and $\Omega_t=\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}\leq 1$
I want to use the change variable for $u=\frac xa, v=\frac yb,w=\frac zc... |
H: Is this Hilbert space construction legal?
Let $X=\ell^2=\{(\xi_i)_{i\in\mathbb{N}}\in\mathbb{R}^\mathbb{N}\, | \, \sum_{i\in\mathbb{N}}|\xi_i|^2<\infty\}$ be the real Hilbert space of square-summable sequences, and for every $j\in\mathbb{N}$ set
$$ X_j=\{(\xi_i)_{i\in\mathbb{N}}\in X\, | \, \forall i\neq j, \xi_i=0... |
H: Extension linear independent set to Hamel basis
If we have a linear independent set, then it is well known that by using Zorn's lemma it can be extended to Hamel basis. My question I have a linear independent set, call it $B_{0}$, I want to extended by transfinite induction to Hamel basis B by the set $ B_{00}=B\se... |
H: Finding the Center of Mass of a disk when a part of it is cut out.
From a uniform disk of radius $R$ a circular disk of radius $\frac{R}{2}$ is being cut out.
The center of the "cut out" disk is at $R/2$ from the venter of the original disk. We have to find the center of mass of leftover body.
I thought that we sh... |
H: Isometry that translates a geodesic has constant displacement
In a Riemannian manifold $M$ we are given an isometry $\alpha$ that translates a geodesic $\gamma$, meaning that $\alpha(\gamma) \subseteq \gamma$, I would like to show that $d(\gamma(t), \alpha\gamma(t))$ is constant for all $t$.
I tried letting $s > 0$... |
H: $\int_0^{\infty} \frac {x.dx}{(x^2+a^2)^{\frac 32}(x^2+b^2)} $
I am solving a problem where I need to find the charge distribution on a conducting plate, and the field due to it. I'm stuck on this integral.
$$\int_0^{\infty} \frac {x.dx}{(x^2+a^2)^{\frac 32}(x^2+b^2)} $$
AI: Hint:
(I leave the intermediate steps fo... |
H: What is the limit of $\sum\limits_{i=1}^N (\frac{i}{N})^l\frac{1}{N}$?
I would like to know the limit of the above summation as $N$ tends to infinity. Thank you guys for hints or solutions. Cheers.
AI: It is the definition of Riemann integral for
$$
\int_{0}^{1}x^kdx
$$ |
H: How many unordered pairs of positive integers $(a,b)$ are there such that $\operatorname{lcm}(a,b) = 126000$?
How many unordered pairs of positive integers $(a,b)$ are there such that $\operatorname{lcm}(a,b) = 126000$?
Attempt:
Let $h= \gcd(A,B)$ so $A=hr$ and $B=hp$, and $$phr=\operatorname{lcm}(A,B)=3^2\cdot 7... |
H: What exactly is a constant angle?
I previously asked a question about what a non constant angle is but it was closed due to lack of clarity and hence, I'm posting a new question. The notation $x^c$ will be used in this question to represent $x$ radians.
So, a proof in my Mathematics textbook is about proving that r... |
H: Let $f:A \rightarrow B$ be a bijective map, and let $P,Q \subseteq A$ be any sets. Then $f(P-Q)=f(P)-f(Q)$
Can you please check my proof of the following theorem?
Theorem: Let $f:A \rightarrow B$ be a bijective map, and let $P,Q \subseteq A$ be any sets. Then $f(P-Q)=f(P)-f(Q)$.
Proof: Let $x \in f(P)-f(Q)$. Hen... |
H: Proving combinatorial identities
Prove $\displaystyle\sum_{k=1}^n kx^k{x\choose k}=nx(1+x) ^{n-1}$
This question can be solved easily (by taking the derivative of the binomial theorem formula), if there was an $\binom{n}{k}$ instead of $\binom{x}{k}$. I mean, the presence of $\binom{x}{k}$ seems a bit fishy. so, ... |
H: In Halmos' Naive Set Theory, how can there be more than 1 successor set?
The axiom of infinity clearly states that there exists a set $A$ containing $0$ and the successor of its elements.
Shortly after introducing this axiom, Halmos goes to say:
Since the intersection of every non-empty family of successor sets is... |
H: Given two polynomials, determine two other degrees through polynomial division
I read in a proof, where it says something in this direction
Divide any polynomial $Q$ of degree $2n-2$ by $P$ (of degree $n$) and get an equation $Q=SP+R$ with $S$ and $R$ polynomials of degree at most $n-2$ and $n-1$, respectively.
I... |
H: Example of $X_n$ which converge a.s. but not in mean
Provide an example of a sequence of random variables which converge a.s. but not in mean.
I know that the random variables $X_n=n\cdot\mathbb{1}_{(0,\frac{1}{n})}$ converge in probability as given any $\varepsilon>0$
\begin{align*}
P(|X_n-0|>\varepsilon)=P(X_n>... |
H: Expected number of coin side changes in a sequence of coin tosses with unfair coin
Suppose with have an unfair coin with probability p for heads and 1-p for tails. In a series of coin tosses(like n times) what is the expected number of times that the coin side changes? for instance if we toss the coin 5 times and t... |
H: Show $f$ can be extended to be analytic in $\mathbb{C}$ except at finitely many poles.
I have attempted the following problem but I am stuck on one part:
Suppose $f$ is analytic on the unit disk and continuous on the boundary of the disk. Also, suppose $|f(z)|=1$ for $|z|=1$. Show that $f$ can be extended to be ana... |
H: Is there a simple function $f(x)$ that follows $2$ rules when $x$ is rational?
Is there a simple function $f(x)$ that follows $2$ rules when $x$ is rational?
$x$'s simplest form is $\frac{a}{b}$ if $x$ is a rational number.
$$f(x) \in \begin{cases} \mathbb{R} \setminus \mathbb{Q}, \ \ \ \ x=\frac{a}{b} \text{ and ... |
H: Matrix of the linear transformation $T$
To the following theorem,
Let $V$ and $W$ be finite-dimensional vector spaces having ordered bases $\beta$ and $\gamma$, respectively, and let $T : V \rightarrow W$ be linear. Then, for each $u \in V$, we have $[T(u)]_\gamma=[T]_\beta^\gamma[u]_\beta$.
the textbook gives th... |
H: Permutation of a number yields a prime?
Given a number $N$ that is constructed only by using these digits: $\{1,3,7,9\}$,
It is not divisible by $3$ (The sum of digits are not divisible by $3$) and thus $3 \nmid N$.
And - it has at least $3$ different digits (maybe it uses only $1,3,9$ or $1,3,7$ or $1,3,7,9$ ...)
... |
H: Can the interval $[0,1]$ be made into a field?
After some cups of coffee with a friend we come up with a non-trivial question to our knowledge and it reads as follows :
Is is possible to define the operations of sum and product on $[0,1]$ so that makes it a field ?
As mentioned, this question is most likely beyon... |
H: What formula could I use to find out how many paperclips my factories could produce?
I'm playing Universal Paperclips and I'm near the end of the game. I currently have 2.3 quadrillion factories. Each factory un-upgraded can produce 100 billion clips per second. The first upgrade increases each factory's performanc... |
H: Prove that $B\cup(\bigcap \mathcal F)=\bigcap_{A\in \mathcal F}(B\cup A)$.
Not a duplicate of
$\cap_{A \in \mathcal{F}}(B \cup A) \subseteq B \cup (\cap \mathcal{F})$
This is exercise $3.5.16.b$ from the book How to Prove it by Velleman $($$2^{nd}$ edition$)$:
Suppose $\mathcal F$ is a nonempty family of sets and $... |
H: Does $A \rightarrow B$ imply $P(A) \le P(B)$?
My intuition is that if you have two events, $A$ and $B$, and you can show that event $A$ implies event $B$, then you should have $P(A) \le P(B)$ because any time A happens so does B, but not necessarily the other way around. Similarly then $A \leftrightarrow B$ implies... |
H: Question about the proof of Theorem 14 of Hoffman and Kunze
While self studying Linear Algebra from Hoffman and Kunze, I have a question in Theorem 14 in the section on unitary operators from Chapter 8.
Here are the relevant images.
My question is in highlighted line of the image. I am unable to get what reasoning... |
H: Proving the connection between limit points and neighborhoods
I am trying to prove the following proposition stated in my general topology textbook:
Let $A$ be a subset of a topological space $(X ,\tau).$ A point $x \in X$ is a limit point of $A$ if and only if every neighborhood of $x$ contains a point of $A$ dif... |
H: Does every continuous random variable have a pdf?
Does every continuous random variable have a pdf?
Is there any random variable which is neither discrete nor continuous?
Here, by continuous random variable I meant those random variables for which probablity of a singleton set is 0.
AI: If $X$ is a random variable ... |
H: Do functions with the same gradient differ by a constant?
Let $f,g:\mathbb{R}^n\to\mathbb{R}$ be such that $\nabla f=\nabla g$.
I believe this implies that $f$ and $g$ only differ by a constant, like in the one-dimensional case. But I'm not sure how to prove it. If it's indeed true, can you give me a hint?
Thanks!... |
H: When is the Lagrange interpolation polynomial exact?
Find the Lagrange interpolation polynomial for data points $x_k=k$ and $f(k)=k^2$, where $k=0,1,2,3$.
Also, find the Lagrange interpolation polynomial for the same data points but with $g(k)= k^4$.
I would like to say that in both cases the Lagrange polynomial (L... |
H: Is there a story proof behind the combinatorial identity $(n-2k)\binom{n}{k} = n\left[ \binom{n-1}{k} - \binom{n-1}{k-1} \right]$?
Is there a "story proof"/combinatorial proof for the following combinatorial identity:
$$(n-2k)\binom{n}{k} = n\left[ \binom{n-1}{k} - \binom{n-1}{k-1} \right]\tag1$$
I know that this i... |
H: Order of a subgroup generated by two elements in $S_5$
Let $G = \langle(12)(34), (15)\rangle$ be a subgroup of $S_5$.
Then I need to show that $G$ has order $12$ and has a non-trivial centre.
I have found thse elements- $$I,(12)(34), (15), (12)(34)(15), (15)(12)(34).$$
If I just keep computing compositions, then th... |
H: How prove that the elementary operations don't change the rank of a matrix
One considers certain operations, called elementary row operations, that are applied to a matrix $A$ to obtain a new matrix $B$ of the same size.
These are the following:
exchange rows $i_1$ and $i_2$ of $A$ (where $i_1\neq i_2$);
replace r... |
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