instruction stringlengths 12 30k |
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**Base Case $n=1$:** if $ax+b=0$ for all values of $x$, then $a=b=0$
\begin{align}
ax+b=0\\
(0)+b=0\\
b=0\\\\
ax+(0)=0\\
ax=0\\
a(1)=0\\
a=0
\end{align}
**Inductive Step:** if $(a_nx^n+a_{n-1}x^{n-1}+...+a_1x^1+a_0=0)\Rightarrow(a_n=a_{n-1}=...=a_0=0)$ for all values of $x$, then $(a_{n+1}x^{... |
Is This Proof For $a_nx^n+a_{n-1}x^{n-1}+...+a_1x^1+a_0=0\Rightarrow a_n=a_{n-1}=...a_1=a_0=0$, $\forall n\forall x$ Okay? |
Is there a description of the group of automorphisms of a product of two simplices depending on the dimension of the simplices?
By automorphisms, I mean simplicial automorphisms of a simplicial complex on $\Delta^p\times \Delta^q$. The simplicial structure on the product is introduced in the standard minimal manner ... |
Problem:
Given that $y = e^{2x}$ is a solution of
$$ (2x+1)y'' - 4(x+1)y' + 4y = 0 $$
find a linearly independent solution by reducing the order. Write the general solution.
Answer:
Let $y = e^{2x}v$. We have:
\begin{align*}
\dfrac{dy}{dx} &= e^{2x} \dfrac{dv}{dx} + 2 e^{2x} v \\
\dfrac{d^2y}{dx^2} &=
... |
When $\mathbb Z_3[x]/\langle x^2+1\rangle$, why is $(2x^2 + 2x + 1 + \langle x^2+1\rangle) = (2x + 2 + \langle x^2+1\rangle)$?
I know $\langle x^2+1\rangle$ has $2x^2+2$ but I'm confused by the fact that how $(2x^2 + 2x + 1 + \langle x^2+1\rangle) = (2x + 2 + \langle x^2+1\rangle)$? |
When $\mathbb Z_3[x] / \langle x^2+1\rangle$, why $(2x^2 + 2x + 1 + \langle x^2+1\rangle) = (2x + 2 + \langle x^2+1\rangle)$? |
The following proof is weaker than the pervious answer since it only works for $\sqrt{2}$.
We prove the following result: if $p, q$ are natural numbers, then
$$\left|\frac{p}{q} - \sqrt{2}\right| > \frac{1}{3q^2}.$$
Denote $r := p/q$. If $q=1$,
$$|r - \sqrt{2}| = |p - \sqrt{2}|\geq \sqrt{2} - 1 > \dfrac{1}{3}.... |
When implementing multiplication in binary finite fields, Barrett reduction allows to perform division with two carryless multiplications and two XOR additions (and some one-time precomputation).
However, because it is always possible to find irreducible polynomials that have a lot of zeros after the first bit (e.g. h... |
I would assume that the chance would be $0%$ as $0=\lim\frac{1}{n}$ but since the equivalent would be something along the lines of $\frac{\aleph_0}{\aleph_1}$ I am not sure.
If $0=\frac{\aleph_0}{\aleph_1}$ would that not mean that anything would be an infinite subset when randomly sampled from an uncountable set, e... |
In Jech's Set Theory, Chapter 11, the universal set is defined as: For each $\alpha \geq 1$, there exists a set $U \subset \mathcal{N}^2$ such that $U$ is $\Sigma_{\alpha}^0$ (in $\mathcal{N}^2$) and that for every $\Sigma_{\alpha}^0$ set $A$ in $\mathcal{N}$, there exists some $a \in \mathcal{N}$ such that $A = \{x:(x... |
I say a field $K$ is cool if every group homomorphism between to $K$-vector spaces is $K$-linear.
$\mathbb{Z}_p$ is cool because multiplication by a scalar is like an iterated sum.
$\mathbb{Q}$ is cool too. Lef $f$ be such a group homorphism. $$bf(\frac a b x)=f(\frac a b x)+\ldots +f(\frac a b x)=f(\frac a b x+... |
I've been struggling to understand how the surjection of a function $f : X \rightarrow Y$ implies that it has a right inverse. My questions basically reside on the application of the axiom of choice to the proof.
**First question**: The axiom of choice states that, for any set $Y$, if $\varnothing \not\in Y$, then ... |
The word "indeed" is often used in proofs, usually to begin a sentence.
What is the meaning and correct usage? |
What does the word "indeed" mean in math? |
\begin{align*}i\frac{d}{dt}a_{n}(t) &= (a_{n+1}(t) + a_{n-1}(t))(1+|a_{n}(t)|^2), \;n \in \mathbb{Z}\end{align*}
Consider the conserved quantity $$P = \sum\limits_{n}g(|a_n|^2)$$
What is the function $g(|a_n|^2)$ which makes $P$ conserved, i.e., $\frac{d}{dt}P = 0$ for all $\{a_{n}(t)\}_{n \in \mathbb{Z}}$?
- - -
... |
Beginner question here... I'm a little unclear on how to go about reversing a transformation of a matrix from the format of Ax=b (or if it's possible for that matter).
Given a 3x3 matrix `A` and a target vector `b`, I can compute the missing vector using `Ax=b`
Now, given a 3x3 matrix `M`, is there a way to use t... |
I've been struggling to understand how the surjection of a function $f : X \rightarrow Y$ implies that it has a right inverse. My questions basically reside on the application of the axiom of choice to the proof.
**First question**: The axiom of choice states that, for any set $Y$, if $\varnothing \not\in Y$, then ... |
In Jech's Set Theory, Chapter 11, the universal set $U$ is defined as: For each $\alpha \geq 1$, there exists a set $U \subset \mathcal{N}^2$ such that $U$ is $\Sigma_{\alpha}^0$ (in $\mathcal{N}^2$) and that for every $\Sigma_{\alpha}^0$ set $A$ in $\mathcal{N}$, there exists some $a \in \mathcal{N}$ such that $A = \{... |
In the book I am reading there is a section about modular arithmetic and modular equations.
First, here is what is stated in the book:
> Theorem: Given the equation $ax \equiv b \mod n$, where $a,b \in \mathbb{Z}$, $n \in \mathbb{N}$.
>
>
> 1. If $gcd(a,n) \nmid b$, then the equations has no solution.
> 2... |
>> If I have $w = f(x,y)$, where $x = u + v$ and $y = u - v$, then why is the following true?
>> $$\frac{\partial}{\partial u}w_x = \frac{\partial w_x}{\partial x}\cdot\frac{\partial x}{\partial u} + \frac{\partial w_x}{\partial y}\cdot\frac{\partial y}{\partial u}$$
>
> Is my assumption that $\frac{\partial}{\part... |
How Do I describe the Lie Algebra of this Symplectic Group? |
I've been struggling to understand how the surjection of a function $f : X \rightarrow Y$ implies that it has a right inverse. My questions basically reside on the application of the axiom of choice to the proof.
**First question**: The axiom of choice states that, for any set $Y$, if $\varnothing \not\in Y$, then ... |
I'm reading "Differential Equations Driven by Rough Paths" by Lyon, Caruana and Lévy and I can't wrap my head around the following part (beginning of Section 1.4.2).
Let $V$ be a finite-dimensional Banach space. For an integer $K \geq 0$ let $P: V \to \mathbb{R}$ be a polynomial of degree $k$. Let $X : [0,T] \to V$ ... |
Beginner question here... I'm a little unclear on how to go about reversing a transformation of a matrix from the format of Ax=b (or if it's possible for that matter).
I started out with a 6x6 matrix `A` target vector `b`, I can compute the missing vector using `Ax=b`
I then applied the points of the solved vecto... |
Beginner question here... I'm a little unclear on how to go about reversing a transformation of a matrix from the format of Ax=b (or if it's possible for that matter).
I started out with a 6x6 matrix `A` and target vector `b`, I can compute the missing vector using `Ax=b`
I then applied the points of the solved v... |
Let $G:=\text{SL}(2,\mathbb R)$ and $\Gamma:=\text{SL}(2,\mathbb Z)$.
Ergodic theory tells us that **for almost every** point $g\Gamma$ in $G/\Gamma$ (with respect to Haar measure), we have $\{u_tg\Gamma\}_{t\ge 0}$ is dense in $G/\Gamma$, where $u_t=\begin{bmatrix}
1 & t \\
0 & 1
\end{bmatrix}$ (Howe-Moore the... |
Examples of specific points $g\in G:=\text{SL}(2,\mathbb R)$ such that the curve $u_tg \Gamma, \Gamma:=\text{SL}(2,\mathbb Z)$ is dense in $G/\Gamma$? |
Do you know that the fundamental group of a CW complex depends only on its 2-skeleton? Then it will no longer be surprising.
There are a few ways to see this. I really like the covering-theoretic exposition [here](https://dmitripavlov.org/notes/topology.pdf) which uses a simplicial analogue of the classical theory t... |
**The problem**
Let $X$ be the space of $C^1[0,1]$ functions with the special property $f(0) = 0$. Consider the inner product defined in the following way:
$$\langle f, g \rangle = \int_0^1 f'(x) \overline{g'(x)} dx$$
I want to show that the space $H$ corresponding to the completion of $X$ under the induced norm m... |
Let $G:=\text{SL}(2,\mathbb R)$ and $\Gamma:=\text{SL}(2,\mathbb Z)$.
Ergodic theory tells us that **for almost every** point $g\Gamma$ in $G/\Gamma$ (with respect to Haar measure), we have $\{u_tg\Gamma\}_{t\ge 0}$ is dense in $G/\Gamma$, where $u_t=\begin{bmatrix}
1 & t \\
0 & 1
\end{bmatrix}$ (Howe-Moore the... |
In Jech's Set Theory, Chapter 11, the universal set $U$ is defined as: For each $\alpha \geq 1$, there exists a set $U \subset \mathcal{N}^2$ such that $U$ is $\Sigma_{\alpha}^0$ (in $\mathcal{N}^2$) and that for every $\Sigma_{\alpha}^0$ set $A$ in $\mathcal{N}$, there exists some $a \in \mathcal{N}$ such that $A = \{... |
Let $G:=\text{SL}(2,\mathbb R)$, $\Gamma:=\text{SL}(2,\mathbb Z)$ and $u_t=\begin{bmatrix}
1 & t \\
0 & 1
\end{bmatrix}, t\ge 0$
Although this paragraph is not really needed to answer this question, I want to point out that ergodic theory tells us that **for almost every** point $g\Gamma$ in $G/\Gamma$ (with res... |
Consider the following bipartite graph $G$, given by $V(G)= \{ s_1,s_2,s_3 \} \cup \{t_1,t_2\}$ and $E(G)=\{ \{ s_1,t_1\},\{s_1,t_2\},\{s_2,t_1\},\{s_2,t_2\},\{s_3,t_2\} \}.$
I want to define the toric edge ring of $G$, that is $K[s_1t_1,s_1t_2,s_2t_1,s_2t_2,s_3t_2],$ in `Macaulay2` ($K$ is a field). Do you know a w... |
Beginner question here... I'm a little unclear on how to go about reversing a transformation of a matrix from the format of Ax=b (or if it's possible for that matter).
I started out with a 6x6 matrix `A` and target vector `b`, I can compute the missing vector using `Ax=b`
$$\left[\begin{matrix}
1 & 0 & 0 & 4 & 7... |
You parked your car on Monday and didn't pay for parking. You leave your car there until the end of Tuesday. You know that you have the same probability of getting a ticket on Monday and Tuesday. You also know that if you get a ticket on Monday, then you won't get one on Tuesday. At the same time, you know that the pro... |
It is not a school-related problem, and pretty much the title. I am looking for a way to solve the system of trigonometric equations:
$$
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... |
Consider the statement:
In a city, $1 \%$ of the population is known to have a rare disease. A test is available to detect this disease, which has a true positive rate of $99 \%$ and a false positive rate of $ 2 \%$.
First, you want to determine the probability that a person has the disease. I try to solve the pr... |
I am trying to analyze a bit how we can extend a differential form to the complexification $V\otimes\mathbb{C}=V_{\mathbb{C}}$ of the vector space. Of course, you can do this for a general $k$-form, but I think the simpler the example the better I can grasp the idea.
So let $V=\{<v_1,\ldots,v_m>\}$ be a real vector ... |
This is similar to my previous model theory question, here: https://math.stackexchange.com/questions/2359185/regarding-the-theory-of-real-addition-with-a-nonzero-constant. Let $r$ and $s$ be any two nonzero real numbers, and consider the structures $(\mathbb{R};-,r)$ and $(\mathbb{R};-,s)$, where $-$ is the binary ope... |
Do any two nonzero real numbers behave the same under subtraction? |
In the book I am reading there is a section about modular arithmetic and modular equations.
First, here is what is stated in the book:
> Theorem: Given the equation $ax \equiv b \mod n$, where $a,b \in \mathbb{Z}$, $n \in \mathbb{N}$. Let d:=gcd(a,n).
>
>
> 1. If $gcd(a,n) \nmid b$, then the equations has ... |
The word "indeed" is often used in proofs, usually to begin a sentence.
What is the meaning and correct usage?
I've usually seen it used in this form
> \[Assertion of $X$ without proof\]. Indeed, \[proof of $X$\].
Is this interpretation correct? |
I know that there are partial binary operations and partial unary operations, but is there such a thing as a partial $0$-ary function (which is not total, of course)? |
Is there such a thing as a partial $0$-ary function? |
In the book I am reading there is a section about modular arithmetic and modular equations.
First, here is what is stated in the book:
> **Theorem:** Given the equation $ax \equiv b \mod n$, where $a,b \in \mathbb{Z}$, $n \in \mathbb{N}$. Let $d := \gcd(a,n)$.
>
> 1. If $\gcd(a,n) \nmid b$, then the equation... |
How many initial and discounted raffle tickets should I try to sell to profit $400?
The initial ticket costs `$10` and each successive ticket costs `$3`.
$$ f(x) = 10 \\ f(y) = 3 $$
The probability of a person buying a $10 ticket is `0.3`.
The probability of any additional purchases start at a probabilit... |
How many initial and discounted raffle tickets should I try to sell to earn $400? |
Take the function $f(x,n)=\frac{10^x-1}{9}$ $mod$ $n$
and the sequence $s_n$ that follows these rules:
1. $x$ is an integer counting up from 1
2. If $f(x,n)$ repeats at some point, $s_n$ is equal to how long the loop is (For the 3. sequence $[1,2,3,4,3,4,3...]$ this number would be 2)
3. If $f(x,n)$ stops (lo... |
I've run into an unexpected problem regarding the following equation:
$$Q = A\cos\phi + \sqrt{A^2\cos^2\phi}$$
A is always a positive number. My first inclination is to just simplify the eqn as,
$$Q = 2A\cos\phi $$
However, I get two different answers if $\cos\phi$ is negative (i.e. if $\phi$ is equal to 1... |
I know that not every recursive function from $\mathbb{N}$ to $\mathbb{N}$ has a range which is a recursive set. I wonder, is there a primitive recursive function $f$ such that the range of $f$ is a general recursive set, but not a primitive recursive set? |
Nearly 8 years after he asked for it, but I'll finally add the answer Mark S requested in the comments on the OP.
The answer to the question depends on what is meant by two things:
- What does it mean to subtract $1$ from $\omega_0$?
- What does it mean to say something is "finite" or "infinite"?
The first ... |
I want to solve the following:
During exam time, in a certain school, only 25% of the teachers warn their students in writing that they are not allowed to get up to ask questions during exam time.
in writing to their students that they are not allowed to stand up and ask questions during the test. However, it has b... |
I'm trying to study and test 32-bit hash functions - specifically probability of collision (repeated results for different inputs). And I'm struggling in defining a threshold for outliers/anomalies, where the *precision of an acceptable random variance* appears to increase as the ideal number of collisions increases.
... |
How many initial and discounted raffle tickets should I try to sell to profit $400?
The initial ticket costs `$10` and each successive ticket costs `$3`.
$$ f(x) = 10 \\ f(y) = 3 $$
The probability of a person buying a $10 ticket is `0.3`.
The probability of any additional purchases start at a probabilit... |
For any $n$, to what group is the commutator subgroup of the dihedral group $D_n$ isomorphic to?
My solution is below. I request verification, feedback, and improvements. In particular, can you help verify the step in the proof which argues that "the commutator subgroup consists _only_ of these elements." This par... |
Is this proof of the isomorphic structure of the commutator subgroup of the dihedral group $D_n$ valid? |
How many initial and discounted raffle tickets should I try to sell to profit $400?
The initial ticket costs `$10` and each successive ticket costs `$3`.
$$ f(x) = 10 \\ f(y) = 3 $$
The probability of a person buying a $10 ticket is `0.3`.
The probability of any additional purchases start at a probabilit... |
How many initial and discounted raffle tickets should I try to sell to profit $400?
The initial ticket costs `$10` and each successive ticket costs `$3`.
$$ f(x) = 10 \\ f(y) = 3 $$
The probability of a person buying a $10 ticket is `0.3`.
The probability of any additional purchases start at a probabilit... |
Given a real square matrix $A \in \Re^{n \times n}$, let's say its eigenvalues have a lovely property: $Re(\lambda^A_i) < 0$. We also have another diagonal real square matrix $K \in \Re^{n \times n}$ whose entries are all positive. Now, if we define $B = KA$, what is the **lower bound** of element $k_{ii}$ that guarant... |
Assume I have a set $H_k$ of all weakly connected directed graphs with $k$ unlabeled vertices. I want to find a directed graph $G$ such that each graph $H \in H_k$ is a subgraph of $G$. Furthermore, what is the minimum number of vertices that $G$ could have?
The trivial solution is of course a graph that is the uni... |
For any $n$, to what group is the commutator subgroup of the dihedral group $D_n$ isomorphic to?
My solution is below. I request verification, feedback, and improvements. In particular, can you help verify the step in the proof which argues that "the commutator subgroup consists _only_ of these elements." This par... |
The word "indeed" is often used in proofs, usually to begin a sentence.
What is the meaning and correct usage?
I've usually seen it used in this form
> \[Assertion of $X$ without proof\]. Indeed, \[proof of $X$\].
Is this interpretation correct?
----
# Update
This question asks a well defined ques... |
so my professor gave us this question to solve, but I don't know have much familiarity with the topic.
Question
--------
The forward difference is first order accurate and is defined to be $$ D_{+} = \frac{f(x+h) - f(x)}{h} $$ and the centered difference is $$ D_{0} = \frac{f(x+h)-f(x-h)}{2h}$$ Consider the finite... |
How do we find the weights for finite differences? |
$\DeclareMathOperator \erf{erf}$
Wolfram Alpha gives the following [$\erf^{-1}(z)$ series](https://www.wolframalpha.com/input?i=series+expansion+inverf%28x%29):
>$$\sum_{n=1}^\infty\frac{z^n}{2\pi n}\int_0^{2\pi}e^{it}\erf(e^{it})^{-n}dt$$
which can be derived via [Lagrange inversion](https://en.m.wikipedia.org/... |
$\DeclareMathOperator \erf{erf}$
Wolfram Alpha gives the following [$\erf^{-1}(z)$ series](https://www.wolframalpha.com/input?i=series+expansion+inverf%28x%29):
>$$\sum_{n=1}^\infty\frac{z^n}{2\pi n}\int_0^{2\pi}e^{it}\erf(e^{it})^{-n}dt$$
which can be derived via [Lagrange inversion](https://en.m.wikipedia.org/... |
How many initial and discounted raffle tickets should I try to sell to earn $400 when the probability of selling discounted decreases per buyer? |
Problem:
Given that $y = e^{2x}$ is a solution of
$$ (2x+1)y'' - 4(x+1)y' + 4y = 0 $$
find a linearly independent solution by reducing the order. Write the general solution.
Answer:
Let $y = e^{2x}v$. We have:
\begin{align*}
\dfrac{dy}{dx} &= e^{2x} \dfrac{dv}{dx} + 2 e^{2x} v \\
\dfrac{d^2y}{dx^2} &=
... |
Recently I encouter an integral of this form $$\int_0^\infty \frac{k^2}{1+k^2}\cos(ak)dk$$ with $a$ a parameter we can tune. As $k\to\infty$ the fraction in front approaches $1$ and the integrand becomes simply $\cos(ak)$, whose integral does not exist. Intuitively the integral should oscillate as a function of $t$ if ... |
Recently I encouter an integral of this form $$\int_0^\infty \frac{k^2}{1+k^2}\cos(ak)dk$$ with $a$ a parameter we can tune. As $k\to\infty$ the fraction in front approaches $1$ and the integrand becomes simply $\cos(ak)$, whose integral does not exist. Intuitively the integral should oscillate as a function of $t$ if ... |
Find $\int_{0}^{\pi}\frac{2t+2\cos{x}}{t^2+2t\cos{x}+1}dx$? |
In Jech's Set Theory, Chapter 11, the universal set $U$ is defined as: For each $\alpha \geq 1$, there exists a set $U \subset \mathcal{N}^2$ such that $U$ is $\Sigma_{\alpha}^0$ (in $\mathcal{N}^2$) and that for every $\Sigma_{\alpha}^0$ set $A$ in $\mathcal{N}$, there exists some $a \in \mathcal{N}$ such that $A = \{... |
Let me apologize ahead of time since I am not at all well versed in the theory of modular forms.
I have seen some nice examples where modular forms are used to study certain interesting numbers. For example, one might like to study the sequence $(h_n)_{n \in \mathbb{Z}}$ where $h_n$ is the class number of $\mathbb{... |
As explained in my comments above, we want to pick semi-norms $L^p(\Omega_n)$ for "larger and larger" $\Omega_n$ in order to make sure that "locally our metric looks like $L^p$". A standard way of doing this (e.g. for Fréchet spaces) is to patch these semi-norms together in the following way
$$ d(f,g) = \sum_{n\geq 1}... |
My professor gave us this question to solve, but I don't know have much familiarity with the topic.
Question
--------
The forward difference is first order accurate and is defined to be $$ D_{+} = \frac{f(x+h) - f(x)}{h} $$ and the centered difference is $$ D_{0} = \frac{f(x+h)-f(x-h)}{2h}$$ Consider the finite di... |
I am trying to solve the following problem from Artin's Algebra.
> Let $I$ be the principal ideal of $\mathbb{C}[x, y]$ generated by the polynomial $y^2 + x^3 - 17$. Which of the following sets generate maximal ideals in the quotient ring $R = \mathbb{C}[x, y]/I$? $(x-1, y-4), (x+1, y+4), (x^3 - 17, y^2)$.
For ... |
Consider a probability distribution functions $f(x)$ with cumulative density function $F(x)$.
**I would like to compute the PDF $g_{ij}$ of the gap $x_j-x_i$ where $x_i$ is the ith smallest sample.**
The rth order statistics i.e the PDF of the rth smallest sample is
$$\frac{n!}{(r-1)!(n-r)!}F(x)^{r-1}\left(1-F... |
In Jech's Set Theory, Chapter 11, the universal set $U$ is defined as: For each $\alpha \geq 1$, there exists a set $U \subset \mathcal{N}^2$ such that $U$ is $\Sigma_{\alpha}^0$ (in $\mathcal{N}^2$) and that for every $\Sigma_{\alpha}^0$ set $A$ in $\mathcal{N}$, there exists some $a \in \mathcal{N}$ such that $A = \{... |
Let $f(z)$ be an entire function (holomorphic function on $\mathbb{C}$) satifying the following condition:
$$|f(z)|\leq \max (e^{\text{im}(z)},1 ),\ \forall z\in\mathbb{C}$$
$$\lim_{\mathbb{R}\ni t\rightarrow+\infty} f(t)=0$$
My question is: can we prove that the following limit exists?
$$\lim_{\mathbb{R}\ni t\ri... |
I am trying to solve the following problem from Artin's Algebra.
> Let $I$ be the principal ideal of $\mathbb{C}[x, y]$ generated by the polynomial $y^2 + x^3 - 17$. Which of the following sets generate maximal ideals in the quotient ring $R = \mathbb{C}[x, y]/I$? $(x-1, y-4), (x+1, y+4), (x^3 - 17, y^2)$.
For ... |
The question is
>Integrate by parts to prove:$$\int_U|Du|^p\;dx\le C \left(\int_U|u|^p\;dx\right)^\frac12\left(\int_U|D^2u|^p\;dx\right)^\frac12$$
for $2\le p< \infty$ and all $u\in W^{2,p}(U)\cap W^{1,p}_0(U)$.
I have managed to get the result for all $v\in C^\infty_c(U)$ and would like to generalise the results... |
As explained in my comments above, we want to pick semi-norms $L^p(\Omega_n)$ for "larger and larger" $\Omega_n$ in order to make sure that "locally our metric looks like $L^p$". A standard way of doing this (e.g. for Fréchet spaces) is to patch these semi-norms together in the following way
$$ d(f,g) = \sum_{n\geq 1}... |
Could someone check my attempt at a proof of a theorem in the lambda calculus? |
**Question:** Given $X\sim\text{Bernoulli}(\alpha)$, $Y\sim\mathcal{N}(0,1)$, and two non-random constants $C_1$ and $C_2$ such that $C_1>C_2$. What can we say about the inequality between the two differential entropies $H(C_1X+Y)$ and $H(C_2X+Y)$? I.e., is it true that
$$
H(C_1X+Y)>H(C_2X+Y).
$$
**Fact:** We kno... |
I've been challenged by a professor to find the roots of $x^2-x+2=0$ in the field $\mathbb{Z}_3[i] = \{a+bi \; \vert \; a,b \in \mathbb{Z}_3\}$. I used the "normal" quadratic formula and got roots of $2+2i$ and $2+i$, and was told there's a way without using the formula. Is there a method for this besides guessing and ... |
As explained in my comments above, we want to pick semi-norms $L^p(\Omega_n)$ for "larger and larger" $\Omega_n$ in order to make sure that "locally our metric looks like $L^p$". A standard way of doing this (e.g. for Fréchet spaces) is to patch these semi-norms together in the following way
$$ d(f,g) = \sum_{n\geq 1}... |
I have a question in the process of marginalizing equation (6) in the DDIM paper (https://arxiv.org/pdf/2010.02502.pdf).
[![enter image description here][1]][1]
When the joint distribution is defined like this (eq(6))
$$
q_{\sigma}(\boldsymbol{x}_{1:T} \mid \boldsymbol{x}_0) := q_{\sigma}(\boldsymbol{x}_T \... |
As explained in my comments above, we want to pick semi-norms $L^p(\Omega_n)$ for "larger and larger" $\Omega_n$ in order to make sure that "locally our metric looks like $L^p$". A standard way of doing this (e.g. for Fréchet spaces) is to patch these semi-norms together in the following way
$$ d(f,g) = \sum_{n\geq 1}... |
**Constructing the metric:**
As explained in my comments above, we want to pick semi-norms $L^p(\Omega_n)$ for "larger and larger" $\Omega_n$ in order to make sure that "locally our metric looks like $L^p$". A standard way of doing this (e.g. for Fréchet spaces) is to patch these semi-norms together in the followin... |
Consider the statemen
A manufacturer of aluminum doors is currently selling 500 doors per week at a price of \$85 each. If the price were lowered to \$80 each, an additional 70 doors per week could be sold.
I wanna find the current elasticity of demand for the doors, and also estimate the current value of the m... |
Let $(X_i)_{i \in \mathbb{N}} \sim \mathcal{N}(0,1)$, where all variables are i.i.d. For the random vector
$$\mathbf{X} = \left [ X_1, \ \ \ \ X_1 + X_2, \ \ \ \ X_2 + X_3, \ \ \ \ \cdots \ \ \ \ X_{n-2} + X_{n-1}, \ \ \ \ X_{n-1} \right ]^T,$$
how can I find $\mathbb{P}(\mathbf{X} \geq 0)?$ In other words, th... |
Let the *multiplicative closure* operator $h$ be defined on $\mathcal{P}(\mathbb{N})$ by setting $hA$ equal to the smallest set containing $A$ that is closed under multiplication.
In L. F. Meyers' solution to Problem E $1941$ on pp. $408$-$9$ of the April $1968$ *American Mathematical Monthly* (https://www.jstor.org/s... |
Here is an exercise of Section 1 of Friedman *Algebraic Surfaces and Holomorphic Vector Bundles*.
>Let $C$ be an irreducible curve whose singular locus is a single ordinary double point $x$, and let $p,q\in\tilde{C}$ be the preimage under the normalization map $\nu$ of $x$. Show that $\nu^*\omega_C=K_{\tilde{C}}\ot... |
The problem stated in the Evan's book is
>Integrate by parts to prove:$$\int_U|Du|^p\;dx\le C \left(\int_U|u|^p\;dx\right)^\frac12\left(\int_U|D^2u|^p\;dx\right)^\frac12$$
for $2\le p< \infty$ and all $u\in W^{2,p}(U)\cap W^{1,p}_0(U)$.
I have managed to get the result for all $v\in C^\infty_c(U)$ and would like ... |
I want to solve the following:
During exam time, in a certain school, only 25% of the teachers warn their students in writing that they are not allowed to stand up and ask questions during the test. However, it has been observed that despite this warning, 20% of students do so. For mentors who do not provide such a ... |
This comes straight from problem 1.5 (p. 12) of *Handbook of Product Graphs, Second Edition* by Hammack, Imrich, and Klavžar.
It even comes with a hint in the appendix:
> **Exercise 1.5** Hint: Show that the homomorphic image of an odd cycle is a closed walk that contains an odd cycle.
But think maybe there's ... |
Let K be the largest value such that (2023! $)^{2023 !}$ is divisible by $42^{\mathrm{K}}$. How many consecutive 0 digits are at the end of $\mathrm{K}$ ?
I'm Stuck literally |
Let $A,B$ be any two events such that $P(A\cap B)>0$ and $A$ is an event which depends on some variable $x$.
> I am trying to check if there exists a constant $C>0$ (which is independent of $x$) such that $P(A\cap B)\geq CP(A)$.
My attempt: It is a weird inequality because we know $P(A\cap B)\leq P(A)$. Also I t... |
Let $n,b\in\mathbb{N}$ and $P_{n_b}$ is polynomial whose coefficients *are digits* of $n$ in base $b$.
Let $Q_{1}(x) \cdot Q_{2}(x) \cdots Q_{m}(x)$ - polynomial factorization (over integers) of $P_{n_b}$.
I'm interested in *coefficients of such factorization*, especially *union of all factor's coefficients* $\ma... |
Let $A,B$ be any two events such that $P(A\cap B)>0$ and $A$ is an event which depends on some variable $x$. Hence $P(A)$ is a function of $x$.
> I am trying to check if there exists a constant $C>0$ (which is independent of $x$) such that $P(A\cap B)\geq CP(A)$.
My attempt: It is a weird inequality because we kn... |
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