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How does this integral based on the Φ function equal x? |
How does this integral based on the $\Phi$ function equal $x$? |
The complex K-theory spectrum $KU$ is equipped with a $C_2$-action coming from conjugation of vector bundles. On homotopy, we have that $\pi_*KU = \mathbb{Z}[u^{\pm 1}]$, and the action is recorded by $u \mapsto -u$.
The smash product $KU \wedge KU$ also has a $C_2$-action: just take the previous action on one of t... |
Consider I have the diffEq `dy/dx + 5y = 20; y(0) = 2`
I solved it like this:
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/nI0v4.png
Here, the value of C changes on different steps.
You could look at the first line and say C is `ln 10`, you could look at the last line and say ... |
For Markov Chains, there are well known formulae which show us how to calculate the Expected Time to Absorption (e.g https://en.wikipedia.org/wiki/Absorbing_Markov_chain). That is, given a transition matrix and the current state - how many iterations (i.e. steps) are required (on average) for the chain to reach a cert... |
How long does it take a Markov Chain to reach its Limiting Distribution? |
Consider I have the diffEq `dy/dx + 5y = 20; y(0) = 2`
I solved it like this:
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/nI0v4.png
Here, the value of C changes on different steps.
You could look at the first line and say C is `ln 10`, you could look at the last line and say ... |
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 tried to find the highest power of \(42\) that divides \(2023!\) Since \( 42 = 2 \times 3 \times 7 \), we need to count the occurrences of \( 2 \), \( 3 \), an... |
For Markov Chains, there are well known formulae which show us how to calculate the Expected Time to Absorption (e.g https://en.wikipedia.org/wiki/Absorbing_Markov_chain). That is, given a transition matrix and the current state - how many iterations (i.e. steps) are required (on average) for the chain to reach a cert... |
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 tried to find the highest power of $42$ that divides $2023!$ Since $ 42 = 2 \times 3 \times 7 $, we need to count the occurrences of $ 2 $, $ 3 $, and $ 7 $ in... |
Let $K_1, K_2$ be two convex closed cones in $\mathbb{R}^n$. If $(-K_1)\cap K_2=${$0$}, show that $K_1+K_2=${$x+y|x\in K_1, y\in K_2$} is closed.
I wanted to say, let $(z_n)^{\infty}_1$ be a series in $K_1+K_2$, then there must exists a series $(x_n)^{\infty}_1$ in $K_1$ and one $(y_n)^{\infty}_1$ in $K_2$, such th... |
How to prove that the sum of two convex closed cones is closed? |
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 = \{... |
This problem is in my book:
>Show that as $z$ traverses a small circle in the complex plane in the
positive (counterclockwise) direction, the corresponding point $P$ on
the sphere traverses a small circle in the negative (clockwise) direction with respect to someone standing at the center of the circle and with... |
For Markov Chains, there are well known formulae which show us how to calculate the Expected Time to Absorption (e.g https://en.wikipedia.org/wiki/Absorbing_Markov_chain). That is, given a transition matrix and the current state - how many iterations (i.e. steps) are required (on average) for the chain to reach a cert... |
Given a real number $x>1$ and a natural number $n$, what can we say about the integer part of $x^n$ in terms of $x$ and $n$?
For simplicity, let us assume $x<2$. For the first few values of $n$, $\left\lfloor x^n\right\rfloor =1$. But, once it crosses $1$, it starts growing quite rapidly. Is it possible to express $... |
How Long does it take to Reach a Limiting Distribution? |
I have the differential equation $\frac{dy}{dx} + 5y = 20$ with $y(0)=2$.
This is my attempt:
$$\ln|20-5y|=-5x+C$$
$$20-5y=Ce^{-5x}$$
$$y=\frac{Ce^{-5x}}{-5}+4$$
Then plugging the initial condition gives $C=10$.
Here, the value of $C$ changes on different steps.
You could look at the first line and say... |
I have a continuous and nondecreasing function $F:\mathbb{R}\to\mathbb{R}$ satisfying following conditions:
1. $F(x) = 0$ for any $x\in (-\infty, 1]$,
2. $\lim\limits_{x\to +\infty} F(x) = 1$,
2. for any $x\in \mathbb{R}$ we have
$$F(x) \;=\; 0.8 F(2x - 1) + 0.2 F(x/2 - 1).$$
Is there any solution to this "fun... |
$\mathbf{Question :}$
Let $n$ be a positive integer greater than $1$. Let $\mathcal{D_n}$ be the set of all proper divisors of $n$.
Let $\mathcal{M}$ = $\{M_1, M_2, \ldots, M_n\}$ be an arbitrary set of real square matrices of order n such that it satisfies $A_iA_j = A_jA_i$ and $A_i \ne A_j$, $\forall$ $i \ne j$... |
I recently found this equivalence between the two propositions in a book
1) $( E )$ is an infinite set
2) $\forall f \in E^E. \exists A \in \mathcal{P}(E) \setminus \{\emptyset, E\} \, f(A) \subset A$
but I don't see a logical connection between the two propositions. I tried to find a way to relate them, and the... |
In [This][1 ] question I tried to solve this problem
>Show that as $z$ traverses a small circle in the complex plane in the
positive (counterclockwise) direction, the corresponding point $P$ on
the sphere traverses a small circle in the negative (clockwise) direction with respect to someone standing at the cent... |
How to express rotation mathematically? |
I recently found this equivalence between the two propositions in a book
1) $( E )$ is an infinite set
2) $\forall f \in E^E. \exists A \in \mathcal{P}(E) \setminus \{\emptyset, E\} \, :f(A) \subset A$
but I don't see a logical connection between the two propositions. I tried to find a way to relate them, and th... |
Suppose that you initially have a container which has the shape of a paraboloid with equation
$ z = a x^2 + b y^2 $ where $ 0 \le z \le h $
Now you tilt this paraboloid by rotating it about any point (let's say the origin) about a horizontal axis whose unit vector is $v = (\cos \varphi, \sin \varphi, 0)$. Let t... |
Why this matrix is PSD? |
Let $K_1, K_2$ be two convex closed cones in $\mathbb{R}^n$. If $(-K_1)\cap K_2=\{0\}$, show that $K_1+K_2=\{x+y \mid x\in K_1, y\in K_2\}$ is closed.
I wanted to say, let $(z_n)^{\infty}_1$ be a series in $K_1+K_2$, then there must exists a series $(x_n)^{\infty}_1$ in $K_1$ and one $(y_n)^{\infty}_1$ in $K_2$, su... |
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 an induced subgraph of $G$. Furthermore, what is the minimum number of vertices that $G$ could have?
The trivial solution is of course a graph that i... |
How to express rotation direction mathematically? |
How to prove: A rotation in complex plane corresponding to opposite rotation in Riemann's sphere (Complex Analysis by Theodore W. Gamelin 1.3.3)? |
I could parametrize the rational $x$ where $\sqrt{1+x^2}$ is rational as:
$y^2-x^2=1,\ y=\frac{a^2+b^2}{a^2-b^2},\ x=\frac{2ab}{a^2-b^2}$.
$\sqrt{9+x^2}$ is similar. However, I cannot find $x$ where "both" are rational. Is there any method except brute force? Thanks.
|
**There are 5 seats vacant in a college, so more candidates are called. Suppose, the next 5 on the waiting list have an equal probability of being a boy or a girl. What is the probability that exactly 3 girls and 2 boys are admitted?**
In this question the answer is given as 5/16
The process=**5C3×1/32**
But I t... |
**There are 5 seats vacant in a college, so more candidates are called. Suppose, the next 5 on the waiting list have an equal probability of being a boy or a girl. What is the probability that exactly 3 girls and 2 boys are admitted?**
In this question the answer is given as 5/16
The process=**5C3×1/32**
But I t... |
When is this indefinite integral $\int\frac{\mathrm dx}{\sqrt{x^n+1}}$ expressible in terms of elliptic integrals? |
$^5C_3$ is not referencing the seats at all. You have a fixed number of seats and an equal number of people waiting.
To have a better understanding, ignore the part about vacant seats completely (since an equal number of candidates are available). Your problem is now equivalent to you having $5$ students and the pro... |
welcome to M.S.E.
This is an amazing problem! You're going right on creating the incircle, tangents, and the altitudes. As a hint, let the four angles at the vertices be equal to $2\alpha,2\beta,2\gamma,2\delta,$ respectively. Let $O$ be the incenter. Then the lines $AO,BO, CO,$ and $DO$ would all bisect the angles ... |
I provided a screenshot to show the problem. Essentially, I don't understand how to get the s_\alpha value from this.
Could someone help me out with a quick explainer? I appreciate you.
[E-frame with s alpha vector relationship][1]
[1]: https://i.stack.imgur.com/APmtm.png |
How would I solve for the s_\alpha value? This is using vector math. |
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 $s\mapsto A_s$ be an increasing right continuous function, $s\geq 0$. Then define $A_s^{-1}:=\inf\{t\geq 0: A_t>s\}$ with the convention $\inf\{\emptyset\}=\infty$. By definition the map $s\mapsto A_s^{-1}$ in increasing so one can define $A_{s-}^{-1}:=\lim_{h\rightarrow 0} A_{s-h}^{-1}$ the left limit. I want to ... |
How can i show this infimum equality $A_{s-}^{-1}=\inf\{t\geq 0: A_t\geq s\}$? |
*There are 5 seats vacant in a college, so more candidates are called. Suppose, the next 5 on the waiting list have an equal probability of being a boy or a girl. What is the probability that exactly 3 girls and 2 boys are admitted?*
In this question the answer is given as $5/16$.
The process:
$${5 \choose 3} \tim... |
Welcome to M.S.E.
This is an amazing problem! You're going right on creating the incircle, tangents, and the altitudes. As a hint, let the four angles at the vertices be equal to $2\alpha,2\beta,2\gamma,2\delta,$ respectively. Let $O$ be the incenter. Then the lines $AO,BO, CO,$ and $DO$ would all bisect the angles ... |
According to Shapiro and Wilk(1965) in lemma 3,
$W$ has lower bound: $na_1^2/(n-1).$
To find this value, they solve the problem:
$$Max\quad y'y$$
$$ s.t.\quad 1'y=0,\quad and\quad a'y = 1.$$
In the paper, they presented the vertices and find the maximum value.
These vertices are
$\left(\frac{n-1}{n... |
For involutant matricies, lets take
$$[A,BC] = ABC - BCA = 0$$
and
$$[AB,C] = ABC - CBA = 0$$
Which holds iff
$$BCA = CBA = ABC$$
Now if $A,B,C$ are all involutant, this is equivalent to $BC = CB = ABCA$.We also have that $ABCA = BC \iff ABC = BCA$. This can be rewritten as:
$$[A,BC] = 0, [B,C]=0$$... |
This question was asked to me by a student of mine and I am not able to make any progress on this question.
>Let $A=${1,2,3,4,5} , then which of the following is a binary operation?
(a) {((1,1),1),((1,2),1), ((1,3),1),((1,4),1) ,((1,5),1))}.
(b){((2,1),2),((2,2),2), ((2,3),2),((2,4),2) ,((2,5),2))}.
(c){((1... |
Which of the following is a binary operation? |
For involutant matricies in $M_n$ lets take
$$[A,BC] = ABC - BCA = 0$$
and
$$[AB,C] = ABC - CBA = 0$$
Which holds iff
$$BCA = CBA = ABC$$
If $A,B,C$ are all involutant, this is equivalent to $BC = CB = ABCA$.We also have that $ABCA = BC \iff ABC = BCA$. This can be rewritten as:
$$[A,BC] = 0, [B,C]... |
According to Shapiro and Wilk(1965) in lemma 3,
$W$ has lower bound: $na_1^2/(n-1).$
To find this value, they solve the problem:
$$Max\quad y'y$$
$$ s.t.\quad 1'y=0,\quad and\quad a'y = 1.$$
In the paper, they presented the vertices and find the maximum value.
These vertices are
$\left(\frac{n-1}{n... |
For involutant matricies in $M_n$ lets take
$$[A,BC] = ABC - BCA = 0$$
and
$$[AB,C] = ABC - CBA = 0$$
Which holds iff
$$BCA = CBA = ABC$$
If $A,B,C$ are all involutant, this is equivalent to $BC = CB = ABCA$.We also have that $ABCA = BC \iff ABC = BCA$. This can be rewritten as:
$$[A,BC] = 0, [B,C]... |
Is my reasoning on the following exercise correct? I am not sure if I correctly used Student t distribution instead of normal distribution and whether my calculation of std makes sense.
*Exercise*:
"Four samples are taken and averaged every hour from a production line and the hourly measurement of a particular impu... |
Question: how to evaluate this logarithm integral?
$$
I=\int_0^1 \frac{x \log ^2(x) \log (1-x)}{1+x^2} d x
$$
My attempt:
$$
\begin{aligned}
I=&\int_0^1 \frac{x \log ^2(x) \log (1-x)}{1+x^2} d x\\
=&\frac{1}{2} \int_0^1 \log ^2(x) \log (1-x) d\left(\log \left(x^2+1\right)\right) \xrightarrow{I B P} \\
\stack... |
For involutant matricies in $M_n$ lets take
$$[A,BC] = ABC - BCA = 0$$
and
$$[AB,C] = ABC - CBA = 0$$
Which holds iff
$$BCA = CBA = ABC$$
If $A,B,C$ are all involutant, this is equivalent to $BC = CB = ABCA$.We also have that $ABCA = BC \iff ABC = BCA$. This can be rewritten as:
$$[A,BC] = 0, [B,C]... |
In [This][1 ] question I tried to solve this problem
>Show that as $z$ traverses a small circle in the complex plane in the
positive (counterclockwise) direction, the corresponding point $P$ on
the sphere traverses a small circle in the negative (clockwise) direction with respect to someone standing at the cent... |
Let $k$ be a number field, $S$ a set of places of $k$, $k_S$ the maximal extension of $k$ unramified outside $S$, $\mathcal{O}_S$ the subring of $k_S$ with $\nu_{\mathfrak{p}}(\alpha)\geq 0$ for all $\mathfrak{p}\not\in S$, and $G_{S}$ the galois group $\text{Gal}(k_S/k)$. I am considering the following exercise at the... |
Unordered probabilities are often difficult. Ordered probabilities are often easier. In order to reason about the unordered 5 vacant spots and how to fill them, it is in my opinion easier to pretend they are ordered, and argue from there.
Thus we list all the possible ways to get two girls and three boys.
- The ... |
Suppose that $(z_n)_{n\in\mathbb N}$ is a sequence in $K_1+K_2$ (so that there exists for each $n\in\mathbb N$ some $x_n\in K_1$ and $y_n\in K_2$ such that $z_n=x_n+y_n$) with $z\equiv\lim_{n\to\infty}z_n$. The claim is that $z\in K_1+K_2$.
I will show that $(\|x_n\|)_{n\in\mathbb N}$ must remain bounded. If not, on... |
Looking at solving:
$$\int_{0}^{2\pi}\frac{1}{\sqrt{2}-\cos x}\,dx \\$$
According to integral tables (such as [this one](https://www.math.stonybrook.edu/~bishop/classes/math126.F20/CRC_integrals.pdf)) the solution is:
$$\frac{2}{\sqrt{a^2 - b^2}}\arctan\frac{\sqrt{a^2 - b^2}\tan(x/2) }{a+b}$$
$$a=\sqrt{2}... |
According to Shapiro and Wilk(1965) in lemma 3,
$W$ has lower bound: $na_1^2/(n-1).$
To find this value, they solve the problem:
$$Max\quad y'y$$
$$ s.t.\quad 1'y=0,\quad and\quad a'y = 1,\quad and \quad y_1 \leq ,\cdots,\leq y_n. $$
In the paper, they presented the vertices and find the maximum value.
... |
Let $k$ be a number field, $S$ a set of places of $k$, $k_S$ the maximal extension of $k$ unramified outside $S$, $\mathcal{O}_S$ the subring of $k_S$ with $\nu_{\mathfrak{p}}(\alpha)\geq 0$ for all $\mathfrak{p}\not\in S$, and $G_{S}$ the galois group $\text{Gal}(k_S/k)$. I am considering the following exercise at the... |
why are statement 1 and statement 2 analogous for disjoint sets?
Statement 1 (Intersection): if $X \indep Y \mid Z$ and $X \indep Z \mid Y$ then $X \indep (Y,Z)$.
Statement 2: if $A \indep B \mid (C \cup D)$ and $A \indep C \mid (B \cup D)$ then $A \indep (B \cup C) \mid D$
Statement 2 is used as conditi... |
why are statement 1 and statement 2 analogous for disjoint sets?
Statement 1 (Intersection): if $X \perp Y \mid Z$ and $X \perp Z \mid Y$ then $X \perp (Y,Z)$.
Statement 2: if $A \perp B \mid (C \cup D)$ and $A \perp C \mid (B \cup D)$ then $A \perp (B \cup C) \mid D$
Statement 2 is used as condition for... |
Suppose that $(z_n)_{n\in\mathbb N}$ is a sequence in $K_1+K_2$ (so that there exists for each $n\in\mathbb N$ some $x_n\in K_1$ and $y_n\in K_2$ such that $z_n=x_n+y_n$) with $z\equiv\lim_{n\to\infty}z_n$. The claim is that $z\in K_1+K_2$.
I will show that $(\|x_n\|)_{n\in\mathbb N}$ must remain bounded. If not, on... |
Given a Markov chain $X$ (discrete time, countable state space) we know that a state $i$ is recurrent iff
$$\sum_{n\geq1}p_{ii}^{(n)}=\infty$$
where $p_{ii}^{(n)}=P(X_n=i|X_0=i)$. Otherwise, $i$ is said to be transient. I would like to prove this using Borel-Cantelli lemmas. I found [here][1] (lemma 2.1) that Borel-... |
You seem to have made a mistake during simplification and/or completing the square.
Substituting $x = -1$ into the given equation, we get
$$\begin{align*}
-1 - 2b + a + b^2 &= 0 \\[0.3cm]
\Rightarrow \;\;\;\;\;\;a^2 + (b-1)^2 &= 2
\end{align*}$$
Since $a^2 \ge 0$,
$$\begin{align*}
(b-1)^2 &\le 2 \\[0.3... |
Is my reasoning on the following exercise correct? I am not sure if I correctly used Student t distribution instead of normal distribution and whether my calculation of std makes sense.
*Exercise*:
"Four samples are taken and averaged every hour from a production line and the hourly measurement of a particular impu... |
You seem to have made a mistake during simplification and/or completing the square.
Substituting $x = -1$ into the given equation, we get
$$\begin{align*}
-1 - 2b + a^2 + b^2 &= 0 \\[0.3cm]
\Rightarrow \;\;\;\;\;\;a^2 + (b-1)^2 &= 2
\end{align*}$$
Since $a^2 \ge 0$,
$$\begin{align*}
(b-1)^2 &\le 2 \\[0... |
According to this [paper][1], theorem 2.5, we have:
$$H_n \cdot \ln^2(n)-\ln(n) \xrightarrow[n\to +\infty]{\mathcal D} X$$
where $X$ has the density function $f_X(x) = g(-x)$ with $g$ defined by the formula $(12)$.
[1]: https://pdf.sciencedirectassets.com/272528/1-s2.0-S0196885800X0067X/1-s2.0-S019688580300023X/... |
Let $\xi^*,\varsigma,\eta:\mathbb{R}\rightarrow (0,\infty)$. You may assume any degree of smoothness required of these functions.
Assume that $\xi^{*}(t)$ and $\varsigma(t)$ have finite positive limits as $t\rightarrow\infty$ and as $t\rightarrow -\infty$.
Define $\delta:\mathbb{R}\rightarrow\mathbb{R}$ by:
$$\d... |
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 am studying Lemma 11 of the [paper][1]
I am having difficulty understanding the first equality on the last step, in particular:
$$u^TSu=u^T(nI_n-zz^T+\sigma [\text{ddiag}(\Delta zz^T-\Delta)])u$$
where $S=\text{ddiag(Azz^T)-A}$. and $\text{ddiag}$ is an operator that
sets all off-diagonal entries of
a matrix ... |
why this equality involving matrix holds true? |
How to calculate the value of this Riemann integral?
$$
\int_{-7 \pi}^{2 \pi} \frac{1}{2 \sin(x) - \cos(x) + 5} dx
$$
I used the universal trigonometric substitution and found the following antiderivative of the integrand function:
$$
\int\frac{1}{2 \sin(x) - \cos(x) + 5} dx = \frac{\arctan \left( \frac{3 \... |
Let function $u:[0,1]^{N+1} \mapsto \mathbb{R}$ be continuously differentiable.
Let $Y_1,\cdots,Y_{N}$ be random variables with support on $[0,1]$ with continuously differentiable joint PDF $f$.
$$v(x)\equiv Pr(u(x,Y_1,\cdots,Y_{N})<0).$$
There is $y^* \in (0,1)$ such that for $Y_1\gtrless y^*$,
$$
u(0,Y_1,... |
I am struggling to find a quick an intuitive solution to this problem:
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, th... |
Let $\xi^*,\varsigma,\eta:\mathbb{R}\rightarrow (0,\infty)$. You may assume any degree of smoothness required of these functions.
Assume that $\xi^{*}(t)$ and $\varsigma(t)$ have finite positive limits as $t\rightarrow\infty$ and as $t\rightarrow -\infty$.
Define $\delta:\mathbb{R}\rightarrow\mathbb{R}$ by:
$$\d... |
I'm dealing with a few sin waves that represent music. An example is the following
f(x)=sin($\frac{1.47x}{440}$)+sin(x)
How would I represent the time period of this superposed wave? Also if the frequency is possible to find (to my knowledge it isn't), can it be represented with a formula? |
Let $G$ be a **non-discrete** Lie Group acting on a Riemannian manifold $M$ by its isometries. When $G$ is discrete, there are plenty of literature on Fundamendal Domains (FD), but **a quick literature search shows *none* on FD when $G$ is *not* discrete, and is say a Lie Group of manifold dimension at least one**.
... |
I am a bit unsure about my following "proof", in particular the rationality of "taking projection".
**Problem**: $R$ is a ring with identity, *not necessarily commutative*.
If $\epsilon\in R$ satisfies the equation
$1+\epsilon+\cdots+\epsilon^{p-1}=0$,
where $p$ is a prime number, show that for all integer $0<c<... |
I'm taking a dynamics class where we are specifically focusing on the transport theorem. I need to first identify the position vector, and then take 2 derivatives for inertial velocity and inertial acceleration. While I understand how that works, there's one step in the solution to the problem below that I do not curre... |
How would I solve for the s_\alpha or s_r vectors using components from the listed E-Frame? Space Vehicle Dynamics |
**Problem definition**
Consider the following bivariate random variable
\begin{equation*}
\overline{z}\triangleq \frac{1}{4} \sum_{j=1}^4 z_j
\end{equation*}
where $z_1, \dots, z_4$ are uniformly distributed over the edges of a square centered in the origin and with edge of unitary length. More precisely,
\begi... |
Let $G$ be a **non-discrete** Lie Group acting ***properly*** on a Riemannian manifold $M$ by its isometries. When $G$ is discrete, there are plenty of literature on Fundamendal Domains (FD), but **a quick literature search shows *none* on FD when $G$ is *not* discrete, and is say a Lie Group of manifold dimension at l... |
How to deal with the optimization problem with non-convex feasible set? |
I am studying Lemma 11 of the [paper][1]
I am having difficulty understanding on the last step, in particular, I have two questions:
**The first question (1)** On the first equality of the last step on page 15, the authors use
$$u^TSu=u^T(nI_n-zz^T+\sigma [\text{ddiag}(\Delta zz^T-\Delta)])u$$
where $S=\text{ddia... |
Let $G$ be a **non-discrete** Lie Group acting ***properly*** on a Riemannian manifold $M$ by its isometries. When $G$ is discrete, there are plenty of literature on Fundamendal Domains (FD), but **a quick literature search shows *none* on FD when $G$ is *not* discrete, and is say a Lie Group of manifold dimension at l... |
In calculus, we have the partial derivatives for example
\begin{align}
\frac{\partial(x^2-y)}{\partial x}(x,y),\ \ \ \frac{\partial(x+e^y)}{\partial y}(x,y),\ \ \ \frac{\partial(x+y)}{\partial(x+y)}(x,y)\ \ ...
\end{align}
But I wonder if these sort of derivatives
\begin{align}
\frac{\partial(ax+by)}{\partial(cx+... |
How to find Probability in this question? It gives different answer based on context.
**Scenario 1:**
Answer = 1
because we are getting Heads always. So probability of getting any number of Head such as be it 1, 2 , 3 or 4 it will be equal to 1.
**Scenario 2 :**
Answer = 0
because None of the times we... |
If a coin is tossed 4 times and we get Head all the time then what is probability of getting exactly 3 heads? |
I am studying Lemma 11 of the [paper][1]
I am having difficulty understanding on the last step, in particular, I have two questions:
**The first question (1)** On the first equality of the last step on page 15, the authors use
$$u^TSu=u^T(nI_n-zz^T+\sigma [\text{ddiag}(\Delta zz^T-\Delta)])u$$
where $S=\text{ddia... |
It is well-known that taking the double integral of a Bivariate Gaussian Distribution across a rectangle aligned with the axes is very trivial.
However, I am currently considering this double integral on a rotated square. Since now the two variables are related during integration, it seems that computing the outer i... |
Let $G$ be a **non-discrete** Lie Group acting ***properly*** on a Riemannian manifold $M$ by its isometries. When $G$ is discrete, there are plenty of literature on Fundamendal Domains (FD), but **a quick literature search shows *none* on FD when $G$ is *not* discrete, and is say a Lie Group of manifold dimension at l... |
In calculus, we are introduced about the partial derivatives, like for example
\begin{align}
\frac{\partial(x^2-y)}{\partial x}(x,y),\ \ \ \frac{\partial(x+e^y)}{\partial y}(x,y),\ \ \ \frac{\partial(x+y)}{\partial(x+y)}(x,y)\ \ ...
\end{align}
But I wonder if these sort of derivatives
\begin{align}
\frac{\partia... |
Let $G$ be a **non-discrete** Lie Group acting ***properly*** on a Riemannian manifold $M$ by its isometries. When $G$ is discrete, there are plenty of literature on Fundamendal Domains (FD), but **a quick literature search shows *none* on FD when $G$ is *not* discrete, and is say a Lie Group of manifold dimension at l... |
In [this](https://github.com/mitmath/18303/blob/master/supp_material/poissonFD.ipynb) notebook from MIT's Intro to Linear PDEs course, it is unclear to me why the Kronecker product is used to formulate the coefficients matrix $A$ for solving the linear system of equations $A u = b$. Given Poisson's equation on the unit... |
Let $f:\mathbb{R}\rightarrow \mathbb{R}$.
Let $f_{\mathrm{max}}=\max_{x\in \mathbb R} f(x)$ and $f_{\mathrm{min}}=\min_{x\in \mathbb R} f(x)$.
Define $r$ as the ratio of $f_{\mathrm{max}}$ to $f_{\mathrm{min}}$.
Intuitively, $r$ measures the "turbulence" of $f$.
Is there a specific terminology for $r$? |
Terminology to measure the ratio of the maximum value to the minimum value of a real-valued function? |
In calculus, we are introduced about the partial derivatives, like for example
\begin{align}
\frac{\partial(x^2-y)}{\partial x}(x,y),\ \ \ \frac{\partial(x+e^y-yz)}{\partial y}(x,y,z),\ \ \ \frac{\partial(x+y)}{\partial(x+y)}(x,y)\ \ ...
\end{align}
But I wonder if these sort of derivatives
\begin{align}
\frac{\p... |
I was working with Rings and recalled that to show that a ring $A$ is subring of a ring $B$, all I need is to show that there is an injection $A\to B$ and then use the first Noether isomorphism theorem. Conversely, given $A\subset B$ subring, the inclusion map is (necessarily) an injective ring map. In essence what I a... |
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.
... |
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