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H: Question about integration of functions
LEt $f$ be a measurable function. Define $\int\limits_E f dm = \sup Y(E,f)$ where $Y(E,f) = \{ \int\limits_E \phi dm : 0 \leq \phi \leq f \}$ and $\phi$ is simple function.
and $E$ is a lebesgue measurable set. I want to show that if $B \subseteq A$, $A$ and $B$ both lebesgue... |
H: If $(x-8)\cdot (x-10) = 2^y$, where $x,y\in \mathbb{Z}$. then $(x,y)$ is
(1) If $(x-8)\cdot (x-10) = 2^y$, where $x,y\in \mathbb{Z}$. Then the no. of ordered pairs of $(x,y)$
(2) If $x^4-6x^2+1 = 7\cdot 2^y$,where $x,y\in \mathbb{Z}$. Then the no. of ordered pairs of $(x,y)$
$\underline{\bf{My\; Try}}::$ for (1) on... |
H: how to prove $E(X|\mathcal G)(\omega)=n\int^{\frac{j}{n}}_{\frac{j-1}{n}}X(s)ds, \omega\in(\frac{j-1}{n},\frac{j}{n}].$
Suppose $\Omega=[0,1]$, and $\mathcal P=$lebesgue Measure , and $\mathcal F=\mathcal B([0,1])$
and also Suppose X is random variable and $\mathcal G$ is $\sigma-$algebra Produced With intervals $ ... |
H: prove that metric and series
Denote $E$ the set of all real sequences $\{a_n\}$ such that $|a_n| \leq 1$ for every positive integers $n$.Let $\{a_n\},\{b_n\} \in E$
Prove that $$d(\{a_n\},\{b_n\})=\sum_{n=1}^\infty\dfrac{|a_n-b_n|}{2^n}$$
defines a metric on E.
AI: $d$ will be a metric if it is a finite non-negativ... |
H: Complex conjugate to the power proof.
How can I proof that:
$$(z^n)^* = (z^*)^n$$
Where:
z is a complex number,
n is a positive whole number
* is the complex conjugate
AI: Write $z = re^{i\theta}$, then $\overline{z} = re^{-i\theta}$. So
$$
\overline{z}^n = r^n e^{-in\theta} = \overline{r^ne^{in\theta}} = \overlin... |
H: Definition Weierstrass $\zeta$-function unclear
The Weierstrass $\zeta$-function is defined as follows for a lattice $\Lambda$, where a lattice is a discrete subgroup of $\mathbb{C}$ containing an $\mathbb{R}$-basis for $\mathbb{C}$.
$$\zeta(z) = \frac{1}{z} + \sum_{\omega\in\Lambda\setminus\{0\}}\left(\frac{1}{z-\... |
H: How is the Borel-Cantelli lemma used in this proof on $\psi$-approximable numbers?
I'm trying to understand a paper called "Almost no points on a Cantor set are very well approximable". In the proof the author uses the Borel-Cantelli Lemma (in the eighth line at the beginning of the proof on the second page). There... |
H: $g_n(z)=z^n$ uniformly on $D={z: |z|<1}$?
I am looking at this example:
$g_n(z)=z^n$ and domain
$D={z: |z|<1}$
I see that every $g_n$ will converge to 0 for $n \rightarrow \infty$. Thus it converges.
Now, how can I show that it is or is not uniformly?
AI: Try to compute the limit of $g_n\left(1-\frac1n\right)$ whe... |
H: What is the lower bound of the subset $2^n,\; n\in\mathbb{N}$
Let:
$$
A = \{2^n,\; n\in\mathbb{N}\},\quad A\subset \mathbb{R}
$$
Is the lower bound:
$(-\infty,0]$
$(-\infty,1]$
$(-\infty,1)$
?
I think it can be the first because $\min_{A}=1,\;\inf_{A}=1$, according to the definition of lower bounds.
AI: Let $... |
H: how to compute this definite integral?
it is the second moment of logistic distribution, i.e.
$$
\int_{-\infty}^{\infty}\frac{x^2 e^x }{(1+e^x)^2}dx
$$
I've been struggling with it for hours. but i fail.
thanks first!
AI: $$\int_{-\infty}^{\infty} \frac{x^{2} e^{x}}{(1+e^{x})^{2}} \ dx = 2 \int_{0}^{\infty} \frac{x... |
H: $ \{ x : f(x) > 0 \} = \bigcup_n f^{-1}([\frac{1}{n}, \infty )) $
$ \{ x : f(x) > 0 \} = \bigcup_n f^{-1}([\frac{1}{n}, \infty )) $
My try: Pick $$x \in \bigcup_n f^{-1}([\frac{1}{n}, \infty )) \implies x \in f^{-1}([\frac{1}{n}, \infty )) \text{ for some $n$ } $$
$$ \therefore f(x) \geq \frac{1}{n} \text{for some ... |
H: Geometric intuition for partial derivatives with a single dependent variable
Having watched an integralCALC video lesson, given
$$w=xe^{y/z}, x = t^2, y = 1-t, z=1+2t$$
which could be rewritten as
$$w=t^2 e^\frac{1-t}{1+2t}$$
How does $dw/dt$ differ from $\partial w/\partial t$? Is there some intuition behind the... |
H: Maximmum and minimum values of function in interval
Hello I would like to learn the following from this site and members.
How to find the intervals ( Increasing and decreasing) of the function and its Maximum and minimum value, where the function: $f(x) = ax^2 + bx + c$.
Thanking you all.
AI: In quadratic functions... |
H: How to solve system of equations depending on parameter
\begin{cases} y=x^4 \\ y+8=a(x+5/4) \end{cases}
How many solutions does this sytem have depending on parameter a?
I need to solve it using derivatives somehow.
Thank you.
I solved using hint which bubba gave me.
Basically, we have this equation: $ x^4 + 8 = a(... |
H: Different implicit definitions
I'm Dutch, and my books are written in Dutch, so appologies if technical terms are incorrectly translated.
$\in$ means 'element of'
$\mathbb{Z}$ is the set of 'whole numbers'
$\leq$ is 'less than or equal to'
$V = \{x \in\mathbb{Z} ~~|~~x \text{ is even}\}$
$W = \{x \in\mathbb{Z} ~~... |
H: Solving equation with absolute value. Two solutions where only one is right.
I have solved this equation:
$$
\sqrt{3x}-\sqrt{4x-5}=0
$$
And I got that the solutions are:
$$
x=\frac{5}{7}, x=5
$$
My question is, did I do it right, because
$$
x=\frac{5}{7}
$$
dont fit in the equation. I mean, if I put 7/5 in equat... |
H: Inequality $|z_1+z_2|^2 \le (1+|z_1|^2)(1+|z_2|^2)$
I have a problem to prove this inequality
$|z_1+z_2|^2 \le (1+|z_1|^2)(1+|z_2|^2)$ $\forall (z_1, z_2)\in \mathbb{C}$.
I tried to take the right hand set and subtract the lfs and after simplification I got this:
$1+(ax)^2+(by)^2 -2(ax+by)+(ay)^2+(bx)^2$ and I co... |
H: Can I approximate sine and cosine without derivatives?
Assuming I don't know derivatives (and Taylor series) can I manage to approximate sine and cosine of a generic given (rational) angle in radians?
AI: I will only use things that are known in basic trigonometry, i.e., the fundamental trigonometric identity ($\si... |
H: Non-recursive way to present $ p_{0}=0$, $p_{n+1}=(e+1)p_{n}+e$ for some $e>0 \in \mathbb{R}$.
Is there a non-recursive way to present this function:
$ p_{0}=0$
$p_{n+1}=(e+1)p_{n}+e$
for some $e>0 \in \mathbb{R}$.
Or at least some estimation from the top would satisfy me.
AI: Note that $p_{n+1}+1=(\mathrm e+1)(p_n... |
H: How to prove $\gcd(a^m,b^m) = \gcd(a,b)^m$ using Bézout's Lemma
The problem is to prove the following
If $\gcd(a,b) = c$, then $\gcd(a^m, b^m) = c^m$
I know that this can be solved easily by proving that $c\mid a \implies c^m \mid a^m$ and $c\mid b \implies c^m \mid b^m$. So the greatest common divisor of $a^m$ a... |
H: How to interpret involutory change of basis transformation?
Just working through an assignment and a change of basis matrix popped up which was involutory - its own inverse.
I am not quite sure how to think about this... Presumably it means that the transformation doesn't 'scale' the basis vectors from one basis to... |
H: How to show that the limaçon has only two vertices.
Question: Show that the limaçon has only two vertices.
I researched what is limaçon. And I reached the following result;
Note that I only know that The limaçon is the parametrized curve
$$\gamma(t) = ((1 + 2\cos t)\cos t, (1 + 2\cos t)\sin t)$$ for $t \in \Bbb... |
H: increasing sequence of sets
suppose $A_n \subseteq \mathbb{R}$ for all $n$. Also if $A_{k} \subseteq A_{k+1} $, does it follow that $\bigcup_{k=1}^{\infty} A_k = \mathbb{R} $?. I know this is obvious but I dont know how to show the other inclusion: that is $ \mathbb{R}$ lives inside the bigcup
AI: Why is that obvio... |
H: Convergence in product topology
Let $x_1,x_2, \ldots$ be a sequence of points of the product space $\prod X_\alpha$. Show that the sequence converges to the point $x$ if and only if the sequence $\pi_\alpha (x_1), \pi_\alpha (x_2)\ldots$ converges to $\pi_\alpha (x)$ for each $\alpha$. Is this fact true for box top... |
H: Prove $A \cap B = A − (A − B)$
$A$ and $B$ are sets, how would I prove the following equality:
$A∩B = A-(A-B)$
Would I be correct in saying that if I take $x$ to be in $A-(A-B)$, it means that $x$ is in $A$, but not in $A-B$. Now, suppose temporarily that $x$ was not in $A∩B$, however, we already know that $x$ is i... |
H: Natural Deduction (FeedBack)
I am looking for feedback to three proofs (alternatively derivations) that I have constructed. The first is:
Theorem. Injectivity does not imply surjectivity.
Proof: Suppose $\{\phi\} \vdash \theta$. Then according to the soundness theorem we have $\{\phi\} \models \theta$, which is t... |
H: How to find a suitable orthogonal matrix
Assume $x,y \in \text{R}^n$ are two vectors of the same length, how to prove that there is an orthogonal matrix $A$ such that $Ax=y$?
Thanks for your help.
AI: Well, if they both have length zero, then take $A$ to be anything. If $\|x\| = \lambda = \|y\|$, then divide by $\l... |
H: Prove that this is an equivalence relation and give all the different equivalence classes
Let $R$ be a relation defined on real numbers by letting $a\mathrel R b$ iff $\cos (a) = \cos (-b)$. Prove that this is an equivalence relation and give all the different equivalence classes. Also show that this is all the cl... |
H: convergence of $\sum \frac {nx^{n-1}} {(1+x^n)^2}$
I want to examine the convergence of this function series $\sum \frac {nx^{n-1}} {(1+x^n)^2}$ for $x \in [2, +\infty)$. I showed pointwise convergence but I'm struggling with uniform convergence.
I tried to apply the Weierstrass M-test but it didn't work for me. I... |
H: Knowing that $b\leq\frac{a}{1-a}$ and $a<0.01$ show that $b \leq 1.01a$
I've been solving a problem in numerical analysis and to finish one of the exercises I need the following result.
Knowing that $b\leq\frac{a}{1-a}$ and $a<0.01$ show that $b \leq 1.01a$.
Now I think that this might be BS, but everything that le... |
H: Prove that there are only 2 solutions of $\frac{n+3}{n-1}=x$ for $n$ natural and $x$ natural.
I know that the solutions are $1) n=3$ and $2) n=5$ but I don't know how to prove that they are only the two.
AI: We have
$$\frac{n+3}{n-1} = 1 + \frac{4}{n-1}.$$
For that to be an integer, $n-1$ must be a divisor of $4$,... |
H: The series $\sum_{n=1}^{\infty}\frac{1}{n+1}$ diverge. What about $\sum_{n=1}^{\infty}\frac{1}{(n+1) \times (n+2)}$?
The series $\sum_{n=1}^{\infty}\frac{1}{n+1}$ diverge. What about $\sum_{n=1}^{\infty}\frac{1}{(n+1) \times (n+2)}$?
AI: Hint: $$\frac1{(n+1)(n+2)} = \frac1{n+1} - \frac1{n+2}.$$ |
H: Are these definitions of a prime ideal equivalent?
I just noticed I have three different definitions of a prime ideal in my notes. So are these definitions equivalent? Are they all correct...I have feeling I might have taken something down wrong.
Let $I$ be an ideal in a commutative ring $R$.
$I$ is prime if -
$a,... |
H: How to prove these propositions?
This is an exercise I don't quite know how to write in a acceptable mathematical form. (No, I haven't found a solution.)
"Prove that if $3 \mid n$ then $3 \mid n^2$.
And that if $3 \nmid n$ then $3 \nmid n^2$. (For the second part consider $n=3a+1$ and $n=3a+2$ in turn.)"
Thank you ... |
H: Prove that if $n \cdot 2^{-t} <0.01$ then $n \cdot 2^{-t} <\frac{1}{101}$
Is the following theorem true?
If $n \cdot 2^{-t} <0.01$ then $n \cdot 2^{-t} <\frac{1}{101}$ for $t,n \in \mathbb{N} $.
I've tried basic induction but that has led me nowhere, same with thinking of a counter-example.
AI: Let's find t such th... |
H: Why are non-separated schemes schemes?
In "the old days", e.g. in the famous texts by Grothendieck and Mumford, a scheme was defined as what we now call a separated scheme. (i.e. a scheme where the image of the morphism $\Delta:X \to X \times X$ is closed)
Nowadays, schemes are usually allowed to be separated. The ... |
H: How would I evaluate this limit? $\lim_{x\to 3+} \frac{x-3}{\sqrt{x^2-9}}$
How would I evaluate the following limit by hand?
$\lim_{x\to 3+} \frac{x-3}{\sqrt{x^2-9}}$
Thanks in advance.
AI: $$\begin{align} \lim_{x\to 3+} \frac{x-3}{\sqrt{x^2-9}} & = \lim_{x \to 3+}\; \frac{\left(\sqrt{x - 3}\;\right)^2}{\sqrt{(x - ... |
H: Coupon selecting problem
Suppose that there are N distinct types of coupons and that each time one obtains a coupon, it is, independently of previous selections, equally likely to be any one of the N types. One random variable of interest is T3, the number of coupons that needs to be collected between the acquisiti... |
H: Proving that a holder continuous function always has a smaller exponent.
According to wikipedia if we have $f:X \rightarrow Y$ which is $\alpha$-Holderian then for all $\beta < \alpha$ the function is also $\beta$-Holderian.
How do we prove this starting from the fact that we have $d_y(f(x),f(y)) \leq K d_x(x,y)^{\... |
H: almost everywhere convergence vs uniform convergence
Let $(\mathbb{R},\mathcal{L},m)$
Can someone explain to me why $f_{n}(x)=\chi_{(0,\frac{1}{n}]}$ converges almost everywhere to $0$ but not uniformly...
also why does $f_{n}(x)=n^{-1}\chi_{(0,n)}$ converge uniformly on $\mathbb{R}$?
AI: (1) For any $x_0\in \mathb... |
H: Continuous Function
Let $ f: \Bbb R^2 \to \Bbb R $ such that :
$ \forall _{y_0 \in \Bbb R\ }: $ function $ x \to f(x,y_0) $ continuous function and increasing
$ \forall _{x_0 \in \Bbb R\ }: $ function $ y \to f(x_0,y) $ continuous function
I mean the continuity of one variable.
Prove the continuity of a functio... |
H: Does there exist an infinite number string without any 'refrain'?
Let us consider an infinite or finite number string which consists of $0,1,2$. Then, let us call an adjacent pair of repeating number(s) 'a refrain'.
For example, we have three refrains in the following string :
$$01\overline{2}\ \overline{2}01202\o... |
H: Help with an elementary set theory proof
"If $A$ and $B$ are two sets, then prove that $A$ is the union of a disjoint pair
of sets, one of which is contained in $B$ and one of which is disjoint from
$B$."
Can somebody help me understand what this question even wants me to prove?
As far as I can understand, it wants... |
H: Evaluate the limit $\lim_{x\to 2} \frac{x-2}{\sqrt{x^2+5}-3}$
I need to evaluate the following limit:
$\lim_{x\to 2} \frac{x-2}{\sqrt{x^2+5}-3}$
I have multiplied both sides by the conjugate $\sqrt{x^2+5}+3$ but am getting $x^2-4$ as the denominator. Is this the correct way to go about it?
AI: Yes indeed, that's th... |
H: Clopen and open sets have the same measure
I was positive I had already asked this one but apparently not. It is simply this:
Show that the Lebesgue outer measure of $[a,b)$ is $b-a$.
Let $\mathscr{K}$ be the collection of open subsets of $\mathbb{R}$ and $\lambda$ a nonnegative setfunction defined on $\mathscr{K... |
H: $\forall$ ${i \in \{1,...n\} }$ $ a_{i}
Here's my exercise:
EDIT:
$v=nu$, not $\nu$ (same Latex code but one is without )
$\forall n \in \mathbb{N}$
$\forall$ ${i \in \{1,...n\} }$ $ |a_{i}|\leq u $ and $nu<0.01$ and $u=2^{-t-1}$ prove that there exists $\eta$ such that $\prod_{i=0}^{n}(1+a_{i})$=1+$\eta$ and $\eta... |
H: Showing Expected Value Does Not Exist
Let X be a uniform random variable on [0,1], and let $Y=\tan\left (\pi \left(x-\frac{1}{2}\right)\right)$. Calculate E(Y) if it exists.
After doing some research into this problem, I have discovered that Y has a Cauchy distribution (although I do not know how to prove this); t... |
H: Initial value problem $x'=x^{2/3}$
I am just analyzing the IVP $x'=x^{\frac{2}{3}}$ with initial condition $x(0)=0$. It is obvious that there is a solution which is not unique, since it is not Lipschitz (not bounded near 0). My book says it has a solution that satifies $x(t)=0$ iff $t\in [t_1,t_2]$ for eached fixed... |
H: Regarding countable sets
Please help to to prove that if a collection of sets $A$ is countable, then the set of all finite intersections of members of $A$ is also countable.
I couldn't find the bijection between them.
AI: Let $\mathcal{A} = \{A_1, A_2, \ldots\}$ be a countable collection. For each $k \in \mathbb{N}... |
H: Given $Re(f(z))$ and the fact that $f(z)$ is analytic, find $Im(f(z))$
The question I'm trying to answer:
Find an analytic function $f(z)$ whose real part $u(x,y)$ is:
$$\frac{y}{x^2+y^2}$$
An analytic function satisfies the Cauchy-Riemann relations. So I thought to differentiate the real function, and then integ... |
H: Subtle question concerning intersection of convex sets
I am attempting to convince myself that if $$\{S_{\alpha}: \alpha \in \mathcal{A}\}$$ is any collection of convex sets, then $$\cap_{\alpha \in \mathcal{A}}S_{\alpha}$$ is convex. This is my proof so far:
Let $C = \cap_{\alpha \in \mathcal{A}}S_{\alpha}$ be an ... |
H: Taylor evaluation in a product solving a limit
I have the following function, which I am supposed to evaluate:
$\lim_{x \to 0}{\frac{(e^{-x^2}-1)\sin x}{x \ln (1+x^2)}}$
My though is to replace sin x by its Maclaurin polynomial, as such:
$\lim_{x \to 0}{\frac{(e^{-x^2}-1)(x+ O(x^3))}{x \ln (1+x^2)}}$
From here I th... |
H: Consider the ring $R=ℂ[X,Y]$ and the ideal $I=(X^2-Y,X^2+Y)$. We find (??) that $R/I ≅ℂ[X]/(X^2)$.
I'm trying to understand a step in an example of my reader about rings.
Consider the ring $R=ℂ[X,Y]$ and the ideal $I=(X^2-Y,X^2+Y)$. We find
that $R/I ≅ℂ[X]/(X^2)$.
As the author doesn't give any more details of... |
H: Does order matter for the convergence of infinite products
Similar to infinite sums, does order matter in the convergence of infinite products? More specifically, I'm interested in the product of all rational numbers in the interval $(0,a]$.
For example, let $a=3$. I assert that the product converges to 0. Since a... |
H: Finding Jordan Basis of a matrix
Having trouble finding the Jordan base (and hence $P$) for this matrix
$A = \begin{pmatrix}
15&-4\\ 49&-13
\end{pmatrix}$
I know that the eigenvalue is $1$, this gives an eigenvector $\begin{pmatrix}
2\\ 7
\end{pmatrix} $
Now to create the Jordan basis and find $P$ (of which its co... |
H: A subgroup of $S_n$ of odd order is contained in $A_n$
I saw this question. The original questioner asserted that the Lagrange's Theorem is sufficient to solve the problem, but I think that the theorem does just say that the order of $H$ divides the order of $A_n$, but not say that $H$ is a subgroup of $A_n$...
Is... |
H: How find Number of integer solutions $x_{1}+x_{2}+\cdots+x_{n}=m$
Question:
let $m$ and $n$ be positive integers,The number of positive integer solutions to the equation
$$x_{1}+x_{2}+\cdots+x_{n}=m,m\ge n,x_{i}\ge 1,1\le x_{1}\le x_{2}\le\cdots\le x_{n},(i=1,2,\cdots,n)$$
is $f(m,n)$,How find this close form $f... |
H: If $x=\prod^{27}_{n=1}(1+\frac{2}{n})$ then find $13x$ - Ramanujan Mathematics olmpiad 2013
I tried this: $$x=\prod^{27}_{n=1}(1+\frac{2}{n})=(1+\frac{2}{1})(1+\frac{2}{2})(1+\frac{2}{3})\ldots(1+\frac{2}{27})=\frac{3}{1}\cdot\frac{4}{2}\cdot\frac{5}{3}\cdots\frac{29}{27}$$ Then the terms cancel out. But I am not g... |
H: vector represented by standard basis
let us consider following problem
i want to clarify one thing,namely vector $v=(a,b,0)$ satisfy yes this condition?or is there any other thing asked? generally every $v$ in $R^3$ is represented by
$v=v_1*i+v_2*j+v_3*k$
i have tried figure out what combination of $m$ a... |
H: Is $f(0, y) = f(y) \implies f_{y} (0,y) = f'(y)$ true?
OK, I think I'm being a bit stupid here, but I need to check whether my reasoning here is correct, as I don't want to mess this up. So my question is:
Is $f(0, y) = f(y) \implies f_{y} (0,y) = f'(y)$ true?
It seems to me that it is, but, for some reason, I've g... |
H: Induction on natural numbers
My textbook, Logic and Discrete Mathematics by Grassman and Tremblay, has an example which I can't wrap my head around (example 3.4; page 127).
It shows that for all $n$, $2(n+2)\le(n+2)^2$.
As the inductive base, we have $P(0): 2(0 + 2) \le (0 + 2)^2 = 4 \le 4$, which is true.
We assum... |
H: If n is such that every element $(\mathbb{Z}/n\mathbb{Z})^{\times}$ is a root of $x^2-1$. Prove that $n$ divides 24.
I have a hard time formulating proofs. For this problem, I can see that if $n$ is equal to $8,$ this statement is true. $(\mathbb{Z}/8\mathbb{Z})^{\times}$ includes elements: $1,3,5,7$, and all o... |
H: Show that there exists a subspace $W \subset \mathcal{P}(4)$ such that $\mathcal{P}(4) = \mathcal{U}(4) \oplus W.$
Let $\mathcal{U}(4)$ be a subspace of $\mathcal{P}(4)$ consisting of polynomials that are even functions. Show that there exists a subspace $W \subset \mathcal{P}(4)$ such that $$\mathcal{P}(4) = \math... |
H: How to prove that $S_n^2 − Var(S_n )$ is a martingale
I would be grateful for some help with the following exercise:
Let $(X_n ,n≥1)$ be a sequence of independent random variables with $E[X_i]=0$, and
$Var(X_i)=σ_i^2<\infty, ∀i ∈\mathbb{N}$. Prove that $S_n^2 − Var(S_n)$ is a martingale, where
$S_n:= \sum\limi... |
H: Does there exist a symmetric tridiagonal matrix with zero determinant?
I will like to know whether there exists a symmetric tridiagonal matrix with zero determinant? I will refer the definition of a tridiagonal matrix to the one found in Wikipedia:
"A tridiagonal matrix is a matrix that has nonzero elements only o... |
H: Eigenvectors of a linear operator
Let $A$ be $m \times m$ and $B$ be $n \times n$ complex matrices, and consider the linear operator $T$ on the space $\mathbb{C}^{m\times n}$ of all $m \times n$ complex matrices defined by $T(M)=AMB$.
(a) Show how to construct an eigenvalue for $T$ out of a pair of column vectors $... |
H: How to prove this similarly but changing the hypothesis?
I'm trying to prove the followong statement, which includes already a proof, but changing the hypothesis $P^2 = P$ by dim Im $P^2= $ dim Im $P$ and assuming $V$ is finite dimensional.
I was trying to prove that dim Im $P^2 = $ dim Im $P$ implies $P^2=P$, but... |
H: no. of quadrilateral in 12 sided polygon
Find the number of quadrilaterals that can be made using the vertices of a polygon of 12 sides as their vertices and having
(1) exactly 1 sides common with the polygon.
(2) exactly 2 sides common with the polygon.
$\underline{\bf{My \; Try}}::$ Let $A_{1},A_{2},A_{3},.......... |
H: Equation of a circle given two points and tangent line
I am given that $P(-3,-1)$ and $Q(5,3)$ are points of the circle. Also, the line $L:0=x+2y-13$ is tangent to said circle. The objective is to find the equation of the circle.
I thought of a way for solving this, but it doesn't seem to be the best one. The optio... |
H: Show $a^2=6k+3 \Rightarrow a = 6n + 3$
Show that if $a^2=6k+3$, for some integer $k$, then also $a = 6n + 3$ for an integer n.
Or in in other words: $a^2=6k+3 \Rightarrow a = 6n + 3$.
Taking the square root, $a=\sqrt{6k+3}$, does not help. I've also looked after factorizations, but I haven't find anything useful.
A... |
H: Definition of Ring Homomorphism
I am using a text right now for abstract algebra ("A Concrete Introduction to Abstract Algebra" by Lindsay Childs) that seems to use a non-standard defn of ring homomorphism. I want to see if others agree and if the difference is a significant one or not.
Let (R,+,$*$) and (R',+',$*'... |
H: If the matrix of $f$ is diagonal, why is $f$ equal to scalar multiple of identity map?
Suppose that $V$ be a finite dimensional vector space, and $f:V \longrightarrow V$ be a non zero linear map. If the matrix of $f$ with respect to any basis of $V$ gives a diagonal matrix, why is $f= \lambda Id. $ where Id is an i... |
H: Prove formally that |N| = | N union a finite set |.
I'd like to show that the cardinality of $\mathbb{N}$ is the same as the cardinality of $\mathbb{N}$ union some other finite set (disjoint from $\mathbb{N}$). For example show that:
$|\mathbb{N}|= | \mathbb{N} \cup \lbrace \sqrt{2},\sqrt{3} \rbrace |$.
To prov... |
H: Can you give me a good alternative to Rotman's Group Theory book?
I've been trying to learn out of Rotman's book "An Introduction To The Theory of Groups" for the last few months, and it's rough going. I've been studying Chapters 7, 10, and 11 in particular, and he's too wordy in many places and makes massive leap... |
H: Why does Wolfram Alpha state that $-\infty/0 = +\infty$?
I ran into a scenario when practicing L'Hôpital's rule which yielded -infinity/0. I broke this down into $-1 \cdot \infty \cdot \frac 1 0$, which I assumed equaled $-1\cdot\infty\cdot\infty$, which simplified to $-1\cdot\infty$ which equals negative infinity.... |
H: Show that $\lim_{x\to \infty}\left( 1-\frac{\lambda}{x} \right)^x = e^{-\lambda}$
Based on the definition of $e: = \lim_{x\to\infty} \left(1+\frac1x \right)^x$, how can we show that
$$\lim_{x\to \infty}\left( 1-\frac{\lambda}{x} \right)^x = e^{-\lambda}?$$
So far I've tried changing variables, $\eta = \frac{-x}{\... |
H: Need help with proof: If R ⊆ S then R^(-1) ⊆ S^(-1)
Here is what I have so far:
If R ⊆ S then R^(-1) ⊆ S^(-1).
Suppose R ⊆ S and that (a, b) ∈ R^(-1). This means that (b, a) ∈ R. Since S is a relation from A to B, that means that (b, a) ∈ S. That would mean that (a, b) ∈ S^(-1), thus proving this statement true.... |
H: Proving a special case of the Squeeze Theorem
Prove that if $\{a_n\}\to0$, $$0\leqslant b_n\leqslant a_n \implies \{b_n\}\to0,$$where $a_n,b_n$ are infinite sequences and $n\in\mathbb{N}$. Then, using the result $$\{a_n\}\to0\iff\{\left|a_n\right|\}\to0,$$prove that if $\{a_n\}\to0$, $$0\leqslant \left|b_n\right... |
H: Problem on inequality
Prove that, $E|X|^p < \infty $ iff $\sum_{k=1}^{\infty}k^{p-1}P\{|X| \geq k\} < \infty$.
Where E is the expectation and P is the usual probability measure. There was a mistake one it's correct.
AI: Let $A_k:=\{\omega,k\leqslant |X(\omega)|\lt k+1\}$. Then switching the order or summation (allo... |
H: Set difference and probability
Trying to work out the following question. If 80% of people like walking, 60% like biking, and everyone likes at least one of these, then what is the fraction of people who like biking, but not walking?
If $P(w)=0.8$, $P(B)=0.6$, and $P(W \cup B)=1$, then what is $P(B \backslash W)$... |
H: Multiplication of linear maps $S$ and $T$
If $T \in \mathcal{L}(U,V)$ and $S \in \mathcal{L}(V,W)$ them we define $ST \in \mathcal{L}(U,W)$ by $$(ST)(v)= S(Tv) \text{ for } v \in U.$$ Verify that the composition $S \circ T$ of functions as defined above is linear.
Is the following verification correct?
$\begin{alig... |
H: Complex double integral
I'm having trouble calculating following (complex) integral.
$$\int_D z^n dA$$
where $D=\{ z \in \mathbb{C} \mid \lvert z \rvert \leq 1 \}$.
I know how to calculate complex (line) integrals and real double integrals but I'm having difficulties combining those.
AI: Hint: Using polar coordinat... |
H: Sum of factorials equation
Could you explain why constant $c \gt 0$ can't satisfy equation $c + \frac{c \cdot n!}{\varphi } \le 1$ , where $\varphi = \sum_{i=0}^{n-1}i!$, where $n \to \infty$
AI: Hint:
$$\frac{n!}{0!+\ldots+(n-2)!+(n-1)!} \ge \frac{n!}{(n-2)!+\ldots+(n-2)!+(n-1)!} = \frac{n!}{(n-1)!+(n-1)!} = \fra... |
H: Can't solve following limit: $\lim_{x \to \infty}x \left( \sqrt[3]{5+8x^3} - 2x\right)$
Need to solve following problem:
$$\lim_{x \to \infty}x \left( \sqrt[3]{5+8x^3} - 2x\right) $$ I've tried to do something like this: $$\lim_{x \to \infty} x\left(\sqrt[3]{5+8x^3} - 2x\right) =\lim_{x \to \infty}x\left( \sqrt[3]{... |
H: exercise on the closed subspaces of an Hilbert spaces
I have a question regarding exercise 3.1.13 of "Analysis Now" by Pedersen volume 118 of the Springer GTM.
The exercise aim to show that any closed subspace $X$ of
$L^2([0,1])\cap L^{\infty}([0,1])]$ is finite dimensional.
I'd like to get whatever proof of this... |
H: How to compute infinite series $\sum_{n=0}^{\infty} ne^{-n}$
I'm trying to compute the infinite series $\sum_{n=0}^{\infty} ne^{-n}$.
I know the answer is $e/(e-1)^2$, but I don't understand how to find this result.
Thanks for the help!
AI: You can differentiate the series $\sum_{n=0}^\infty e^{-nx} = \frac{1}{1-e^... |
H: Limit Comparison Test Problem confusion
I have a problem where I'm supposed to find if the series converges. I'm looking at the solutions manual and I'm really confused. I know it uses the limit comparison test, but one moment it goes from $n+n^2+n^3$ to $1/n^2+1/n+1$ and I feel like I don't understand how they did... |
H: Chance of a double in three dice
What is the chance of rolling doubles in three six sided dice?
The answer I have is:
$$
\frac{1}{6}•\frac{1}{6}•\frac{3}{2} = \frac{1}{24}
$$
AI: The number of elements in the sample space = 216 or (6*6*6)
Of the three numbers two of them to be same is (1,1) (2,2) (3,3) (4,4) (5,5)... |
H: What is the cardinality of an element of an free ultrafilter?
Let $U$ be a free ultrafilter on a set $X$.
I want to prove that the cardinality of every element $u\in U$ is equipotent to $X$. Is that true? Or does it lack some hypothesis?
AI: No, this is not provable because it can be false.
Consider an ultrafilter... |
H: Prove the column span of $A$, an $m\times n$ matrix, is $F^m$ iff $\exists$ an $n\times m$ matrix $B$ such that $AB = I_m$
This problem has a second part, about the columns of $A$ being linearly independent in $F^m$ iff $\exists n \times m$ matrix $B$ such that $BA = I_n$ but I think if I understand the first part,... |
H: Discrete Distribution Help
If p(n) = c(5/8)^n, 3 <= n <= infinity is a p.m.f. for a discrete random variable X, find (a) c, (b) the probability P(6 <= X <= 16), (c) the mean and (d) the variance
Here's my work. I think got c wrong and haven't gone past it.
(a) Since the sum of p(n) = 1, I got c(5/8)^1 + c(5/8)^2 + ... |
H: Matrix calculation from an equation
I am trying to find matrix y from an equation
For example, I know (python)
>>> x = matrix([[5,1],[2,4]])
>>> y = matrix([[1,5],[3,3]])
>>> print x
[[5 1]
[2 4]]
>>> print y
[[1 5]
[3 3]]
>>> print x*y % 7
[[1 0]
[0 1]]
Assuming, I don't know what y is. But I know xy = I mod7 ... |
H: Reflection across a line?
The linear transformation matrix for a reflection across the line $y = mx$ is:
$$\frac{1}{1 + m^2}\begin{pmatrix}1-m^2&2m\\2m&m^2-1\end{pmatrix} $$
My professor gave us the formula above with no explanation why it works. I am completely new to linear algebra so I have absolutely no idea ... |
H: Please help me interpreting Elementary Set Theory question
I have to prove something for homework, but I can not for the life of me figure out what the theorem I'm supposed to prove is. Again, please: I am not asking for you to prove it for me (that's my hw and I want to do it myself), I am just asking you to help ... |
H: Checking understanding of concept
I want to check if I have understood a concept correctly.
Problem: Describe geometrically the action of an orthogonal $3$ x $3$ matrix with determinant -1.
My solution: The orthogonal $3$ x $3$ matrix with determinant $-1$ is an improper rotation, meaning it is a reflection combine... |
H: Homeomorphism preserving partitions
Let $X$ and $Y$ are homeomorphic topological spaces.
Consider a equivalent relation $R_X$ and $R_Y$ that partition $X$ into $X_1,\ldots,X_n$ of and $Y$ into $Y_1,\ldots,Y_n$ respectively.
$X_i$ is homeomorphic to $Y_i$ for all $1\leq i\leq n$.
The quotient topologies $X/R_X$ an... |
H: determining abelian groups of a certain size up to isomorphism
Say the size is 360. My book uses this as an example. It says there are 6 distinct groups up to isomorphism:
$\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_3 \times \mathbb{Z}_3 \times \mathbb{Z}_5$
$\mathbb{Z}_2 \times \mathbb... |
H: Does $1/n$ satisfy the Cauchy criterion?
I have been told that if a series satisfies the Cauchy criterion, it converges. I was not quite convinced on this, however, since some of the posts on this website seemed to imply otherwise. I came up with what I thought was a counterexample: I argued that $1/n$ satisfies th... |
H: Need your help with the integral $\int_0^\infty\frac{dx}{e^{\,e^{-x}} \cdot e^{\,e^{x}}}$.
Is it possible to evaluate this integral in a closed form?
$$\int_0^\infty\frac{dx}{e^{\,e^{-x}} \cdot e^{\,e^{x}}}$$
AI: This is clearly equivalent to
$$\int_0^{\infty}e^{-2\cosh x}dx=K_0(2).$$
Here $K_0(x)$ denotes Macdonal... |
H: When is the Cayley transform of a matrix $J$-orthogonal?
The (real) general linear group is defined $GL(n)=\{A \in \mathbb{R}^{n\times n} \mid \operatorname{det}(A) \neq 0\}$. It is a matrix Lie group. Let $J$ be a constant $n$-by-$n$ real matrix. The so-called $J$-orthogonal group is defined $O_J(n)=\{A \in GL(n... |
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