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I want to find solutions $u$ to the equation
$$b(x,y,z)\cdot\nabla u(x,y,z)=0\quad (x,y,z)\in D$$
$$u(1,0,z)=1,\, u(0,y,0)=0$$
where $D$ is the domain (a simplex) defined by
$$D=\{(x,y,z)\in[0,1]^3: 0\leq x+y\leq 1,\quad 0\leq z\leq x\}$$
The vector field $b$ is given by
$$b(x,y,z)=\begin{pmatrix}z(1-x)-xy\\(1-x... |
Assume that $\mu$ a quasi invariant measure on a homogenous space $M$. Pick $k\in M$. The Dirac distribution $\delta_k$ at $k$ associated with $\mu$ is defined as the distribution such that $\langle \delta_k ,\phi \rangle\equiv \phi (k)$ for any text function $\phi$ on $M$. Equivalently, it is introduced as a "Dirac de... |
For instance, the Tor (or any other derived functor of an additive functor) long exact sequence associated to a short exact sequence of modules. The naturality is stated in many books as "for any commutative diagram of short exact sequences the induced diagram of long exact sequences is commutative".
I see clearly ... |
Is the homology long exact sequence natural? |
$\phi(3^2)=\phi(9)=6; \phi(4^2)=\phi(16)=8; $ and $\phi(5^2)=\phi(25)=20$.
Can you show (using the Chinese remainder theorem) that $8^{20}\equiv176\bmod400$
and the same holds for $8^{20m}$, and that $6^{176}\equiv256\bmod1000$?
(For the last congruence, it helps to know that $6^{25}\equiv1\bmod{125}$.) |
I want to find solutions $u$ to the equation
$$b(x,y,z)\cdot\nabla u(x,y,z)=0\quad (x,y,z)\in D$$
$$u(1,0,z)=1,\, u(0,y,0)=0$$
where $D$ is the domain (a simplex) defined by
$$D=\{(x,y,z)\in[0,1]^3: 0\leq x+y\leq 1,\quad 0\leq z\leq x\}$$
The vector field $b$ is given by
$$b(x,y,z)=\begin{pmatrix}\delta z(1-x)-\... |
Im looking for an answer and I found many similar answers but the difference is that you do not know if first or second throw contains H.
When I think normally if you have one H and toss then you have 1/2 probability of having T and 1/2 of another H so: HH if XH or HX (also how to simply describe such probability??).
... |
**Context**
With some effort we can show that Ramanujan's fastest series implies:
\begin{align}
\frac{8E(k_{58})K(k_{58})}{\pi^2}-\frac{aK^2(k_{58})}{\pi^2}=\frac{\sqrt{58}}{29\pi},\tag{1}
\end{align}
with: $a=\left(\frac{489227532 \sqrt{58}}{{29}}- \frac{691872192\sqrt{29}}{29}+90847272\sqrt{2}-128477440\right)... |
Can every divisibility problem solved by Lagrange's theorem? |
Can every divisibility problem be solved by Lagrange's theorem? |
**Corollary**. For each abelian Lie algebra $\mathfrak{g}$, its *universal enveloping algebra* is given by $(\mathfrak{S(g)},i)$, where:
$\cdot\mathfrak{S(g)}$ is the symmetric algebra of $\mathfrak{g}$, i.e.
$$\mathfrak{S(g)}:=T(\mathfrak{g})/I, $$ where $I \subset T(\mathfrak{g})$ is the *two-sided ideal generat... |
If $m\mid n$ can we find a group $G$ of order $n$ with a subgroup $H$ of order $m$? |
Can every divisibility problem be solved by Lagrange's theorem? |
If $m\mid n$ can we find a group $G$ of order $n$ with a subgroup $H$ of order $m$? |
Can $m\mid n$ be established using group theory & Lagrange? That is, if $m\mid n$, can we find a group $G$ of order $n$ and $H\le G$ of order $m$? |
I can't understand the red line well. Seemly, if $\tilde \gamma_1\cap \tilde \gamma_2$ has at least two points, it will contradict with the simple connectivity. I want to know why it is ?
Picture below is from the 260th page of [do Carmo's Riemannian Geometry][1].
>[![enter image description here][2]][2]
... |
Why invariant geodesics of translation can't have at least two intersections? |
If $m\mid n$ can we find a group $G$ of order $n$ with a subgroup $H$ of order $m$, i.e. can every divisibility be proved group theoretically? |
Suppose the intersection contains just one point, $P$. Where must that point go *after* the translation, i.e., what is $f(P)$? It must lie on both geodesics, because we're assuming both are (setwise) invariant under $f$. But the only point on both geodesics is $P$. So we get $f(P) = P$, i.e., $P$ is a fixed-point of $f... |
By plugging the integral in [Wolfram alpha](https://www.wolframalpha.com/input?i=integral+sin%5E2%285x%29%2Fx+) we have
$$
\int \dfrac{\sin^2(5x)}{x} dx = \frac{1}{2}\left(\log\left(5x\right)-\operatorname{Ci}(10x)\right)
$$
Where $Ci(x)$ is the cosine integral that vanishes at infinity (see [here](https://en.wik... |
$\require{AMScd}$Firstly, naturality of the long exact sequence and uniqueness of the long exact sequence are two different things. That the long exact sequence is natural is a fairly simple consequence of the construction and the naturality of the long exact sequence in homology for an exact sequence of nonnegative co... |
I'm cross-posting this from Mathoverflow. Since I'm asking for recent developments, it seems best to have answers in both sites.
---
The website [Formalizing 100 Theorems](http://www.cs.ru.nl/~freek/100/) by [Freek Wiedijk](http://www.cs.ru.nl/~freek/) contains a list of some theorems that were chosen at some poi... |
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). 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 the condit... |
As per Zoltan Kocsis' [suggestion](https://math.stackexchange.com/questions/2205154/are-there-any-recent-advances-in-formalizing-the-undecidability-of-mathitch?noredirect=1#comment10417757_2205154), I'm answering the question with an updated on our developments (formerly, this was an edit to the OP).
---
We finis... |
I was looking at the proof of Euler product at Mathworld https://mathworld.wolfram.com/EulerProduct.html
First we expand the product, this I understand, then “we write each term as a geometric series.” But I see that we don’t write “each term” as a geometric series. The whole term is $\frac{1}{1-\frac{1}{p_k^s}}$ bu... |
I'm trying to do the integral:
$$\int{\frac{1}{\sqrt{e^{-2x}-1}}}dx$$
So I try two ways to do it, the first method I used is to multiply $e^x$ on both sides first.
$$\int{\frac{1}{\sqrt{e^{-2x}-1}}}dx$$
$$=\int{\frac{e^x}{\sqrt{1-e^{2x}}}}dx$$
$$=\int{\frac{\sin{\theta}}{\sqrt{1-\sin^2{\theta}}}\times\frac{\cos{... |
The classic $e^{x^2}$ is known for not being able to be integrated analitically but there are some ways of evaluating its Riemann integral.
I know that often people use residues theorem for calculating it bringing it to the complex plane, but I remember when I was taking a 2-variables course that the professor manag... |
Calculating $\int_0^{\infty} e^{x^2}dx$ using two variable integration? |
I was looking at the proof of Euler product at Mathworld https://mathworld.wolfram.com/EulerProduct.html
First we expand the product, this I understand, then “we write each term as a geometric series.” But I see that we don’t write “each term” as a geometric series. The whole term is $\frac{1}{1-\frac{1}{p_k^s}}$ bu... |
I'm trying to do the integral:
$$\int{\frac{1}{\sqrt{e^{-2x}-1}}}dx$$
So I try two ways to do it, the first method I used is to multiply $e^x$ on both sides first.
$$\int{\frac{1}{\sqrt{e^{-2x}-1}}}dx$$
$$=\int{\frac{e^x}{\sqrt{1-e^{2x}}}}dx$$
$$=\int{\frac{\sin{\theta}}{\sqrt{1-\sin^2{\theta}}}\times\frac{\cos{... |
We know that every finitely generated metabelian group $\Gamma$ has a faithful representation over a finite product of fields.
Denoting $C_n$ the cyclic group of order $n$, the restricted wreath product $C_6\wr C_\infty$ is linear over a product of two fields ( a field of char. 2 and a field of char. 3). May I ask h... |
Let $M$ be a compact and connected smooth manifold with a symplectic form $\omega$.
$Ham(M, \omega)$ denotes the space of hamiltonian symplectomorphisms of $(M,\omega)$.
I have the following statement in my lecture notes:
> Using Darboux’s theorem one can show that the action of $Ham(M, \omega)$ on M is trans... |
Why I cannot find the inverse matrix of A with the second method? |
The classic $e^{-x^2}$ is known for not being able to be integrated analitically but there are some ways of evaluating its Riemann integral.
I know that often people use residues theorem for calculating it bringing it to the complex plane, but I remember when I was taking a 2-variables course that the professor mana... |
Calculating $\int_0^{\infty} e^{-x^2}dx$ using two variable integration? |
I've already seen http://math.stackexchange.com/questions/107341 but I'm looking into a similar but distinct problem, one I don't even know how to approach besides simply enumerating all possible results.
As a concrete example, if the two 12-sided dice come up 4 and 12, that's ${1,2,2}$ that they share.
Is it eas... |
Probability distribution and expectated number of factors shared by two dice with 12 sides? |
Consider the function
$$
f(t)=e^{-at}
$$
with $a>0$, $t \in [0, \infty)$ and let $\rho$ a smooth function on $[0, \infty)$ such that
$$
\int_0^\infty \rho(t)=1
$$
and let $\rho_\delta(t):= \frac{1}{\delta} \rho\left(\frac{x}{\delta}\right)$ so that
$$
\lim_{\delta \to 0} f * \rho_\delta(t)= f(t)
$$
I'm tr... |
Your mistake is when you write
> Since f is convex, by definition holds that:<br>
$f(tx + (1-t)y) < tf(x) + (1-t)f(y), \forall x,y \in \Re, t \in (0,1)$<br> And g is concave, so:<br> $g(tx + (1-t)y) < tg(x) + (1-t)g(y), \forall x,y \in \Re, t \in (0,1)$<br>
you put the same inequality sign (<), though they must... |
Is this proof vigorous enough? Any suggestion/improvement would help. (If it's better to split the claims into separate questions, please let me know).
(A centre of a connected graph $G$ is a vertex $v$ with the property that the maximum of the distances between $v$ and all other vertices of $G$ is as small as possi... |
Let $p > 2$ and $I: W^{1,p}_0(\Omega) \to \mathbb{R}$ be a functional given by
$$I(u) = \dfrac{1}{p}\int_{\Omega} |\nabla u|^p \, \mathrm{d}x.$$
Is it possible to show that it is convex? In other words, for all $u,v \in W^{1,p}_0(\Omega)$ and $t \in [0,1]$, the inequality holds
$$I(tu + (1-t)v) \leq tI(u) + (1-t)I(v... |
I(u) = \int_{\Omega} |\nabla u|^p \mathrm{d}x is convex for $p > 2$? |
$I(u) = \int_{\Omega} |\nabla u|^p \mathrm{d}x$ is convex for $p > 2$? |
Let's say I have a stochastic optimization problem of the type ($x \in \mathbb{R}^n$):
$$J(x,t)= \max_u E\bigg [\int_t^{+\infty} g(x,u)e^{-\rho s}ds\bigg]$$
The SHJB equation for this is
$$\rho J = \max_{u} \bigg [ g + \langle \mu, \nabla J \rangle + \frac{1}{2}\sum_{ij}\frac{\partial^2 J}{\partial x_i \partial x_... |
**Task:** Find the number of ways to distribute 10 customers (distinguishable) to 7 salesmen (dist.) with each having at least 1 customer.
**Attempt**: We first pick 7 customers (unordered: $C(10,7)$) and give 1 to each salesman (order matters thus $P(7,7)$ ways to do this) for a total of $C(10,7) \cdot P(7,7) = P(1... |
How to compensate for overcount? |
I'm reading Munkres Topology and I'm stuck in lemma 68.5 as you can see he uses the theorem 68.4 in order to imply that there is a isomorphism between $G$ and $G'$, but in order for this theorem to be applied we must have that $\{i_{\alpha}(G_{\alpha})\}$ and $\{i'_{\alpha}(G_{\alpha})\}$ generate $G$ and $G'$ which fo... |
Hi I was looking for $f_n : [0,1] \to \mathbb{R}$ such that every $f_n$ is continuous and $f_n$ converges pointwise to a function $f : [0,1] \to \mathbb{R}$ that is continuous as well. Then I wanted to find a series with $x_n \to a \in [0,1] $ such that :
$\lim_{n \to \infty} f_n(x_n) \neq f(a)$.
I was wondering ... |
How to apply a Central Limit Theorem to Pólya Urns? |
I am learning unification (Martelli and Montanari 1982 paper). I have a question about how the algorithm (Algorithm 1) deals with constants. Take these two equations:
$$
f(a) = c \\
f(X) = c
$$
where $a$ and $c$ are constants (zero-arity functions), $f$ an unary function and $X$ a variable. Note that there are... |
> Let $\{u_n\}_{n\geq 0}$ be an arithmetic sequence defined by $u_n=an+b
\quad (a,b\in\mathbb{N}^*)$.
>
> a) Let $a=6$. Give 2 values of $b\leq 6$ s.t for each such value of $b$, $\{u_n\}$ has
> infinite primes.
>
> b) Let $a=10$. Prove that there are
> infinite values of $b$ s.t for each such value of $b$, $\... |
Let four random variables form the Markov chain
$${\displaystyle W\rightarrow X\leftarrow Y\rightarrow Z},$$
such that the conditional distribution of $X$ depends only on $W$ and $Y$ and is conditionally independent of $Z$ given $Y$. What is the relationship between the mutual informations I(W;X|Y) and I(W;X|Z) ... |
I am currently working with magnetic field sensors for educational purpose. Those sensors require a calibration phase in order to transform the "raw" un-calibrated ellipsoid into a unit sphere ; the calibration itself, with known data points, is functional.
During the calibration phase I need to get enough well disp... |
Does the opposite direction of alpha congruence in applications hold? |
I am currently trying to get my hands on double coset rings. I use the definition given in [wikipedia][1], namely if $G$ is a finite group and $H\subset G$ is a subgroup, then for any double coset $HgH$ we can form the element $[HgH]=\sum_{x\in HgH}x\in \mathbb{Z}[G]$, and the subgroup generated by the $[HgH]$ is actua... |
How to actually find the stream function for a simple Laplace problem? |
Theorem: The set of compact operators that map to any Banach space is closed.
Proof: Let $\{\mathcal{T}_n\}$ be a sequence of compact operators from $\mathcal{X}_1$ to
$\mathcal{X}_2$, where $\mathcal{X}_2$ is a Banach space. Our goal is to show that if $\|\mathcal{T}_n - \mathcal{T}\| \to 0$
for some $\mathc... |
I have come across an integral that involves a hypergeometric function, which can be expressed as follows:
$$I = \int_0^1 x^{1/2}(1-x)^{\epsilon-1} {_{2}F_1}(\frac{1}{2}+\epsilon,1+\epsilon;\frac{3}{2};x) dx.$$
Here, $\epsilon$ is a small complex quantity where $|\epsilon|\ll1$.
I found an integral formula i... |
I have been learning how to transform from one basis to another. I don't have any issues when both basis are orthogonal because I use a different formula then the one below. An issue I am having is when the new basis is not orthogonal. From searching the internet, I have the following formula for finding an inverse of ... |
<h4>Question Summary (to Make Easier to Reference in the Answer)</h4>
$$
\text{If }x^2-16\sqrt{x}=12 \text{ what is the value of }f=x-2\sqrt{x}
\tag{Eq. 1}$$
<h4>Checking for Consistency by Plugging in the $f=x-2\sqrt{x}=2$</h4>
Check $ x - 2\sqrt{x}-2=0$ implies $(\sqrt{x}-1)^2=2+1=3$ so $\sqrt{x}=1+\sqrt{3... |
Is this proof vigorous enough? Any suggestion/improvement would help. (If it's better to split the claims into separate questions, please let me know).
(A centre of a connected graph $G$ is a vertex $v$ with the property that the maximum of the distances between $v$ and all other vertices of $G$ is as small as possi... |
Short answer: no, the solution set cannot be expressed with four inequalities. In fact, any system of linear inequalities that express this solution set, must contain these six inequalities, or positive multiples of them.
The longer answer is long, so here is the structure:
1. Analysis of illustrative examples in... |
This is exercise 2.16.10 in Kunen's *Foundations of Mathematics*. How to make use of the hypothesis the group is of exponent $p$ to prove that the theory of infinite abelian group of exponent $p$ is model-complete?
The following lemma is given as a hint in the book:
Let $\mathfrak{A} \subseteq \mathfrak{B}$ and i... |
I saw the following example of an abstract vector space: $V$ is the vector space made up of vectors of the form $(a, b)$ with $a+b=1$. The way we add and scale vectors in $V$ is:
$$(a_1, b_1) + (a_2, b_2)=(a_1+a_2-1, b_1 + b_2)$$
$$c(a, b)=(ac-c+1,bc)$$
It's easy to check this is indeed a vector space with $(1,0)$ a... |
I am currently studying the textbook *Digital Design by Mano*, and learned that the Boolean function can be expressed algebraically from a given truth table by forming a minterm for each combination of the variables that produces a 1 in the function and then taking the OR of all those terms.
If the following truth ... |
Let's assume you are faced with the following random variables
$$ X_i = \sum_{k}\alpha_{i,k}Z_k + \epsilon_i$$
where $Z_k,\epsilon_i$ are i.i.d. standard normal random variables and $\alpha_{i,k} >0$ are non-null coefficients. You can think of the $Z_k$ as common risk factors and the $\epsilon_i$ as the idiosyncratic... |
Let's assume you are faced with the following random variables
$$ X_i = \sum_{k}\alpha_{i,k}Z_k + \epsilon_i$$
where $Z_k,\epsilon_i$ are i.i.d. standard normal random variables and $\alpha_{i,k} >0$ are non-null coefficients. You can think of the $Z_k$ as common risk factors and the $\epsilon_i$ as the idiosyncratic... |
Are these two logical formulas logically equivalent:
∀x ∈ U(P(x) ∨Q(x)), and ¬(∃x ∈ U(¬P(x)) ∧ ∃x ∈ U(¬Q(x))), where we let U be a domain.
For the 2nd statement, I simplified the expression
¬(∃x ∈ U(¬P(x)) ∧ ∃x ∈ U(¬Q(x)))
= ¬∃x ∈ U(¬P(x)) v ¬∃x ∈ U(¬Q(x) )
= ∀x ∈ U(P(x)) v ∀x ∈ U(Q(x) )
= ∀x ∈ U(P(x)... |
Why would these 2 predicate logics not be equivalent? |
Are these two logical formulas logically equivalent:
∀x ∈ U(P(x) ∨Q(x)), and ¬(∃x ∈ U(¬P(x)) ∧ ∃x ∈ U(¬Q(x))), where we let U be a domain.
For the 2nd statement, I simplified the expression
¬(∃x ∈ U(¬P(x)) ∧ ∃x ∈ U(¬Q(x)))
= ¬∃x ∈ U(¬P(x)) v ¬∃x ∈ U(¬Q(x) )
= ∀x ∈ U(P(x)) v ∀x ∈ U(Q(x) )
= ∀x ∈ U(P(... |
The category of commutative rings is not balanced (the inclusion $\mathbb{Z} \to \mathbb{Q}$ is monic and epic but not an isomorphism).
So, is there a category $C$ and a functor $\mathbf{CRing} \to C$ that is faithful but not conservative (isomorphism-reflecting)? If so, then there should be such a functor that maps... |
From Complex analysis by Serge Lang.
>**Definition:** $C_1,C_2:[a,b]\mapsto U$ are two closed paths in open set $U\subset \mathbb C$.Then we say $C_1$ is homotopic to $C_2$ if there exist a continuous function $h: [a,b]\times [c,d]\mapsto U$ such that
>
>$$ h(t,c)=C_1(t) ~\text{and} ~h(t,d)=C_2(t)$$ for all $t\in... |
This answer ended up being kind of long. So if you want to see the process, jump to the summary at the end.
#### Denominators of the form $2^n - 1$
When we write a real number $x \in [0, 1]$ as a sequence of bits $(b_1, b_2, \dots)$, where each $b_i \in \{0, 1\}$, what we are really doing is expressing it as the ... |
Let's assume you are faced with the following random variables
$$ X_i = \sum_{k}\alpha_{i,k}Z_k + \epsilon_i$$
where $Z_k,\epsilon_i$ are i.i.d. standard normal random variables and $\alpha_{i,k} >0$ are non-null coefficients. You can think of the $Z_k$ as common risk factors and the $\epsilon_i$ as the idiosyncratic... |
I would also like to point out to one more interpretation, which is more intuitive and which was quite helpful for me to fix the meaning behind everything said about the problem. You can think of
\begin{align}\frac{df}{dg}\end{align}
as a slope of a tangent line to the parametric curve defined by points $(g(t),f(t))$... |
I'm playing a programming game (jsbattle.jmrlab.com) in which script-controlled tanks drive around and try to shoot each other until only one side remains. Because of the way the steering works, the tanks often follow circular trajectories, so leading a shot in those circumstances is very useful.
I have developed an... |
Lets consider a set of 3D points `P`. All points in `P` are distributed in a non-random (but still not known) way over an ellipsoid surface `Er` which parameters are unknown.
NOTE: the points in `P` are obtained in a linear sequence over the ellipsoid surface (as if you were drawing a continuous line over the surfac... |
I'm looking at the following problem from Pg. 47 of Miles Reid's Undergraduate Algebraic Geometry (https://homepages.warwick.ac.uk/staff/Miles.Reid/MA4A5/UAG.pdf):
> 2.11 (Group law on cuspidal cubic.) Consider the curve
> $$C:(z=x^3)\subset k^2;$$
$C$ is the image of the bijective map $\varphi\colon k\to C$ by ... |
I am currently studying the textbook *Digital Design by Mano*, and learned that the Boolean function can be expressed algebraically from a given truth table by forming a minterm for each combination of the variables that produces a 1 in the function and then taking the OR of all those terms.
If the following truth ... |
I'm looking at the following problem from Pg. 47 of Miles Reid's Undergraduate Algebraic Geometry (https://homepages.warwick.ac.uk/staff/Miles.Reid/MA4A5/UAG.pdf):
> 2.11 (Group law on cuspidal cubic.) Consider the curve
> $$C:(z=x^3)\subset k^2;$$
$C$ is the image of the bijective map $\varphi\colon k\to C$ by ... |
I have a question regarding quasiconvexity and its usage in CVXPY.
I have the following optimization problem.
\begin{equation*}
\begin{aligned}
\min_{x} \quad & \sqrt x\\
\textrm{subject to:} \quad & 1 \leq x \leq 2\\
\end{aligned}
\end{equation*}
The solution is trivial $x^* = 1, \ p^* = 1$ but I want to... |
Solving a convex problem with quasiconvexity with CVXPY? |
We all know that a circle can be drawn with the trigonometric functions $x=\cos(t), y=\sin(t)$. If we define the sine and cosine functions in terms of triangles (like we do in high school), then this is quite obvious.
But then later on in our education, we learn that the solution to a simple harmonic oscillator is ... |
Is there a simple geometric proof of why two simple harmonic oscillators draw a circle? |
Consider an arbitrary function of two variables $f \colon \mathbb N \times \mathbb R^+ \to \mathbb R$. My question is very simple: what does one mean by writing that
$$ \lim_{N \to \infty} \, f(N,r) = 0 $$
uniformly in $r \in (0,R]$, where $R > 0$ is an arbitrary fixed constant ?
**My intuition.** Intuitivel... |
What is the meaning of "uniformly" in the limit of a function of two variables? |
Lets consider a set of 3D points $P$. All points in $P$ are distributed in a non-random (but still not known) way over an ellipsoid surface $E_r$ which parameters are unknown.
NOTE: the points in $P$ are obtained in a linear sequence over the ellipsoid surface (as if you were drawing a continuous line over the surfa... |
# Set up
Consider the example in section 2 of [Ferguson (1982)](http://staff.ustc.edu.cn/~zwp/teach/Math-Stat/MLE2.pdf).
I.e., let $X_1, \ldots, X_n$ be i.i.d. with a distribution which with probability $\theta$ is the $U(-1, 1)$, and with probability $(1-\theta)$ is equal to the triangular distribution on $(\theta -... |
I have random variables $X_1, \ldots, X_N$ and two functions: a bounded function $|f|\leq M$ and another function $|g|\leq 1$. I would like to upper bound the variance
$$
\mathbb{V}\left[\sum_{i=1}^N f(X_i)g(X_i) \right]
$$
by a bound that does not depend on $f$ (happy for it to depend on $g$). Is it possible?
... |
I'm playing [a programming game][1] in which script-controlled tanks drive around and try to shoot each other until only one side remains. Because of the way the steering works, the tanks often follow circular trajectories, so leading a shot in those circumstances is very useful.
I have developed an equation that se... |
The dodecahedral graph is the Platonic graph corresponding to the connectivity of the vertices of a dodecahedron, with 20 nodes and 30 edges. Assuming all edges have a length (weight) of 1, I want to find the shortest path that
- visits every node at least once,
- traverses every edge at least once (regardless of d... |
How would I solve this.
$$I= \int_0^{\frac{\pi}{4}} \sqrt{\sec^2(x)+\sin^2(x)}\ dx$$
The approximate value of the integral:
$$I = 0.937723050585$$
I am trying to solve this integral to find the arc lenght of a parametric equation but do not see any trig substitution or other method to proceed.
I tried to use w... |
How to solve this integral trigonometric integral or prove that is does not have a closed form solution? |
Consider an $m \times n$ grid of unit squares, indexed by
$(i, j)$ with $1 \leq i \leq m$ and $1 \leq j \leq n$.
Then why is the number of possible paths from $(1,n)$ to $(m,1)$ such that you are only going down and to the right equal to $\binom{m+n-2}{m -1}$?
could someone give me some intuition for this?
T... |
I have random variables $X_1, \ldots, X_N$ and two functions: a bounded function $|f|\leq M$ and another function $|g|\leq 1$ with $g \geq 0$. I would like to upper bound the variance
$$
\mathbb{V}\left[\sum_{i=1}^N f(X_i)g(X_i) \right]
$$
by a bound that does not depend on $f$ (happy for it to depend on $g$). Is ... |
I have random variables $X_1, \ldots, X_N$ and two functions: a bounded function $|f|\leq M$ and another function $0 \le g \le 1$. I would like to upper bound the variance
$$
\mathbb{V}\left[\sum_{i=1}^N f(X_i)g(X_i) \right]
$$
by a bound that does not depend on $f$ (happy for it to depend on $g$). Is it possible?... |
If you have any two matrices A and C such that A$\vec{x}$ = $\vec{b}$ and C$\vec{x}$ = $\vec{b}$ have the same general solution for all $\vec{b}$ then why is that A = C.
I can understand this when $\vec{x}$ is always unique, but I don't know how to prove this in general. Does this equation imply that A and C are... |
My research is mainly in Harmonic analysis, so I'm not sure if I'm missing some subtleties with the following Differential Geometry argument:
The unit sphere has periodic geodesic flow, of period $2 \pi$. By the Killing Hopf Theorem, every manifold with constant positive sectional curvature must be a quotient of the... |
I have random variables $X_1, \ldots, X_N$ and two functions: a bounded function $|f|\leq M$ and another function $0 \le g \le 1$. I would like to upper bound the variance
$$
\mathbb{V}\left[\sum_{i=1}^N f(X_i)g(X_i) \right]
$$
by a bound that does not depend on $f$ (happy for it to depend on $g$). Is it possible?... |
> For each prime $p>5$, prove that there always exists an integer $a$
> s.t the two following equations $$x^2+py+a=0\quad\text{and}\quad
x^2+py+a+2=0$$ both have integer solutions.
Here are my thoughts for this problem:\
$x^2+py+a=0\iff x^2+a\equiv 0\pmod{p}$ and $m^2+pn+a+2=0\iff m^2+a+2\equiv 0\pmod{p}.$\
So i... |
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