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This is a statement from Rudin's *Functional Analysis* section 1.46:
Given an open set $\Omega \subset \mathbb{R}^n$, choose compact sets $K_i$ such that $\Omega = \bigcup K_i$ and $K_i \subset K_{i+1}^{\mathrm{o}}$. Then define seminorms for $\phi \in C^\infty(\Omega)$:
$$
p_N(\phi) = \max\{ |D^\alpha \phi(x)| ... |
Two players play a coin-tossing game where they flip a fair coin and record the outcomes. Each player has a distinct victory condition that involves the sequence of the last two flips. Player 1's condition for winning is obtaining two consecutive 1 while player 2's condition for winning is getting two consecutive 0. Th... |
Consider a propositional language over the set of propositional variables $\{p^+,p^-,q^+,q^-,\ldots\}$ and connectives $\{\wedge,\vee,\rightarrow,\equiv\}$ (conjunction, disjunction, implication, equivalence). We call a formula $\phi$ *monotone* if it only contains $\wedge$ and $\vee$. Additionally, we call $\phi^\part... |
I am trying to calculate on average how many keystrokes will it take a monkey hitting keypad of four characters
(ABCD) to write DAC for the first time. For example, ABCDAAHDAC it's 10 keystrokes before the first appearance of DAC. I thought it's 64 but my professor is insisting it's not. Why isn't it? |
The context is that I am trying to construct a holomorphic map with certain properties.
$f(x,y,z,w)=\begin{cases}
xw-yz & \text{ if } (x,y,z,w)\in \mathbb{C}^4\setminus{S^3}\\
0& \text{ if } (x,y,z,w)\in S^3
\end{cases}$
The function above has the properties that I want, but it seems far away from been holo... |
If a sentence $P$ holds in first-order classical logic, then you can derive $\vdash P$ using the rules present in the usual Gentzen sequent calculus LK. No other rules are necessary. You don't need to add _any_ extra rules, in particular you don't need to add rules like the one you suggest. That should answer your ques... |
Let $p$ be a prime, $d$ a proper divisor of $p-1$, and $\sigma\in S_{p-1}$ of the form (everything is modulo $p$):
$$\sigma=(1,x,\dots,x^{d-1})(i_2,i_2x,\dots,i_2x^{d-1})\dots(i_k,i_kx,\dots,i_kx^{d-1})\tag1$$
(hence $|\sigma|=d$) where $kd=p-1$ and, $\forall i,j=1,\dots,p-1$ such that $i+j\not\equiv_p0$:
$$\sigma(i... |
**Question**
Write $(2n\times 2n)$-matrices in block form
$A=$
$\begin{bmatrix}a&b\\c&d\end{bmatrix}$
$\in Mat_{\mathbb{R}}(2n)$
where each of $a,b,c,d$ is an $n\times n$ block. Define
$J=$
$\begin{bmatrix}0&1\\-1&0\end{bmatrix}$
where 1 is a shorthand for an $n\times n$ identity matrix. Let
$... |
How Do I Show this Differential is Surjective? |
**Hint:** two circles are tangent at a point $P$ means that $P$ is on the line connecting the two centers.
What does that mean? For example, $M, O_3, N$ are on the same line. Given the radii $R$ and $r$ of the two initial circles, the radius $x$ for the small circle, and the distance $d$ between the original center... |
Denote the curve $V(y-e^x)\subset \mathbb A_{\mathbb C}^2$ by X, and consider its closure $\bar X$ in $\mathbb P_{\mathbb C}^2$. By GAGA, we know $\bar X$ is algebraic. I think this is amazing. I want to ask what is the algebraic equation of $\bar X$? |
What is the compactification of $y=e^x$ in $\mathbb P_{\mathbb C}^2$? |
**Question**
Write $(2n\times 2n)$-matrices in block form
$A=\begin{bmatrix}a&b\\c&d\end{bmatrix} \in Mat_{\mathbb{R}}(2n)$
where each of $a,b,c,d$ is an $n\times n$ block. Define
$J=\begin{bmatrix}0&1\\-1&0\end{bmatrix}$
where 1 is a shorthand for an $n\times n$ identity matrix. Let
$Sp(2n)= \{ A\in... |
**Question**
Write $(2n\times 2n)$-matrices in block form
$A=\begin{bmatrix}a&b\\c&d\end{bmatrix} \in Mat_{\mathbb{R}}(2n)$
where each of $a,b,c,d$ is an $n\times n$ block. Define
$J=\begin{bmatrix}0&1\\-1&0\end{bmatrix}$
where 1 is a shorthand for an $n\times n$ identity matrix. Let
$Sp(2n)= \{ A\in... |
Let $X$ and $Y$ be two random variables, each with range $R\subset\mathbb{N}$ of size $|R|=n\in\mathbb{N}$. Consider the sample space $\Omega$ "generated" by the two variables, i.e. an atomic algebra with atoms $X=k\wedge Y=k'$ for all $k,k'\in R$, closed under (finite) conjunction and negation. Let $p$ be a probabilit... |
I am new to math. stack exchange, I am really not sure how I am supposed to ask this but I need a logical explanation and a way to logical way to approach questions like these. I tried doing it myself. Thank you.
The question is as follows:
$$\text{Let } A = \{a, b, c, d, f, g, h, k, u, x, z,y\}. \\$$
... |
What's a good reference for the classification of finite-dimensional algebraic representations of $\mathrm{GL}(n,k)$ when $k$ is an arbitrary field of characteristic zero? For $k = \mathbb{C}$, Fulton does it using Young diagrams in Chapter 8 of his book *Young Tableaux*. Other treatments for $k = \mathbb{C}$ are e... |
Classifying representations of GL(n): what's a good reference? |
If the base function for the transformed function
f(x)= -4(3x)^2+5 is f(x)=x^2, then: is k=3 or is k=9? |
Let $R = \mathbb Z[X,Y]$. Construct an exact sequence of $R$-modules
$$0 \to R\to R\oplus R \to R\xrightarrow{f} \mathbb Z \to 0,$$
where $f(g(X,Y)) = g(0,0)$ for all $g \in R$. Here $\mathbb Z$ is viewed as an $R$-module via $X\cdot 1 = Y\cdot 1 = 0$.
The only choice of $f$ is to actually *define* $f(g(x,y)) = ... |
The law of tangents still has its uses! But most modern school syllabuses seem to have ditched it, solving non-right-angled triangles using the law of sines and law of cosines instead, so your question is a fair one. There's a misconception here:
> I was thinking about it and anything I could think about using the l... |
I am new to math. stack exchange, I am really not sure how I am supposed to ask this but I need a logical explanation and a way to logical way to approach questions like these. I tried doing it myself. Thank you.
The question is as follows:
$$\text{Let } A = \{a, b, c, d, f, g, h, k, u, x, z,y\}. \\$$
... |
If the base function for the transformed function
$$f(x)= -4(3x)^2+5$$.
Is $f(x)=x^2$, then: is $k=3$ or is $k=9$? |
Consider a function $f(x)$ that is defined as$$f(x)=x+\sin(x+\sin(x+\sin(x+\cdots)))$$Now, say we wanted to find the value of $f(1)$ to $5$ decimal places. My question is, how would I go about doing this?
Here is what I have tried:
Note that we have $$f(1)=1+\sin(1+\sin(1+\sin(1+\cdots)))=1+\sin(f(1))$$So we need... |
Given $f(x)=x+\sin(x+\sin(x+\sin(x+\cdots)))$, how would I go about finding the value of $f(1)$ to $5$ decimal places? |
Why does $f(t)\le g(t)$ imply $\int_a^b f(t)\; dt\le \int_a^b g(t)\; dt$? |
I am new to math. stack exchange, I am really not sure how I am supposed to ask this but I need a logical explanation and a way to logical way to approach questions like these. I tried doing it myself. Thank you.
The question is as follows:
$$\text{Let } A = \{a, b, c, d, f, g, h, k, u, x, z,y\}. \\$$
... |
I am new to math. stack exchange, I am really not sure how I am supposed to ask this but I need a logical explanation and a way to logical way to approach questions like these. I tried doing it myself. Thank you.
The question is as follows:
$$\text{Let } A = \{a, b, c, d, f, g, h, k, u, x, z,y\}. \\$$
... |
If the base function for the transformed function
$$f(x)= -4(3x)^2+5$$.
Is $f(x)=x^2$, then: is $k=3$ or is $k=9$?
By comparing the transformed function to: af(k(x-d)+c, you can pinpoint the factors which affect the function. In this case, a=-4, k=(3 or 9), d=(doesn't apply) and c=5. My notes imply that for the t... |
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/2tq7X.png
In the book Adaptive Algorithms and Stochastic Approximations Part 2, P.216,
the author assume $|H(\theta, x)| \leq C_1(1 + |x|^{q1})$.
I am now reading a paper that applies the concepts from this book. While I grasp the prac... |
In convergence of adaptive algorithm, why we have $|H(\theta, x)| \leq C_1(1 + |x|^{q1})$? |
One day a long time ago I stumbled across the Law of Tangents which states
$$\frac{\tan{\frac{A-B}{2}}}{\tan\frac{A+B}{2}}=\frac{a-b}{a+b}$$ Now, as my question states, what possible use could I have for this?
I was thinking about it, and anything I could think about using the Law of Tangents for could just be do... |
I'm trying to prove that if a|b then $a^n|b^n, \forall n \in \mathbb{N}$.
I know $a|b \rightarrow b = aq, q \in \mathbb{Z}$ by the definition of divisibility.
I can then (I think) exponentiate both sides to n, since they are equivalent.
$b^n = (aq)^n$
$b^n = a^nq^n$
From here I would assume that I use i... |
In convergence of adaptive algorithm, why we assume $|H(\theta, x)| \leq C_1(1 + |x|^{q1})$? |
[![enter image description here][1]][1]
In the book Adaptive Algorithms and Stochastic Approximations Part 2, P.216,
the author assume $|H(\theta, x)| \leq C_1(1 + |x|^{q1})$.
I am now reading a paper that applies the concepts from this book. While I grasp the practical aspects, the theoretical underpinnings e... |
Let $X$ be compact metric space, and $\mu_n$ a sequence of finite Borel measures on $X$ such that $\sup_n \mu_n(X)<\infty$. Prove
(a) For every $f \in C(X)$, there exists a subsequence $n_j$ such that $\int f d\mu_{n_j}$ converges.
(b) Let $B\subset C(X)$ be countable dense, show there exists a subsequence $n_j$... |
Not a school-related problem, and pretty much the title. I am looking for a way to solve the system of equations:
$$
A\cos(x) + B\cos(y) = C
$$
$$
A\sin(x) + B\sin(y) = D
$$
I have already solved the system for the specific case of $A=B$ and am looking to generalize this to $A \neq B$.
For reference, my sol... |
Let $X$ and $Y$ be two random variables, each with range $R\subset\mathbb{N}$ of size $|R|=n\in\mathbb{N}$. Consider the sample space $\Omega$ "generated" by the two variables, i.e. an atomic algebra with atoms $X=k\wedge Y=k'$ for all $k,k'\in R$, closed under (finite) conjunction and negation. Let $p$ be a probabilit... |
# Problem
I want to find the geodetic coordinates (longitude, latitude) of the ellipse defined by the intersection between a plane and ellipsoid.
[![Problem diagram][1]][1]
# Questions
1. Did I perform the derivation correctly?
2. What if I don't know $d$ upfront, is there any way to solve this?
- In m... |
I have a fairly simple discrete Markov Chain with a transition probability matrix that looks like this:
\begin{bmatrix}0&{n_1}&0&0&0&...\\
{n_2}&0&{n_3}&0&0&...\\
0&{n_4}&0&{n_5}&0&...\\
...&...&...&...&...&...\\
...&0&0&0&n_k&0\end{bmatrix}
Basically, just a long line of states, all of which can go into a ne... |
# Problem
I want to find the geodetic coordinates (longitude, latitude) of the ellipse defined by the intersection between a plane and ellipsoid. Point $P$ is an observer away from Earth, and the ellipse defines the boundary between the visible earth at $P$ and the not-visible portion.
That is, a ray of light com... |
I am new to math. stack exchange, I am really not sure how I am supposed to ask this but I need a logical explanation and a way to logical way to approach questions like these. I tried doing it myself. Thank you.
The question is as follows:
$$\text{Let } A = \{a, b, c, d, f, g, h, k, u, x, z,y\}. \\$$
... |
It is well known how to parameterize rational solutions to $x^2+y^2=a$ for $a\in \mathbb N^+$: first we know which [$a$ admit integer solutions][1]; our characterization is strong enough to tell us that theses are [exactly the $a$][2] admitting rational solutions; then [if we have one rational solution][3] we can get a... |
Let $X$ be compact metric space, and $\mu_n$ a sequence of finite Borel measures on $X$ such that $\sup_n \mu_n(X)<\infty$. Prove
(a) For every $f \in C(X)$, there exists a subsequence $n_j$ such that $\int f d\mu_{n_j}$ converges.
(b) Let $B\subset C(X)$ be countable dense, show there exists a subsequence $n_j$... |
# Problem
I want to find the geodetic coordinates (longitude, latitude) of the ellipse defined by the intersection between a plane and ellipsoid. Point $P$ is an observer away from Earth, and the ellipse defines the boundary between the visible earth at $P$ and the not-visible portion. That is, a ray of light coming... |
$$\sum_{k=1}^{r}\left\lfloor r^2-k^2 \right\rfloor$$
where $r$ is a constant
---
I apologize if I'm not able to contribute a lot to this problem as the floor function is really weird to me. I have never seen this inside a summation and don't know what to make of this problem. If anything, can one at least pla... |
Is there a nice form to $\sum_{k=1}^{r}\left\lfloor r^2-k^2 \right\rfloor$? |
Edit: Thanks to those commenting, I fixed my issue and now have reached the below function, as I had aimed. My remaining question would be how I might describe this function. I have taken the function from (https://math.stackexchange.com/questions/2885002/function-for-sine-wave-between-two-exponential-cuves) where they... |
Here I explained how the two-dimensional recurrence can be solved directly without knowing the answer.
Let us first fix $x=1$. Then, we have the following recurrence:
$$P(1,n) = P(1,n-1)\left(\frac{n-2}{n-1}\right).$$
The [solution][1] of this is given by
$$P(1,n) = \frac{c}{n-1},$$
where from $P(1,2)=1... |
Given a CL term $E$, can there exist multiple non-equivalent fixed points for $E$?
I think: any fixed point of $E$ can be expressed as $Y(E)$, this expression cannot reduce to multiple non-equivalent forms, due to confluence. So, I think that any CL term can have only 1 fixed point.
Is my logic correct? All ideas... |
Can a CL-term have multiple fixed points? |
$$\sum_{k=1}^{n}\left\lfloor\sqrt{ r^2-k^2 }\right\rfloor$$
where $r$ is a constant (not necessarily an integer)
---
I apologize if I'm not able to contribute a lot to this problem as the floor function is really weird to me. I have never seen this inside a summation and don't know what to make of this proble... |
Is there a nice form to $\sum_{k=1}^{n}\left\lfloor\sqrt{ r^2-k^2 }\right\rfloor$? |
Given a CL term $E$, can there exist multiple non-equivalent fixed points for $E$?
I think: any fixed point of $E$ can be expressed as $Y(E)$, this expression cannot reduce to multiple non-equivalent forms, due to confluence. So, I think that any CL term can have only 1 fixed point.
Is my logic correct? All ideas... |
Move either the terms with $A$ or $B$ to the other side and square:
$$B^2\cos^2y=A^2\cos^2x+2AC\cos x+C^2\\B^2\sin^2y=A^2\sin^2x+2AD\sin x+D^2$$
Now add them together:
$$B^2=A^2+C^2+D^2+2AC\cos x+2AD\sin x$$
Now write $C=R\cos\phi$, $D=R\sin\phi$. So $R^2=C^2+D^2$.
$$B^2-A^2-C^2-D^2=2\sqrt{C^2+D^2}(\cos x\cos\phi+... |
>A function $f: {\mathbb{R}^m} \to {\mathbb{R}^n}$ is said to be differentiable at $x$ if there exists a linear function $L(x)$ such that $f(x+h)-f(x)=L(x)h + o(h)$, as $h\to0$.
I'm abit confused about the use of little $o$ here. Which one of the interpretation is correct?
1. There exists a function $\alpha(t) = o... |
What does adding a little o mean in differentiability? |
Given a CL term $E$, can there exist multiple non-equivalent fixed points for $E$?
I think: any fixed point of $E$ can be expressed as $Y(E)$, this expression cannot reduce to multiple non-equivalent forms, due to confluence. So, I think that any CL term can have only 1 fixed point.
Is my logic correct? All ideas... |
I wanted to prove that 9 divides the cubes of 3 consecutive integers. I represented this as:
For $n \in \mathbb{Z}, 9|n^3 + (n+1)^3 + (n+2)^3$
By the definition of divisibility, if $9|n^3 + (n+1)^3 + (n+2)^3$ then there would exist an integer, let's say m such that $n^3 + (n+1)^3 + (n+2)^3 = 9m$
$$n^3 + (n+1)^... |
I'm studying proofs trying to use logic before start with the proof.
A direct proof can be written as $P\implies Q$, by forcing $P$ to be true, we have to force $Q$ to be true so the statement stays true.
But in a case of the form $P\implies (Q \lor R)$, why we have to proof both propositions to be true ($Q, R$... |
If the base function for the transformed function
$$f(x)= -4(3x)^2+5$$.
Is $f(x)=x^2$, then: is $k=3$ or is $k=9$?
By comparing the transformed function to: $af(k(x-d)+c$, you can pinpoint the factors which affect the function. In this case, $a=-4, k=3$ or $k=9$, $d=$(doesn't apply) and $c=5$. My notes imply that... |
If the base function for the transformed function
$$f(x)= -4(3x)^2+5$$
is $f(x)=x^2$, then: is $k=3$ or is $k=9$?
By comparing the transformed function to: $af(k(x-d))+c$, you can pinpoint the factors which affect the function. In this case, $a=-4, k=3$ or $k=9$, $d=$(doesn't apply) and $c=5$. My notes imply that... |
If the base function for the transformed function $f(x)= -4(3x)^2+5$ is $f(x)=x^2$, then: is $k=3$ or $k=9$? |
If $A$ is normal then closure of numerical range is the convex hull of spectrum of $A$?
I am reading book byPeter Rosenthal on "Introduction to Operators on the Hilbert Hardy space".
showing the convex hull of spectrum is contained in closure of numerical range part is done by me.
But i am getting the idea to sh... |
Someone commented under one of my previous posts that, intuitively, a generic set isn't supposed to have any "conspicuous properties". I wonder what the precise meaning of that comment was and whether we could provide some kind of proof of this.
For reference, I'm working with these definitions:
> A dense set $D$... |
What are generic ideals and dense sets, intuitively? |
Let $A$ be a algebra over a set $E$, and $\mu$ be a premeasure over $A$. We can define a outer measure $\mu^*$ over $2^E$, and a $\sigma-$algebra $M$ over $E$ (by Caratheodory theorem). In fact $\mu^*\mid_M$ is a measure $m$. (cf. Real Analysis by Folland P29, 30, 31).
I'm thus wondering whether there is a character... |
Move either the terms with $A$ or $B$ to the other side and square:
$$B^2\cos^2y=A^2\cos^2x-2AC\cos x+C^2\\B^2\sin^2y=A^2\sin^2x-2AD\sin x+D^2$$
Now add them together:
$$B^2=A^2+C^2+D^2-2AC\cos x-2AD\sin x$$
Now write $C=R\cos\phi$, $D=R\sin\phi$. So $R^2=C^2+D^2$.
$$B^2-A^2-C^2-D^2=-2\sqrt{C^2+D^2}(\cos x\c... |
[![enter image description here][1]][1]
In the book Adaptive Algorithms and Stochastic Approximations Part 2, P.216,
the author assume $|H(\theta, x)| \leq C_1(1 + |x|^{q_1})$.
I am now reading a paper that applies the concepts from this book. While I grasp the practical aspects, the theoretical underpinnings ... |
In convergence of adaptive algorithm, why we assume $|H(\theta, x)| \leq C_1(1 + |x|^{q_1})$? |
Suppose x'= Hx, with x' and x homogeneous coordinates in an image and H a homography matrix. However, H is the result of calculations that take into account the uncertainty of x' and x, meaning H~{$\mu_H$, $\Sigma_{hh}$}, with h=vec(H). I follow the approach in the book Photogrammetric Computer Vision chapter 10, Reaso... |
Let $A \in \mathbb{R}^{m \times n }; A=U-V$ be a splitting such that $R(A)=R(U), N(A)=N(U)$ where $R(A),N(A)$ denote the range space and null space $A.$
Definition of Moore-Penrose inverse of a matrix
> $A^\dagger y=x$ if $Ax=y, x \in R(A^T)$ and
$A^\dagger y=0$ if $y \in N(A^T).$
>Doubt: Let $A=U-V$ be a... |
If a sentence $P$ holds in first-order classical logic, then you can derive $\vdash P$ using the rules present in the usual Gentzen sequent calculus LK. No other rules are necessary. You don't need to add _any_ extra rules, in particular you don't need to add rules like the one you suggest. That should answer your ques... |
Let $A \in \mathbb{R}^{m \times n }; A=U-V$ be a splitting such that $R(A)=R(U), N(A)=N(U)$ where $R(A),N(A)$ denote the range space and null space $A.$
Definition of Moore-Penrose inverse of a matrix
> $A^\dagger y=x$ if $Ax=y, x \in R(A^T)$ and
$A^\dagger y=0$ if $y \in N(A^T).$
https://math.stackexchange... |
Suppose $a,b\in\mathbb{Z}$. If both $ab$ and $a+b$ are even, then both $a$ and $b$ are even
Proof by contrapositive.
Propositions:
- $P$: $ab$ is even
- $Q$: $a+b$ is even
- $R$: $a$ is even
- $S$: $b$ is even
Then logically we have $(P\land Q)\implies (R\land S)$.
We have to negate $R\land S$, so $\n... |
How to calculate definite integral of $e^{\frac{-y^2}{2}}$? |
How to calculate definite integral of $e^{-\frac{y^2}{2}}$? |
I'm studying proofs trying to use logic before starting with the proof.
A direct proof can be written as $P\implies Q$, by forcing $P$ to be true, we have to force $Q$ to be true so the statement stays true.
But in a case of the form $P\implies (Q \lor R)$, why do we have to prove both propositions to be true (... |
Given matrices $A \in \mathbb{R}^{m \times n}$, $B \in \mathbb{R}^{p \times n}$, and $H \in \mathbb{R}^{q \times n}$, I want to prove that exactly one of the following 2 systems has a solution:
1) $Ax < 0$, $Bx \leq 0$, and $Hx = 0$
2) $A^{T}u + B^{T}v + H^{T}w = 0, u \geq 0, v \geq 0, e^{T}u = 1$ where $e^{T}$ is ... |
I have seen many calculus books to use the limits $\lim_{x\to0}\sin x=0$ and $\lim_{x\to0}\cos x =1$ without their proofs. Are these limits obvious? I think no. These two limits are essentially telling us that the functions $\sin x$ and $\cos x$ are continuous at $x=0$. But how do we know this?
In fact, it is a st... |
We know that the Y-combinator is defined as:
$$\text{Y}:=\lambda f.(\lambda x.f(xx))(\lambda x.f(xx))$$
Wikipedia says :$$\text{Y}:=\text{S(K(SII))(S(S(KS)K)(K(SII)))}$$
Now the question is: **What logical steps can we take to convert the first definition to the second?**
While it is easy to show the equivale... |
I'm currently working on an analytical philosophy problem, and I think formal logic tools could help me.
First of all, I introduce some definitions. Let us give ourselves a language $\mathcal{L}$ and a theory $\mathcal{T}$. Unless otherwise stated, we place ourselves within this theory, and do not deviate from it.
... |
Existence and uniqueness of "minimal p-object"? |
A group $G$ is said to be capable if there is some group $H$ for which $H/Z(H)$ is isomorphic to $G$.
It is known that the only capable cyclic group is the trivial group. So, if $n$ is a cyclic number (i.e., the cyclic group is the only group of order $n$), then there is no group whose center has index $n$.
But i... |
Does the Annihilator for a differential equation have to be injective? |
It is not a school-related problem, and pretty much the title. I am looking for a way to solve the system of trigonometric equations:
$$
A\cos(x) + B\cos(y) = C \tag{1}
$$
$$
A\sin(x) + B\sin(y) = D \tag{2}
$$
I have already solved the system for the specific case of $A=B$ and am looking to generalize this... |
We know that, If $f$ and $g$ are functions then their convolution is defined as,
$(f*g)(x) =\int_{-∞}^{∞} f(t)g(x-t) dt$
(This is the convolution structure for Fourier transform)
**But, what if $f$ and $g$ are functions of two variables?** say $f(x, y)$ & $g(x, y)$ then how we define the convolution in this ... |
A group $G$ is said to be capable if there is some group $H$ for which $H/Z(H)$ is isomorphic to $G$.
It is known that the only capable cyclic group is the trivial group. So, if $n$ is a cyclic number (i.e., the cyclic group is the only group of order $n$), then there is no group whose center has index $n$.
But i... |
Let $A$ be an algebra over a set $E$, and $\mu$ be a premeasure over $A$. We can define an outer measure $\mu^*$ over $2^E$, and a $\sigma-$algebra $M$ over $E$ (by Caratheodory theorem). In fact $\mu^*\mid_M$ is a measure $m$. (cf. Real Analysis by Folland P29, 30, 31).
I'm thus wondering whether there is a charact... |
What is the steady state of a Markov chain with two absorbing states? |
I am new to math. stack exchange, I am really not sure how I am supposed to ask this but I need a logical explanation and a way to logical way to approach questions like these. I tried doing it myself. Thank you.
The question is as follows:
$$\text{Let } A = \{a, b, c, d, f, g, h, k, u, x, z,y\}. \\$$
... |
Consider the IVP consisting of the homogeneous heat equation
\begin{equation}
v_t(x,t)=\alpha v_{xx}(x,t), \quad (x,t)\in(-\infty,\infty)\times(0,\infty), \tag{1}
\end{equation}
subject to the initial condition
\begin{equation}
v(x,0)=f(x), \quad x\in (-\infty,\infty). \tag{2}
\end{equation}
Here $\alpha>0$ and... |
I've been trying to understand a question in Barrett O'Neill's Elementary Differential Geometry that goes as follows:
"Show that the geometric surface in Exercise 2 is isometric to a helicoid. (Hint: use x(u,v) = (f(u)cos(v), f(u)sinv, v).)"
Exercise 2 references what the book calls a conformal structure (an abst... |
I've been trying to understand a question in Barrett O'Neill's Elementary Differential Geometry that goes as follows:
"Show that the geometric surface in Exercise 2 is isometric to a helicoid. (Hint: use x(u,v) = (f(u)cos(v), f(u)sinv, v) for a suitable function f(u).)"
Exercise 2 references what the book calls a... |
I saw this challenging problem: If $f(x) =\sin^{1200}(x) \ln(1+x)^{500}\arctan^{300}(x)$ how to find $f^{(2000)}(0)$.
Applying Leibniz rule seems to be the only way to solve this, but it quickly became a nightmare finding a pattern:
$$f'(x)=1200 \sin^{1199}(x) \ln(1+x)^{500}\arctan^{300}(x)\cos(x)+\sin^{1200}(... |
How do I compute the eleventh roots of unity algebraically, expressed with only square roots and fifth roots? |
I know that there are a lot of posts calculating the Frulani's integral but I really know how to do it on myself. Here is the problem:
>Lef $f:[0,\infty)\to\mathbb{R}$ a $C^{1}$ function such that $\displaystyle\int_{1}^{\infty} \frac{f(u)}{u}\,d u$ converges. Prove, using the Fubuni's theorem that if $0<a<b$ then $... |
Given a Poisson process with parameter $\lambda$, what is the probability $P(T_2 \leq 2 | N(3) = 1)$, where $T_2$ is the interarrival time between the first and second arrivals? In words, given that there is one arrival in the first 3 time units, what is the probability that the second arrival occurs within 2 time unit... |
I know that there are a lot of posts that derive Frulani's integral but I would really like to know how to do it for myself. Here is the problem:
>Lef $f:[0,\infty)\to\mathbb{R}$ a $C^{1}$ function such that $\displaystyle\int_{1}^{\infty} \frac{f(u)}{u}\,d u$ converges. Prove, using the Fubuni's theorem that if $0<... |
Suppose $a,b\in\mathbb{Z}$. If both $ab$ and $a+b$ are even, then both $a$ and $b$ are even
Proof by contrapositive.
Propositions:
- $P$: $ab$ is even
- $Q$: $a+b$ is even
- $R$: $a$ is even
- $S$: $b$ is even
Then logically we have $(P\land Q)\implies (R\land S)$.
We have to negate $R\land S$, so $\n... |
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