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H: Understanding Cantor diagonalisation in binary
Until now I have studied and understood Cantor's proof. My problem comes when looking at binary representation:
integer binary representation encoding for diag proof
1 1 10000...
2 10 01000...
3 ... |
H: In this textbook explanation of needing partial derivatives, how is this partial derivative not an indeterminate form?
$$ f(x,y) = x^\frac{1}{3}y^\frac{1}{3} $$
$$\frac{\partial f}{\partial x}(0,0) = \lim_{x \to 0} \frac{f(h,0)-f(0,0)}{h}= \lim_{x \to 0} \frac{0-0}{h} = 0$$
"and, similarly, $\frac{\partial f}{\par... |
H: Given $S=1+3+5+\cdots+2017+2019$, find $\frac{1}{1010}S-1008$.
I'm wrong, or the answer in the book is wrong:
Given $S=1+3+5+\cdots+2017+2019$, find $$\frac{1}{1010}S-1008$$
My attempt is:
$S=1+3+5+\cdots+2017+2019$
$S=\frac{2019-1}{2}(1+2019)$
$S=2038180$
Now I find
$$\frac{1}{1010}S-1008=\frac{1}{1010}2038180-100... |
H: both $A$ and $B$ has eigenvalues other than $0,1$ and $rkA＋rkB＝n$
Let $A$ and $B$ be diagonalizable $n$-dimensional square matrices. Suppose both $A$ and $B$ has eigenvalues other than $0, 1$, and and $rkA＋rkB＝n$. Show that such $A$ and $B$ do not exist.
Any help would be appreciated, thank you.
In my original prob... |
H: Show that $|\cos(x)| \geq 1 - \sin^2 (x), \forall x \in \mathbb{R}$.
Show that $|\cos(x)| \geq 1 - \sin^2 (x), \forall x \in \mathbb{R}$.
I'm using the graph of both $f(x) = |\cos(x)|$ and $f(x) = 1 - \sin^2 (x)$ only for showing, but I think it's doesn't enough. For $>$, some $x$ holds, for example: $x=\frac{2\pi}... |
H: Given $\sum\limits_{i=1}^m(a_i+b_i)=c,$ what is the maximal value of the expression $\sum\limits_{i=1}^ma_ib_i?$
Given $2m$ non-negative numbers $\{a_i\}_{i=1}^m$ and $\{b_i\}_{i=1}^m$ with $\sum\limits_{i=1}^m(a_i+b_i)=c,$ what is the maximal possible value of the expression $\sum\limits_{i=1}^ma_ib_i?$
When $m=1$... |
H: Prove that a set is Borel(and hence Lebesgue)
I'm trying to practice for the real-analysis final exam and I found this...Could you please help?
For $n$ $\in$ $\mathbb{N}$, define the following subsets of $\mathbb{R}$:
$$
A_n=\begin{cases} (0,1]\cup[n,n+1) & , n-even \\
(0,1]\cup[n,n+2)... |
H: $X = f^{-1}(f(X))$ if and only if $X = f^{-1}(Z)$ for some $Z \subseteq B$
In my study of functions, I found this result in ”Proofs and Fundamentals” by Ethan D. Bloch that I’m attempting to prove. First, I already now that $X \subseteq f^{-1}(f(X))$ and $f(f^{-1}(Y)) \subseteq Y $ and I’m using this two results i... |
H: Isn't $V^n$ a $\Bbb K$-vector space?
Suppose $V$ is a $\Bbb K$-vector space. Let $n \in \Bbb N.$ Can't we say that $V^n$ is also a $\Bbb K$-vector with respect to component wise addition and component wise scalar multiplication? If so what can we say about $\dim (V^n)$ in terms of $\dim (V)$?
AI: Yes, this is indee... |
H: Convex polyhedron with $3$ vertices, $2$ faces and $3$ edges
Can a convex polyhedron with $3$ vertices, $2$ faces and $3$ edges exist? If so, how does it look like or what's its name? My imagination just fails here...
I was reading a proof of Euler's formula (https://plus.maths.org/content/eulers-polyhedron-formula... |
H: $f:(X,\tau) \mapsto (Y,\tau')$ is continuous and $ \tau'$ is T2 Why is $ \{p \in X\ f(p)=q\}=f^{-1}(\{p\}) $ closed?
Let $f:(X,\tau) \mapsto (Y,\tau')$ be continuous and $\tau' $ is Hausdorff.
$ \forall q \in Y$, we have that $\{p \in X\ f(p)=q\}=f^{-1}(\{p\})$ is closed
I don't know how to prove this. I wanted to ... |
H: How to show that $J_{n+1} = \frac{3n-1}{3n} J_n$?
Let $$J_n := \int_{0}^{\infty} \frac{1}{(x^3 + 1)^n} \, {\rm d} x$$
where $n > 2$ is integer. How to show that $J_{n+1} = \frac{3n-1}{3n} J_n$?
AI: Hint:
Let $y=\dfrac x{(x^3+1)^m}$
$$\dfrac{dy}{dx} =\dfrac1{(x^3+1)^m}+\dfrac{(-m)x(3x^2)}{(x^3+1)^{m+1}} =\cdots =\df... |
H: All solutions of $f(x)f(-x)=1$
What are all the solutions of the functional equation $$f(x)f(-x)=1\,?$$
This one is trivial: $$f(x)=e^{cx},$$
as it is implied (for example) by the fundamental property of exponentials, namely $e^a e^b=e^{a+b}$. But there is another solution:
$$f(x)=\frac{c+x}{c-x}.$$
Are there any... |
H: Prove: $\tan{\frac{x}{2}}\sec{x}= \tan{x} - \tan{\frac{x}{2}}$
I was solving a question which required the above identity to proceed but I never found its proof anywhere. I tried to prove it but got stuck after a while.
I reached till here:
To Prove: $$\tan{\frac{x}{2}}\sec{x}= \tan{x} - \tan{\frac{x}{2}}$$
But I ... |
H: Reduce the differential equation $y= 2px+p^{2}y^{2}$ to Clairaut’s form
Reduce the following differential equation to Clairaut’s form by using the substitution and hence solve:
$y= 2px+p^{2}y^{2}$ where $p={dy\over dx}$
I used $y^{2}=v$ then I get
$v-2p_{1}x + {(x p_{1})^{2}\over v}= ({p_{1}\over2})^{4}$ where $p_{... |
H: Volume of the region of sphere between two planes.
I want to find the volume of the region of the sphere $x^2+y^2+z^2=1$, between the planes $z=1$ and $z=\frac{\sqrt{3}}{2}$
I have used triple integral for calculating this
$$\int _0^{2\pi }\int _{0}^{\frac{\pi }{6}}\int _{0 }^1\:\rho ^2sin\phi \:d\rho \:d\phi \:d\t... |
H: Is my proof of an upper bound $u$ is the supremum of $\mathit{A}$ iff $\forall(\epsilon>0)$ $\exists a\in\mathit{A}$ such that $u-\epsilon
I have attempted to prove that an upper bound $u$ is the supremum of $\mathit{A}$ if and only if for all $\epsilon>0$ there exists an $a\in\mathit{A}$ such that $u-\epsilon<a$.
... |
H: Tensor Algebra. Finding a well-defined linear map from Functional's.
Let $V$ and $W$ be finite dimensional $K$ vector spaces. Prove that for $\varphi \in V^*$ and $\psi \in W^*$ exists a well-defined map: $$P_{\varphi , \psi} : V \otimes W \to K, v\otimes w \mapsto \varphi(v) \psi(w)$$
I would like some help in und... |
H: $\cos\theta\cos2\theta\cos3\theta + \cos2\theta\cos3\theta\cos4\theta + ...$
Evaluate: $$\cos\theta\cos2\theta\cos3\theta + \cos2\theta\cos3\theta\cos4\theta + …$$ upto $n$ terms
I tried solving the general term $\cos n\theta\cos (n+1)\theta\cos (n+2)\theta$.First, I applied the formula $2\cos\alpha\cos\beta = \... |
H: Does a non-negative polynomial of three variables have minimum?
I was wondering, does a non-negative polynomial of three variables (in $\mathbb{R}^3$) have a minimum point?
I understand that for example $(0,0,0)$ is a minimum point for some of them, but what could be the answer in the general case?
AI: Such a polyn... |
H: Determine Lebesgue integral of a function containing floor function
I'm practicing for the real-analysis exam and I've got stuck at this integral... Could you help me, please?
Determine: $$ \int_{[0,\infty)}\dfrac{1}{\lfloor{x+1}\rfloor\cdot\lfloor{x+2}\rfloor}d\lambda(x).$$
It is ok to split in two integrals and t... |
H: Problem with general progressions
Question: Let $a_1,a_2,a_3,a_4$, and $a_5$ be such that $a_1,a_2,a_3$ are in an $A.P.$ and $a_3,a_4,a_5$ are in $H.P.$ Then prove that $\log{a_1},\log{a_3},\log{a_5}$ will be in $A.P.$
My approach:
As $a_1,a_2,a_3$ are in an $A.P.$, $$2{a_2}={a_1+a_3}$$
Let's call this equation $I$... |
H: Doubts about series convergence/divergence and properties of compound functions.
Here are some questions about series and functions.
The task is to provide a counterexample for false statements and a proof for true statements (which are at most two).
-> Questions in image format <-
/Question in text format/
-(I) Le... |
H: Why is $(a,+\infty)$ part of the topology generated by the base $\mathfrak{B}=\{B \subseteq \mathbb{R}\ | B=[a, +\infty), a \in \mathbb{R}\}$?
If we consider
$\mathfrak{B}=\{B \subseteq \mathbb{R}\ | B=[a, +\infty), a \in \mathbb{R}\}$ is the base of a topology over $\mathbb{R}$, whose open sets are the positive ha... |
H: If $\pi$ is a permutation, how many permutation $\sigma$ can be reached from $\pi$ exchanging $r$ indices?
This problem comes from my Master Thesis in combinatorial optimization.
Let $\pi \in \mathbf{S}_n$ be a permutation of $n$ elements and $r \in \{2,3,\dots,n\}$. Define the neihborhood $\mathbf{N}_r(\pi)$ cente... |
H: Why does $\sqrt a\sqrt b =\sqrt {ab}$ only hold when at least one of $a$ and $b$ is a positive number?
I've just been introduced to complex numbers, and I have found it surprising that the radical rule apparently holds even when one of $a$ and $b$ is a negative number. However, if both $a$ and $b$ are negative, the... |
H: $P(X-EX \geq t) \leq P((M-m)S \geq 2t)$. Is this inequality true? And if so, how does one prove it?
Let $X \in [m,M]$ be a random variable and $S$ be the Rademacher random variable (e.i $P(S=1)=P(S=-1)=1/2$). Is the following inequality true?
$$
P(X-EX \geq t) \leq P((M-m)S \geq 2t)
$$
This inequality showed up whi... |
H: Confusion about Suprema Properties and Spivak's Proof of the Intermediate Value Theorem
If it's possible, I'm wondering if someone can clarify the following for me as part of the proof of the Intermediate Value Theorem by Spivak in the 4th edition of his Calculus (proof and auxiliary theorem given at the bottom of ... |
H: Calculate $\int \left(1+\ln \left(1+\ln (...+\left(1+ \ln(x))\right)\right)\right) dx$.
This is not particularly a useful integral or one asked in an exam or anything. I just really enjoy doing random integrals and derivatives. With that being said, how can I find:
$$\int \left(1+\ln \left(1+\ln (...+\left(1+ \ln(x... |
H: Why is expectation of a exponential family equal to $\frac{\partial A(\eta)}{\partial \eta}$?
My understanding: A member of the exponential family is any well defined distribution of the form $h(x)\exp[\eta \boldsymbol{\cdot} T(x)-A(\eta)]$ where $T:\mathbb{R}^n \to \mathbb{R}^n$ , $A:\mathbb{R}^n \to \mathbb{R}$,... |
H: What is the motivation behind Binomial Distribution?
If there are $N$ repetitions of a "random experiment" and the "success"
probability is $θ$ at each repetition, then the number of "successes" $x$ has a binomial
distribution:
$$p(x|θ) = {N\choose k}θ^x (1 − θ)^{N−x} $$
Now I am wondering what ${N\choose k}θ^x (1 ... |
H: Is the operation $i = x + y * \text{width}$ reversible?
In game programming, there is this very common operation we do to index positions in a two-dimensional matrix over a one-dimensional array. We basically associate the value $a_{xy}$ of some matrix $A$ with width $w$ to the index $i = x + y * w$ of the array. T... |
H: Can a ratio of two members of a positive sequence converge to zero?
Can this happen $\lim_{n\to\infty}\frac{a_{n+1}}{a_n}=0$ for a sequence $a_n>0$?
AI: Yes. For example letting
$$a_n = \frac{1}{n!}$$
yields
$$\lim_{n\to\infty}\frac{a_{n+1}}{a_n} = \lim_{n\to\infty}\frac{1}{n+1} = 0$$ |
H: Prove that lim $\int_{E_n}f d\mu = 0$
Well they give me the following statement:
Be $f$ integrable in $(X,F,\mu)$ and {$E_n$} $\subset F$ as {$E_n$} $ \downarrow E$ with $\mu(E) = 0$. Prove that $\lim_{n \to \infty} \int_{E_n} f d\mu = 0$
Well my idea to prove this is use the DCT (Dominated Convergence Theorem) a... |
H: Doubt in understanding definition of random variable
I have recently started studying probability. Kindly explain me following
Random variable means a function from $\Omega \to R$(set of real numbers). This is what i understood from my school books. But when i started reading Ross, it is given as in addition to abo... |
H: Convergence/divergence of $\sum_{n=1}^{\infty} \frac{{(-1)}^n \tan{(n)}}{n^2}$
Convergence/divergence of $$\sum_{n=1}^{\infty} \frac{{(-1)}^n \tan{(n)}}{n^2}$$
I thought diverge because some value $n \approx \frac{\pi}{2}+\pi k \implies \tan{n}=\pm \infty$ but key says that this never holds because $\pi$ is irratio... |
H: What's a fair way to share fees in a group road trip with a personal and a rental car?
I'm planning vacations with a group of friends (12 people), and it involves a ~1200km return trip by car. Only one of us owns a suitable car (4 pax), so we've rented a minivan to transport the other 8, and we're debating on how b... |
H: Extending bounded functions to unbounded ones
In a proof I read - which I will omit here since it does not contribute to the question - something has to be proven for a general function $h(x): IR \rightarrow IR^+$ where $h$ is a Borel function.
At the beginning, the proof was restricted to a bounded function $h(x)$... |
H: Is a function from $\Bbb{R}^+\times\Bbb{R}^+\rightarrow\Bbb{R}^+\times\Bbb{R}^+$ injective and/or surjective?
The function is defined on $\Bbb{R}^+\times\Bbb{R}^+$ as $f(x,y)=(2x+y, xy)$
I know the usual way to prove $f$ is injective is to assume $f(a,b)=f(c,d)$, and then show that this implies $(a,b)=(c,d)$.
That ... |
H: Series equivalent to harmonic series
Question: In the accepted answer, In this link given here
Another simple series convergence question: $\sum\limits_{n=3}^\infty \frac1{n (\ln n)\ln(\ln n)}$
What is mean by last line "which is essentially the harmonic series"?
Is that mean, $n\ln^2 2 + \ln2\cdot\ln\ln 2 ≤n$ for ... |
H: Two definitions of $C_0(X)$. Do they coincide?
Let $X$ be a topological space. Then we can define
$$C_0(X):=\{f \in C(X)\mid \forall \epsilon > 0: \exists K \subseteq X \mathrm{\ compact}: \forall x \notin K: |f(x)| < \epsilon\}$$
If $X$ is locally compact, I have also seen the following definition, if $X$ is local... |
H: Delta approximating function integral
I was dealing with a problem where a delta approximating function is given. Apart from the fact that this delta approximating functions has the form $$g_\epsilon(x) = \epsilon^{-3}g(\epsilon^{-1}x)\tag{1}$$ and that $g\in L^1(\mathbb{R}^3)\cap L^2(\mathbb{R}^3)$ with $\int g(x)... |
H: Venn Diagram Set Theory Question
So Given that Set A and B are disjoint, then A intersection C and B intersection C are also disjoint.
Now I'm having trouble figuring out how to exactly draw That A intersection C and B intersection C are also disjoint. Would I just draw Set C in the middle of Set A & B and then A i... |
H: Matchings in bipartite graph
I was given the following statement: Be $G=(X \cup Y, E)$ a bipartite graph connected with $|X|=|Y|=4$ $|E|=7$ , all maximal matching in G is maximum.
I must say if it is true or false and justify.
By testing examples I deduced that it is true.
But I don't know how I could justify it.
A... |
H: Does a left group action induce an open continuous map?
Let $X$ be a topological space and $G$ a topological group with an action $f:G\times X\to X$ so that $f(g,x)$ is denoted by $g\cdot x$. Let us fix $g\in G$, I want to know if given an open set $U\subset X$ the set $g\cdot U=\{g\cdot x\mid x\in U \}$ is homeomo... |
H: Can an $n \times n$ matrix satisfy an $n$ degree polynomial equation other than its characteristic polynomial equation?
Can an $n \times n$ matrix satisfy an $n$ degree polynomial equation other than its characteristic polynomial equation?
I was curious if the characteristic polynomial equation is the only $n$ de... |
H: Calculate $\frac{d}{dx}\left(x^x+x^{2x}+x^{3x}+...\right)$.
Calculate:
$$\frac{d}{dx}\left(x^x+x^{2x}+x^{3x}+...+x^{nx}\right), n \in\mathbb{N_{\geq 1}}$$
If I have $x^x$ as my first case, then I get $$\frac{d}{dy}x^x=x^x\left(\ln \left(x\right)+1\right)$$ Likewise, for $n=2$ I get: $$\frac{d}{dx}\left(x^x+x^{2x}... |
H: Open subset of C[0,1]?
Let us define $\|u\|_{\infty}=\sup_{x\in[0,1]}|u(x)|$ in the space $C[0,1]$. So we are working in the normed vector space $(C[0,1], \|\cdot\|_{\infty})$. $\;$Let: $$\boldsymbol{F}=\{f\in C[0,1]: f(x)>0,\; \forall\; x\in [0,1]\}.$$
Is $\boldsymbol{F}$ open?
Intuition says yes it is open becaus... |
H: Solving a system with tensor quantities
I have an equation that involves
$\sum_{abcd} x_ax_bx_cx_d M^{abcd}+\sum_{ab}x_ax_bN^{ab}=0$ where M, N are known.
M has the property that $M^{abcd}=M^{cdab}$. N is symmetric in its indices.
I would like to do something like this (although I think it is not formal):
Define ma... |
H: Evaluate $\int\frac{dx}{(a+b\cos(x))^2},(a>b)$
Evaluate $$\int\frac{dx}{(a+b\cos(x))^2},(a>b)$$
I tried to write 1 in numerator as $p'(x)(a+b\cos(x))-p(x)(a+b \cos(x))'$,making something like quotient rule but did not get after that.
AI: $$I=\int \frac{dx}{(a+b\cos x)^2} \\=\int\frac{dx}{\left(a+b\cdot \frac{1-\ta... |
H: Question about finite sequences
Suppose $(a_1,\dots,a_n)$ is a sequence of real numbers such that $$a_1\leq a_2\leq \dots \leq a_n.$$
If $(b_1,\dots b_n)$ is a rearrangement of the sequence $(a_1,\dots,a_n)$ such that $$b_1\leq b_2\leq \dots \leq b_n,$$
then does it follow $a_1=b_1,\dots,a_n=b_n$?
I know that if th... |
H: Frobenius endomorphism not surjective in a ring
I'm trying to find a simple counterexample that proves that, given a ring A with prime characteristic, the Frobenius endomorphism is not surjective in general. Are there any elegant examples?
AI: You can think about $\mathbb{F}_p(X)$. Then $X$ is not in the image of t... |
H: Let $a,b,c,d$ be $∈ ℝ$ and let A=$\begin{pmatrix}a & b \\ c & d\end{pmatrix}$
Background
Find values of a,b,c,d such at $A^T$=-A and $det(A)≠0$
My work so far
$$\begin{pmatrix}-3 & -5 \\ -7 & -9\end{pmatrix}^T$$
which equals
$$\begin{pmatrix}-3 & -7 \\ -5 & -9\end{pmatrix}$$
and $det(A)=-8$, which satisfies $A^T$=-... |
H: Projection onto a closed convex set in a general Hilbert space
Let $E$ denote a real Hilbert space and suppose $G \subset E$ is a nonempty closed convex set. I know that in this case, there is a unique nearest point in $G$ to each $x \in E$. Call this point $P_G(x)$.
I am trying to prove the following proposition:
... |
H: Conservation laws with source term
Consider the IVP
\begin{eqnarray}
u_t+F(x,u)_x=S(x,u)\\
u(x,0)=u_0(x)
\end{eqnarray}
If $S(x,u)=0$ and $u\in C([0,T],L^1(\mathbb{R})),$ then we have $$\int\limits_{\mathbb{R}}u_0(x)dx=\int\limits_{\mathbb{R}}u(x,t)dx.$$ (physically which can be interpreted as conservation of mass... |
H: GLn group action rank
Having the group action $GL(n,K) \times K^{n\times m} \to K^{n\times m} \; (g,A)↦gA $. The rank is an invariant but not an separating one (why?), how then does the orbits look like? If we acted also from the other side it would be JNF, but so its applying Gauss and for me it seems like only t... |
H: Limit of function of two variables
Let
$$f(x,y)=\frac{x^4y}{x^2+y^2}$$
for $(x,y)\ne (0,0)$. I want to prove that $\lim_{(x,y)\to (0,0)}\frac{x^4y}{x^2+y^2}=0$ by definition. So, I was wondering if there is any useful bound for
$$\frac{x^4|y|}{x^2+y^2}$$
in order to use the definition of limit for proving this stat... |
H: I was wondering if a general identity of $\cos^{2m}(x)=\,$[in terms of] $\cos^{2m-1}$ exists?
I was wondering if a general identity of $\cos^{2m}(x)=\,$[in terms of] $\cos^{2m-1}$ exists?
One particular example where $m=1$ is:
$$\cos^2(x)=\frac12(\cos(2x)+1).\tag{1}$$
but can this be generalised to an even power of... |
H: Why is $ \frac{5}{64}((161+72\sqrt{5})^{-n}+(161+72\sqrt{5})^{n}-2)$ always a perfect square?
I'm working on a puzzle, and the solution requires me somehow establishing that
$$ f(n):=\frac{5}{64}\Big(\big(161+72\sqrt{5}\big)^{-n}+\big(161+72\sqrt{5}\big)^{n}-2\Big)$$
is a perfect square for $n\in \mathbb{Z}_{\geq 0... |
H: Evaluate $\lim_{x\to+\infty} \frac{x-\sin{x}}{x+\sin{x}}$
Evaluate $$\lim_{x\to+\infty} \frac{x-\sin{x}}{x+\sin{x}}$$
$\sin{x}$ is a bounded function, but I still have no clue what to do. Any hint?
AI: Use:
$$
\frac{x-\sin x}{x+\sin x}=\frac{x+\sin x-2\sin x}{x+\sin x}=1-\frac{2\sin x}{x+\sin x}
$$
It should be eas... |
H: What is the Fourier transform of functions of the type $p(x)e^{-x^2}$ $(p \in \mathbb{C}[x])$?
First some context to my question:
I have two sets $M=\{p(x)e^{-x^2}:p\in \mathbb{C}[x]\}$ and $N=\{\hat{f}:f\in M\}$. Both are left modules of the Weyl algebra $A_1$. There are a few other technical details that I will n... |
H: Integrating $ \int_{0}^{\pi} \sin^4 (x) \cos(kx) dx$
Caveat: $ k \in N$
so after some 'repeated integration by parts I get and some 'evaluations',
$ k^2 G = 4G - 3 \int_{0}^{\pi} \sin^2 (2x) \cos(kx) dx$
so this becomes,
$ G = \frac{3}{4-k^2} \int_{0}^{\pi} \sin^2 (2x) \cos(kx) dx$
now this result is already weir... |
H: Question about some term in Sage while using GF(9)
I tried to define an elliptic curve over $GF(9)$ in Sage, and some term $z2$ appeared, see below (click on the image if the font is too small):
I know that it has something to do with the definition of $GF(9)$ - probably it describes how it works as a $GF(3)$-spac... |
H: Given $f:\mathbb R \backslash \{-1,0\} \to \mathbb R$, $f(x) = \frac{ |x \sin x| }{x + x^2}$: what values $c \in\mathbb R$ makes $f$ has limit.
I tried to demonstrate that is known that $x$, $sin x$, $x^2$ and $|x|$ are continuous functions. So the unique values that could not have limit are $c = -1$ or $c = 0$ but... |
H: At most one connected component of $\{z: |f(z)| < M \}$
I am trying to show that if $f$ is an entire function, then there is at most one connected component of the complement of $\widehat{\mathbb{C}}$ of the set $\{ z: |f(z)| < M \}$.
Based on the post At most one connected component of unbounded portion of entire ... |
H: Show that $E\exp(-tX_i) \leq \frac{1}{t}$
This is exercise 2.2.10 present in the book High-Dimensional Probability, by Vershynin.
Let $X_1,\ldots,X_n$ be non-negative independent r.v with the densities bounded by $1.$ Show that the MGF of $X_i$ satisfies
$$
E \exp(-tX_i)\leq \frac{1}{t}
$$
After that, deduce that f... |
H: One property of limits
How to prove this property of limits:
$ \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f(x) - f(a)}{g(x) - g(a)}$?
Also, is there intuition for this result?
Note. I have seen this when learning L'Hospital rule so I am not sure if this is true always.
AI: This is not true in general. ... |
H: Why is any ring homomorphism from a ring of integers to an algebraically closed field (char=0) injective?
Let $\mathcal{O}_L$ be the ring of integers in a number field $L$. Let $K$ be an algebraically closed field of characteristic zero, and let $f:\mathcal{O}_L \to K$ be a ring homomorphism. Why must $f$ be inject... |
H: The sequence $(x_n)$ diverges, then $(\sqrt[3]{x_n})$ diverges
I need to prove if it is true or not. I really tried with some definitions and propositions and I could not reach the answer.
AI: Suppose $\sqrt[3]{x_n}$ converges. Then we know, that we can multiply two converge sequence and obtain $\sqrt[3]{x_{n}^2}$ ... |
H: Solving system of equations with three unknowns
I need to solve an equation of a line using three known coordinate pairs (x0, y0), (x1, y1), and (x2, y2).
The equation of the plane is, of course, ax + by + c = 0.
I'm writing a little piece of code to calculate the position of a point w.r.t a line as it changes, so ... |
H: Argument matrix row operations (3x4)
Background
Given the argument matrix
$$A=\begin{bmatrix}1 & 3 & -5 & 3\\4 & 10 & -6 & -4\\-4 & -14 & -4 & -5\end{bmatrix}$$
perform each row operation in the order specified and enter the final result.
My work so far
a) First: $R2→R2-4R1$
$$\begin{bmatrix}0 & -2 & 14 & -16\end... |
H: $L^\infty(\mathbb{R}^n)$ function that is also homogenous with degree zero
Consider a homogeneous function $m$ in $\mathbb{R}^n$ with degree zero, ie
$$m(\lambda \xi) = m(\xi), \;\;\;\;\;\; \forall \lambda >0.$$
Is it true that $m \in L^\infty(\mathbb{R}^n)$ if, and only if, $m \in L^\infty(S^{n-1})$??
Attempt: [E... |
H: Is $A^2$ the same thing as $A^TA$?
Assume A is a matrix. Is $A^2$ the same thing as $A^TA$? I keep on seeing $A^2$ but it's tough to find a walkthrough of calculating $A^2$.
AI: Here's a counterexample:
Let $A = \begin{pmatrix}
1 & 1 \\
0 & 1 \\
\end{pmatrix}$. We have:
$$
A^2 =
\begin{pmatrix}
1 & 1 \\
0 & 1 \\
... |
H: Central limit theorem; shooter hitting a target
A shooter hits a target with probability $0.4$ and shoots at target $150$ times. Find at least one interval in which, with probability $0.8$, will be the number of shooter's hits of target.
A random variable $S_{150}: B(150;0.4)$ represents the number of hits. We are ... |
H: Evaluate $\lim_{x\to+\infty} \frac{3x^3+x\cos{\sqrt{x}}}{x^4\sin{\frac{1}{x}}+1}$
Evaluate $$\lim_{x\to+\infty} \frac{3x^3+x\cos{\sqrt{x}}}{x^4\sin{\frac{1}{x}}+1}$$
My attempt: $$\lim_{x\to+\infty} \frac{3x^3+x\cos{\sqrt{x}}}{x^4\sin{\frac{1}{x}}+1}=\lim_{x\to+\infty} \frac{3x^2+\cos{\sqrt{x}}}{x^3\sin{\frac{1}{x}... |
H: Using Bernoulli's Inequality to prove $n^{\frac{1}{n}} < 2-\frac{1}{n}$
I was trying to prove that $$n^{\frac{1}{n}}<2-\frac{1}{n}$$ for all natural numbers $n \ge 2$.
The base case of n = 2 was trivial.
Looking at the $n+1$ case, I wrote that $$(n+1)^{\frac{1}{1+n}} \ge 1 + \frac{n}{n+1}$$ for some $n>2$, but I wa... |
H: Strong induction and mistake
what is the fault in this reasoning by strong induction
For all $ A $ and $ B $ of $ M_p (K) $ and
all integer $n$ we have: $ A ^ n B = B $
The proof :
Denote $\forall n\in \mathbb N,\quad P (n) $ : $ A ^ n B = B $
The property is true at rank $ n = 0 $ because $ A ^ 0
B = I_n B =... |
H: Relationship between image of a linear transformation and its support
Suppose I have a linear transformation $T: V \rightarrow V$.
The kernel of the transformation is the subspace spanned by the vectors $v\in V$ such that $Tv = 0$. The orthogonal complement to the kernel is called the support of $T$. Finally, the i... |
H: Solve the inequality $|3x-5| - |2x+3| >0$.
In order to solve the inequality $|3x-5| - |2x+3| >0$, I added $|2x+3|$ to both sides of the given inequality to get $$|3x-5| > |2x+3|$$ Then assuming that both $3x-5$ and $2x+3$ are positive for certain values of $x$, $$3x-5 > 2x+3$$ implies $$x>8$$ If $3x-5$ is positive ... |
H: Value of $\lim_{n \to \infty} \sqrt[n^2]{\sqrt{3!!}\cdot \sqrt[3]{5!!} \ldots \sqrt[n]{(2n-1)!!}}$
$$L=\lim_{n \to \infty} \sqrt[n^2]{\sqrt{3!!}\cdot \sqrt[3]{5!!} \ldots \sqrt[n]{(2n-1)!!}}$$
It turns out that this limit equals $1$. The solution key uses Stolz-Cesaro theorem and I was wondering if this could be e... |
H: Closed subset of metric spaces
Let $X$ be a metric space with $p \in X$ a point, $C \subset X$ a subset.
Show $C$ is closed iff $C \cap \overline{B_R(p)}$ is closed for any $R>0$.
Supposing $C$ is closed is pretty easy as intersecting it with closed ball is still closed.
So then assume $C \cap \overline{B_R(p)}$ is... |
H: Is a finite union of countable sets at most countable?
I thought of this result:
A finite union of countable sets is at most countable.
which I also tried to prove:
Let $A_1, A_2, \dots, A_n$ be a finite collection of countable sets. Then, each $A_i$ must be enumerable, that is, we can write
\begin{align*}
... |
H: Laurent Series of $\frac{1}{z(1-z)}$ in neighborhood of $z=1$ and $z=0$
So, the question is: Laurent Series of $\frac{1}{z(1-z)}$ in neighborhood of $z=1$ and $z=0$.
I know I can find Laurent series' all over MSE, but in an effort to build my own intuition, and to see the entire process, I'm just trying to show all... |
H: Rational numbers as a series of rationals
Any real number $0<x\leq 1$ can be written as
$$
x = \sum_{n=1}^\infty \frac{1}{p_1\dots p_n},
$$
where $p_1\leq p_2\leq\dots$ is a unique sequence of integers $>1$. The number $x$ is a rational if and only if for some $n_0\in\mathbb{N}$, $p_n=p_{n_0}$ for all $n>n_0$. I kn... |
H: Proving symplectic identity
Let $\Lambda$ be a skew-symmetric matrix and $Q$ a symmetric matrix. Let $\text{Id}$ be the identity matrix and $h > 0$ a real number. I am trying to prove the following identity:
$$
(\text{Id} + \frac{h}{2} \Lambda Q) \Lambda (\text{Id} + \frac{h}{2} \Lambda Q)^\top = (\text{Id} - \frac... |
H: Evaluate $\lim_{x\to+\infty} \frac{\sqrt{x}(\sin{x}+\sqrt{x}\cos{x})}{x\sqrt{x}-\sin({x\sqrt{x})}}$
Evaluate $$\lim_{x\to+\infty} \frac{\sqrt{x}(\sin{x}+\sqrt{x}\cos{x})}{x\sqrt{x}-\sin({x\sqrt{x})}}$$
My attempt: $$\lim_{x\to+\infty} \frac{\sqrt{x}(\sin{x}+\sqrt{x}\cos{x})}{x\sqrt{x}-\sin({x\sqrt{x})}}=\lim_{x\to+... |
H: Is the $\arg\min$ of a strictly convex function continuous?
Let $X\subset \mathbb{R}^n$ and $Y\subset \mathbb{R}^m$ be compact and convex sets, and let $f:X\times Y\rightarrow \mathbb{R}$ be a continuous function. Suppose that for each $y$, $f(x,y)$ is strictly convex.
Define the function $g : Y \to X$ as follows:
... |
H: Show that $A^3+4A^2+A=I_3$ for 3x3 matrix
Let $A$ be the matrix
$$\begin{pmatrix}-1 & 2 & 1 \\ 2 & -4 & -4 \\ 2 & 0 & 0 \end{pmatrix}$$
Show that $A^3+4A^2+A=I_3$
How would I go about doing this? The first step would suffice, but I'm having difficulty starting this off.
AI: Hints: try to find out characteristics eq... |
H: Integrating the exponential over the area bounded by the functions $y=x$ and $y=x^3$
Can someone please help me solve the following problem below? Thank you
Compute the integral of the function over the area bounded by the functions $y=x$ and $y=x^3$ $$f(x,y) = e^{x^2}$$
AI: HINT
You can think about it in terms of ... |
H: Evaluating $\lim_{x\to+\infty} \frac{\sqrt{x}\cos{x}+2x^2\sin\left({\frac{1}{x}}\right)}{x-\sqrt{1+x^2}}$
Evaluate $$\lim_{x\to+\infty} \frac{\sqrt{x}\cos{x}+2x^2\sin({\frac{1}{x}})}{x-\sqrt{1+x^2}}$$
My attempt: $$\lim_{x\to+\infty} \frac{\sqrt{x}\cos{x}+2x^2\sin\left({\frac{1}{x}}\right)}{x-\sqrt{1+x^2}}=\lim_{x\... |
H: Defining a multiset
Can the multiset $A=\{1,1,1,2,2,2,3,3,3,...,n,n,n\}$ be represented as
$$A=\bigcup_{i=1}^{n}\{n,n,n\}$$
where $n$ is a positive integer.
Or am I using the union notation completely incorrectly? If so, is how would I define set $A$?
Note:
To clarify my experience with this area of maths, I am goi... |
H: one person winning 5 tickets odds
In a raffle with 90 tickets, 9 people buy 10 tickets each. There are 5 winning tickets which are drawn at random.
Find the probability that one person gets all 5 winning tickets?
P(person A wins 1st ticket) = $\frac{10}{90}$
P(person A wins 2nd ticket) =$ \frac{9}{89}$
P(person A w... |
H: Understanding an example about Cauchy's integral formula
I have two questions about the following example taken from Palka's "An introduction to complex function theory". I highlighted with red the parts that I don't understand.
Why does the first equality in 5.11 hold?
Since $r \to \infty$, won't we have that eve... |
H: Separable Differential Equation, finding the constant C
I have a question about the Separable Differential Equation theorem.
According to my textbook, this is the theorem.
A differential equation of the form $dy/dx=f(y)g(x)$ is cal separable. We separate the variables by writing it in the form
$1/f(y) dy =g(x) dx.$... |
H: Deriving the Laplacian in spherical coordinates by concatenation of divergence and gradient.
In earlier exercises, I have derived the formula of divergence in spherical coordinates as $$\textrm{div }\vec{v}= \frac{1}{r^2}\frac{\partial (r^2 v_r)}{\partial r}+\frac{1}{r \sin \vartheta}(\frac{\partial(v_{\vartheta}\s... |
H: Multiplying $A\preceq B$ with a matrix
I have a matrix inequality,
$$A\preceq B,$$
where $\preceq$ means that $B-A$ is psd.
update: How can I show that if $M$ is a positive definite matrix, then the inequality above is equivalent
$$M A M^\ast \preceq M B M^\ast.$$
AI: Your "only if " is not ok here, for counter exa... |
H: Does the convergence to 0 in $L^2(0,T;L^2(K))$ for all compact $K \subset \mathbb{R}^{d}$ imply the convergence in $L^2(0,T;L^2(\mathbb{R}^{d}))$?
Let $(f_n)$ be a sequence in $L^2(0,T;L^2(\mathbb{R}^{d}))$ such that:
$\|f_n\|_{L^2(0,T;L^2(\mathbb{R}^{d}))} \leq C_T$ for all $n \in \mathbb{N}$;
$f_n \rightarrow 0$... |
H: What does fibre-wise mean?
I am doing some exercises in Lagrangian systems in the book Quantum Mechanics for Mathematicians. One exercise says:
Let $f$ be a $C^\infty$ function on a manifold $M$. Show that the Lagrangian systems $(M,L)$ and $(M,L+df)$ (where $df$ is fibre-wise linear function on $TM$) have the sam... |
H: how to construct this equalizer
If X is a topological space and (U$_i$)$_i$$_\in$$_I$ is a family of open subsets of X, write U = $\cup$$_i$$_\in$$_I$ U$_i$ and U$_i$$_j$ = U$_i$ $\cap$ U$_j$ for i, j $\in$ I.
(i) if f, g: U $\to$ R are continuous functions such that f|U$_i$ = g|U$_i$, then f = g
(ii) if (f$_i$: U$... |
H: Why does $\frac{a}{b}<0$ imply $ab<0$?
I'm not sure if this was asked before, but my question is: why does $\frac{a}{b}<0$ imply $ab<0$? How do you prove it both intuitively and rigorously(using math)? I think I understand it intuitively: it's becuase for $\frac{a}{b}$ to be negative, exactly one of $a$ or $b$ has ... |
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