instruction stringlengths 12 30k |
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"Best" Subordinate norm? |
I have $y = Ax$ where x, y are vectors and $A$ is a matrix.
I want to get the best K such that $||y|| \leq K||x||$. Clearly K is a matrix norm. Especually, K is a subordinate matrix norm. I tried with Induced norm and Frobenius norms and they provide a very loose bound for K. Are there studies im this direction?
... |
I have $y = Ax$ where x, y are vectors and $A$ is a matrix.
I want to get the best $K$ such that $||y|| \leq K||x||$. Ideally, $K$ is a matrix norm. Especually, K is a subordinate matrix norm. I tried with Induced norm and Frobenius norms and they provide a very loose bound for K. Are there studies im this direction?... |
I have $y = Ax$ where x, y are vectors and $A$ is a matrix.
I want to get the best $K$ such that $||y|| \leq K||x||$. Ideally, $K$ is a matrix norm. Especually, $K$ can be a subordinate matrix norm. I tried with Induced norm and Frobenius norms and they provide a very loose bound for K. Are there studies im this dire... |
I have $y = Ax$ where x, y are vectors and $A$ is a matrix.
I want to get the best $K$ such that $||y|| \leq K||x||$. Ideally, $K$ is a matrix norm. Especually, $K$ can be a subordinate matrix norm. I tried with Induced norm and Frobenius norms and they provide a very loose bound for $K$. Are there studies im this di... |
I have $y = Ax$ where x, y are vectors and $A$ is a matrix.
I want to get the best $K$ such that $||y|| \leq K||x||$. Ideally, $K$ is a matrix norm. Especually, $K$ can be a subordinate matrix norm. I tried with Induced norm and Frobenius norms and they provide a very loose bound for $K$. Are there studies in this di... |
"Best" Submultiplicative / Subordinate norm? |
I have $y = Ax$ where x, y are vectors and $A$ is a matrix.
I want to get the best $K$ such that $||y|| \leq K||x||$. Ideally, $K$ is a matrix norm. Especially, $K$ can be a subordinate matrix norm. I tried with Induced norm and Frobenius norms and they provide a very loose bound for $K$. Are there studies in this di... |
What is the actual further meaning/application of studying about unitary Cayley graphs? I hope it is not only to connect rings (or algebra in general) with graph theory. Are there any published documents about this aspect? |
I have $y = Ax$ where x, y are vectors and $A$ is a matrix.
I want to get the best $K$ such that $||y|| \leq K||x||$. Ideally, $K$ is a matrix norm. Especially, $K$ can be a subordinate matrix norm. I tried with Spectral norm and Frobenius norms and they provide a very loose bound for $K$. Are there studies in this d... |
I am trying to understand parts of the authors solution given to the following question:
>The generators of $\mathrm{SO}(3)$ can be chosen as
>$$t^1=\begin{pmatrix}0 & 0 & 0\\ 0 & 0 & -i \\\ 0 & i & 0 \\ \end{pmatrix},\ t^2=\begin{pmatrix}0 & 0 & i\\ 0 & 0 & 0 \\ -i & 0 & 0 \\ \end{pmatrix},\ t^3=\begin{pmatrix}0 & ... |
When I first learnt calculus I was so surprised that there is a meaningful mathematical operator $D$ that $D(x^n)= n x^{n-1}$ this seem to be a very random thing to multiply the exponent with the function and take one from the exponent this was very strange for me when I first learnt about derivatives I mean if I was ... |
I am trying to understand parts of the authors solution given to the following question:
>The generators of $\mathrm{SO}(3)$ can be chosen as
>$$t^1=\begin{pmatrix}0 & 0 & 0\\ 0 & 0 & -i \\\ 0 & i & 0 \\ \end{pmatrix},\ t^2=\begin{pmatrix}0 & 0 & i\\ 0 & 0 & 0 \\ -i & 0 & 0 \\ \end{pmatrix},\ t^3=\begin{pmatrix}0 & ... |
> Let $S:\mathbb{N}\to\mathbb{Z}^+$ satisfy
$$
\left\{
\begin{aligned}
S(n) &= S(A^n \bmod n) + S(B^n \bmod n) \\
S(0) &= 1
\end{aligned}
\right.
$$
My question is, is this function always surjective for all values of $A, B \geq 2$ with $A \neq B$?
(I realized odd numbers are unobtainable whenever $A$ is... |
I am trying to understand parts of the authors solution given to the following question:
>The generators of $\mathrm{SO}(3)$ can be chosen as
>$$t^1=\begin{pmatrix}0 & 0 & 0\\ 0 & 0 & -i \\\ 0 & i & 0 \\ \end{pmatrix},\ t^2=\begin{pmatrix}0 & 0 & i\\ 0 & 0 & 0 \\ -i & 0 & 0 \\ \end{pmatrix},\ t^3=\begin{pmatrix}0 & ... |
I have $y = Ax$ where x, y are vectors and $A$ is a matrix.
I want to get the best $K$ such that $||y|| \leq K||x||$. Ideally, $K$ is a matrix norm. Especially, $K$ can be a subordinate matrix norm. I tried with Spectral norm and Frobenius norms and they provide a very loose bound for $K$. Are there studies in this d... |
**Question:**
There is a game that involves $n$ ordered boxes each with a hidden value associated with it. The value is sampled from a probability distribution density function $P(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2}x^2}$. You observe each box’s value in order from box 1 to $n$. After observing any given box’... |
I give a proof over open upward parabola, which can be generalised to open downward parabola too (treat it as exercise!)
**Lemma.** Let $f(x)=(x+h)^2+k$ be an open upward parabola such that $f(x)=0$ has roots $a\pm bi$, where $a,b\in\mathbb{R}$. Define $g(x):=k-(x+h)^2$, then $g(x)=0$ has roots $a\pm b$.
*Proof.*... |
This may be simple, but I want to know if my reasoning is ok. I came across a problem whose essential set up is:
let $f_k$ be a sequence of functions in $L^1(\mathbb{R})$ (Lebesgue integrable functions on $\mathbb{R}$). Suppose that
$$\displaystyle \lim_{t \to \infty} f_k(t) = 0 \text{ for all } k \in \mathbb{N}... |
I am not sure if the complete answer is known, but the interior and the exterior of the image $E$ of the exponential map are known:
1. The interior of $E$ consists of all matrices in $SL(n,\mathbb R)$ which have no negative eigenvalues.
2. The exterior of $E$ consists of all matrices in $SL(n,\mathbb R)$ which h... |
Let $T$ be the one-sided shift on sequences with $k$ symbols. What are the sufficient conditions for $\mu_n \to \mu$ where $\mu_n$ are all invariant measures?
Is it sufficient to show that for each cylinder $A$, $\mu_n(A)\to\mu(A)$? I think yes, since I think characteristic functions of cylinders $\chi_A$ are dense ... |
Show that x = sec θ + tan θ and y = 2 sec θ satisfy $x^{2}-xy+1=0$
I would sub but are there alternative methods to consider? |
I apologize in advance if my title is not super descriptive, I'm not exactly sure if "algebraic" is the correct term.
Let's say I have two sequences, $A$ and $B$, where $A_i$ is the $i$th element, 1 indexed. $n$ is the size of $A$, and $m$ is the size of $B$
If I have the following statement about these sequences... |
Possibly a more detailed approach(?) that doesn't involve long-page calculation:
Firstly, observe that the coefficients of the terms $x ^ 4, x ^ 3, x$ and the constant term 1 seem to have some secrets hidden about them. So we'd like to separate them out, and it turns out this is indeed correct.
Secondly, notice t... |
Given potential range of velocity for matter, what is the minimum speed of computational speed in order for a "loop" to exist? |
> Show that $x = \sec\theta + \tan\theta$ and $y = 2\sec\theta$ satisfy $x^{2}-xy+1=0$
I would sub, but are there alternative methods to consider? |
Showing that $x = \sec\theta + \tan\theta$ and $y = 2\sec\theta$ satisfy $x^{2}-xy+1=0$ without substitution? |
Let $f:A\to \Bbb R$ where $A\subseteq \Bbb R$. We say that, $f$ is uniformly continuous on $A$ if for any $\epsilon\gt 0$ there exists $\delta(\epsilon)=\delta\gt 0$ such that for any $x_1,x_2\in A$ and satisfying $|x_2-x_1|\lt \delta$ we have, $|f(x_2)-f(x_1)|\lt\epsilon.$
Now, my question is: Say, for a particular... |
This is a question on one part from Tao's note: https://terrytao.wordpress.com/2015/01/04/254a-supplement-4-probabilistic-models-and-heuristics-for-the-primes-optional/. Refer to Prediction $3$ (Riemann hypothesis).
Construct a random model ${{\mathcal P}'}$ of the primes ${{\mathcal P}}$ to be a random subset of th... |
> Let $S:\mathbb{N}\to\mathbb{Z}^+$ satisfy
$$
\left\{
\begin{aligned}
S(n) &= S(A^n \bmod n) + S(B^n \bmod n) \\
S(0) &= 1
\end{aligned}
\right.
$$
My question is, is this function always surjective for all values of $A, B \geq 2$ with $A \neq B$?
(When $A$'s and $B$'s set of prime factor are completely... |
If we consider the following linear predictor of $Y$ based on $X$:
$$
Y_{\mathbf{b}}=\boldsymbol{\Sigma}_{Y, \mathbf{X}} \boldsymbol{\Sigma}_{\mathbf{X}}^{-1}\left(\mathbf{X}-\boldsymbol{\mu}_{\mathbf{X}}\right)+\mu_Y,
$$
where $\mu_Y=\mathbb{E}[Y]$.
1) It is well known that this is the best linear predictor of ... |
Is the expectation of the error of any projection of $\mathbb{E}[Y\mid X]$ onto subspace zero? |
During my studying linear algebra, I came across the following problem:
> Let $A$ be a square matrix with real entries, $k$ be a positive integer, and assume $\operatorname{rank}(A^k)=\operatorname{rank}(A^{k+1})$.
Then, prove $\operatorname{rank}(A^k)=\operatorname{rank}(A^m)$ for any integer $m \ge k$.
Below i... |
### Progressive Tax Rate Explanation
Progressive taxation works by taxing income within a certain bracket at different rates. For example:
| Bracket # | % tax rate within bracket | Min amount (exclusive) | Max amount (inclusive) |
| --------- | -------------------------------------- | ------------ | ------------ |... |
$F$ is a function from $\mathbb{R}^2$ to $\mathbb{R}^2$. For every line $L$ in $\mathbb{R}^2$, $F(L)$ is a line, too. Also it maps origin to origin. Then, is $F$ a linear transform? |
Suppose we consider a variant of [twenty questions](https://en.wikipedia.org/wiki/Twenty_questions) game, where the answerer chooses two answers. On a query from the questioner, the answerer replies $0, 1$ or $2$, depending on the number of answers that satisfy the question.
The questioner queries the answerer in th... |
If $U \sim U(0, 1)$, does $2U - \left\lfloor 2U\right\rfloor \sim U(0, 1)$?
If $g(x) = 2x - \left\lfloor 2x\right\rfloor$, $\epsilon \in (0, 1)$, then $g(\frac{1}{2} - \epsilon) = 1 - 2\epsilon$. But $g(\frac{1}{2}) = 0$, making $g$ not continuous, so I can't use the result before Example 4 [here](https://www.math.a... |
If U is uniformly distributed on [0, 1], is 2U - floor(2U) also? |
So, the works of [Bobkov et al. (1999)][1], [Otto et Villani (2000)][2], and even more recently [Otto et Reznikoff (2006)][3], contain elements of solution to the questions I asked. I'll provide a quasi self-contained answer, hoping it will be helpful to a general public (like myself) without kungfu-panda-level knowled... |
I've been trying to prove the following: Let $V$ and $W$ be vector spaces over a field $\mathbb{K}$. Let $\{v_{1},\dots, v_{n}\}\subseteq V$ be a linearly independent subset and $n\in\mathbb{N}$. A linear map $\Phi: V\rightarrow W$ is injective iff $\{\Phi(v_{1}),\dots, \Phi(v_{n})\}\subseteq W$ is linearly independent... |
This is from an exercise in Migdal's "Qualitative Methods in Quantum Theory".
For the case for where b>>a, we can arrive at an estimate by re-writing the integral in the following way:
$$
\frac{1}{\sqrt{b}}\int_0^{\infty} e^{-z^2}sin(\frac{a}{b}z^2)dz \approx \frac{a}{b^{3/2}}\int_0^{\infty} e^{-z^2}z^2dz = a\s... |
Let $B \subset \mathbb{R}^n$ be an open ball in $\mathbb{R}^n$. Find a $C^\infty$ function $f : \mathbb{R}^n \rightarrow \mathbb{R}$ such that:
$f^{-1}(0, \infty)$ and $f^{-1}[0, \infty)$ are $B$ and its closure $\overline{B}$ respectively.\
The linear functional $f'(x) : \mathbb{R}^n \rightarrow \mathbb{R}$ is non... |
Is it possible to color the edges of K6 by three colors so that there is no monochromatic cycle? I tried many times but always end up getting monochromatic cycle. |
If there are $n$ objects, there are $\frac {n(n-1)}2$ pairs and you need to find one pair. Assuming you can make maximum use of each question, splitting the remaining possibilities evenly among the three answers, there are $3^{20}$ sets of answers, so we need to solve
$$\frac {n(n-1)}2 =3^{20}$$
which [Alpha][1] sa... |
I´m trying to solve the following problem:
Let $a$ be a scalar. Consider the set $\left\{ v_{1},v_{2},...,v_{n} \right\} \subseteq \mathbb{R}^{n}$ such that the $i$th element of $v_{i}$ is equal to 1 and the rest of the elements are equal to $a$ ($i=1,2,...,n)$. The scalar $a$ needs to meet 2 conditions for the set ... |
Below I am trying to make the asked points clear, computations do not stay in focus. First of all a picture:
[![mse 43881764][1]][1]
The problem starts somehow with the middle, making the least number of words in a random language count for exposing the problem. This is not the way we think, and not the way we st... |
Let $f$ be a holomorphic function of $n$ complex variables in region $\Omega \subset \mathbb{C}^n$, with $\Omega \supset \mathbb{R}^n$.
Let $\lambda$ be the Lebesgue measure on $\mathbb{R}^n$.
Is the following statement true?
For any set $A \subset \mathbb{R}^n$ with $\lambda(A) = 0$,
$$
\lambda\left(
f^{-1}(f(... |
I´m trying to solve the following problem:
Let $a$ be a scalar. Consider the set $\left\{ v_{1},v_{2},...,v_{n} \right\} \subseteq \mathbb{R}^{n}$ such that the $i$th element of $v_{i}$ is equal to 1 and the rest of the elements are equal to $a$ ($i=1,2,...,n)$. The scalar $a$ needs to meet 2 conditions for the set ... |
Can we tell $\int_{0}^{\pi/5} \sin^2 x \ dx < \int_{0}^{\pi/5} \sin^2 5x \ dx $ without integrating both like from plot?
|
I wonder that for
> Given scenario 1: A ≥ B ≈ C
> Given scenario 2: A ≈ B ≥ C
Can we conclude that the relationship between A and C to be A ⪆ C for both of the scenario?
I can recall that I saw a symbol with "≈" on top of ">", does this one the same as ⪆?
|
Math relationship for the real-value A, B, and C? |
**An attempt for approximating the logarithm function $\ln(x)$: Could be extended for big numbers?**
*PS: Thanks everyone for your comments and interesting answers showing how currently the logarithm function is numerically calculated, but so far* **nobody is answering the question I am asking for,** *which is* *rel... |
There probably is no simple expression, but here is a recurrence relation that will let you compute this number using dynamic programming.
Imagine that you are observing the rolls, as they arrive, one at a time, and keeping track of just enough information to determine whether it contains at least $n$ consecutive nu... |
This is perhaps a very basic problem regarding stochastic calculus but I am going off the deep end in trying to do some of my homework problems and I think I don't understand some small basic concept. If I have a weak solution to a stochastic differential equations of the form:
$$dX_t=a(X_t)dt+\sigma(X_t)dB_t$$
Wher... |
I am reading an interesting paper [One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational](https://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=rm&paperid=427&option_lang=eng) by Zudilin. We fix odd numbers $q$ and $r$, $q\geq r+4$ and a tuple $\eta_0,\eta_1,...,\eta_q$ of positive integer parameters satisfying... |
If $V=\{v_{1},v_{2},...,v_{n}\}$ is an orthogonal set, determine if $\sum_{i=1}^{n}\left\| v_{i} \right\|=\left\| \sum_{i=1}^{n}v_{i} \right\|$ is true or not.
I don´t even know where to start. However, before trying to solve this one, I solved the following:
Prove that: $$\sum_{i=1}^{n}\left\| v_{i} \right\|\le... |
According to Aschbacher's theorem, if $H$ is a maximal subgroup of $SL(n,q)$, then either it belongs to one of the classes $C_1 - C_8$, or $H$ is absolutely irreducible and $H/(H\cap Z(SL(n,q)))$ is almost simple.
I know that representatives of the classes $C_1 - C_8$ are described for $SL(n,q)$ in the general case,... |
$$ \sum_{n=0}^\infty\prod_{m=1}^n\frac{x-m+1}{km} = \sum_{n=0}^\infty\prod_{m=1}^n\frac{x+m-1}{(k+1)m} $$
I noticed this identity that relates the falling and rising factorial using a MacLaurin series. How can I derive this identity?
I noticed we can use the product $ \prod_{m=0}^n{x+m} $ to build a triangle simi... |
Here, $D_8$ is the dihedral group of order $8$.
From the First Isomorphism Theorem, I'm trying to find a solution for this by considering the quotient groups of $D_8$. If $\phi$ is a surjective homomorphism from $D_8$ to $H$ then it's image is isomorphic to the quotient group of $D_8$ with the kernel of $\phi$.
... |
$$ \sum_{n=0}^\infty\prod_{m=1}^n\frac{x-m+1}{km} = \sum_{n=0}^\infty\prod_{m=1}^n\frac{x+m-1}{(k+1)m} $$
I noticed this identity that relates the falling and rising factorial using a MacLaurin series. How can I derive this identity?
I don't know how to find the radius of convergence, but for k>=1, both expressio... |
[![enter image description here][1]][1]
I'll explain what I am finding a bit hard using the first two $m$s.
For $m=0$, (1) shows that both the $i_k$ and $j_k$ indices start from $k=1$ and end at $k=m$, but $m$ starts from zero, and, in the second equation, it seems like they are considering the case $i_0$ $j_0$ a... |
This is a linear programming problem I was given in my semester examinations. The question is attached as an image. Given its size , I couldn't type it out.
[LPP problem][1]
[1]: https://i.stack.imgur.com/dxnGf.jpg
So basically we have to come up with all the inequalities for this problem given the conditi... |
$$ \sum_{n=0}^\infty\prod_{m=1}^n\frac{x-m+1}{km} = \sum_{n=0}^\infty\prod_{m=1}^n\frac{x+m-1}{(k+1)m} $$
I noticed this identity that relates the falling and rising factorials using a power series. How can I derive this identity?
I don't know how to find the radius of convergence, but for k>=1, both expressions ... |
$$ \sum_{n=0}^\infty\prod_{m=1}^n\frac{x-m+1}{km} = \sum_{n=0}^\infty\prod_{m=1}^n\frac{x+m-1}{(k+1)m} $$
I noticed this identity that relates the falling and rising factorials using a power series. How can I derive this identity?
For k>=1, both expressions converge to $$ \frac{(k+1)^x}{k^x} $$If k>1, it seems to... |
[![enter image description here][1]][1]
I'll explain what I am finding a bit hard using the first two $m$s.
For $m=0$, (1) shows that both the $i_k$ and $j_k$ indices start from $k=1$ and end at $k=m$, but $m$ starts from zero, and, in the second equation, it seems like they are considering the case $i_0$ $j_0$ a... |
This is a linear programming problem I was given in my semester examinations. The question is attached as an image. Given its size , I couldn't type it out.
[LPP problem][1]
[1]: https://i.stack.imgur.com/dxnGf.jpg
So basically we have to come up with all the inequalities for this problem given the conditi... |
The question arose while trying to derive the moment of inertia of solid vs hollow sphere.
In calculating the surface area of sphere, the integral used is
$\int{2\pi r×Rd\theta}$.
In calculating the volume of sphere the integral is $\int{\pi r^2 dx}$.
After reading from some related [questions: (volume calcu... |
We can write $$A=\begin{pmatrix} a & b & \ldots & b & b \\ -b & a & \ldots & b & b \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ -b & -b & \ldots & a & b \\ -b & -b & \ldots & -b & a \end{pmatrix}=aI_n+bX$$,
where
$$X=\begin{pmatrix} 0 & 1 & \ldots & 1 & 1 \\ -1 & 0 & \ldots & 1 & 1 \\ \ldots & \ldots & \ldots ... |
> . A man and a woman agree to meet at a certain location about 12:30
> P.M. If the man arrives at a time uniformly distributed between 12:15
> and 12:45, and if the woman independently arrives at a time uniformly
> distributed between 12:00 and 1 P.M., find the probability that the
> first to arrive waits no longe... |
I was self studying blow up in algebraic geometry. My question is what is the blow up of compact surfaces? Does there exists any relation of Euler characteristics of base space and the blown up space? Can we get any compact surface by blowing up a suitable sub space of sphere? In particular
Can we obtain torus as a bl... |
> . A man and a woman agree to meet at a certain location about 12:30
> P.M. If the man arrives at a time uniformly distributed between 12:15
> and 12:45, and if the woman independently arrives at a time uniformly
> distributed between 12:00 and 1 P.M., find the probability that the
> first to arrive waits no longe... |
I am trying to figure out if the restriction of a differential form on a manifold $M$ onto an open subspace of $M$ is the same as the original form? <br><br>
Take, for example, the $1$-form $\alpha=-ydx+xdy$ on $S^1$ and the open $U = S^1 \setminus \left\{(0,1)\right\}$ of $S^1$. We know that $\alpha$ is not exact on ... |
Restriction of a differential form along an inclusion (example)? |
I've read that rerolling 1s pushes your average from 3.5 to about ~3.9. Does rerolling 6s along with 1s once renomalize the average?
If you add dice does this change the bell curve?
What does the curve look like if you must continue to reroll 1s and 6s until you have none? |
What are the effect of reroll 1s and 6s once on 1d6 die? |
> A man and a woman agree to meet at a certain location about 12:30
> P.M. If the man arrives at a time uniformly distributed between 12:15
> and 12:45, and if the woman independently arrives at a time uniformly
> distributed between 12:00 and 1 P.M., find the probability that the
> first to arrive waits no longer ... |
I wonder that for
> Given scenario 1: A ≥ B ≈ C
> Given scenario 2: A ≈ B ≥ C
Can we conclude any relationship between A and C for each of the scenario?
Thanks to the fruitful comments, I now add an extra condition that: |B - C| < for some small > 0 in Scenario 1, and |A - B| < for some small > 0 in Sc... |
Math relationship for the real-value A and C given the medium B? |
The answer is no, and we give some counter-examples below: two in $2$ and one in $3$ and more dimensions. References for the two-dimensional examples are [this paper](https://doi.org/10.1088/1367-2630/15/7/073045) and [this paper](https://arxiv.org/abs/2403.19643), and the three-dimensional example also appears in a sl... |
Let $E/K$ be an elliptic curve and $m$ be a positive integer such that char$(K)\nmid m$. I know that the Weil pairing defined on elliptic curves is a bilinear & non-degenerate pairing. However is the pairing perfect? Do we have,
$$E[m]\cong \text{Hom}(E[m],\mu_m)$$
If so, how? |
> A man and a woman agree to meet at a certain location about 12:30
> P.M. If the man arrives at a time uniformly distributed between 12:15
> and 12:45, and if the woman independently arrives at a time uniformly
> distributed between 12:00 and 1 P.M., find the probability that the
> first to arrive waits no longer ... |
In order to calculate the Fourier transform of a formula as follow:
$$g(x, y) = \exp\left(-\frac{(x-x_0)^2 + (y-y_0)^2}{W^2}\right)e^{-il\tan^{-1}(\frac{y}{x})} $$ Where $B$ and $l$ are constants. $$G(f_x, f_y)=\iint g(x, y)\exp\{-2\pi(xf_x +yf_y)\}dxdy$$ Where $F\{\}$ is the Fourier transform.
After transform th... |
We have unlimited balls from 8 colors. Balls from the same color are identical. How many options are there to arrange them in a line of 10, so that line will include exactly 4 different color? |
Is there any work linking information channel theory to statistical inference? |
**Question:**
There is a game that involves $n$ ordered boxes each with a hidden value associated with it. The value is sampled from a probability distribution density function $P(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2}x^2}$. You observe each box’s value in order from box 1 to $n$. After observing any given box’... |
(1) Even the maximum of each term of the form $\prod_{j=1}^k X_{n_j}$ is $O(x^{o(1)})$, so the expectation certainly is as well.
(2) Are you able to bound the variance of $\sum_{2<n\le x} X_n$ with the given bound? What happens when you input that bound into Chebyshev's inequality? |
By equating the sine-cosine and exponential Fourier series for the function $f$:
$$f(x) = \sum_{n = 0}^\infty f_n \cdot e^{inx} = \frac{a_0} {2} + \sum_{n=1}^{\infty} (a_n\cos(nx) + b_n\sin(nx))$$
and using Euler's formula, how can we find the explicit relation between the coefficients $f_n$ and the $a_n$, $b_n$? |
I was just messing with numbers when I created this formula.
But to understand the formula, you have to understand this: Multiplication is repeated addition, as $8*6$ is more or less $8+8+8+8+8+8$, or vice-versa. In the same way, exponents are repeated multiplication, as $5^3$, can be written as $5*5*5$, which can ... |
Does my formula work? |
Consider the second-order nonlinear dynamical system
\begin{align*}
\dot{x}&=x^3-y\\
\dot{y}&=x-x^2y
\end{align*}
The (0,0) equilibrium is obviously a center but I cannot find a way to prove this.
[![Streamplot][1]][1]
I tried using some reversibility argument but this does not seem to work. Certainly the inv... |
I've got this theorem right here. It involves two arbitrary lines $AB$ and $CD$ and two transversals $AC$ and $BD$.
[![The drawing of the theorem / lemma][1]][1]
The simplest proof of which I know is based on the [Sum of Angles of Triangle equals Two Right Angles][2]: two transversals intersecting in between two ... |
Does this theorem about lines and two transversals have a name? |
**Question:**
There is a game that involves $n$ ordered boxes each with a hidden value associated with it. The value is sampled from a probability distribution density function $P(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2}x^2}$. You observe each box’s value in order from box 1 to $n$. After observing any given box’... |
I learnt from the Nash Embedding theorem that every Riemannian manifold can be isometrically embedded into some Euclidean space. But is there some pratical way to construct the embedding?
A toy example, suppose the metric of the manifold is g=m*diag(3), where m varies over the manifold. How to embed it into Euclidean ... |
>The diagram below shows nine circles each tangent to all its neighbours. One of
the circles is labelled 1. The remaining circles are to be labelled with 1, 2, 3, 3, 3, 4,
4 and 4, such that no two tangent circles have the same label. In how many
different ways can this be done?
> <img src="https://i.stack.imgur.... |
I have the following problem:
Consider complex Euclidean spaces H and H′
such that $dim(H) ⩽ dim(H′)$.
Let ${|v_i⟩}^{k}_{i=1} ⊂ H$ and ${|w_i⟩}^{k}_{i=1} ⊂ H′$ denote orthonormal sets of vectors.
Show that there exists an isometry $V : H → H′$such that $V |v_i⟩ = |w_i⟩$ for all i ∈ {1,. . . , k}.
I know that... |
I've got this theorem right here. It involves two arbitrary lines $AB$ and $CD$ and two transversals $AC$ and $BD$.
The theorem (or lemma, I don't know) in question states that $\alpha+\beta = \varphi + \theta$.
[![The drawing of the theorem / lemma][1]][1]
The simplest proof of which I know is based on the [S... |
Does this theorem about two lines and two transversals have a name? |
**Question:**
There is a game that involves $n$ ordered boxes each with a hidden value associated with it. The value is sampled from a probability distribution density function $P(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2}x^2}$. You observe each box’s value in order from box 1 to $n$. After observing any given box’... |
Comment: See my comment on this question:' Given $a$ in $\gcd(a,b)$ with $a > b > 0$, how can I find $b$ which give the maximum number of steps($l$) for the Euclidean algorithm?'
I found the size of $a^2+b^2$ depends on the number of steps for the Euclidian algorithm. Look at this example from my previous comment:
... |
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