text stringlengths 83 79.5k |
|---|
H: A linear algebra related detail in a proof of Index Theorem
Here is a clipping from Milnor's Morse Theory. Since this question about linear algebra, I will present my question below so that no prior knowledge of the materials in the book required to answer this question.
The only relevant thing here is the matrix ... |
H: Are real numbers enough to solve simpler exponential equations such as $2^x=5$, $(1/e)^x=3$, and $\pi^x=e$?
How to prove that solutions of simpler exponential equations (*) are real numbers?
In other words, how to prove that set of real numbers is enough to solve something like $2^x = 5,$ or $(\frac{1}{e})^x = 3$ o... |
H: Show that the group in the center of this short exact sequence is abelian
Suppose that $$1\to A\xrightarrow{f} B\xrightarrow{g}C\to 1$$ is an exact sequence of groups, where $A$ has order $85$ and $C$ has order $9$. Show that $B$ is abelian.
I have proved earlier that for a Sylow $3$-subgroup $S$ of $B$, we have th... |
H: Hessian form of a real valued function on a submanifold of $\mathbb{R}^{n+m}$
I recently came across a result in a text in the field of differential geometry and I'm wondering why it is true. Let $M\subset\mathbb{R}^{n+m}$ be an $n$-dimensional submanifold and $w:M\longrightarrow\mathbb{R},\,w(x)=u(x)-\langle x,z\r... |
H: Formal way of proving that the ring $2 \Bbb Z_{10}$ has an additive inverse
I'm trying to find a formal way of proving that the ring $2 \Bbb Z_{10}$ has an additive inverse. I understand that 8+2 = 0 mod 10 and 6+4 = 0 mod 10, etc. Is there a way to formally prove this is true?
AI: Since $ \mathbb{Z}_{10} $ is a ri... |
H: How can a subset be undecidable?
A subset of a set can have an undecidable member relation. Though how can you determine if $A$ is actually a subset of $B$ if the member relation of $A$ is not decidable?
That feels contradictory because the definition of the subset relation uses the member relation: “If all the mem... |
H: Is 3D space just a surface with an infinite number of handles?
I have recently come across the concept of a genus, and I was wondering, is 3D space mathematically equivalent to a surface with an infinite number of handles? I ask this because I asked a question a few weeks ago on whether a graph can be 'non-planar' ... |
H: classify stable and unstable equilibrium points for differential equation $\frac{dx}{dt} = x(\lambda -x)(\lambda + x)$
I am doing exercise to find equilibrium points and classify them as stable/unstable for the following differential equation:
$\frac{dx}{dt} = x(\lambda -x)(\lambda + x)$
With $\frac{dx}{dt} = 0$, I... |
H: Finding a joint distribution given a marginal and conditional distribution
My question is given a marginal distribution $p_X(x)$ and conditional distribution $p_{X|Y}(x|y)$, am I guaranteed to be able to find a joint distribution $p_{X,Y}(x,y)$.
In almost every form of this question I have seen asked or discussed,... |
H: What is this integral with the reciprocal of sine?
$$f(z) = \int\limits_{-\infty}^{\infty} \frac{x}{\sin(x z)} dx$$
What is the function $f(z)$? This doesn't converge for real $z$ but when $z$ is purely imaginary then $\sin$ becomes $\sinh$. So for example $f(i)=-\frac{i}{2}\pi^2$. But that's as far as I got.
AI: F... |
H: Show that $\int_{E_n} f d \mu \to \int f d \mu$, when $n \to \infty$
Given a measure space $(\Omega, \mathcal{F}, \mu)$ and $f : \Omega \to \mathbb{R}$ integrable. Consider
$$E_n = \left\{x \in \Omega : |f(x)| \geq \frac{1}{n} \right\}, n \geq 1.$$
I have shown that $\mu(E_n) < \infty$, for all $n \geq 1$. However,... |
H: Suppose $U \sim Unif(0,1)$ and $Z \sim Unif(U,3+U)$. How can I find the pdf for $U + Z$?
Suppose $U \sim Unif(0,1)$ and $Z\mid U \sim Unif(U,3+U)$. I would like to find the pdf for $U + Z$, which in my process on $Z$ is a continuation of $U$. Is there a straightforward way to derive this?
AI: Looks like we have ano... |
H: Prove that $u(x,t)\leq 6t+|x|^2$ for all $(x,t)\in U_T$. Here $U_T=U\times(0,T]$
let $U$ be the unit ball in $R^3$. Suppose $u$ solves the heat equation
$$
\left\{\begin{matrix}
u_t-\Delta = 0 \text{ in } U_T\\
u(x,t)=6t \text{ when } |x|=1\\
u(x,0)=g(x)
\end{matrix}\right.
$$
Suppose that $g\leq0$. Prove that ... |
H: Is the volume of a cube the greatest among rectangular-faced shapes of the same perimeter?
My child's teacher raised a quesion in class for students who are interested to prove. The teacher says that the volume of a cube is the greatest among rectangular-faced shapes of the same perimeter and asks his students to p... |
H: Prove that $\mathrm{Ker}(T - \lambda I_V)^n = {0}$
I´m having trouble proving this statement, I already tried induction but I failed miserably.
Let $V$ a $K$-vectorial space and $T: V\to V$ endomorphism. Let $\lambda\in K$ that it's not a eigenvalue of $T$. Prove that $\mathrm{Ker}(T - \lambda I_V)^{n} = \{0\}$ for... |
H: Formula for unique orderings of N items from k bins sorted contiguously.
I'm looking for a formula that will tell me how many unique orderings I can get if I pick out $N$ items from $k$ bins that those items are divided into in continuous segments. For example, for $N=4$ and $k=2$, the $2$ bins are: $\{0,1\}$ and $... |
H: Analysis on $y = x$ line for Equilibria Points
I am having trouble analyzing what happens when I set the parameters to $(A, B) = (0,0)$ for the phase portrait in reference to this graph below. The first phase portrait graph solutions look undefined, but I wanted to know what it meant terms of the ab graph where $x=... |
H: Ultrafilter with finite set
While I was working in some exercise about filters, a question came to my mind: let $X$ a set and $F\subseteq X$ a non empty finite set. How many ultrafilters $U$ there are such that $F\in U$? I think that there exist a unique ultrafilter that contains $F$ but I can't see why or how to p... |
H: Proof with invertible matrices
Let $A,B,C$ be matrix of the same size, and suppose A is invertible, Prove that
$(A-B)C=BA^{-1}$ then $C(A-B)=A^{-1}B$
I tried to prove it as following.
$(A-B)C=BA^{-1}$ implies
$AC-BC=BA^{-1}$ so
$ACA-BCA=B$ taking $A^{-1}$
$CA-A^{-1}BCA=A^{-1}B$
Any hint will be appreciated
AI: Assu... |
H: Is there a closed form of $\sum_{n=0}^{\infty} \frac{(-1)^n}{(4n+1)!!}$?
This may be an impossible problem. But I imagine it's worth asking still.
What is the closed form of the sum:
$$\sum_{n=0}^{\infty} \frac{(-1)^n}{(4n+1)!!}$$
Perhaps there isn't a closed form. Double factorials are WAY out of my comfort zone... |
H: Proving an equality associated with symmetric positive definite matrices
Let $Q$ be an $n \times n$ symmetric positive definite matrix, $\vec{a}, \vec{b} \in \Bbb R^n$ be two random vectors. Prove that $a^TQ(ba^T-ab^T)Qb$ is non-positive.
Since $Q$ has $n$ independent eigenvectors with positive eigenvalues, I've t... |
H: Finding modulus of a complex number
The question is as follows:
if $|a|=|b|=|c|=|b+c-a|=1$ where $a$,$b$,$c$ are distinct complex numbers , find $|b+c|$.
My attempt: By observation that $b=i$, $c=-i$, $a=1$ satisfy the following conditions
thus $|b+c| =|i-i|=0$
I realise that this is not a generalised method. Is ... |
H: Prove that $2\cos^2(x^3+x) = 2^x + 2^{-x}$ has exactly one solution
I've been stuck on this question for some time now:
Show that there is exactly one value of $x$ which satisfies the equation $2\cos^2(x^3+x) = 2^x + 2^{-x}$.
Now this is obviously intuitively correct — I've modelled the equation with a function $... |
H: yes/ No :Is $f$ is uniformly continious .?
let $(X,T)$ be the subspace of $\mathbb{R}$ given by $X= [0,1] \cup [2,4] $. Define $f :(X, T) \rightarrow \mathbb{R}$ be given by
$f(x)= \begin{cases} 1 , \text {if x} \in [0,1] \\ 2 \ \text {if x} \in [2,4] \end{cases}$.
Is $f$ is uniformly continious ?
My attempt ... |
H: The expected value of the Ito integral of functions in $\mathcal{V}$ is zero, $\mathbb{E}[\int_S^T f dB_t] = 0$ for $f\in\mathcal{V}$
In Oksendal's Stochastic Differential Equations, the set $\mathcal{V}(S,T)$ is defined to be all functions $f:[0,\infty)\times \Omega \to\mathbb{R}$ such that
$(t,\omega)\to f(t,\om... |
H: Help with Baby Rudin Theorem 5.5 Proof
I have a question about the last sentence. I know $s \rightarrow y$ as $t \rightarrow x$; hence $u(t) \rightarrow 0$ as $t \rightarrow x$. But I'm confused that why the term $v(s)$ disappears. That is, why $v(s) \rightarrow 0$ as $t\rightarrow x$?
I know $s \rightarrow y$ as... |
H: Is there a tighter bound on $\prod_{i=1}^I x_i$?
If $x_i\in[0,1]$ for all $i=1,2,\cdots,I$, then we have a simple bound on $\prod_{i=1}^Ix_i$, i.e.,
$$\prod_{i=1}^I x_i \le \sum_{i=1}^I x_i.$$
I wonder if there is a tighter bound than the one above?
AI: The tightest you can get bounding by one element is $\bar{x} =... |
H: Why can't you lose a chess game in which you can make $2$ legal moves at once?
So here is the Problem :-
Consider a normal chess game in an $8*8$ chessboard such that every player makes $2$ legal moves at once alternatively . Now imagine that you was asked to play with Magnus Carlsen .Then Prove that it's impossibl... |
H: How $|f(a) |= e^{-i\alpha}f(a) ?$
I have some confusion in maximum modulas theorem , my confusion marked in red box given below
How $|f(a) |= e^{-i\alpha}f(a) ?$
My thinking : here we have taken $\alpha = Arg (f(a))$
By definition of Arg we have
$arg(f(a))= -i \log \frac{f(a)}{|f(a)|}$
Now if u put the value $\a... |
H: For which $n \in N$ is the following matrix invertible?
For which $n \in N$ is the following matrix invertible?
$$\left[\begin{array}{[c c c]}
10^{30}+5 & 10^{20}+4 & 10^{20}+6 \\
10^{4}+2 & 10^{8}+7 & 10^{10}+2n \\
10^{4}+8 & 10^{6}+4 & 10^{15}+9 \\
\end{array}\right]$$
My attempt:
For the matrix to be invertib... |
H: Interpret convolutions with different arguments. For example, $f(t) * \delta(t-\alpha)=f(t-\alpha)$
I'm trying to learn signal processing and I don't know how to process this.
My textbook says that the pure time shift LTI system that goes $y(t) = x(t-t_0)$ has an impulse response $h(t) = \delta(t-t_0)$, which means... |
H: Finding $\iint_D \frac{10}{ \sqrt {x^2 + y^2}}dx\,dy$ for $D = \{(x,y) \mid 0 \leq x \leq 1, \sqrt{1-x^2} \leq y \leq x\}$
As I said title,
$$\iint_D \frac{10}{ \sqrt {x^2 + y^2}}\,dx\,dy$$ for $$D = \{(x,y) \mid 0 \leq x \leq 1, \sqrt{1-x^2} \leq y \leq x\}$$
I tried it using integration by substitution by $(x,y) ... |
H: Why is $[\mathbb{Q}(\sqrt{2},\sqrt{3}):\mathbb{Q}] \le 4$
I am working on the following exercise:
Show that $\mathbb{Q}(\sqrt{2},\sqrt{3})$ has degree $4$ over $\mathbb{Q}$ by showing that $1,\sqrt{2},\sqrt{3}$ and $\sqrt{6}$ are linearly independent.
I have shown that $1,\sqrt{2},\sqrt{3}$ and $\sqrt{6}$ are lin... |
H: If $\omega$ vanishes in a surface, so do $d\omega$
It seens like an easy question, but I am newbie in differential forms.
Let $\omega$ a $C^\infty$ $r$-form in an open set $U\subset\Bbb{R}^n.$ If $\omega$ vanishes over the tangent vectors of a surface contained in $U$, so do $d\omega$.
First things first, I want to... |
H: If $n > 1$, there are no non-zero $*$-homomorphisms $M_n(\Bbb{C}) \to \Bbb{C}$
If $n > 1$, there are no non-zero $*$-homomorphisms $M_n(\Bbb{C}) \to \Bbb{C}$. A $*$-homomorphism is an algebra morphism $\varphi: M_n(\Bbb{C}) \to \Bbb{C}$ with $\varphi(\overline{A}^T) = \overline{\varphi(A)}$
I tried to show that eve... |
H: Is $\operatorname{Aut}(D_{12})\simeq D_{12}$?
Let $D_{12}$ be the dihedral group of order 12. Then
$$|\operatorname{Aut}(D_{12})|=6\phi(6)=12=|D_{12}|,$$
and the standard method of proof for
$$\operatorname{Aut}(D_6)\simeq D_{6}\qquad\mbox{and}\qquad \operatorname{Aut}(D_8)\simeq D_{8}$$
seems to also work for
$$\o... |
H: How to show that $\begin{pmatrix} 0 & -x \\\ 1/x & 0\end{pmatrix}$ is conjugate to a rotation?
Let $x >0$, and set $A=\begin{pmatrix} 0 & -x \\\ 1/x & 0\end{pmatrix}$.
Question: How to show that $A \in \operatorname{SL}_2(\mathbb R)$ is conjugate to an element of $\operatorname{SO}(2)$? That is, I am trying to show... |
H: Are all distributions with conjugate priors exponential families?
The Wikipedia page for conjugate priors lists several examples. Of the ones I'm immediately familiar with, all are exponential families. This leads me to wonder whether all families distributions that admit conjugate priors are exponential families.
... |
H: Why is the second derivative of Cumulant Generating Function positive?
Wikipedia states (without reference) that the cumulant generating function of a random variable has the property
Its first derivative ranges monotonically in the open interval from the infimum to the supremum of the support of the probability d... |
H: Help in proving that $\int_{0}^{\infty} \frac{dx}{1+x^n}=\int_{0}^{1} \frac{dx}{(1-x^n)^{1/n}},n>1$
I wanted to prove that $$\int_{0}^{\infty} \frac{dx}{1+x^n}=\int_{0}^{1} \frac{dx}{(1-x^n)^{1/n}}, n>1.$$ I converted them to Gamma functions but I could not prove it. Please help me.
AI: $$I=\int_{0}^{\infty} \frac... |
H: How to define a morphism from the Spec of the completion of $O_{Y,y}$ to $Y$?
Let $Y$ be a noetherian scheme and $y \in Y$. We denote by $\hat{O}_{Y,y}$ the completion of the local ring $O_{Y,y}$. I want to define a morphism
$$
\operatorname{Spec} \hat{O}_{Y,y} \to Y
$$
which sends the closed point $\hat{m}_y$ to ... |
H: Use trapezoidal rule to find $\lim _{n\rightarrow \infty }\frac{2^{2n}e^{-n}n^{n}n!}{\left( 2n\right) !}$
When $f\left( x\right) =\log x$ and section $\left[ 1,2\right]$
$$\lim _{n\rightarrow \infty }n\left[ \int _{a}^{b}f\left( x\right) dx-\dfrac{b-a}{n} \left\{\dfrac{f\left( a\right) +f\left( b\right) }{2}+\sum ... |
H: For What positive values of x ; the below mentioned series is convergent and divergent?
$$ \sum \frac{1}{x^{n}+x^{-n}}$$
My attempt
$$
\begin{aligned}
&\begin{aligned}
\therefore & u_{n+1}=\frac{x^{n+1}}{x^{2 n+2}+1} \\
\therefore & \frac{u_{n+1}}{u_{n}}=\frac{x^{n+1}}{x^{2 n+2}+1} \cdot \frac{x^{2 n}+1}{x^{n}} \\
... |
H: How to evaluate $\sum_{n=1}^{\infty}\:\frac{2n+1}{2n(n+1)^2}$?
Note: Similar questions have been asked here and here, but this is quite different.
I am trying to evaluate
$$\sum_{n=1}^{\infty}\:\frac{2n+1}{2n(n+1)^2} \quad (1)$$
I re-wrote the fraction as $$ \frac{2n+1}{2n(n+1)^2} = \frac1{2(n+1)} \cdot \frac{2n+1... |
H: Can we prove we know all the ways to prove things?
The things like induction and contradiction, they're all ways we prove things. Is that set of ways to prove things complete? Does the self referential nature of this question make it unprovable with something related to Gödel's incompleteness theorem? Has it been p... |
H: Evaluate integral $\int (x^2-1)(x^3-3x)^{4/3} \mathop{dx}$
How can I evaluate this integral $$\int (x^2-1)(x^3-3x)^{4/3} \mathop{dx}=\;\;?$$
My attempt:
I tried using substitution $x=\sec\theta$, $dx=\sec\theta\ \tan\theta d\theta$,
$$\int (\sec^2\theta-1)(\sec^3\theta-3\sec\theta)^{4/3}
\sec\theta\ \tan\theta d... |
H: What type of polygon will fit this description?
A particle moving with constant speed turns left by an angle of 74° after travelling every 1m distance. It returns back to the starting point in 18s and we are required to find out the speed of the particle.
My attempt: I know that the particle will return back to its... |
H: Find $a$, $b$ such that $x^2 - x -1$ is a factor of $ax^9 + bx^8 + 1$
Find $a$, $b$ such that $x^2 - x -1$ is a factor of $ax^9 + bx^8 + 1$
The second polynomial can be rewritten as
$$ax^9 + bx^8 + 1 = f(x)(x^2 - x - 1)$$
The roots of this polynomial are $\frac{1 \pm \sqrt 5}{2}$. Substituting one of these roots ... |
H: Number of solutions of $2011^x$ $+$ $2012^x$ $+$ $2013^x$ $=$ $2014^x$
The problem is : To find number of real solutions of $ \ \ 2011^x$ $+$ $2012^x$ $+$ $2013^x$ $=$ $2014^x$
My attempt : I first tried to see if the equation has zero solutions ;i.e ,the LHS of the equation was even and so was the RHS so I couldn'... |
H: Why is the convergence point of $ \sum _{n=1}^{\infty }\frac{1}{2^n}-\frac{1}{2^{n+1}} $ negative?
I am trying to evaluate $\frac1{2^1} - \frac1{2^2} + \frac1{2^3} - \frac1{2^4} + \cdots$
I re-wrote the sum using sigma notation as:
$$ \sum _{n=1}^{\infty } \left( \frac{1}{2^n}-\frac{1}{2^{n+1}} \right) \quad (1) $$... |
H: Throwing dice: Compute the probability of strictly increasing values in the first $n$ throws
Let a die be thrown $m$ times. What is the probability of strictly increasing values in the first $n< m$ throws? In other words, compute the probability $P(X=n)$ with $X\,\widehat{=}$"The first $n$ face values are strictly ... |
H: How to prove the statement about the rank of a block matrix?
Let $A$ and $B$ be real matrices with the same number of rows. Prove that:
$$\mbox{rank} \begin{bmatrix} A & B\\ 2A & -5B\end{bmatrix} = \mbox{rank}(A) + \mbox{rank}(B)$$
I have no idea how to approach the problem. Could you give me a hint?
AI: Subtract ... |
H: dealing infinities in the book frank jones lebesgue integration on euclidean spaces
I am currently reading the book "Frank Jones : Lebesgue Integeartion on Euclidean Spaces". The writing is not completely rigorous. for example measure can be $\infty$ but he doesn't tell what to do in such cases.
what does the follo... |
H: The meaning of $\frac{d^2 \sin x}{d (\cos x)^2}$
Just wondering if there is a meaning in the the following $$\frac{d^2 \sin x}{d (\cos x)^2}$$
I know that the Leibinz notation is some kind of symbolic representation of derivative.
But is there any formalism behind this?
Actually let me tell the story from the begi... |
H: RMM 2015 /P1: Does there exist an infinite sequence of positive integers $a_1, a_2, a_3, . . .$
Does there exist an infinite sequence of positive integers $a_1, a_2, a_3, . . .$ such that $a_m$ and $a_n$ are coprime if and only if $|m - n| = 1$?
My Progress:This is a very beautiful problem ! I think I have got a co... |
H: $71\nmid a$ and $x^7\equiv a\pmod {71}$ has an integer solution. How many positive integer solution there is for this equation that lower than 71?
I need a little help with the following question:
If $a$ is an integer such as $71\nmid a$ and $x^7\equiv a\pmod {71}$ has an integer solution. How many positive intege... |
H: Contrapositive of a statement involving integers
If we need to write a contrapositive and negation for c and d are integers, then cd is an integer. So does it mean that c and d can be fractions or natural numbers for negation? I understand that contrapositive is making it to negative and back to positive but I can'... |
H: Show $(\Bbb R\setminus\{−1\},*)$ is an Abelian group, where $a*b:=ab+a+b$. Solve $3 * x * x = 15$ in that group.
I have stumbled across the following problem and cannot understand how to solve it. Especially part b. I understand that part a must show that the rules of a Abelian group must apply for it to be an Abel... |
H: If $f$ is a open and continuous function then $f$ is injective.
I think I managed to prove:
"If $f:\mathbb{R} \rightarrow \mathbb{R} $ is continuous and open,( i.e., any open $\mathcal{A}\subset\mathbb{R}$ then $f(\mathcal{A})$ is open,) then $f$ is injective."
but I have some doubts especially in the end of the pr... |
H: How can I describe this domain?
I solved an equation on a 3D-domain which is the unitary sphere where a torus has been removed.
How can I describe it with proper mathematical words?
AI: If the solid sphere is given by $S(x,y,z)\le0$ and the solid torus is given by $T(x,y,z)\le0$, then your region is given by $S(x,y... |
H: Chain rule proof confusion
Here is a common informal proof for the chain rule:
If $S(a)=f(g(x))|_{x=a}$, then $S'(a)$ is given by
\begin{align}
\lim_{x \to a}\frac{S(x)-S(a)}{x-a}&=\lim_{x\to a}\frac{f(g(x))-f(g(a))}{x-a} \\
&=\lim_{x \to a}\frac{f(g(x))-f(g(a))}{g(x)-g(a)}\cdot\frac{g(x)-g(a)}{x-a} \\
&=\lim_{x \t... |
H: $Q(n)-Q(n-1) = T(n)$ Prove that $Q(n)$ degree is $k+1$
I was given this problem and I've been thinking a lot of time and still I have nothing.
$Q:ℕ↦ℕ$
$Q(n)-Q(n-1) = T(n)$
$T(n)$ degree is $k$
Prove that $Q(n)$ degree is $k+1$
any idea?
Thank you
AI: Let $Q(n)$ be a $r^{th}$ degree polynomial.
Thus $Q(n)=a_0n^r+a_1... |
H: Relating actions of intersections of subgroups of a finite group.
Suppose that we have a finite group $G$ with three subgroups $A,B,C$. I am interested in relating the action of $A \cap B \cap C$ on $B \cap C$ with the action of $A \cap B$ on $B$ (both via right multiplication). In particular, I am interested in re... |
H: Problem with the power of functions' set and number of discontinuity points
I am considering functions $f:\mathbb{R}\rightarrow \mathbb{R}$ with property that $\forall_{r\in\mathbb{R}}$ exists a limit $\lim_{x\rightarrow r}f(x)$ (it doesn't have to be equal to $f(r)$). I have two problems.
The first with showing t... |
H: How to find the minimum of $f(\mathbf{x}) = \| \mathbf{A}\mathbf{x}-\mathbf{b} \|^2_\mathbf{p} + \| \mathbf{x}-\mathbf{c} \|^2_\mathbf{q}$
Let $\mathbf{A} \in \mathbb{R}^{M \times N}$, $\mathbf{x} \in \mathbb{R}^{N}$, $\mathbf{b} \in \mathbb{R}^{M}$, $\mathbf{c} \in \mathbb{R}^{N}$, $\mathbf{p} \in {\mathbb{R}^+}^{... |
H: What is wrong with my approach in converting a complex equation into polar form?
The equation I wanted to convert was $|z^2-1|=1$. This is a very easy example but I have no idea where I made my mistake.
Putting $z=re^{i\theta}$, we have $z^2=r^2e^{2i\theta}$. Square both sides on the equation above and use the fact... |
H: How to find a probability that sum of geometric variables is less than a number
Let $X_i, i=1, \ldots, n$ be Geometric i.i.d random variables, which represent the number of fails, with parameter $p$.
Calculate or estimate from above and below:
$$
P(\sum_{i=1}^n X_i\leq A), \quad A \in N.
$$
I know that sum of the ... |
H: Let $E_1 \subset E_2$ both be compact and $m(E_1) = a, m(E_2) = b$. Prove there exists a compact set $E$ st $m(E) = c$ where $a < c < b$.
Exercise 1.27 (Stein & Shakarchi): Suppose $E_1$ and $E_2$ are a pair of compact sets in $\mathbb{R}^d$ with $E_1 \subset E_2$ and let $a = m(E_1)$ and $b = m(E_2)$. Prove that ... |
H: Help with proving/disproving an inequality
$\textbf{Question:}$Let $x_1,x_2,x_3,x_4 \in \mathbb{R}$ such that $(x_1^2+1)(x_2^2+1)(x_3^2+1)(x_4^2+1) =16 $. Is it then true that $x_1x_2+x_1x_3+x_1x_4+x_2x_3+x_2x_4+x_3x_4- x_1x_2x_3x_4 \le 5$, with equality $\iff x_1=x_2=x_3=x_4=\pm 1$?
Rough calculations seem to sugg... |
H: Why is the definite integral from $a$ to $b$ is negative of integral from $b$ to $a$ graphically?
I was studying properties of definite Integrals, and I came across this property which was easy to prove.
$\int_{b}^{a}f(x) \,\mathrm{d}x = -\int_{a}^{b}f(x) \,\mathrm{d}x$
However, I had a problem with the graphical a... |
H: Integral of Series over Domain of Convergence
I'm looking for a little help regarding integration of series where the domain of integration gets very close to the edge of the series' domain of convergence. My particular case is the logistic function and it's Maclaurin series (via the geometric series expansion arou... |
H: Why does $\sum_{n=1}^{\infty}\frac{1}{(2n-1)(2n)} = \ln(2)$?
While proving some results on series I encountered that, one of those result implies that
$$\sum_{n=1}^{\infty}\frac{1}{(2n-1)(2n)}$$
is convergent and it has sum equal to sum of alternating harmonic series. (And we know that alternating harmonic series... |
H: Indefinite integral of $\sin^8(x)$
Suppose we have the following function:
$$\sin^8(x)$$
We have to find its anti-derivative
To find the indefinite integral of $\sin^4(x)$, I converted everything to $\cos(2x)$ and $\cos(4x)$ and then integrated. However this method wont be suitable to find the indefinite integral ... |
H: Separability of a $\sigma$-algebra generated by an algebra
Let $X:(Ω,\mathcal{F})$ be a measurable set and $\mathcal{F}=\sigma(C)$ where $C$ is an algebra .
Now We define $B_\mathcal{w}=\underset{\mathcal{w}∈A,A∈C}{\cap}A$, which suggests the intersection of all sets in $C$ that contain $\mathcal{w}$.
Similarly, we... |
H: Prove that $1<\int_{0}^{\frac{\pi}{2}}\sqrt{\sin x}dx<\sqrt{\frac{\pi}{2}}$using integration.
Prove that $$1<\int_{0}^{\frac{\pi}{2}}\sqrt{\sin x}dx<\sqrt{\frac{\pi}{2}}$$ using integration.
My Attempt
I tried using the Jordan's inequality
$$\frac{2}{\pi}x\leq\sin x<1$$
Taking square root throughout
$$\sqrt{\frac{2... |
H: How do you find the area of the 'floors' in a torus?
Take a Torus such that 'r', the radius of the circle of the torus's cross-section (see fig. 1), is $300$ m.
Now, given that there are two 'floors' in said Torus and the sum of their areas is $864,000 \; m^2$, find R. (see fig. 2) [Floors highlighted.]
This is ... |
H: How can I prove that each component of a 2-regular graph is a cycle?
I tried to look at the similar questions asked before, however, they all assume that if a graph is simply one vertex then that also constitutes a cycle which is an assumption I'm not allowed to make. I did a lot of cases and my question statement ... |
H: Find the eccentricity of the conic $4x^2+y^2+ax+by+c=0$, if it tangent to the $x$ axis at the origin and passes through $(-1,2)$
Solving this would require three equations
(1) Tangent to x axis at origin
Substituting zeroes in all $x$ and $y$ gives $c=0$
(2) Passes through (-1,2)
$$4(1)+4-a+2b=0$$
$$-a+2b=-8$$
How ... |
H: Transitive models of $V≠L$ within L
Suppose $V=L$. Can there be transitive models of $ZFC+V≠L$?
Let $M$ be a transitive model of ZFC. If $x\in M$, then $x\in L_\alpha$ for some $\alpha$ because $V=L$, but it's not evident to me that $\alpha\in M$.
Such an $M$ would necessarily have to be a set, since the only inner... |
H: How to find parameters of negative binomial random variable?
Consider $X_i \sim NB(r_i, p)$, where Let $X_i$ be i.i.d. for $1, \ldots, N$. Let $Y_i\sim Geometric (p)$.
Then,
$X_1\sim NB(r_1,p)$ satisfies $X_1 = Y_1 + \cdots +Y_{r_1}$,
$X_2\sim NB(r_2,p)$ satisfies $X_2= Y_{r_1+1} + \cdots + Y_{r_1+r_2}$,
$\ldots$
$... |
H: Existence of an analytic function in a unit disc
Let $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$.Then, is it true that there exists a holomorphic function $f:\mathbb{D} \to\mathbb{D}$ such that $f(\frac{1}{2})=-\frac{1}{2}$ and $f'(\frac{1}{4})=1$?
Can we use Schwarz's lemma to solve it?
AI: This is a double applicatio... |
H: Can an integer that is $3\pmod 7$ be expressed as a sum of two cubes?
I was going back through my past exams in a Discrete Mathematics Course and came across this problem which I failed to solve- Does there exist $x$ and $y$ s.t $x^3+y^3 \equiv 3\pmod 7$? Give a convincing proof of your assertion.
I went through so... |
H: Why isn't Universal enveloping algebra graded?
Given a Lie algebra $L$, define $U(L) = T(L)$ mod $I(L)$ where $T(L)$ is the tensor algebra of $L$ and $I(L)$ is the two sided ideal of $T(L)$ generated by all elements of the form $xy-yx-[x,y]$ where $x,y \in L$. Can somebody explain to me why the generators of $I(L)$... |
H: How do I find integers $x,y,z$ such that $x+y=1-z$ and $x^3+y^3=1-z^2$?
This is INMO 2000 Problem 2.
Solve for integers $x,y,z$: \begin{align}x + y &= 1 - z \\ x^3 + y^3 &= 1 - z^2 . \end{align}
My Progress: A bit of calculation and we get $x^2-xy+y^2=1+z $
Also we have $x^2+2xy+y^2=(1-z)^2 \implies 3xy=(1-z)^2-(... |
H: how many $\sqrt{2}^{\sqrt{2}^{\sqrt{2}\ldots }}$
Use $y=x^{\frac{1}{x}}$ graph and think the following calculate.
$$\sqrt{2}^{\sqrt{2}^{\sqrt{2}\ldots }}$$
I want to know how many $\sqrt{2}^{\sqrt{2}^{\sqrt{2}\ldots }}$ will be.
When $\sqrt{2}^{\sqrt{2}^{\sqrt{2} }}$ =$2$
I can't solve this calculate with using gra... |
H: Creating a graph within specific vertex/degree parameters
Suppose that T is a tree with four vertices of degree 3, six vertices of degree 4, one vertex of degree 5, and 8 vertices of degree 6. No other vertices of T have degree 3 or more. How many leaf vertices does T have?
So this seemed simple enough at first; ev... |
H: If $A$ is a positive definite real symmetric matrix and $\lambda$ its eigenvalue then $A^r v = \lambda^r v$ for all real $r>0$ .
Suppose $A$ is a positive definite real symmetric matrix with eigen value $\lambda$ and $v$ an eigenvector corresponding to $\lambda$. How to prove that
$$A^rv= \lambda^r v$$
holds for al... |
H: When is it possible to "move" an exponent out of a radical?
It seems when a value in a radical is positive it's valid to "move" the exponent out of the radical.
Consider the function $\sqrt{x^2}$. When $x\geq0$ then
$\sqrt{x^2} = (\sqrt{x})^2$
For example, when $x = 5$
$\sqrt{5^2} = (\sqrt{5})^2$
$\sqrt{25} = \... |
H: What are the set theoretic properties of vector spaces?
I am reading introductory Quantum Mechanics, where it says-
For a classical system, the space of states is a set (the set of
possible states), and the logic of classical physics is Boolean. The space of states of a quantum system is not a mathematical set; it... |
H: Multiplication group for $\mathbb Z_n$ modulo $n$
By definition:
Let $\mathbb Z^+_n = \{[0],[1],[2],\ldots,[n−1]\}$
$\mathbb Z^+_4 = \{[0],[1],[2],[3]\},$
but
how $\mathbb Z^*_{12}$ is $\{[1],[5],[7],[11]\}$ ?
how $\mathbb Z^*_{7}$ is $\{[1],[2],[3],[4],[5],[6]\}$ ?
AI: You should distinguish between the additive g... |
H: Calculate the sum of the series $\sum_{n\geq 0}^{}a_{n}x^n$ with $(a_n)$ periodic
Let $(a_{n})_{n\geq0}$ be a periodic sequence of period $T$ satisfying for all $n\geq0$: $a_{n+T}=a_{n}.$
Calculate the sum of the series of coefficients $a_{n}$ with
$$S(x)=\sum_{n=0}^{\infty}a_{n}x^n$$ for $|x|<R $, where $ R $ i... |
H: If $\int_{-2}^{3} [2f(x)+2]\,dx = 18$, and $\int_1^{-2} f(x)\,dx=-8$, then $\int_1^{3} f(x)\,dx$ is equal to what?
If $\int_{-2}^{3} [2f(x)+2]\,dx = 18$, and $\int_1^{-2} f(x)\,dx=-8$, then $\int_1^{3} f(x)\,dx$ is equal to what?
So I've tried the regular sum rules and tried plugging in everything I can for any ca... |
H: Converse of Deduction Theorem
I have a basic question about natural deduction and deduction theorem. I learn from my textbook that the deduction theorem
$$\textit{If }\ \Gamma,A\vdash B,\ \textit{ then }\ \Gamma\vdash A\rightarrow B.$$
corresponds to the introduction rule of $\rightarrow$ in natural deduction. This... |
H: Which open sets are invariant under rotations?
Let $Q \in \operatorname{SO}(2)$, and let $U \subseteq \mathbb R^2$ be an open, bounded, connected subset. Suppose that $QU = U$.
Is it true that $Q$ must be a disk, or the interior of a regular polygon (if $Q$ is a rotation by $2\pi/n$ then a regular $n$-gon would be ... |
H: Do algebraic fields in mathematics have any connection to the fields in physics?
Basic question, but is there some sort of connection between the two, or are they just separate definitions.
AI: They are different things. Algebraically fields are just a number system. Good examples are the rational, real, and comple... |
H: Two different definitions for a model in first order logic?
So, if $T$ is a theory in a first order language $\mathcal L$, I thought a model for $T$ is
a set $M$ with interpretations for all the constant, function and relation symbols of $\mathcal L$, in which all statements in $T$ are true.
But recently, I had s... |
H: A problem to show that a certain field extension is not normal
Problem: Let $\alpha$ be a real number such that $\alpha^4=5$. Then show that $\mathbb Q(\alpha +i\alpha) $ over $\mathbb Q$ is not a normal extension.(where $i^2=1$)
My approach: I could show that the minimal polynomial of $(\alpha+i\alpha)$ over $\m... |
H: When is a limit of measure preserving maps also measure preserving?
Take some compact space, for simplicity take the interval $[0,1]$.
Let $f_n:[0,1] \to [0,1]$ be measure preserving, i.e. $\mu (f_n^{-1}(A))=\mu(A)$ for all lebesgue measurable $A \subset [0,1]$.
The question is under what kinds of convergence $f_n ... |
H: Showing divergence of $\sum_{n=2}^{\infty} \frac{1}{n\ln n + \sqrt{\ln^3n}}$
$$\sum\limits_{n=2}^{\infty} \frac{1}{n\ln n + \sqrt{\ln^3n}}$$
As $n \to \infty$, we see that $n \ln n \gg (\ln n)^{3/2}$. Hence
$$\sum\limits_{n=2}^{\infty} \frac{1}{n\ln n + \sqrt{\ln^3n}} \sim^{\infty} \sum\limits_{n=2}^{\infty} \frac{... |
H: Why $I^2$ a vector space where $I$ is the space of differentiable functions vanishing at a point $x$
I am reading the following wikipedia article on tangent spaces, in particular, this subsection on the definition via the cotangent space. Here is a paraphrasing of the first two sentences of the first paragraph at t... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.