instruction stringlengths 12 30k |
|---|
Let $\{e_1, e_2, ..., e_n\}$ be the set of elements. Let $\{s_1(e_1), s_1(e_2), ..., s_1(e_n)\}$, $\{s_2(e_1), s_2(e_2), ..., s_2(e_n)\}$ be the scores of elements after applying two ranking functions $s_1$, $s_2$. Let $\{w\cdot s_1(e_1) + (1-w)\cdot s_2(e_1), w\cdot s_1(e_2) + (1-w)\cdot s_2(e_2), ..., w\cdot s_1(e_n)... |
Let's take the following random variables $X,Y,Z_1,...,Z_K$, on the same probability space (or associated with the same experiment), where we know that
$$ Y = \sum_{k=1}^K\beta_kZ_k$$
Let's say I am interested in computing the conditional variance of $X$ knowing $Y$, similarly as in this [question][1]. Applying th... |
Consider a linear dynamic system of the form:
x' = Ax + Bu
Some of the parameters used in forming the system matrix A are random variables with known probability density functions.
Is there a way to determine the distribution (or at least some features of the distribution) of the system eigenvalues?
O... |
The modelling probability space is the space of all infinite tuples with entries among $1,2,3,4,5,6$. There is a stopping time $\tau$, the first occurrence of the pattern
$$
a<b<c
$$
with three last rolls $a,b,c$. When such a pattern is seen, the "game stops". We have $\tau<\infty$ with probability one.
So we br... |
Suppose that there is an $n$ such that the set of $L$-formulas in variables $x_1,\dots,x_n$ up to $T$-equivalence is finite. That is, there is no finite set of formulas $\{\varphi_i\}_{i\in I}$ (with $I$ finite) such that for all $L$-formulas $\psi(x_1, x_2, \ldots, x_n)$ there is a $i \in I$ such that $$T \models \for... |
Consider a convex function $f(x)$ and the line $y=px$. Let $x_p$ be the value of $x$ that maximizes $xp-f(x)$. This allows us to define the Legendre transform of $f$ as a function of $p$:
$g(p) = x_pp - f(x_p)$
This definition is so nice and simple, and many important properties of the Legendre transform can be d... |
Consider a convex function $f(x)$ and the line $y=px$. Let $x_p$ be the value of $x$ that maximizes $xp-f(x)$. This allows us to define the Legendre transform of $f$ as a function of $p$:
$g(p) = x_pp - f(x_p)$
This definition is so nice and simple, and many important properties of the Legendre transform can be d... |
I was checking the subadditivity of the function $f\colon x\mapsto \min(1, x)$, *i.e.*, whether $f(x + y)\le f(x) + f(y)$ and was expecting that the entire first quadrant would be colored. However, the following was outputted:
[![enter image description here][1]][1]
But clearly, for nonnegative $x, y$ satisfying ... |
Desmos not plotting the obvious inequality correctly? |
Consider the symmetric tensor algebra of a vector space $V$
$$
\mathrm{Sym} (V) =\oplus_{m\ge 0} \mathrm{Sym}^m (V)
$$
Suppose $V$ happens to be a chain complex with a differential $d:V\to V$ such that $d\circ d=0$.
Then, is $\mathrm{Sym}(V)$ also a chain complex? If so, how does the differential look like? |
Suppose we have two coins. The probability that the first coin lands on heads is $p_1(n)$ and depends on $n,$ the $n-$th coin toss of this first coin. Similarly, the probability that the second coin lands on heads is $p_2(n)$ and depends on $n,$ the $n-$th coin toss of this second coin.
If $\displaystyle\lim_{n\to\i... |
Two coins, each with P(Head)$\to 0$ as $n\to\infty,$ where $n=n-$th coin toss. Under what conditions is P(both coins Head) infinite times $<1?$ |
It turns out I misunderstood the question, so my answer addresses a different, but related, problem. I'll leave it up anyway, since it might be of interest to some readers.
----------
Every non-empty set of cardinals has a least element, but it is not true that every non-empty set of cardinals has a greatest e... |
Let $G$ be a finite group. Let $A<G$.
When is it true that $G$ has only a single maximal subgroup containing (or equal to) $A$?
The necessary condition is that there must be $x \in G$ such that $\left<A \cup \{x\}\right>=G$ and the order of $\left<x\right>$ is a power of a prime.
In many cases, this is also t... |
When does a finite group $G$ only have a single maximal subgroup containing given $A<G$? |
I prefer the notation $D(P_{Y|X}\|Q_{Y|X}|P_X),$ since this makes the law over $X$ explicit.
For a pair of laws $P_{XY},Q_{XY},$ the chain rule for KL divergence is $$ D(P_{XY}\|Q_{XY}) = D(P_X\|Q_X) + D(P_{Y|X}\|Q_{Y|X}|P_X).$$
Now, if $P_X = Q_X$ as in the question, then the first term is $0$. But exchanging ... |
This can efficiently be done using generating functions.
Summing the coefficient sequence corresponds to multiplying the generating function by $\frac1{1-x}=\sum_{k=0}^\infty x^k$. There are some complications because you’re not including the first term as the first difference, but that can be accounted for by some ... |
The answer is no, and we give some counter-examples below: two in $2$ and one in $3$ and more dimensions. A version of the two-dimensional example is mentioned on page 24 of [this paper](https://doi.org/10.1088/1367-2630/15/7/073045), and the three-dimensional example also appears in a slightly different form in [these... |
Suppose we have two coins. The probability that the first coin lands on heads is $p_1(n)$ and depends on $n,$ the $n-$th coin toss of this first coin. Similarly, the probability that the second coin lands on heads is $p_2(n)$ and depends on $n,$ the $n-$th coin toss of this second coin.
If $\displaystyle\lim_{n\to\i... |
The formula satisfied by all formulas that are not self-applicable, and only by them, is self-applicable if and only if it is not self-applicable.
Because, if it is self-applicable, it is inconsistent with its definition, because it should be satisfied only by formulas that are not self-applicable.
And if it is n... |
Suppose $B_t$ is a standard Brownian motion on $\mathbb{R}$ and let $L_t$ be its local time at zero. Let $p_t(x,y)$ be the transition density of $B_t$, i.e. $p_t(x,y) = \frac{1}{\sqrt{2\pi}}\exp\left(- \frac{(x-y)^2}{2t} \right)$ and let $u^\alpha(x,y)$ be the potential density of $B_t$, i.e. $u^\alpha(x,y) = \int_0^\i... |
Is there a general way to invert (solve for $u$) this?
$$\sum_{ij}R_{ijk}a_iu_j = -x_k$$
With $a,u,x \in \mathbb{R}^N$. $R_{ijk}$ is symmetric in the last two indices.
So really I'm trying to invert this:
$$A= \begin{bmatrix} a'R_1 \\ a'R_2 \\ ...\\ a'R_N \end{bmatrix}$$
Could there be a nice formula for this inve... |
This feels like a very specific problem, but I hope there already is a method to achieve what I want. There is a random process from which I can draw samples of non-numerical, variable sized objects (basically labelled point clouds, but this is probably irrelevant now). I denote these samples with $\boldsymbol{x}$. I a... |
How to construct a maximum-information embedding of sampled objects using a binary function? |
Is there a general way to invert (solve for $u$) this?
$$\sum_{ij}R_{ijk}a_iu_j = -x_k$$
With $a,u,x \in \mathbb{R}^N$. $R_{ijk}$ is symmetric in the last two indices.
So really I'm trying to invert this:
$$A= \begin{bmatrix} a'R_1 \\ a'R_2 \\ ...\\ a'R_N \end{bmatrix}$$
Could there be a nice formula for this inve... |
I am working with Marsdens/Hoffmanns book "Basic Complex Analysis" (3rd edition). They define the contour integral and show on page 99 that it if you integrate a function along a representation and along a reparametrization of a curve you will get the same answer for both integrals.
Then they state that this result ... |
Suppose a function
$$
g(x) = a*q(x) + b*f(x)
$$
where $q(x)$ is an arbitrary quadratic function, $f(x)$ is an arbitrary function that's decreasing and $f(x) <= 0$ fopr all x.
Is it possible to show that g(x) only has 0,1,or 2 roots? Here a and b are arbitrary positive scalars. |
Suppose a function
$$
g(x) = a\cdot q(x) + b\cdot f(x)
$$
where $q(x)$ is an arbitrary quadratic function, $f(x)$ is an arbitrary function that's decreasing and $f(x) \leq 0$ for all $x$.
Is it possible to show that $g(x)$ only has 0,1,or 2 roots? Here a and b are arbitrary positive scalars. |
Given a tool that can draw a sphere by given center and a point on it and a surface by given 3 points, is the constructable set of the tool equivalent to the streightedge and compass constructable numbers? In simpler words, is there anything you can construct in 3d and not in 2d? And why? |
I revisited the [old post of mine](https://math.stackexchange.com/questions/4873704/double-negation-in-sequent-calculus) and I am confused with the answer. The cut elimination theorem states that for any sequent that is derived with cut rule, there exists a derivation of the same sequent without using cut rule. It seem... |
Deriving $\Phi\vdash\Delta$ from $\lnot\lnot\Phi\vdash\Delta$ without cut rule? |
Let $G$ be a finite group. Let $A<G$.
When is it true that $G$ has only a single maximal subgroup containing (or equal to) $A$?
The necessary condition is that there must be $x \in G$ such that $\left<A \cup \{x\}\right>=G$ and the order of $\left<x\right>$ is a power of a prime.
In many cases, this is also t... |
Investigate whether the following set is closed or open in $\mathbf{R}^2$. Provide precise justification.
$$A = \{(x, y) \in \mathbf{R}^2 : y^2 = x\}.$$
To show that $A^C$ is open, which is equivalent to demonstrating that $A$ is closed, I tried to utilize the theorem stating that $A^C$ is open if and only if the... |
It is not *precisely* true that every vertex lies in exactly $q$ sets; however, we can reduce to that case without loss of generality.
First, what do we know? If we take each $S \in \mathcal S_0$ with multiplicity $p_S$, then for every vertex $v$, we have $$\sum_{S \ni v} f(S) \ge 1 \iff \sum_{S \ni v} \frac{p_S}{q}... |
Let $G$ be a finite group. Let $A<G$.
When is it true that $G$ has only a single maximal subgroup containing (or equal to) $A$?
The necessary condition is that there must be $x \in G$ such that $\left<A \cup \{x\}\right>=G$ and the order of $\left<x\right>$ is a power of a prime.
In many cases, this is also t... |
A builder claims that heat pumps are installed in $70\%$ of all homes being constructed today in Chennai. Would you agree with this claim if a random survey of new homes in this city showed that $8$ out of $15$ had heat pumps installed? Use a $10\%$ level of significance.
Here the null hypothesis(H$_0$) is $p = 0.... |
In our book we have the manipulation
\begin{align*}
b\int\limits_0^{2\pi}\sqrt{\left(1-\left(1-\frac{a^2}{b^2}\right)\sin^2(t)\right)}~\!dt=4b\int\limits_0^{\frac{\pi}{2}}\sqrt{\left(1-\left(1-\frac{a^2}{b^2}\right)\sin^2(t)\right)}~\!dt
\end{align*}
where $a,b>0$. Where does $4$ come from and why the change of th... |
Let $G$ be a finite group. Let $A<G$.
When is it true that $G$ has only a single maximal subgroup containing (or equal to) $A$?
The necessary condition is that there must be $x \in G$ such that $\left<A \cup \{x\}\right>=G$ and the order of $\left<x\right>$ is a power of a prime.
In many cases, this is also t... |
Let $G$ be a finite group. Let $A<G$.
When is it true that $G$ has only a single maximal subgroup containing (or equal to) $A$?
The necessary condition is that there must be $x \in G$ such that $\left<A \cup \{x\}\right>=G$ and the order of $\left<x\right>$ is a power of a prime. In many cases, this is also suff... |
In class we were given this definition: $Div (\vec{F}):= lim_{r \rightarrow0} \oint_{C} \vec{F} \cdot \vec{n} \:ds$ (where r is the radius of the circle C and $\vec{n}$ is the outward pointing normal of the curve C).
How do i show that defining divergence in this way also implies that
$Div(\vec{F}) = \partial_1 F... |
How do i prove that the these two definitions of divergence are equivalent? |
What is the pointwise, absolute and uniform convergence of the $\sum_{n=1}^\infty\left[\arcsin\left(\frac{2x^2-1}{x}\right)\right]^n$? |
Given a double sequence $(a_{n,m})_{n,m \in \mathbb{N}}$ with $0 \leq a_{n,m} \leq C$ for some constant $C > 0$, and knowing that $\liminf_{n \to \infty} a_{n,m} = 0$ for any fixed $m$, does it follow that
$$
\liminf_{n \to \infty} \left( \limsup_{m \to \infty} a_{n,m} \right) = 0?
$$
I am looking for a proof or co... |
An answer, which needs at the end a complementary study to be fully rigorous.
First of all, it is clear that setting $y=2x$ in the implicit equation gives $0$. Therefore, it is not surprizing that the following graphical representation of the implicit equation produced by GeoGebra has 2 branches : the straight line ... |
[Picture][1]
[1]: https://i.stack.imgur.com/N88ay.jpg
The picture shows an example to solve the integer problem with a decomposition method. However, what I am trying to ask is about the dual function part, instead of the integer part. As you can see, the integer variables are relaxed. But then I cannot find ... |
I first learned of the below identity from [MathWorld][1] and the works of Ramanujan, but it's completely crazy with polygammas and Laplace transforms of hyperbolic trig. It seems weird that the Laplace Transform of $\frac{t}{\cosh{t}}$ at $s=\sqrt5$ should yield such an interesting continued fraction; why this functio... |
Let $f\in L^{2}(0,T;L^{2}(\Omega))$ where $T$ is a finite number and $\Omega\subset \mathbb{R}^{3}$ is a bounded domain. I'm consider the average function with respect to time that is
\begin{equation*}
f_{\kappa}(t)=\frac{1}{\kappa}\int_{t-\kappa}^{t}f(\tau) d\tau,
\end{equation*}
after extending $f$ to zero outsid... |
Equation $x^4+ax^3+bx^2+ax+1=0$ has at least 1 real root. $a,b \in \mathbb{R}$\
What's the minimum value of $(a^2+b^2)$?
Ok so here I tried to divide by $x^2$ and get this :
$$x^2+\frac{1}{x^2}+a\left(x+\frac{1}{x}\right)+b=0$$
Here with a subtle $t$ substitution: $$x+\frac{1}{x}=t\Rightarrow x^2 + \frac{1}{x^2}... |
Equation $x^4+ax^3+bx^2+ax+1=0$ has at least 1 real root. $a,b \in \mathbb{R}$ What's the minimum value of $(a^2+b^2)$? |
I am solving the following exercise:
Five balls are dropped on a table that has 7 holes. Assuming each ball falls independently into any of the 7 holes with equal probability, find the probability that a hole contains more than one ball, and the probability that two holes contain more than one ball.
We can define... |
I have a vector $v1$, and a rotated vector $v2$. I want to find two rotation matrices $Rx$ and $Ry$, which are rotation matrices around x-axis and y-axis, respectively, so that $Rx \times Ry \times v1 = v2$. My intuition tells me both $Rx$ and $Ry$ are unique if I restrict $\theta$, which corresponds to $Rx$ as between... |
I have a vector $v1$, and a rotated vector $v2$. I want to find two rotation matrices $Rx$ and $Ry$, which are rotation matrices around x-axis and y-axis, respectively, so that $Rx \times Ry \times v1 = v2$. My intuition tells me both $Rx$ and $Ry$ are unique if I restrict $\theta$, which corresponds to $Rx$ as between... |
I have a vector $v1$, and a rotated vector $v2$. I want to find two rotation matrices $Rx$ and $Ry$, which are rotation matrices around x-axis and y-axis, respectively, so that $Rx \times Ry \times v1 = v2$. My intuition tells me both $Rx$ and $Ry$ are unique if I restrict $\theta$, which corresponds to $Rx$ as between... |
I want to find the maximum of $|g(\kappa)|$ defined as follows
For some $A>0$, $\alpha\in (0,\infty)$, and $\kappa \in [-\kappa^{*},\kappa^{*}]\subset [-\pi,\pi]$, define $g(\kappa )$
\begin{align}
g(\kappa ) &= -\sqrt[]{\Big | \lambda - 4A^{2}\zeta(1+\alpha) -2(1+A^{2})\sum\limits_{m\in\mathbb{Z}_{>0}}\frac... |
I have two generating functions.
$A(x)= \frac{x^{\frac{m \left(m-1\right)}{2}}}{\left(1-x\right)\cdot \left(1-x^{2}\right)\cdot ...\cdot \left(1-x^{m}\right)}$
$B(x)= \frac{A(x)}{x^t} = \frac{x^{\frac{m \left(m-1\right)}{2}}}{\left(1-x\right)\cdot \left(1-x^{2}\right)\cdot ...\cdot \left(1-x^{m}\right) \cdot x^t... |
Let $f\in L^{2}(0,T;L^{2}(\Omega))$ where $T$ is a finite number and $\Omega\subset \mathbb{R}^{3}$ is a bounded domain. I'm consider the average function with respect to time that is
\begin{equation*}
f_{\kappa}(t)=\frac{1}{\kappa}\int_{t-\kappa}^{t}f(\tau) d\tau,
\end{equation*}
after extending $f$ to zero outsid... |
I am solving the following exercise:
Five balls are dropped on a table that has 7 holes. Assuming each ball falls independently into any of the 7 holes with equal probability, find the probability that a hole contains more than one ball, and the probability that two holes contain more than one ball.
We can define... |
Suppose we have two coins. The probability that the first coin lands on heads is $p_1(n)$ and depends on $n,$ the $n-$th coin toss of this first coin. Similarly, the probability that the second coin lands on heads is $p_2(n)$ and depends on $n,$ the $n-$th coin toss of this second coin. All coin tosses are mutually ind... |
$ \frac{y}{x}+ (y^3-\log(x))\frac{dy}{dx}=0 \\ y(1)=\sqrt{2}$
I have the IVP above and I'm trying to solve it, I know that $y=k\log(|x|); k \in \mathbb{R}$ is a solution of the similar linear-homogeneous equation $\frac{y}{x} -\log(x)\frac{dy}{dx}=0$ but this doesn't seem to help me find the solution of this prob... |
**Setup:**
Let $\pi:(M,g)\to (N,h)$ be a surjective *Riemannian* submersion,
* i.e. $\forall p\in M, D\pi_p$ is surjective between the respective tangent spaces and that .
* $T_pM=H_pM \oplus V_pM$ ($g_p$-orthogonal direct sum), and
* $H_pM$ is isometric to $T_qN, q:=\pi(p)$ via $D\pi_p, i.e. h_q(D\pi_p(v),... |
Is there a fast, reasonable way to compute the approximate value of $\log - \log x$? |
**Setup:**
Let $\pi:(M,g)\to (N,h)$ be a surjective *Riemannian* submersion,
* i.e. $\forall p\in M, D\pi_p$ is surjective between the respective tangent spaces and that .
* $T_pM=H_pM \oplus V_pM$ ($g_p$-orthogonal direct sum), and
* $H_pM$ is isometric to $T_qN, q:=\pi(p)$ via $D\pi_p, i.e. h_q(D\pi_p(v),... |
Let's consider the following deterministic optimal control problem, where $c(t)$ is the control, and $x(t)$ and $y(t)$ are the state variables:
\begin{align}
J(t) = \inf_{c(t)} \ &\int_0^\infty e^{-\delta u} \left( x^2(u) + \lambda y^2(u) \right) du
\end{align}
where the dynamics for the states are:
\begin{align}... |
In Sharpe's book on Cartan geometry, he mentions in kind of an offhand way that Cartan atlases extend to unique maximal Cartan atlases, but I'm feeling skeptical. For a Klein pair $(G,H)$ with Lie algebras $(\mathfrak{g},\mathfrak{h})$, a Cartan atlas is defined as a family of $\mathfrak{g}$-valued $1$ forms on an open... |
Do (ineffective/locally effective) Cartan geometries have a unique maximal atlas? |
I have a vector $v1$, and a rotated vector $v2$. I want to find two rotation matrices $Rx$ and $Ry$, which are rotation matrices around x-axis and y-axis, respectively, so that $Rx \times Ry \times v1 = v2$. My intuition tells me both $Rx$ and $Ry$ are unique if I restrict $\theta$, which corresponds to $Rx$ as between... |
I was reading proof of Binet's first expression for $\log \Gamma(z)$; that is for $\Re z > 0$,
$$
\log \Gamma(z) = \left(z - \frac 1 2\right)\log z - z + \frac 1 2 \log (2\pi) + \int_0^\infty \left(\frac 1 2 - \frac 1 t + \frac 1 {e^t - 1} \frac {e^{-tz}}t \right) \operatorname{d}\!t.
$$
One of the steps was usin... |
> Prove that for a fixed $w\in D(0,1)$, the function
> $$F(z)=\dfrac{w-z}{1-\overline{w}z}$$ is holomorphic.
My attempt:\
Since $|\overline{w}z|<1$, we have $1-\overline{w}z\neq 0$. Because if it was, then $|\overline{w}z|=1$ (Contradiction!)\
Can I conclude that $F$ is holomorphic here? |
Can I conclude $f$ is holomorphic? |
I want to numerically calculate the low temperature limit of the following [Matsubara sum](https://en.wikipedia.org/wiki/Matsubara_frequency)
$$ S(\Omega) = \pi T \sum_{\omega_n} \frac{4\Delta^2+\Omega^2}{s_1 s_2 (s_1 + s_2)},$$ with $\omega_n = \pi T (2n+1)$, $s_1(\omega_n) = \sqrt{\omega_n^2 + \Delta^2}$ and $s_2... |
> Prove that for a fixed $w\in D(0,1)$, the function
> $$F(z)=\dfrac{w-z}{1-\overline{w}z}\quad\forall z\in D(0,1)$$ is holomorphic.
My attempt:\
Since $|\overline{w}z|<1$, we have $1-\overline{w}z\neq 0$. Because if it was, then $|\overline{w}z|=1$ (Contradiction!)\
Can I conclude that $F$ is holomorphic here? |
In order to set up the apparatus for some work I'm doing, I'd like to construct a field with certain properties. I'm hoping this is a standard construction, though I've been unable to find it in the literature, and I hope someone can just point me to a good reference for it.
For the complex numbers $\mathbb{C}$ "al... |
**Setup:**
Let $\pi:(M,g)\to (N,h)$ be a surjective *Riemannian* submersion,
* i.e. $\forall p\in M, D\pi_p$ is surjective between the respective tangent spaces and that .
* $T_pM=H_pM \oplus V_pM$ ($g_p$-orthogonal direct sum), and
* $H_pM$ is isometric to $T_qN, q:=\pi(p)$ via $D\pi_p, i.e. h_q(D\pi_p(v),... |
**Setup:**
Let $\pi:(M,g)\to (N,h)$ be a surjective *Riemannian* submersion,
* i.e. $\forall p\in M, D\pi_p$ is surjective between the respective tangent spaces and that .
* $T_pM=H_pM \oplus V_pM$ ($g_p$-orthogonal direct sum), and
* $H_pM$ is isometric to $T_qN, q:=\pi(p)$ via $D\pi_p, i.e. h_q(D\pi_p(v),... |
I am reading the paper "[Choosability and fractional chromatic numbers][1]".
It concerns the fractional chromatic numbers of a graph $G$: let $\mathcal{S}(G)$ be the collection of independent sets in $G$. A fractional coloring is a function $f:\mathcal{S}(G)\rightarrow\mathbb{R}_{\ge 0}$ such that $\sum_{S\in\mathca... |
Why is every vertex in $q$ sets? |
Consider the following statements regarding union and their equivalent statements ($\alpha$ indexes a family of sets $A_\alpha$):
- $x \in \bigcup_\alpha A_\alpha \iff x \in A_\alpha \text{ for some } \alpha$
- $x \notin \bigcup_\alpha A_\alpha \iff x \notin A_\alpha \text{ for all } \alpha$
And the following fo... |
Let $f\in L^1(\mathbb{R})$, and let $r_n$ be a sequence of numbers converging to $0$. I want to show that $f_n(x) = f(x+r_n)$ converges to $f$ in $L^{1}(\mathbb{R})$.
I think this problem can be solved by using the density of the step function. Choose a step function $\psi$ such that $\| f-\psi \|_{L^{1}} \leq \epsi... |
I am self-studying number theory and came across the question asked by another user in this post [here][1]. Specifically part (b).
My initial solution is to use the fact that the Legendre Symbol is completely multiplicative to split $(\frac{-5}{p})$ = $(\frac{-1}{p})*(\frac{5}{p})$, then follow the same steps to com... |
I think a possible clue to your misunderstanding lies in the following sentence (quoting the OP, with emphasis added):
> given that the cardinality of the real numbers is strictly greater
> than the cardinality of the rational numbers, *it doesn't follow that
> the rationals can be dense in the reals*
The struc... |
So going off of Kurt G, for $\alpha = 1$ on the boundaries, I should get:
$$\oint_C 1\nabla \phi_i \cdot \hat{\textbf{n}} ds = \int\int_D \nabla \cdot \nabla\phi_i dA$$
And computing the double integral, I should get:
$$-2(n\pi)^2\int_0^1 \int_0^1 \sin(n\pi x)\sin(n \pi y)dxdy = \\ 2n\pi \int_0^1 \Big(\cos(n\p... |
For simplicity, assume we are working with simple regression where the predictor $x\in\mathbb{R}$.
First write $y=E[y \mid x]+u$, where the variance of $u$ is a constant, and $E[u|x]=0$. I understand $E(y \mid x)$ minimizes the mean square prediction error (MSPE). We call $E[y \mid x]$ the true regression.
I und... |
According to this answer:
https://math.stackexchange.com/a/2582334/1098426
We know $\forall x \ \exists y \ \forall z$ differs from $\forall x \\forall z\ \exists y$ insofar as $y$ depends on $x$ versus $y$ depends on $x$ and $z$.
I traditionally translate the definition of uniform continuity on some set $A\su... |
How can I chain rotate and scale operations (each with different origins) while keeping each operation origin in relation to the original local component space?
Using matrix denotation (i.e. *T* for translation matrix, *R* for the rotation matrix, and *S* for the scaling matrix) I'm performing the following operatio... |
I was trying to solve Exercise 3.13.15 in Cameron's Combinatorics: Topics, Techniques, Algorithms. It goes like this:
> (a) Let $n = 2k$ be even, and $X$ a set of $n$ elements. Define a *factor* to be
a partition of $X$ into $k$ sets of size $2$. Show that the number of factors is equal to $(2k-1)!!=1.3.5...(2k-1)$... |
Let random variable $X$ be defined over alphabet $X = \{-2, 0, 2\}$.
a) Find the distribution $p(x)$ that maximizes the entropy $H(X)$ while maintaining $E\{|X|\} = \theta$, where $\theta \in [0, 2]$. What is this maximum $H(X)$?
b) Find distribution $q(x)$ that minimizes the divergence $D[q||p]$, with respect to... |
Problem Statement: Let $f$ be a function with domain $[-1, 1]$ such that the coordinates of each point $(x, y)$ of its graph satisfy $x^2 + y^2 = 1$. What is the total number of points at which $f$ is necessarily continuous?
My confusion: I've been treating the first sentence as a roundabout way of saying that we h... |
According to this answer:
https://math.stackexchange.com/a/2582334/1098426
We know $\forall x \ \exists y \ \forall z$ differs from $\forall x \forall z\ \exists y$ insofar as $y$ depends on $x$ versus $y$ depends on $x$ and $z$.
I traditionally translate the definition of uniform continuity on some set $A\sub... |
I'm currently working on dynamic system identification and performing Monte Carlo runs with different noise realisations at a given $SNR$ ratio. Once the parameters converge, I save them in a matrix which contains all the converged parameters for the runs. My question is really basic (maybe even dumb). If I'm willing t... |
Dick Dastardly is taking part in an infinite wacky race. What is infinite about it, you ask? Well, just everything! There are infinitely many racers, every one of which can run infinitely fast and the finish line is infinitely far away.
To be more precise, the race will take place on the natural numbers, starting at... |
Find the area of the portion of the plane $2x+3y+4z=28$ lying above rectangle $1\le\ x \le 3$, $2 \le\ y\le5$ in the xy-plane.
How come I cannot simply use $<2,3,4>$ for the normal in the formula $\int_ {1}^{3} \int_{2}^{5} ||N(x,y)|| \,dx\,dy$ ? |
> Let $f$ be holomorphic on $\mathbb{C}$,
> $f(z)=\sum\limits_{n=0}^{\infty}a_nz^n$ be its Taylor expansion about
> $z=0$. Show that if there exists $M>0$ s.t
> $$\iint\limits_{\overline{D}(0,R)}|f(x+iy)|^2dxdy\leq M,\quad\forall
> R>0$$ then $f$ is constant.
My attempt:\
I already proved the following equality... |
What is the origin of the terms fences, funnels, and separatrix in differential equations?
I am not asking what they mean but rather who introduced them and how. They seem to be very recent terms, not used in the classical literature; at the same time, they seem to be taken for granted in some contemporary sources.... |
What is the origin of the terms fences, funnels, and separatrix in differential equations? |
I'm trying to understand the proof of the following theorem (Gamelin; CA)
> If $u(z)$ is a harmonic function on a domain $D$, and if the disk $\{ |z-z_0| < \rho\}$ is contained in $D$, then
$$
u(z_0) = \int_{0}^{2\pi} u(z_0 + re^{i\theta})\frac{d\theta}{2\pi}, \quad 0 < r < \rho
$$
Proof (so far)
If $u$... |
Find the area of the portion of the plane $2x+3y+4z=28$ lying above rectangle $1\le\ x \le 3$, $2 \le\ y\le5$ in the xy-plane.
I know this is solved by $||N|| = \sqrt{1+g_x+g_y}\ $ where $g(x,y)=(x,y,7-\frac{x}2\ -\frac{3y}4\ )$.
But how come I cannot simply use $<2,3,4>$ for the normal in the formula $\int_ {1}^{3}... |
> Let $f$ be holomorphic on $\mathbb{C}$,
> $f(z)=\sum\limits_{n=0}^{\infty}a_nz^n$ be its Taylor expansion about
> $z=0$. Show that if there exists $M>0$ s.t
> $$\iint\limits_{\overline{D}(0,R)}|f(x+iy)|^2dxdy\leq M,\quad\forall
R>0$$ then $f$ is constant.
My attempt:\
I already proved the following equality ... |
We know that if $f(z)$ is analytic at $a \in \mathbb{C}$, then by Taylor, we know
$$f(z) =\sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(z-a)^n$$
By Cauchy, we also know that
$$f^{(n)}(a) = \frac{n!}{2 \pi i } \int_C \frac{f(z)}{(z-a)^{n+1}} \, dz$$
Combining these two, I'm imagining that it would then follow... |
Let $E/\Bbb C : y^2=x^3+ax+b$ be an elliptic curve over $\Bbb C$. Let
$$E(\Bbb C)=\{(x,y)\in\Bbb C^2:y^2=x^3+ax+b\}$$
be the affine variety generated by $E$. Finally, let
$$I(E(\Bbb C))=\{f(x,y)\in\Bbb C[x,y]: f(p)=0,\, \forall p\in E(\Bbb C)\}.$$
I am trying to show that $I(E(\Bbb C))=I_E$ is the ideal generat... |
The code below is to use the simultaneous iteration method to calculate the eigenvalues of the 1d Laplacian operator. The exact eigenvalues are known analytically.
It is strange that there is no convergence even after 4000 iterations. **Is it because of the degeneracy of the eigenvalues?**
If the matrix $H $ is... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.