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H: How can I solve this definite integral: $\int_{0}^{a}\frac{x^4dx}{\sqrt{a^2-x^2}}$
Evaluate $$\int_{0}^{a}\dfrac{x^4dx}{\sqrt{a^2-x^2}}$$
I tried taking $t$ as
$$t = \sqrt{a^2-x^2}$$
Thus my final integral became
$$\int_{0}^{a}(a^2-t^2)^{3/2}dt$$
but I couldn't go any further in solving this integral.
I also trie... |
H: Is an open bounded subset of $\mathbb{R}^n$ a Banach space?
Let $\Omega$ be an open bounded subset of $\mathbb{R}^n$. Is $\Omega$ a Banach (sub)space?
AI: Subspaces of $\mathbf R^n$ are all unbounded and closed. |
H: What is the remainder when $2019^{2019}-2019$ is divided by $2019^2+2020$
After some time I gave up and cheated using Wolfram Alpha and got the result $4076363$.
I played around with the general statement
What is the remainder when $x^x-x$ is divided by $x^2+x+1$ where $x$ is an integer.
After trying few values
I... |
H: Find numbers $A,B$ such that the function is differentiable at $x=0$
I have the following function:
And the statement says:
Find $A,B$ such that $f$ is differentiable at $x=0$
My attempt was:
$f$ will be differentiable at $0 \iff \lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$ exists.
Solving the lateral limits, i got:
$\li... |
H: Small Confusion in definition of a limit of function
Consider the definition of a limit of function. Suppose that $E\subseteq \mathbb{R}$ and the function $f:E\to \mathbb{R}$ and $x_0$ is the limit point of $E$.
Definition: We say that $A=\lim \limits_{x\to x_0} f(x)$ iff $\forall \varepsilon>0$ $\exists \delta=\de... |
H: Epsilon delta definition with restricted epsilon
Here, I tried to prove:
$\lim\limits_{x \to 4} \sqrt{x} = 2$
$2 -\epsilon < \sqrt{x} < 2 + \epsilon$
Edit:
Mu Prime Math has told me that before squaring both sides of the inequality must be positive:$2 - \epsilon \geq 0$
$2 \geq \epsilon$
$\epsilon>0$
$0 < \epsilon ... |
H: $eSe$ in a finite semigroup
Where is the following argument going wrong?
Let $S$ be a finite semigroup. There exists $e\in S$ such that $ee=e$. The subsemigroup $eSe = \{ese \mid s\in S\}\subseteq S$ is a monoid with the identity $e$. The map $ese\mapsto s$ is an injection from $eSe \to S$. Therefore $eSe = S$. Thu... |
H: How do we know that the P versus NP problem is an NP problem itself?
I have been doing some research on the P versus NP problem. On multiple occasions, I have seen people say that the problem itself is an NP problem. I have been curious about how we know this. If we know that the problem is NP, then has anyone c... |
H: How Lebesgue integration solved the problem of changing the order of integration will change the value of integration?
Our professor started a course in measure theory by stating the problems of Riemann integration. One of the problems he\she stated is the following double integration:
$\int_{0}^{1}\int_{0}^{1} \fr... |
H: What do the open sets in the Urysohn Metrization Theorem look like?
I am following Munkres' Topology and I am a bit confused about the Urysohn metrization theorem construction (Theorem 34.1). The proof goes as follows:
Show that there is a countable collection of continuous functions $f_n : X \to [0,1]$ having the... |
H: What is the Solution to this sum $\sum \limits_{n=1}^{\infty}(1-(-1)^{\frac{n(n+1)}{2}})(\frac{1}{2})^n$
what is the value of this series $\sum \limits_{n=1}^{\infty}(1-(-1)^{\frac{n(n+1)}{2}})(\frac{1}{2})^n$ ?
Anything what's solid and that i got so far is only
$\sum \limits_{n=1}^{\infty}(1-(-1)^{\frac{n(n+1)}... |
H: If an abelian group has subgroups of relatively prime orders r and s(which are cyclic), there exists a subgroup of order rs?
Task is:
Let $G$ be an abelian group and let $H$ and $K$ be finite cyclic subgroups with $|H| = r$ and $|K| = s$.
Show that if $r$ and $s$ are relatively prime then $G$ contains a cyclic sub... |
H: Implicit differential of diameter
In my physics course there is a problem where the volume "V" of a sphere is filled with a gas. The sphere is released in a liquid, therefore the amount of gas in this volume "V" decreases because of concentration differences. If you evaluate a time dependent mass-balance the follow... |
H: Galois correspondence of subgroups of $D_4$ with subfields of $\mathbb Q (\sqrt[4]{2},i)$
The Galois group of $\mathbb Q (\sqrt[4]{2},i)$ over $\mathbb Q$ is the Dihedral group $D_4$ = {$id, \sigma, \sigma^2, \sigma^3, \tau, \sigma\tau, \sigma^2\tau, \sigma^3\tau $}
Denoting $\sqrt[4]{2}$ as $\theta$, the action o... |
H: Prove that if $A=A^2$, and $0\ne \bar{v}\in \text{Col} A$. then $\bar{v}$ is an eigenvector corresponding to the eigenvalue $1$.
Suppose $A=A^2$ for $A\in \Bbb{M}_{n\times n}^{(\Bbb{R})} \ ,$ and $0\ne \bar{v}\in \text{Col} (A)$. then $\bar{v}$ is an eigenvector corresponding to the eigenvalue $1$.
AI: So, $v=Ax$ f... |
H: What does $[5.9]$ mean?
I came across this notation in the CAA module 0 sample questions. See photo:
It looks like it means lower bound but not sure. Can’t find any info. on google either. Anyone come across this notation before in this context? If so, what does it mean?
AI: It looks like the greatest integer (als... |
H: Is it true that the following definition of a convex function cannot be used in a more general way?
One definition for convex functions I found on Wikipedia was that 'the line segment between any two points on the graph of the function always lies above or on the graph'. That makes sense. However, am I right in sug... |
H: Is a differentiable non constant real function with continuous derivative strictly monotone on a interval?
Let $f : [0,1] \to \mathbb{R_+}$ be a differentiable non constant real function with continuous derivative
My question:
Is it true that $\exists \lambda \in \mathbb{R_+}$ such that $f$ is strictly monotone on ... |
H: Proving $(x_1 x_2 \cdots x_n)^{-1} = x_n^{-1} x_{n-1}^{-1} \cdots x_2^{-1}x_1^{-1}$ for $x_i $ in group $G$
Let $x_1, x_2, \ldots, x_n \in G$ for some group $G$. We wish to prove that
$$(x_1 x_2 \cdots x_n)^{-1} = x_n^{-1} x_{n-1}^{-1} \cdots x_2^{-1} x_1^{-1}.$$
I'm not sure if the correct way to proceed is by sho... |
H: How does $ f_{( T_1, T_2 )} (t_1, t_2 ) = \frac {\partial ^2 }{ \partial t_1 \partial t_2 } \mathbb P ( T_1 > t_1 , T_2 > t_2 )$?
I saw that expression in a paper :
$$ f_{( T_1, T_2 )} (t_1, t_2 )= \frac {\partial ^2 }{ \partial t_1 \partial t_2 } \mathbb P ( T_1 > t_1 , T_2 > t_2 )$$
And it seems to me that it is... |
H: Answer true or false: For A and B sets, A ∩ B = B ∩ B'
Answer true or false: For sets $A$ and $B:$ $A \cap B = B \cap B'.$
The statement is false. Let $A$ and $B$ be non-empty sets with $A = B$ and let $X = \{ a , b , c \}.$ Then
$A \cap B = \{ a \} \cap \{ a \} = \{ a \} $ and $B \cap B'= \{ a \} \cap \{ b , c... |
H: Eisenstein criterion on f(x+1)
I need to show that the polynomial $f(x)=x^6+x^5+x^4+x^3+x^2+x+1$ is irreducible in $\mathbb{Z}[X]$ and in $\mathbb{F}_2[X]$. As we can't find a prime number p satisfying the conditions for the Eisenstein criterion, I did not know how to solve it. I looked into the solutions and they ... |
H: Calculate a vector that lies on plane X and results in vector b when projected onto plane Y.
I'm working on a computer program and ran into this problem that I'm struggling to figure out.
Given:
two planes X and Y that belong to $\mathbf{R^3}$ that both pass through the origin and are defined by perpendicular vect... |
H: After applying a sequence of involutory real matrices to a vector, is the norm of this vector bounded from below?
For $n,N \in \mathbb{N}$, let $A_1, \ldots, A_n$ be a finite sequence of involutory $(N \times N)$-matrices over $\mathbb{R}$, i.e. $A = A^{-1}$.
We know, that the eigenvalues of any involutory matrix l... |
H: Proyection of a subspace.
Let $W \subset \mathbb{R}^{4}$ a subspace generated by two vectors
$$W := span \left\lbrace \begin{pmatrix}
1\\
1\\
0\\
0
\end{pmatrix},\begin{pmatrix}
1\\
1\\
1\\
2
\end{pmatrix}
\right\rbrace.
$$
Find $w \in W$ wich minimize $||w-v||$ where $v= \begin{pmatrix}
1\\
2\\
3\\
4
\end{pmatrix}... |
H: Where did 3.5 come from?
This is a homework question.
The problem. I know the solution, but I don't know where it came from. The videos say nothing. The equation is $d(v) = (2.15v^2)/(64.4f)$ I need to solve for $f$, so I tried plugging in the numbers from the table into the equation and solving. I got approximatel... |
H: Why is $C^5[a,b]$ infinite dimensional
Let $V$ denote the vector space $C^5[a,b]$ over $R$.
How to show it is infinite dimensional?
I know that we can write:
$C^5[a,b]$ = { $f\in$ $C[a,b]$ : $5$th derivative exists and is continuous}
How to show that there does not exist a linearly independent subset of $V$ which s... |
H: Question on proof of existence of a maximum of a continuous function on a closed set. - Proof inspiration
I'm trying to get stronger in constructing my mathematical arguments and so through that process I attempt to prove as much as possible the theorems that are presented in the textbook I'm reading from, in this ... |
H: The ratio of moments in a normal distribution
I'm reading a paper where they (Mann and Whitney) want to show the limiting distribution they get is normal. They do this by looking at a ratio of moments. They do a computation then conclude the limiting distribution is normal by a "well known theorem". Can someone pro... |
H: Example such that dimension of subspace is 24
If $S$ and $T$ are two subspaces of a vector space $\Bbb R^{24}$ of dimensions $19$ and $17$ respectively. Then what $S$ and $T$ can I choose such that the $\dim(S+T)=24$?
Help, please!
AI: Since $S$ is a 19-dimensional subspace of $\Bbb R^{24}$, the Orthogonal compleme... |
H: Intuitive steps can be used during a proof?
Im trying to prove that: if $A \cup C = B \cup C$ and $A \cap C = B \cap C$ then $A = B$
I started assuming that $A \neq B$ then in a intuitive way i can see that $$((A \cup C = B \cup C) \land A \neq B) \Rightarrow A \subseteq C \land B \subseteq C$$
And from this in an ... |
H: Radius of convergence and absolute convergence
Say I have a power series $\sum_{n=1}^{\infty} a_nx^n$ with a known radius of convergence $-1< x \leq 1$. What can I generally know about the series convergence in $x=1$? I'm asking because I've seen it can't converge absolutely in $x=1$ and I don't understand why.
AI:... |
H: Finite group of even order has an element $g \neq e$ such that $g^ 2 = e$
I am trying to prove the following result.
Let $G$ be a finite group of even order. Prove that there exists $g \in G$ where $g^2 = e$ and $g \neq e$.
Here is my attempt.
Since $G$ has even order, $|G| \geq 2$. Hence, there exists some $g \... |
H: Show that if $\phi_{X}(t)=1$ in a neighborhood of $0$, then $X=0$ a.s.
Let $\phi(t),t\in\mathbb{R}$, be the characteristic function of a random variable $X$. Show that if $\phi(t)=1$ in a neighborhood of $0$, then $X=0$ a.s.
The problem comes with the following hint: Show that $1-Re(\phi(2t))\le4(1-Re(\phi(t)))$ ... |
H: If $\alpha$ is algebraic number then so is $\alpha+1$
I have to prove that if $a$ is algebraic number then so is $\alpha+1$. I've tried to construct the polynomial for $\alpha+1$ using the polynomial for $\alpha$ but that didn't lead me to anything.
Let the $W(x)=a_n x^n+a_{n-1} x^{n-1} +\dots + a_1x+a_0$ be a poly... |
H: Proof Verification: Equivalent Definition for Locally Compact Hausdorff Space
The main theorem is as follows. I think most people are familiar with that:
Theorem. Let $X$ be a Hausdorff space. Then $X$ is locally compact if and only if for every $x\in X$ and every open set $U$ containing $x$, there exists a neighb... |
H: If the limit of the difference of two random variables goes to 0, are the limits of their expectations the same?
I have two discrete random variables $X_n$ and $Y_n$ and a relation between them that looks like, for $n\geq 1$
$$X_n = a_nP\{A_n\} + Y_n P\{{A_n}^c\},$$
where $A_n$ is an event and $a_n$ depends on $n$ ... |
H: Covariance of a uniform distribution
I have the following problem:
The variables $X$ and $Y$ have the joint probability: $f(x,y)=2 $ for
$0 \le y \le x \le 1$. What is the covariance between $X$ and $Y$?
The answer is 1/36
I know that $\operatorname{Cov}(x,y) = E[XY] - E[X]E[Y]$
First I calculated the marginal PD... |
H: Combinatorics of binning data with repetitions
I'm trying to model random arrival times in discrete time bins.
Suppose I have $n$ (integer) arrival times, which are between $1$ and $m$, with $m$ possible time bins. I randomly draw $n$ integers between $1$ and $m$, and I place every one of the (possibly alike) rand... |
H: The curl operator: Why does it map $C^k$ functions in $\mathbb{R}^3$ to $C^{k-1}$ functioins in $\mathbb{R}^3$?
My Understanding of the curl operator is the following: it maps continuously differentiable functions
$f$: $\mathbb{R}^3$ $\to$ $\mathbb{R}^3$ to continuous functions $g$: $\mathbb{R}^3$$\to$ $\mathbb{R}^... |
H: Coin Flipping Game - Wins at 20 Heads
Game Rules
Let's say you have a coin, with $50/50$ chance of lending on Heads or Tails.
You win the game when you get $20$ Heads.
Question
Now, knowing we already threw the coin $50$ times, what are the odds that we have less than $10$ throws left to win.
Answer
The first thing... |
H: Understanding Wittgenstein's proof of Infinitude of prime
Can someone please tell me why the last claim "It is thus the case..." is true?
I tried considering negation of the last claim. But it didn't help.
Any help would be appreciated. Thanks in advance.
AI: The product on the right consists of the sum of $\frac 1... |
H: How to evaluate $\int_{-\pi/2}^{\pi/2} \tan x \cos (A \cos x +B \sin x) \, dx$?
$$\int_{-\pi/2}^{\pi/2} \tan x \cos (A \cos x +B \sin x) \, dx$$
Is it possible to calculate this? Both A and B are non-zero and assumed to be real numbers.
I tried Integrate[Tan[x]*Cos[A*Cos[x]+B*Sin[x]],{x,-Pi/2,Pi/2},PrincipalValue->... |
H: Find inverse element of $1+2\alpha$ in $\mathbb{F}_9$
Let $$\mathbb{F}_9 = \frac{\mathbb{F}_3[x]}{(x^2+1)}$$ and consider $\alpha = \bar{x}$. Compute $(1+2 \alpha)^{-1}$
I think I should use the extended Euclidean algorithm: so I divide $x^2 +1 $ by $(1+2x)$:
$$x^2 + 1 = (1+2x)(2x+2)+2$$
$$(2x+2)(1+2x) + 2(x^2+1... |
H: Understanding surjectivity proof of $f(n)=2^n$.
Working on the book: Richard Hammack. "Book of Proof" (p. 252)
Let $B=\{2^n:n \in \mathbb{Z}\}$. Show that the function $f: \mathbb{Z}\to B$, defined as $f(n) = 2^n$ is bijective. Then find $f^{-1}$.
The author proves surjectivity:
The function $f$ is surjective ... |
H: Two different cases of uniform hypothesis testing
I have two different p-value uniform-distribution problems. I know that the definition of the p-value is: The probability of observing a new $X$ at least as extreme or more extreme than the initial $X$.
Problem I:
$X$ has a uniform distribution on interval $[0, z]$... |
H: Question about integral in measure theory
I have a question About excersice 4.E of Bartle's the elements of integration and measure theory, the problem says: If $f\in M^+ (X,\mathbb{X})$ and
$$\int fd\mu < +\infty,$$
then for every $\varepsilon>0$ there exist a set $E \in \mathbb{X}$ such that $\mu(E)< +\infty$ and... |
H: Contention of arbitrary families of sets
Apparently for $\{ A_{ij} \}_{(i,j)\in I \times J}$,
$$\bigcup\limits_{j \in J} \left(\bigcap\limits_{i \in I} A_{ij}\right) \subseteq \bigcap\limits_{i \in I} \left(\bigcup\limits_{j \in J} A_{ij}\right).$$
I'm having a hard time proving that contention. Here's what I've go... |
H: counterexample of reflexive space not hilbert
We know that all Hilbert spaces are reflexive. My problem is to show that the reciproque is not true: But I can't find a counterexample. An idea please.
AI: $\ell^{p}, L^{p}([0,1])$ with $1 <p <\infty$ are reflexive spaces which are not inner product spaces. |
H: Simplification of $\sqrt{2\zeta^2-1+2\zeta\sqrt{\zeta^2-1}}+\sqrt{2\zeta^2-1-2\zeta\sqrt{\zeta^2-1}}$
If I try to evaluate $\sqrt{2\zeta^2-1+2\zeta\sqrt{\zeta^2-1}}+\sqrt{2\zeta^2-1-2\zeta\sqrt{\zeta^2-1}}$ numerically for real $\zeta$, it looks like it is just equal to $2|\zeta|$ for $\zeta \ne 0$ and $2j$ for $\z... |
H: State Diagrams Probability Question
A bug is sitting on vertex A of a regular tetrahedron. At the start of each minute, he randomly chooses one of the edges at the vertex he is currently sitting on and crawls along that edge to the adjacent vertex. It takes him one minute to crawl to the next vertex, at which point... |
H: If the function does not depend on the indicated parameter, why is the derivative zero?
If we have the derivative $\dfrac{dy}{dx}$ but $y$ doest not depend on $x$, why is $\dfrac{dy}{dx} = 0 ?$
I think that a possible correct thought is that if we see the derivative as rate of change, is clear that since the variab... |
H: Showing an operator is not bounded.
There are two spaces $C^1 [0,1]$ and $C[0,1]$ with supremum norm, which is defined by
$$ \|f\| = \sup_{x\in[0,1]} |f(x)|$$
for any $f$. I have to show that if the operator $A:C^1[0,1] \rightarrow C[0,1]$ is defined by $Af=f'$, then $A$ is not bounded.
I tried to find some counter... |
H: Area between $5e^x$ and $5xe^{x^2}$ using substitution
I need to find the answer to this problem:
and was told to substitute $u=x^2$. I tried that and couldn't get to the correct answer. When substituting, I get $\frac{5}{2}\; du = 5x$, and I pull out $\dfrac{5}{2}$ from the integral, and my integrand turns out to... |
H: Integration of $\frac{1}{u^4 + (4\zeta^2-2)u^2 + 1}$
I am trying to compute
$$I(\zeta) = \int_{-\infty}^{\infty} \frac{1}{u^{4} + \left(4 \zeta^{2} - 2\right)u^{2} + 1}\, du$$
for positive real $\zeta$. Can anyone help?
I'm way out of practice for integrals except for simple stuff like $\int 1/(1+u^2)\, du = \tan^... |
H: IS $(\mathbb{Z}_4,+) \rightarrow (\mathbb{Z}_5^{*},\cdot), n\pmod 4 \mapsto 2^n \pmod 5 $ well-defined??
For the following relation
$(\mathbb{Z_4},+) \rightarrow (\mathbb{Z_5^{*}},\cdot), n\bmod 4 \mapsto 2^n \bmod 5 $
Determine if it is well-defined (so that it is a mapping)
Can someone show me how to do it?
So... |
H: I have this identity that I'd like to prove. $\sum_{k=0}^{n}\left(\frac{n-2k}{n}\binom{n}{k}\right)^2=\frac{2}{n}\binom{2n-2}{n-1}$
I have this identity that I'd like to prove.
$$\displaystyle{\sum_{k=0}^{n}\bigg(\dfrac{n-2k}{n}\binom{n}{k}}\bigg)^2=\dfrac{2}{n}\binom{2n-2}{n-1}$$
Here's what I have done so far: (u... |
H: For Galois extension $L:K$, does $L = K(\alpha)$ imply $\{\sigma_1(\alpha), \dots, \sigma_n(\alpha)\}$ is a basis for $L$ over $K$?
For Galois extension $L:K$ with Galois group $\{\sigma_1, \dots, \sigma_n\}$, does $L = K(\alpha)$ imply $\{\sigma_1(\alpha), \dots, \sigma_n(\alpha)\}$ is a basis for $L$ over $K$?
Th... |
H: Generating function of recursive algorithm with random subcalls
I was presented with the following algorithm. As input the algorithm gets an array of length $n \geq 0$. If $n \geq 2$ then for each $k \in \{1, 2, ..., n\}$ the algorithm calls itself recursively with the probability $\frac{1}{2}$ with the array of le... |
H: Find a sequence of $\alpha(t)$ such that $\sum_{t=1}^\infty\alpha(t)=\infty$ while $\sum_{t=1}^\infty{\alpha(t)}^2<\infty$
As described in the title, can we find a $\alpha(t)$ sequence that satisfies those two requirements?
AI: As suggested in the comment, a good choice is $a(t) = \frac{1}{t}$, since we know from c... |
H: Prove that for A, B and C sets, A - ( B - C ) = ( A - B ) ∪ ( A ∩ C ) (Alternative to the proof given)
My proof:
Let A, B and C be arbitrary sets.
As a means to prove such a statement we are going to verify that
x ∈ A - ( B - C ) ⇔ x ∈ ( A - B ) ∪ ( A ∩ C )
Note that
x ∈ A - ( B - C ) ⇔ x ∈ A ∧ x ∉ ( B - C ) ⇔ x ∈ ... |
H: Is $( \mathbb{ Z}_{10}^{*},\cdot) \rightarrow (\mathbb{Z}_5^{*},\cdot), n\pmod {10} \mapsto n \pmod 5 $ well-defined?
Is $( \mathbb{ Z}_{10}^{*},\cdot) \rightarrow (\mathbb{Z}_5^{*},\cdot), n\pmod {10} \mapsto n \pmod 5 $ well-defined?
So what I think is that it is not because the odd multiples of 5 in $\mathbb{ Z}... |
H: What kind of problem is the membership problem of a recursive enumerable language?
Is it correct that the membership problem (i.e. the characteristic function) of a recursive language is a decidable i.e. computable problem?
What kind of problem is the membership problem of a recursive enumerable language, as oppose... |
H: Is there another way to prove this expression over $1/(1-z)$
I came across the following relationship:
$$
\frac{1}{1-z} = (1+z)(1+z^2)(1+z^4)(1+z^8)...
$$
If induction is used, the statement can be proven given that:
$$
(1+z)(1+z^2)=1+z+z^2+z^3
$$
and
$$
(1+z)(1+z^2)(1+z^4)=1+z+z^2+z^3+z^4+z^5+z^6+z^7
$$
and so on ... |
H: Enumerate the possible combinations
We have three windows of opportunity, say W1, W2, W3. And we have 4 competitors, say, C1, C2, C3, C4.
Each competitor wants to be allocated at least two windows (so will get either 2 or all 3). Each window can be allocated to any number of competitors (so, from 1 to 4; or even 0 ... |
H: Use Slutsky's theorem to show that: $\sqrt{n}(e^{\frac{S_n}{n}}-e^{\mu}) \xrightarrow{d} \sigma e^{\mu}Z$
Let $\{X_n\}_{n\ge1}$ be a sequence of i.i.d. random variables with common mean $\mu$ and variance $\sigma^2 \in (0,\infty)$. Use Slutsky's theorem to show that:
\begin{align}
\sqrt{n}(e^{\frac{S_n}{n}}-e^{\mu... |
H: How to evaluate $\int \sqrt{1-n\cos(\omega t)}\,dt$?
How can I evaluate $$\int \sqrt{1-n\cos(\omega t)}\,dt$$
I don't even know if this is even elementary. Just found it on an astronomy problem. If necessary, evaluate from $0$ to $2\pi/\omega$.
AI: I think the answer involves incomplete elliptic integral of second ... |
H: Does there exist a gradient chain rule for this case?
My question comes from this article in Wikipedia. I noticed that there is a chain rule defined for the composition of $f:\mathbb{R}\to\mathbb{R}$ and $ g: \mathbb{R}^n \to \mathbb{R}$ given by
$$
\nabla (f \circ g) = (f' \circ g) \nabla g \tag{1}
$$
My question ... |
H: A complete DVR $A$ is a $\Bbb Z_p$ module, Serre's local field
I am trouble understanding how one obtains a $\Bbb Z_p$ action in the last line in this statement in pg. 36 of Serre's Local fields
In particular
Observe that $\Bbb Z$ injects into $A$ and
by continuity $\Bbb Z_p$ injects into $A$.
Maysome one elabor... |
H: Is $GL_{n}(\Bbb{C})$ isomorphism to a subspace of $GL_{2n}(\Bbb{R})$
A problem in the Algebra by Artin.
Is $GL_n(\Bbb{C})$ isomorphism to a subspace of $GL_{2n}(\Bbb{R})$
I think there is an isomorphism. Because I know that when $n=1$, $$\{A\mid A=\left(\begin{matrix}a&b\\-b&a\end{matrix}\right)\}\simeq\mathbb{C}... |
H: Does the Borel-Cantelli Lemma imply countable additivity?
Let $(\Omega, \mathcal F, P)$ be a finitely additive probability space. If $P$ is not only finitely additive but also countably additive, then it satisfies the Borel-Cantelli Lemma:
For all sequences $A_1, A_2,...$ in $\mathcal F$, if $\sum_n P(A_n) < \inft... |
H: Question about "commutation" relation
I'm reading Dummit & Foote, Abstract Algebra and they briefly mention something about a ' "commutation" relation', such as $xy = yx^2$. In general, if we have the relation $xy = y^i x^j$, where $x,y$ generate the (finite) group $G$, it seems to me that any element of the group ... |
H: How to build unambiguous binary expression with property
Is there a strategy for building unambiguous binary expressions that fit some property like "no consecutive 1's", "can't start with 0", "Blocks of 1 can only be divisible by 5", etc.?
I can't seem to figure out a list of steps I want to follow after practicin... |
H: Decomposition of symmetric matrices over $\mathbb{F}_2$
Can every $n\times n$ symmetric matrix over $\mathbb{F}_2$ be decomposed into $$\sum_{v=1}^k v_i v_i^T$$ for vectors $v_1, \dots,v_k\in\mathbb{F}_2^n$ and integer $k$?
As far as I know, for symmetric real matrices this is true and these vectors are orthogonal,... |
H: Power series approximation for $\ln((1+x)^{(1+x)}) + \ln((1-x)^{(1-x)})$ to calculate $ \sum_{n=1}^\infty \frac{1}{n(2n+1)} $
Problem
Approximate $f(x) = \ln((1+x)^{(1+x)}) + \ln((1-x)^{(1-x)})$ and then calculate $ \sum_{n=1}^\infty \frac{1}{n(2n+1)} $
My attempt
Let
$$f(x) = \ln((1+x)^{(1+x)}) + \ln((1-x)^{(1-x... |
H: Upper bound on probability of binomial exceeding expectation
For iid $X_i$ taking values in $\{0, 1\}$ with parameter $E[X_i]$ show that when $nE[X_i] > 1$:
$$P\left(\frac{1}{n}\sum_i^n X_i > E[X_i]\right) \leq1/4$$
This inequality is from the proof of Lemma 4.1 in Vapnik's Statistical Learning Theory.
My first tho... |
H: Give an example of two sets A and B such that |A| = |B|, and a function f : A → B such that f is one-to-one but not onto
Actually, I am a new student in discrete mathematics and this question appeared in my text book. And just before that I read that if |A|=|B| then the function must be one-one, onto, Invertible if... |
H: Laplace equation with boundary condition
Solve Laplace's equation in polar coordinates $$ \frac {1}{r} \frac {\partial u} {\partial r} + \frac {\partial^2 u} {\partial r^2} + \frac {1} {r^2} \frac {\partial^2 u} {\partial \theta^2} = 0$$
on the disk $$ {{(r, \theta) | 0 \leq r \leq R , 0 \leq \theta \leq 2}} $$
sub... |
H: Is an invariant or reducing subspace necessarily the image of a spectral projection?
In the following, the section numbers I mention are from Rudin's Functional Analysis text,
Chapter 12.
Let $T$ be a bounded normal operator in the (not necessarily separable) Hilbert space $\mathfrak{H}$.
Let $E$ be the resolution ... |
H: If $\mathbb{R^k}= \cup^{\infty} F_n $ where $ F_n $ is closed, then at least one $ F_n $ has non empty interior.
Question: If $\mathbb{R^k}= \cup^{\infty} F_n $ where $ F_n $ is closed, then at least one $ F_n $ has non empty interior.
closed definition: a set E is closed if every limit point of E is a point of E... |
H: Find integers $1+\sqrt2+\sqrt3+\sqrt6=\sqrt{a+\sqrt{b+\sqrt{c+\sqrt{d}}}}$
Root numbers Problem (Math Quiz Facebook):
Consider the following equation:
$$1+\sqrt2+\sqrt3+\sqrt6=\sqrt{a+\sqrt{b+\sqrt{c+\sqrt{d}}}}$$
Where $a,\,b,\,c,\,d$ are integers. Find $a+b+c+d$
I've tried it like this:
Let $w=\sqrt6,\, x=\sqrt... |
H: How to solve $\int_{-\infty}^{\infty} exp(-\sqrt{2\pi}x-\dfrac{(y-x)^2}{2})dy$
What i tried was:
$\int_{-\infty}^{\infty} exp(-\sqrt{2\pi}x-\dfrac{(y-x)^2}{2})dy = -\dfrac{1}{y-x}exp(-\sqrt{2\pi}x-\dfrac{(y-x)^2}{2})|_{-\infty}^{\infty}$
The exp dominates the expression so it goes faster to $0$ for $y$ that approac... |
H: How to construct a linear system that has no sol
Construct a linear system that has no solution. Unknown Variable counts must be more than the equation count. Is it possible an equation like this? What are needed?
AI: Here is an example which might be useful.
$$
\begin{bmatrix}
1 & 1 & 1\\
2 & 2 & 2\\
\end{bmatrix}... |
H: Show that $p \to q$ is a tautology if and only if $P \subseteq Q$.
I really don't know how to approach this.
I wrote out the truth table for $p' + q$ in which each row has a value of $1$ apart from when $p = 1$ and $q = 0$.
Intuitively if $q = 0$ then $Q$ is a null set. And if $P\subseteq Q$ then that means $\{1\}... |
H: What are the values of $a$ for which this integral converges?
What are the values of a for which this integral converges?
$$I = \int_{0}^{\infty} \frac{\sin x}{x^a}\,dx.$$
I tried comparing it with the integral $$\int_{0}^{\infty} \frac{1}{x^a}\,dx.$$ but I couldn't get anything out of it.
Any help would be appreci... |
H: Find the area between $(y-x+2)^2=9y, \ \ x=0, \ \ y=0.$
Find the area between $$(y-x+2)^2=9y , \ \ x=0, \ \ y=0.$$ The graph is attached below.
The area between these lines is
$$A=\int_0^2 ydx$$
From $(y-x+2)^2=9y \ \ (*)$ we get $$x=y+3\sqrt{y}+2 \implies dx=(1+\frac{3}{2\sqrt{y}})dy$$ Thus,
$$A=\int_0^2 ydx=\in... |
H: Why is grouping $2n$ students into pairs NOT equal to $^{2n}C_2$?
I just don't understand.
The way to couple $2n$ people into pairs $\dfrac{2n!}{(n!(2!))^n}$
I get this reasoning. $2n!$ is the rearrangement of the $2n$ students, divide it $n!$ and $2!$ to get rid of repeated groupings.
Shouldn't $^{2n}C_2$ give me ... |
H: question on Natural Log, $\lim \limits_{n\to∞ }(1+\frac{1}{n} + \frac{1}{n^2})^n $
I'm curious what is the solution of this. Is this just same as ordinary natural log?
$\lim \limits_{n\to∞ }(1+\frac{1}{n} + \frac{1}{n^2})^n =e?$
A few people says the $\frac{1}{n^2}$ just goes to $'0'$, so it's same as $'e'$.
But w... |
H: Prove $\frac{a^2}{(a+b)^2} \geqslant \frac{4a^2-b^2-bc+7ca}{4(a+b+c)^2}$
Let $a,\,b,\,c$ are positive numbers. Prove that
$$\frac{a^2}{(a+b)^2} \geqslant \frac{4a^2-b^2-bc+7ca}{4(a+b+c)^2}. \quad (*)$$
Note. My proof is use sos. Form $(*)$ we get know problem
$$\frac{a^2}{(a+b)^2}+\frac{b^2}{(b+c)^2}+\frac{c^2}{(c+... |
H: Prove that V is an identity operator
Recently I have attended linear algebra course, unfortunately it was more focused on practice, when I've discovered that I decided to focus more on theory to boost up my skills during summer vacation. Yesterday I've came across following task and literally have no idea how to so... |
H: A question about convergence of a sequence
Theorem. Let $X$ a uniformly convex Banach space. Let $\{x_n\}$ be a sequence in $X$ such that $x_n \rightharpoonup x$ and $$\limsup\lVert x_n\rVert\le\lVert x\rVert$$ then $x_n\to x.$
Proof. We assume that $x\ne 0$. Set $\lambda_n=\max\{{\lVert x_n\rVert,\lVert x \rVert}\... |
H: On a real square matrix of order $10$
Let $M_{10}$ be the set of $10×10$ real matrices; if $U\in M_{10}$, then let $\rho(U)=rank(U)$.
Which of the followings are true for every $A\in M_{10}$?
$(1)\rho(A^8)=\rho(A^9)$
$(2)\rho(A^9)=\rho(A^{10})$
$(3)\rho(A^{10})=\rho(A^{11})$
$(4)\rho(A^8)=\rho(A^7)$
I am able to di... |
H: What if $\epsilon$ is infinity in the $\epsilon$-$\delta$ definition of limits?
The epsilon delta definition of limits says that if the limit as $x\to a$ of $f(x)$ is L, then for any $\delta>0$, there is an $\epsilon>0$ such that if $0<|x-a|<\delta$, then $|f(x)-L|<\epsilon$.
But the problem is that this definition... |
H: f is a continuous real valued function with period $2 \pi$. Determine which of the following cases are always true.
This is a NBHM Phd 2019 question.
f is a continuous real valued function of period $2 \pi$ then which of the cases are true.
Case 1: $\exists $ $t_0 \in \mathbb{R}$ such that $f(t_0) = f(t_0 + \frac{\... |
H: why the basis generates the ideal as an R-module?
This is from Page 82 of Rotman's homological algebra book.
Definition: Let k be a commutative ring. Then a ring R is a k-algebra if R is a k-module satisfying: a(rs) = (ar)s = r(as) for all a in k and r,s in R.
If k is a field and R is a finite-dim k-algebra, then e... |
H: Verification of a logarithmic inequality
Verify the inequality
$ \frac{(\log (x) + \log (y))}{2} \le \log\frac{(x+y)}{2}$, where $x,y>0$
I'm still struggling how to solve the inequality, I have tried AM-GM and Bernoulli, without any success.My suggestion is that the solution is very elementare, but I can't see it.
... |
H: Variance of Univariate Gaussian Mixture
Let $\mathbb{P}_1,\dots,\mathbb{P}_n$ be univariate Gaussian measures with respective means $m_1,\dots,m_n \in \mathbb{R}$ and respective variances $\sigma_1,\dots,\sigma_n$. Let $r_1,\dots,r_n$ be numbers in $(0,1)$ which sum to $1$. Is the variance of a random variable di... |
H: Dimensional property of kernel for sum of two linear maps
Let $T: V \longrightarrow V, S: V \longrightarrow V$ be two linear operators. Let $P: V \longrightarrow V$ be another linear operator. Suppose $P \circ S=S \circ P$, then prove or disprove that
$$
\operatorname{dim}(\operatorname{ker}(T \circ S+P))=\operator... |
H: $X_i \sim^{iid}\operatorname{Ber}(p)$ and $Y_m = \sum_{i=1}^{m}X_i$. find $E[Y_m|Y_n]$
I have a math problem regarding condition expectancy.
Let there be $$X_i \sim^{iid}\operatorname{Ber}(p), Y_m = \sum_{i=1}^{m}X_i$$.
Now we know that $$Y_m\sim \operatorname{Bin}(m,p), m \leq n$$
Im tring to find $E[Y_m\mid Y_n]$... |
H: Inequality 6 deg
For $a,b,c\ge 0$ Prove that $$4(a^2+b^2+c^2)^3\ge 3(a^3+b^3+c^3+3abc)^2$$
My attempt: $$LHS-RHS=12(a-b)^2(b-c)^2(c-a)^2+2(ab+bc+ca)\sum_{sym} a^2(a-b)(a-c)$$
$$+\left(\sum_{sym} a(a-b)(a-c)\right)^2+14\left(\sum_{sym} a^3b^3+3a^2b^2c^2-abc\sum_{sym} ab(a+b)\right)+\sum_{sym} c^2(a-b)^2[2(a+b)^2+(a-... |
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