instruction stringlengths 12 30k |
|---|
It is an open problem whether the partition function is even half the time. Inspired by this, I wrote some Sage/Python code to check how many times $p(n)$ hits each residue class:
```
def partitionmod(n, k):
bins = [0,] * k
for i in range(n):
bins[mod(Partitions(i+1).cardinality(), k)] += 1
... |
As part of a personal project, I found the paper ["On $C^2$-smooth Surfaces of Constant Width" by Brendan Guilfoyle and Wilhelm Klingenberg][1], which discusses deriving equations for a 3D surface I want to parameterize. However, despite my efforts to decipher the paper, I can't follow everything it is describing. The ... |
Let $\Omega$ be an open, bounded domain in $\mathbb{R}^n$ (NOT necesasrily with smooth boundary).
Now, let $C_c^\infty(\Omega)$ be the space of smooth functions on $\Omega$ with compact support.
Then
> How to prove that $C_c^\infty(\Omega)$ dense in $L^p(\Omega)$ for each $p \in [1,\infty)$?
[This post](ht... |
Density of $C_c^\infty(\Omega)$ in $L^p(\Omega)$ for a bounded open set $\Omega$ - any detailed proof? |
I have a question about this problem posted [here](https://math.stackexchange.com/questions/3828930/example-of-schauder-basis-to-ell2). Essentially they are trying to prove that the sequence $\{x_n\}_{k = 1}^\infty = \{\alpha e_n + \beta e_{n+1}\}_{n=1}^\infty$ (here $\{e_n\}_{n = 1}^\infty$ is the canonical orthonorma... |
It's not true that $ \int _ { \bigcup _ i A _ i } f = \sum _ i \int _ { A _ i } f $ unless the $ A _ i $ are all disjoint (or at least have intersections of measure zero). And even then, I'm not sure how you would prove that integrals decompose like this without already knowing at least the $ \Longrightarrow $ directi... |
Let $a_{i}$ be a set of positive real numbers such that $\sum a_{i}³=3$ and $\sum a_{i}⁵=5$ for all $i=1$ to n.
Prove that $\sum a_{i} > \frac{3}{2}$
My attempt:
Using Titu's lemma,
$\frac{a_{1}⁶}{a_{1}}$ + $\frac{a_{2}⁶}{a_{2}}$ + ...≥ $\frac{(a_{1}³ + a_{2}³+...)²}{a_{1} + a_{2}+..}$
Giving
$5 ≥ \frac{3²}{... |
Oscar Lanzi has already given quite a nice overview over a general method to attack a certain set of polynomials trying to disprove eventual divisibility. Let's explore this idea a bit further and give a nice summary of the method:
We start with defining $q(x) := x^2+ax+b$ and examine solutions to $q(x) = x$ and $q... |
In a bialgebra $H$, subspace of primitive elements are $P(H)=\{x\in H:\Delta (x)=x\otimes 1+1\otimes x\}$. I know that if $x,y$ are primitive, then $[x,y]=xy-yx$ is also primitive, so $P(H)$ is a lie subalgebra under the bracket. I also know if $x_1,\cdots, x_n$ are primitive, then $$\Delta(x_1\cdots x_n)=1\otimes x_1\... |
There's a graph $G$ which is conjectured to exist. We know it's $k$-regular, and we know its spectrum. (I don't think the exact problem is important for this question). $G$ is much too large to attack directly by computation, however, some time ago I was able to produce (with computer assistance) a graph $H$ such th... |
In a bialgebra $H$, subspace of primitive elements are $P(H)=\{x\in H:\Delta (x)=x\otimes 1+1\otimes x\}$. I know that if $x,y$ are primitive, then $[x,y]=xy-yx$ is also primitive, so $P(H)$ is a lie subalgebra under the bracket. I also know if $x_1,\cdots, x_n$ are primitive, then $$\Delta(x_1\cdots x_n)=1\otimes x_1\... |
Given the demand equation
$$p=\frac{240}{q+15}$$
Where $15 \leq q \leq 105$, for what value of $q$ is $|\eta|$ a maximum? for what value is it a minimum?
Well, i try the following. Note that:
$$\eta=\frac{dq}{dp} \frac{p}{q}$$
As $p=\frac{240}{q+15}$, then $q=\frac{240-15p}{p}$ and $\frac{dq}{dp}=\left(\frac{2... |
Consider a function $f(x)$ with a period of 2L defined within the range $(L_1, L_2)$.
Will the Fourier series remain the same, regardless of the range in which I define the function $f(x)$?
For example, I can define $f(x)$ in the range $(0, 2L)$, $(\frac{L}{2}, \frac{3L}{2})$, or $(-L, L)$, but the Fourier series... |
Here is a statement of the question, which is taken from Chaisson and McMillan's Astronomy: A Beginner's Guide to the Universe.
*Assume that the number of (technological) civilisations in the Milky Way Galaxy is equal to the average lifetime of a civilisation. Assuming that civilisations are uniformly spread over a... |
Evaluate $\int \frac{\sqrt{\sin\sqrt x}\cos \sqrt x}{1+x^2} dx$? |
In a bialgebra $H$, subspace of primitive elements are $P(H)=\{x\in H:\Delta (x)=x\otimes 1+1\otimes x\}$. I know that if $x,y$ are primitive, then $[x,y]=xy-yx$ is also primitive, so $P(H)$ is a lie subalgebra under the bracket. I also know if $x_1,\cdots, x_n$ are primitive, then $$\Delta(x_1\cdots x_n)=1\otimes x_1\... |
Suppose $x'' \in \ker(d)$. As you note, we then know that here is some $x \in M$ and $y \in N'$ such that $u'(y) = f(x)$, $v(x) = x''$, and $y \in \operatorname{img}(f')$. We want to find some $a \in M$ such that $v(a) = x''$ and $f(a) = 0$.
Whatever $a$ ends up being, we will have $v(a-x) = v(a) - v(x) = x'' - x'' ... |
In my PhD thesis, I'm interpreting some of my code in Category Theory. I ended up modeling some stuff as a slice category $\mathbf{Set}/C$, and ended up with the question whether I could define a functor $F$ from $\mathbf{Set}$ to $\mathbf{Set}/C$. To my surprise, it seems that this would not be possible... Am I corre... |
Is there a functor between $\mathbf{Set}$ and $\mathbf{Set}/C$? |
> Prove that the following definitions of a strongly convex function $f: \mathbb{R}^n \to \mathbb{R}$ are equivalent:
>
>\begin{gather*}
f(y) \ge f(x) + \nabla f(x)^T (y-x) + \frac{\mu}{2} {\lVert y-x \rVert}_2^2, \quad \forall x,y \in \mathbb{R}^n \tag{A} \\
\end{gather*}
>
> \begin{gather*}
(\nabla f(x) - ... |
$x,y \in \mathbb{N}$
$\exists x \exists y (x + y = 0) \lor (x * y = 0)$
In this propositional function, is it true that the $x$ and $y$ in $(x * y = 0)$ are free variables?
If so, am I allowed to assign a random value to the $x$ and $y$ so that I can get the truth value of the propositional function? |
How to determine the truth value of a propositional function with a free variable? |
In a bialgebra $(H,m,u,\Delta,\epsilon)$, subspace of primitive elements are $P(H)=\{x\in H:\Delta (x)=x\otimes 1+1\otimes x\}$. I know that if $x,y$ are primitive, then $[x,y]=xy-yx$ is also primitive, so $P(H)$ is a lie subalgebra under the bracket. I also know if $x_1,\cdots, x_n$ are primitive, then $$\Delta(x_1\cd... |
Just after Theorem 10.13 in the book Multiplicative number theory I: Classical theory by Hugh Montgomery, Robert C. Vaughan the following two statements are assumed to be without proofs. Perhaps they are simple to prove but I have no idea of the proofs:
Since $N(T) ≪ T \log T$ then $\sum_{\rho} |\rho|^{-A} < \infty... |
How to show that $\sum_{\rho} 1/|\rho| = \infty$? |
This answer is partial.
The brief search showed several related papers and even questions at Mathematics StackExchange.
I list them below as the references. Unfortunately, now I have no time to study them. But I shall try to briefly survey some of them.
It seems that all of these references concern natural solut... |
Just after Theorem 10.13 in the book Multiplicative number theory I: Classical theory by Hugh Montgomery, Robert C. Vaughan the following two statements are assumed to be without proofs. Perhaps they are simple to prove but I have no idea of the proofs:
Since $N(T) ≪ T \log T$ then $\sum_{\rho} |\rho|^{-A} < \infty... |
How to determine the truth value of a propositional function with free variables? |
Just after Theorem 10.13 in the book Multiplicative number theory I: Classical theory by Hugh Montgomery, Robert C. Vaughan the following two statements are assumed to be without proofs. Perhaps they are simple to prove but I have no idea of the proofs:
Since $N(T) ≪ T \log T$ then $\sum_{\rho} |\rho|^{-A} < \infty... |
$x,y \in \mathbb{N}$
$\exists x \exists y (x + y = 0) \lor (x * y = 0)$
In this propositional function, is it true that the $x$ and $y$ in $(x * y = 0)$ are free variables?
If so, am I allowed to assign a random value to the $x$ and $y$ so that I can get the truth value of the propositional function?
edit: ... |
I have a question about this problem posted [here](https://math.stackexchange.com/questions/3828930/example-of-schauder-basis-to-ell2). Essentially they are trying to prove that the sequence $\{x_n\}_{k = 1}^\infty = \{\alpha e_n + \beta e_{n+1}\}_{n=1}^\infty$ (here $\{e_n\}_{n = 1}^\infty$ is the canonical orthonorma... |
I want to solve the following optimization problem:
Let $S \in \mathbb{R}^{p \times p}$ be a symmetric matrix, and fix $n \in \mathbb{R}$ such that $n$ is significantly less than $p$ and the rank of $S$ is at most $n$. Now, I want to find a positive-definite real matrix $R \in \mathbb{R}^{p \times p}$ that minimizes... |
The traditional, most common proof of $_nP_n = n!$ is by the multiplication rule:
> There are $n$ choices for the first position, $n-1$ for the second, and so on until there is only one choice for the last position. By the multiplication rule, there are $n(n-1)\dots(2)(1)=n!$ permutations.
Some sources label this... |
Why is the traditional proof of the formula for permutations by the multiplication rule not formal? |
How to solve $\int_0 ^1 \left( \frac{x^2}{1+x^2} \right)\frac{1-x\tan(x)+ \tan(x)-x}{1 -x\tan(x)-\tan(x)-x}dx$? |
*"The" transfer principle or "a" transfer principle?*
****
In general, a transfer principle has two components: the type of "structure-transformation" whose input and output we want to transfer between, *and* the logic in question which determines the "transferrable" properties. The classical "reals-to-hyperreals... |
Consider a probability space $(\Omega,\mathcal{F},P).$ Let $f \in C^{\infty}_{c}(\mathbb{R}^d,\mathbb{R}),p \geq 2.$ $(X_r^{y})_{(r,y) \in \mathbb{R}_+ \times \mathbb{R}^d}$ is a stochastic process such that for every $(\omega,r) \in \Omega \times \mathbb{R}_+,y \to X_r^y(\omega)$ is continuous and for every $y \in \ma... |
This answer is partial.
The brief search showed several related papers and even questions at Mathematics StackExchange.
I list them below as the references. Unfortunately, now I have no time to study them. But I shall try to briefly survey some of them.
It seems that all of these references concern natural solut... |
I have been trying and struggling to come up with a generalized recurrence relation for the number of $L$-bit strings with no runs of $w$ consecutive zeros. I have a relation that gives the right answers for $L = 3$ but it gives the wrong answers for $L = 4$.
I'll use 3-bit strings for a concise example. The 3-bit s... |
This answer is partial.
The brief search showed several related papers and even questions at Mathematics StackExchange.
I list them below as the references. Unfortunately, now I have no time to study them. But I shall try to briefly survey some of them.
It seems that all of these references concern natural solut... |
**Is this construction of an example of embedding correct?**
We want to construct an example for the embedding $S^1\times S^1\rightarrow S^3$.
We define $f:S^1\times S^1\rightarrow S^3$ by
$f(\theta, \phi)=(\frac{1}{\sqrt{2}}cos\theta,\frac{1}{\sqrt{2}}sin\theta, \frac{1}{\sqrt{2}}cos\phi, \frac{1}{\sqrt{2}}s... |
Is this Construction of an Example of Embedding Correct? |
* Solution is apparently 6.
Current attempt, although, I am not sure whether this the right approach:
Let $f(x):=x$. We require $f(63)=63x\equiv 0 \mod 147$; that is, $63x=147k\iff 7\mid x$. So we have 21 distinct homomorphisms.
If we consider a few, say,
$f_7(a)=7a\in \mathbb Z_{147}\implies |img(f_7)|=\l... |
Consider the language F = {$a^i$$b^j$$c^k$ | i, j, k ≥ 0 and if i = 1 then j = k}.
1. Show that F is not regular.
2. Show that F acts like a regular
language in the pumping lemma. In other words, give a pumping length
p and demonstrate that F satisfies the three conditions of the
pumping lemma for... |
Does pumping lemma ever fail? |
* Solution is apparently 6.
Current attempt, although, I am not sure whether this the right approach:
Let $f(x):=x$. We require $f(63)=63x\equiv 0 \mod 147$; that is, $63x=147k\iff 7\mid x$. So we have 21 distinct homomorphisms.
If we consider a few, say,
$f_7(a)=7a\in \mathbb Z_{147}\implies |img(f_7)|=\l... |
I am stuck with this question. Suppose a stock price follow Brownian motion starting at price $p$. Denote by $\tau_r$ the passage time reaching level $r$ with $r<p$. What would be the distribution of the stock at time $t$ given that $\tau_r \geq t$ (given that the stock's price never went below price $r$)?
|
Conditional distribution of Brownian motion given the passage time? |
Consider the language F = {$a^i$$b^j$$c^k$ | i, j, k ≥ 0 and if i = 1 then j = k}.
1. Show that F is not regular.
2. Show that F acts like a regular
language in the pumping lemma. In other words, give a pumping length
p and demonstrate that F satisfies the three conditions of the
pumping lemma for... |
Consider the language F = {$a^i$$b^j$$c^k$ | i, j, k ≥ 0 and if i = 1 then j = k}.
1. Show that F is not regular.
2. Show that F acts like a regular
language in the pumping lemma. In other words, give a pumping length
p and demonstrate that F satisfies the three conditions of the
pumping lemma for... |
The textbook says that given $n$ atomic wff, there are $2^{2^n}$ truth functions. Is it correct to say that since the set of all the atomic wff is denumerable, there are $2^{2^{\aleph_0}}$ truth functions (that is, truth functions are uncountable)? |
How many truth functions are there? |
Consider the language F = {$a^i$$b^j$$c^k$ | i, j, k ≥ 0 and if i = 1 then j = k}.
1. Show that F is not regular.
2. Show that F acts like a regular
language in the pumping lemma. In other words, give a pumping length
p and demonstrate that F satisfies the three conditions of the
pumping lemma for... |
[Kinematic Equations][1]
[1]: https://i.stack.imgur.com/66Wp6.png
So for the fourth equation, final velocity squared is found without t, time.
My question is it possible for this equation to be derived or explained visually/graphically?
For the first three equations it is easy to come up with a visual ... |
Is there a way to represent the 4th kinematic equation of velocity squared visually? |
I came across linear programming while trying to find a solution to a problem. I didn't use LP before so pardon if it is naive question. I hope this forum can help.
There are n tasks (x1, x2 .., xn) with current assigned values (r1, r2 .. rn) and new target values (t1, t2 ... tn). Each of the tasks has certain know... |
Kinematic Equations:
\begin{align}
v &= v_0+at \\
\Delta x &= \left( \frac {v+v_0} 2 \right) t \\
\Delta x &= v_0t + \frac12 at^2 \\
v^2 &= v_0^2 + 2a\Delta x
\end{align}
So for the fourth equation, final velocity squared is found without $t$, time.
My question: is it possible for this equation to be der... |
**Definition** A real number is *irrational* if it is not rational.
According to this definition, does "$\sqrt{2}$ is irrational" mean $\lnot$($\sqrt{2}$ is rational)?
This seems right to me, but an example that is why I am not confident is that $\lnot$($i$ is rational) is true, but "$i$ is irrational" is false. ... |
$\sqrt{2}$ is irrational =$\lnot (\sqrt{2}$ is rational)? |
Let $n$ be a large positive integer and let $X$ be a random element of hypercube $\{0,1\}^n$ such that $\mathbb P(|X|= \ell) \propto \ell^{-\beta}$ for all $\ell \in \{0,1,\ldots,n\}$. Here $\beta \ge 0$ is a constant. Fix $x \in \{0,1\}^n$, $r \in (0,1)$, and let $B_n(x;rn)$ be the Hamming ball of radius $rn$ around ... |
Let $a_{i}$ be a set of positive real numbers such that $\sum_{i=1}^n a_{i}^3=3$ and $\sum_{i=1}^n a_{i}^5=5$.
Prove that $\sum_{i=1}^n a_{i} > \frac{3}{2}$.
My attempt: Using Titu's lemma,
$$\frac{a_{1}^6}{a_{1}} + \frac{a_{2}^6}{a_{2}} + \cdots \ge \frac{(a_{1}^3 + a_{2}^3+\cdots)^2}{a_{1} + a_{2}+\cdots} \, ,$$... |
$\sqrt{2}$ is irrational $\equiv \lnot (\sqrt{2}$ is rational)? |
I am reading through ["State Estimation for Robotics" by Timothy Barfoot][1] and I came across a line that I don't understand in pg 242, equation (7.136)
Suppose we have the following definitions:
$$
C = (C_2C_1^T)^{\alpha}C_1, \alpha \in [0,1]
$$
$$
C , C_1, C_2 \in SO(3),
$$
That is, we have a rotation ma... |
Can anyone explain a compact set in a general topological space in an intuitive way? |
lim x->0 (1+x)e^(-(1/|x| + 1/x))
obviously lhopital is inapplicable here, i guess it can be done by saying that e^-infinity is almost zero so the limit is zero but whats the formal way of doing this? |
Suppose that $(A, ∥ · ∥_A)$ and $(B, ∥ · ∥_B)$ are two Banach algebras such that $B$
is an ideal in $A$ and $∥·∥_A ≤ ∥·∥_B$. We know that each Banach $A$-bimodule $X$
is also a Banach $B$-bimodule. But the converse is not true in general!
Do you have an example of a $B$-bimodule $X$ that is not an $A$-bimodule, wh... |
This answer is partial.
The brief search showed several related papers, which I listed below as the references. Unfortunately, now I have no time to study them all, but I shall try to briefly survey some of them.
It seems that all of these references concern natural solutions of the equation.
For each natural ... |
I was trying to prove $L^\infty$ is complete with $(X,\mathcal{A},\mu)$ given and $\mu$ is $\sigma$-finite.
Outline of my attempted proof:
(i) Assume Cauchy in norm - let $\{f_n\}$ be a Cauchy sequence which converges in $L^\infty$ norm. That is, let $\epsilon>0$, and assume $$\left\Vert f_n-f_m \right\Vert_\infty<... |
$\lim\limits_{x\to 0} (1+x)e^{-\left(\frac{1}{|x|} + \frac{1}{x}\right)}$
obviously L'hospital is inapplicable here, I guess it can be done by saying that $e^{-\infty}$ is almost zero so the limit is zero but what's the formal way of doing this? |
$\lim\limits_{x\to 0} (1+x)e^{-\left(\frac{1}{|x|} + \frac{1}{x}\right)}$
obviously L'hopital is inapplicable here, I guess it can be done by saying that $e^{-\infty}$ is almost zero so the limit is zero but what's the formal way of doing this? |
In general, if $R$ is a unital ring (commutative for simplicity), $I$ is an ideal of $R$ and $M$ is an $R$-module, how do we know if $M$ is also an $R/I$-module? Assuming it is, choose any $r\in R$ and any $m\in M$. We want $(r+I)m = rm$, from which we may deduce that $Im = 0$ and hence $IM = 0$. In other words, the co... |
Let $x_1=2$ and define $$ x_{n+1}:=\frac{1}{2}\left(x_n+\frac{2}{x_n}\right).$$Show that $x_n^2\geq 2$.\
This was my working \
Suppose inductively that $x_n^2\geq 2$ for some $n\in \mathbb N$. Then
$$x_{n+1}^2=\left(\frac{x_n+\frac{2}{x_n}}{2}\right)^2\underbrace{\geq}_{\text{AM-GM}} x_n\left(\frac{2}{x_n}\right)=2.... |
Abbott: Which type of proof is this? |
I came across linear programming while trying to find a solution to a problem. I didn't use LP before so pardon if it is naive question. I hope this forum can help.
There are n tasks (x1, x2 .., xn) with current assigned values (r1, r2 .. rn) and new target values (t1, t2 ... tn). Each of the tasks has certain know... |
Let $ X $ be a locally compact Hausdorff space, and $ C_c(X) $ be the space of all complex-valued continuous functions with compact support on $ X $.
As far as I know, there is two formulations of Riesz–Markov–Kakutani representation theorem.
**Formulation 1 (e.g., D. L. Cohn, *Measure Theory*, 2nd edition, Theor... |
I came across linear programming while trying to find a solution to a problem. I didn't use LP before so pardon if it is naive question. I hope this forum can help.
There are n tasks (x1, x2 .., xn) with current assigned values (r1, r2 .. rn) and new target values (t1, t2 ... tn). Each of the tasks has certain know... |
Being rigorous, your proof is not by induction. When we are doing proofs by induction, we need to do "three steps". In order:
1. Prove the base, i.e., prove that the assertion is true for $n=0$ (or n=1 in some cases)
2. Suppose that the assertion is true for $n$.
3. Using the fact that is true for $n$, prove ... |
Let $(\mathbb{R}, \mathcal{F}, \mu)$ be a measure space. Let $F_n$ be measurable subsets of $\mathbb{R}$. Show that $\lim_{n\to \infty} I_{F_n}(x)=0$ a.e. for $\in \mathbb{R}$ if and only if $\mu(\lim \sup F_n)=0$.
---
My proof is as follows.
Note that
$$
\{x:\lim I_{F_n}\neq 0\}=\cup_{k\ge 1}\cap_{N\ge 1}\... |
I was trying to prove $L^\infty$ is complete with $(X,\mathcal{A},\mu)$ given and $\mu$ is $\sigma$-finite.
Outline of my attempted proof:
(i) Assume Cauchy in norm - let $\{f_n\}$ be a Cauchy sequence which converges in $L^\infty$ norm. That is, let $\epsilon>0$, and assume $\left\Vert f_n-f_m \right\Vert_\infty... |
I was trying to prove $L^\infty$ is complete with $(X,\mathcal{A},\mu)$ given and $\mu$ is $\sigma$-finite.
Outline of my attempted proof:
(i) Assume Cauchy in norm - let $\{f_n\}$ be a Cauchy sequence which converges in $L^\infty$ norm. That is, let $\epsilon>0$, and assume $\left\Vert f_n-f_m \right\Vert_\infty... |
Sorry for dumb question and possibly duplicate but I can't get my head around order of combining transformation matrices. I've read that transformation matrices should be combined by multiplication and when multiplying transformations are "applied" from right to left.
Let’s say there are two matrices $T$ and $S$ whe... |
# Setup
Assume the following formation
$$
\bar{X} \overset{P}{\to} 0 \text{ as } n\to\infty.
$$
# Lemma
We want to show
$$
\sqrt{n}\cdot\left(\bar{X}\right)^2 \overset{P}{\to} 0.
$$
This is indeterminate form ($\infty \times 0$).
It seems to me that the sample mean would win because it is sensibly multip... |
This answer is partial.
The brief search showed several related papers, which I listed below as the references. Unfortunately, now I have no time to study them all, but I shall try to briefly survey some of them.
It seems that all of these references concern natural solutions of the equation. So let $a(n)$ be
t... |
I was trying to prove $L^\infty$ is complete with $(X,\mathcal{A},\mu)$ given and $\mu$ is $\sigma$-finite.
Outline of my attempted proof:
(i) Assume Cauchy in norm - let $\{f_n\}$ be a Cauchy sequence which converges in $L^\infty$ norm. That is, let $\epsilon>0$, and assume $\left\Vert f_n-f_m \right\Vert_\infty... |
We want to know if a smooth surface under a $C^{\infty}$ diffeomorphism (infinitely differentiable with a infinitely differentiable inverse) map to a smooth surface too?
Further, it would be nice to understand what happens if the diffeomorphism was $C^k$ for some $k\geq 1$.
So far I have looked at these books:
... |
I was trying to prove $L^\infty$ is complete with $(X,\mathcal{A},\mu)$ given and $\mu$ is $\sigma$-finite.
Outline of my attempted proof:
(i) Assume Cauchy in norm - let $\{f_n\}$ be a Cauchy sequence which converges in $L^\infty$ norm. That is, let $\epsilon>0$, and assume $\left\Vert f_n-f_m \right\Vert_\infty... |
Let $n$ be a large positive integer and let $X$ be a random element of hypercube $\{0,1\}^n$ such that $\mathbb P(|X|= \ell) \propto (\ell + 1)^{-\beta}$ for all $\ell \in \{0,1,\ldots,n\}$. Here $\beta \ge 0$ is a constant. Fix $x \in \{0,1\}^n$, $r \in (0,1)$, and let $B_n(x;rn)$ be the Hamming ball of radius $rn$ a... |
There are two issues with your proof. Firstly, apriori to $\lim\sup$ and $\lim\inf$ of a real sequence existing and being equal, the term "$\lim$" does not make sense. So you should say $\{x:\lim\sup I_{F_{n}}\neq 0\}$. Also note that since this is a sequence of $0$s and $1$'s, the limit exists and is $0$ if and only i... |
There are two issues with your proof. Firstly, apriori to $\lim\sup$ and $\lim\inf$ of a real sequence existing and being equal, the term "$\lim$" does not make sense. So you should say $\{x:\lim\sup I_{F_{n}}\neq 0\}$. Also note that since this is a sequence of $0$s and $1$'s, the limit exists and is $0$ if and only i... |
While finding the Domain of this expression
√(2x-8)(x-1)
I got this:
(2X-8)(x-1)≥0
2x-8≥0 v x-1≥0
X≥4 v x≥ 1
So the Domain is:
X€[1; +∞[
But the real domain is
X€]-∞;1] U [4;+∞]
So it means that X≤1
Can someone please explain this? |
I was trying to prove $L^\infty$ is complete with $(X,\mathcal{A},\mu)$ given and $\mu$ is $\sigma$-finite.
Outline of my attempted proof:
(i) Assume Cauchy in norm - let $\{f_n\}$ be a Cauchy sequence which converges in $L^\infty$ norm. That is, let $\epsilon>0$, and assume $\left\Vert f_n-f_m \right\Vert_\infty... |
While finding the Domain of this expression
$$\sqrt{(2x-8)(x-1)}$$
I got this:
$(2X-8)(x-1)≥0$
$2x-8\geq0 \vee x-1\geq0$
$x\geq4 \vee x\geq 1$
So the Domain is:
$x\in [1, +\infty)$
But the real domain is
$x\in (-\infty,1] \cup [4,+\infty)$
So it means that $x\leq1$
Can someone please expl... |
Why $S_{\gamma'(0)}(J(0))=0$ for the geodesic sub-manifold? |
> $(2X-8)(x-1)≥0$
>
> $2x-8\geq0 \vee x-1\geq0$
An error occurs here. In writing $\lor$, you are saying "or," but this is not the case for this. In general, $ab \ge 0$ only if both are positive, both are negative, or both are zero. Hence, you would instead write
$$\Big( 2x - 8 \ge 0 \text{ and } x-1 \ge 0 \Big... |
While finding the Domain of this expression
$$\sqrt{(2x-8)(x-1)}$$
I got this:
$(2x-8)(x-1)≥0$
$2x-8\geq0 \vee x-1\geq0$
$x\geq4 \vee x\geq 1$
So the Domain is:
$x\in [1, +\infty)$
But the real domain is
$x\in (-\infty,1] \cup [4,+\infty)$
So it means that $x\leq1$
Can someone please expl... |
I was trying to prove $L^\infty$ is complete with $(X,\mathcal{A},\mu)$ given and $\mu$ is $\sigma$-finite.
Outline of my attempted proof:
(i) Assume Cauchy in norm - let $\{f_n\}$ be a Cauchy sequence which converges in $L^\infty$ norm. That is, let $\epsilon>0$, and assume $\left\Vert f_n-f_m \right\Vert_\infty... |
Consider $U$ to be a unitary $n \times n$ matrix and let $p \in \Lambda^{*} =\frac{2\pi}{L}(\mathbb{Z}+\frac{1}{2})$ be fixed. Suppose $f_{ij}$ (the subindices $i,j$ don't play any important role here) is a real-valued function which satisfies the following equality:
$$\frac{1}{2}\int_{-L}^{L}dx e^{ipx}f_{ij}(x) = \su... |
I should probably study the classical viewpoint first. I haven't yet and I will eventually but let us stick to $\infty$-operads for this question. I'm following the [lecture notes from Hebestreit-Wagner][1]. Let me recall the terminology first.
One can define an operad $\mathbb{A}\mathrm{ssoc} \to \mathbb{\Gamma}^{\... |
I have a question about the convergence of $(f_n)_{n \in \mathbb{N}}$ with
$$
f_n(x) := \begin{cases} (x-n+1)(n+1-x): n-1 < x < n+1\\ 0: x \leq n-1 \lor x \geq n+1 \end{cases}.
$$
Does this sequence of functions converge and if so what is its limit function? Does it converge point wise or also uniformly?<br>
**Wha... |
What is the limit function of $f_n(x) := \begin{cases} (x-n+1)(n+1-x): n-1 < x < n+1\\ 0: x \leq n-1 \lor x \geq n+1 \end{cases}$? |
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