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I started studying differentials on schemes (Hartshorne II.8), and read Eisenbud to gain more intuition and algebra. After vaguely understanding cotangent bundle and conormal sequence, I tried to calculate fiber of $\Omega_{S/k}$, which I expected to be the cotangent space $m/m^2$.
Let $S=k[x_1,\dots,x_r]/I$ where ... |
In the Comments you write:
> … it's exactly a 50:50 chance that I either selected the car or the "set of all other goats and the car", isn't it? Because the car can only be in one of two sets, regardless of the number of goats in it and therefore the chance that the car is in one of two sets is 50%.
You seem to b... |
I was reading the definition of CW-complex in terms of pushouts given by Lück's *Algebraische Topologie: Homologie und Mannigfaltigkeiten* (Chapter 3). It is stated (though not proven) that such a topological space is always Hausdorff. I guess this follows readily by induction if we can prove that the adjunction space ... |
**What set of points on the $256 \times 256$ integer grid maximizes the number of vertices in its convex hull?**
For the full context, I was given an assignment to test algorithms that find the vertices of the convex hull of a set of points in $2D$. One natural thing to test is when we set the input set such that al... |
In the Comments you write:
> … it's exactly a 50:50 chance that I either selected the car or the "set of all other goats and the car", isn't it? Because the car can only be in one of two sets, regardless of the number of goats in it and therefore the chance that the car is in one of two sets is 50%.
You seem to b... |
**My question is, is my proof correct in concluding "If the premises are false, then the conclusion is true"?**
Is my proof of [If the premises are false, the conclusion is true] correct?
1. If the premises are false, the conclusion is false.
Statement 1 is false
because
I am cup. cup is an anima... |
I was reviewing Folland's Real Analysis.
Given the Algebra of unions of disjoint half-open intervals (a,b].
And given an increasing, right continuous $F$.
$\mu\left(\bigcup_1^n(a_j,b_j] \right) = \sum_1^n\left(F(b_j)-F(a_j)\right)$
is a premeasure on that Algebra.
The Lebesgue-Stieltjes measure comes fro... |
Apparent contradiction on Lebesgue-Stietjes measure? |
Let F be a differentiable function of space and time.
Apparently, a first-order Taylor expansion of F yields:
$$\delta\,F=F(x+\delta\,x,t+\delta\,t)-F(x,t)=(\delta\,x\nabla)F+(\frac{\partial\,F}{\partial\,t})\delta\,t$$
What do I need to know in order to understand that?
What type of function is $\delta\,F$?
... |
On a 2-dimensional complex manifold consider two functions which are meromorphic with singularities along two divisors which meet at a point. There is a residue which is a 2-form at the point of intersection.
We could also blow up the point of intersection to $\mathbb{CP}^1$. What connection is there between the... |
I am trying to solve this problem.
>Let $I$ be a non trivial ideal of a ring $A$, and let $r(I)$ be its radical. Prove that $I=r(I)$ iff $I$ is an intersection of prime ideals.
I got the right to left implication:
If $I$ is an intersection of prime elements, let $x\in I$. Since $x^1\in I$, we know $x\in r(I)$,... |
Given a projective $R$-module $M$, how does one find another $R$-module $N$ such that $M \oplus N$ is a free $R$-module?
>In my case, $R = {\mathbb{C}[x, y]}/{(y^2 - x^3 + x)}$, and $M$ is the ideal generated by $\{x, y\}$ over $R$. Since $M$ is projective I know such $N$ exists, but how do I find it?
Here is my i... |
Suppose that $X$ is a well-pointed topological space (i.e. the basepoint is cofibered, not necessarily closed). Is the Moore loop space/space of measured looks $\Omega_MX$ well-pointed? I have seen this claimed in multiple places, but have not been able to locate or produce a proof.
To recap, the Moore loop space is... |
Suppose $M$ is a Poisson random measure on $(E,\mathcal{E}) \equiv (\mathbb{R}_+\times\mathbb{R}^d, \mathcal{B}(\mathbb{R}_+\times\mathbb{R}^d))$ with mean measure $\nu\equiv Leb\times\lambda$.
This means that:
1. $\forall A \in \mathcal{E}$, the random variable $N(A)$ has Poisson distribution with mean $\nu(A)$;
... |
A software company conducted a test on their new platform by exposing their users to two versions of the same product.
Number of users that were given version A: 4000
Number of user that were given version B: 5000
Number of users that experienced a bug: 3000
Number of users with version B that experienced a... |
**Background**
I've recently been formulating a game that incorporates elements from <a href="https://en.wikipedia.org/wiki/Egyptian_fraction">Egyptian fractions</a>, <a href="https://en.wikipedia.org/wiki/Magic_square">magic squares</a>, and <a href="https://en.wikipedia.org/wiki/Tessellation">tilings</a>. It is a ... |
Let $ X $ be a locally compact Hausdorff space, and $ C_c(X) $ be the space of all complex-valued continuous functions with compact support on $ X $.
As far as I know, there are two formulations of Riesz–Markov–Kakutani representation theorem.
**Formulation 1 (e.g., D. L. Cohn, _Measure Theory_, 2nd edition, Theo... |
**My question is, is my proof correct in concluding "If the premises are false, then the conclusion is true"?**
**Please see if my proof is correct**
Is my proof of [If the premises are false, the conclusion is true] correct?
1. If the premises are false, the conclusion is false.
Statement 1 is false
... |
You are on the right track, except for one mistake.
$$P(Y|X) = \frac{1500}{5000} = \frac{3}{10}$$
The denominator will be $5000$ because it's the probability of a user experiencing a bug if they had version B. How many had version B? $5000$, right. How many of those experienced a bug? $1500$. See why it should be... |
Consider the model $y_t = F_{S_t} x_t + \varepsilon_{S_t}$ and $x_t = A_{S_t} x_{t-1} + \nu_{S_t}$, where $\varepsilon_{S_t}, \nu_{S_t} \sim N(0, R_{S_t})$ and $N(0, Q_{S_t})$ and $S_t$ is Markov chain with $\mathbb{P}(S_t = i | S_{t-1} = j) = p_{i,j}$.
We want to estimate the parameters of the model using EM algor... |
A software company conducted a test on their new platform by exposing their users to two versions of the same product.
Number of users that were given version A: 4000
Number of user that were given version B: 5000
Number of users that experienced a bug: 3000
Number of users with version B that experienced a... |
Firstly I am answering my own question after 2 days because I found a very cool solution for finding the best 'u' value in this approximation:
> $log_2(1+x)≈x+u$
My solution starts with finding the error value with an integral
> $f_{error}(x,u)=log_2(1+x)-(x+u)$\
$f_{totalErr}(u)=∫_{0}^{1}|f_{error}(x,u)|dx$ ... |
In my differential geometry course, the $(1,3)$-riemann curvature tensor $R$ is defined by
$$R(X,Y)Z:=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z$$
and the $(0,4)$-riemann curvature tensor Rm by
$$Rm(X,Y,\color{red}Z,\color{blue}W):=g(R(X,Y)\color{blue}W,\color{red}Z)$$
In coordinates $Rm_{ij\ell k}=g_{\ell... |
Let $X:=[X_1,\dots, X_n]$ be a random vector with $X_i \in (0,2)$ and having a joint distribution $F_X$. Take a constant vector $a:=[a_1,\dots, a_n]$ with $a_i \in [0,1]$ and $\sum_{i=1}^n a_i = 1$. Consider the variance
$$
v(a):={\rm var}( \log a^\top X)
$$
Is it possible to tell if the function $v(a)$ is convex o... |
I know I have to use Lagrange's Theorem.
$H \cap K$ is a subgroup of H, so its cardinality divides that of $H$. The same for $K$. So then the cardinality of $H \cap K$ is a common divisor of the cardinalities of $H$ and $K$, but those are coprime, so it must be one.
Is this correct?
|
Suppose H and K are subgroups of a group G. If |H| = 12 and |K| = 35, what is $|H \cap K|$? |
Legendre polynomials can be given by several expressions, but perhaps the most compact way to represent them is by Rodrigues' formula as
$$P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} (x^2 - 1)^n.$$
I was wondering if there exists a multidimensional variant of the Legendre polynomials, e.g. an analogous $P_n(x,y,z... |
Multidimensional Legendre polynomials? |
I've noticed that the definition of a winding number is rather different in Stewart & Tall's _Complex Analysis_ than in Ahlfors' _Complex Analysis_:
Let $\gamma:[a,b]\to\mathbb{C}$ be an arc and let $a\in\mathbb{C}$ lie outside its image.
> **Definition (Stewart, Tall):** the winding number $w(\gamma,a)$ is defi... |
Using the method of finite induction show that $2^{(n-1)} \leq n! \forall n$ belong to $N$ (Natural Number)
Can someone please prove this? |
I've noticed that the definition of a winding number is rather different in Stewart & Tall's _Complex Analysis_ than in Ahlfors' _Complex Analysis_:
Let $\gamma:[a,b]\to\mathbb{C}$ be an arc and let $a\in\mathbb{C}$ lie outside its image.
> **Definition (Stewart, Tall):** the winding number $w(\gamma,a)$ is defi... |
From Ahlfors' _Complex Analysis_ (by 'analytic' he means 'complex-differentiable' or 'holomorphic'; he does not mean that the function can be expanded as a power series, as that is what he sets out to prove):
> **Theorem 7.** Suppose that $f(z)$ is analytic in the region $\Omega'$ obtained by omitting a point $a$ fr... |
How is Ahlfors applying Cauchy's Integral Formula when deriving Taylor's Theorem? |
Your card playing rival Sleazy Silas owns a card shuffling device, which always uses the exact same procedure to shuffle cards, the knowledge of which he unfairly leverages to his advantage. You want to level the playing field, so, when he is not looking, you feed the machine the spades in order (ace to king). Right af... |
How is it possible for the L² norm of f − g to measure the area between the graphs of f and g? |
The concept of Shannon entropy as a basic measure of uncertainty for a random
variable was introduced by Shannon. Let X be an absolutely continuous
non-negative random variable having probability density function (pdf) f and
cumulative distribution function (cdf) F. In Reliability Theory, X represents
the lifetime ... |
My integral is $\int \cos(x^{\frac{3}{4}})dx$. I did the substitution $u=x^{\frac{3}{4}}$ then
$\frac{4}{3}u^{\frac{1}{3}}du=dx$ and my new integral is
$$ \frac{4}{3}\int \cos(u)u^{\frac{1}{3}}du$$.
Now my intuition is to try with integration by parts, $a=u^{\frac{1}{3}}$ and $db= \cos(u) du$ then $da= \frac{1}{3... |
Is there any online source for proofs than $(1+\frac{1}{n})^n$ and $(1+\frac{1}{x})^x$ is $e$? |
I want to compute $Var[A+B+C]$ where $A$, $B$, and $C$ are not independent of each other. In particular, I don't know how to compute $Cov[A,B]$, $Cov[A,C]$, and $Cov[B,C]$. The model specifications follow.
Let:
- $A=X_1(-\lambda_1X_1-\lambda_1\delta_1+\nu_1-\overline{\nu}_1)$
- $B=X_2(-\lambda_2X_2-\lambda_2\d... |
Variance of sum of products of normally distributed random variables? |
The question Equation 1 (is repeated here to easier reference) is:
$$
\left(2 n \right)^{\log 2} = \left( 5 n \right)^{\log 5}
\tag{Eq. 1}$$
Then,
$$\begin{align*}
\left(2 \right)^{\log 2}\left(n \right)^{\log 2} &= \left( 5 \right)^{\log 5} \left( n \right)^{\log 5}
\tag{Eq. 2}\\
\frac{\left(n \right)^{\log ... |
I am studying Hodge theory on complex manifolds; Hoering's notes suggest, as an exercise and I guess as a counterexample to Hodge decomposition when our manifold is not compact, that $ \mathbb{C}$ with standard Kahler structure doesn't admit a decomposition of its first cohomology group, in symbols $H^1(\mathbb{C},\mat... |
We know the fact that any type II$_1$ factor contains the hyperfinite II$_1$ factor $R$.
My question: If we let $M$ be an arbitrary type II$_1$ von Neumann algebra. can we have $R\subseteq M$? |
It is a known result that an infinite one-dimensional random walk almost surely crosses every point (according to Wikipedia at least). Can this result be generalized to random walks with probabilities depending on the current position?
Formally speaking, given a function from the integers to the reals $p_n$, does a ... |
So, I was asked to do this integral using the limit method (or the Riemann Sum)
$$\int_0^3 x{\sqrt {3-x}}\;dx $$
And, I do it like this:
$$\int_0^3 x{\sqrt {3-x}}\;dx $$
Firstly, I determine the $\Delta x$ and $c_i$
$$\Delta x = \frac{b-a}{n}$$
$$\Delta x = \frac{3-0}{n}$$
$$\Delta x = \frac{3}{n}... |
So, I was asked to do this integral using the limit method (or the Riemann Sum)
$$\int_0^3 x{\sqrt {3-x}}\;dx $$
And, I do it like this:
$$\int_0^3 x{\sqrt {3-x}}\;dx $$
Firstly, I determine the $\Delta x$ and $c_i$
$$\Delta x = \frac{b-a}{n}$$
$$\Delta x = \frac{3-0}{n}$$
$$\Delta x = \frac{3}{n}... |
I want to prove the identity:
$$p(y|D,x)=\int p(y|W,x)p(W|D)dW$$
using the conditional independences given by one of the following two graphical models:
[![enter image description here][1]][1]
Is the identity provable assuming (B), or do we need to assume (A)? The identity is intuitively very obvious but I am h... |
How do you "un-marginalize" this conditional distribution? |
$f(x,y)$ is a linear polynomial in $x,y$. if $f(x,y)=\frac 1{xy}$ for $x,y$ $\epsilon$ {3,4}, then $f(5,5)=?$ |
Wikipedia's proof of Leibiz Integral Theorem begins as follows:
> We use Fubini's theorem to change the order of integration. For every $x$ and $h$, such that $h > 0$ and both $x$ and $x+h$ are within $[x_0,x_1]$, we have:
$$\int_x^{x+h}\int_a^bf_x(x,t)\text{d}t\text{d}x = \int_a^b\int_x^{x+h}f_x(x,t)\text{d}x\tex... |
Why is uniform convergence needed in this proof of Leibniz Integral Rule? |
Background: I'm an engineering student with no experience with a real investment account or anything of that sort.
If we were to extend the concept of compound interest strictly mathematically, the formula would be:
$$\text{Amount} = \text{Principal} \times \text{rate} ^ t$$
where $t$ is time in number of compou... |
The result can be derived from the theorem below on surface integrals, by setting the integrand function $f$ to unity.
I have used the following conventions below :
1. The physics convention $(r, \theta, \varphi)$ for spherical coordinates, with $\theta$ the polar angle and $\varphi$ the azimuthal angle [SPH]. ... |
$f(x,y)= Ax +By +Cxy +D$. if $f(x,y)=\frac 1{xy}$ for $x,y$ $\epsilon$ {3,4}, then $f(5,5)=?$ |
If I have understood correctly the question you have
$$
\begin{align}
f(3,4)=f(4,3)=\frac{1}{12}\\
f(3,3)=\frac{1}{9}\\
f(4,4)=\frac{1}{16}\\
\end{align}
$$
and
$$
f(x,y)=ax+by+cxy+d
$$
Solving the system of equation with your favourite methods yelds
$$
f(x,y)=\frac{1}{144}\left(-7x-7y+xy+49\right)
$$... |
I want to prove the identity:
$$p(y|D,x)=\int p(y|W,x)p(W|D)dW$$
using the conditional independences given by one of the following two graphical models:
[![enter image description here][1]][1]
I believe I have a proof that assumes model (A).
$$p(y|D,x)=\int p(y,W|D,x)dW=\int \frac{p(y,W,D,x)}{p(D,x)}dW$$
$$... |
What assumptions are needed to "un-marginalize" this conditional distribution? |
Let $f(x,y)= Ax +By +Cxy +D$. if $f(x,y)=\frac 1{xy}$ for $x,y \in \{3,4\}$, then what is the value of $f(5,5)$?
here in this question i tried substituting $y=x$ thus giving,
$f(x,x) = \frac 1x^2$ for $x$ $\epsilon$ {3,4}
where $f(x,x)$ is a polynomial in $x$ of degree 2. so $x^2f(x,x)$ should be a polynomial ... |
$f(x,y)= Ax +By +Cxy +D$. if $f(x,y)=\frac 1{xy}$ for $x,y \in \{3,4\}$, then $f(5,5)=?$ |
I am looking at repeated operations, and it seems really hard to *precisely* define multiplication.
Of course, for integer $b$ and real number $a$, we use the grade school definition we all know:
$$ab = \underbrace{a + a + a + \cdots + a}_{b\text{ times}}$$
but what about for real numbers $a$ and $b$?
For ... |
The result can be derived from the theorem below on surface integrals, by setting the integrand function $f$ to unity.
I have used the following conventions below :
1. The physics convention $(r, \theta, \varphi)$ for spherical coordinates, with $\theta$ the polar angle and $\varphi$ the azimuthal angle [SPH]. ... |
In the case where the surface can be expressed in both spherical and Cartesian forms the result can be obtained by transforming the Cartesian integral into spherical coordinates. The theorem below gives a proof of this in terms of surface integrals - to obtain the result for surface areas choose an integrand function $... |
You have
$$
\begin{align}
f(3,4)=f(4,3)=\frac{1}{12}\\
f(3,3)=\frac{1}{9}\\
f(4,4)=\frac{1}{16}\\
\end{align}
$$
and
$$
f(x,y)=ax+by+cxy+d
$$
Solving the linear system of equation with your favourite methods yields
$$
f(x,y)=\frac{1}{144}\left(-7x-7y+xy+49\right)
$$
see [here](https://www.wolframalph... |
I am looking into repeated operations, and it seems really hard to *precisely* define multiplication.
Of course, for integer $b$ and real number $a$, we use the grade school definition we all know:
$$ab = \underbrace{a + a + a + \cdots + a}_{b\text{ times}}$$
but what about for real numbers $a$ and $b$?
Fo... |
So I know how to calculate percentile on a simple data set:
lets say student grades are 38, 47, 49, 58, 60, 65, 70, 79, 80, 92
Number of scores below 70 = 6
Using the percentile formula,
Percentile = (Number of Values Below “x” / Total Number of Values) × 100
Percentile of 70
= (6/1... |
percentile calculation on group of data? |
The title of the question [Solving $(2n)^{\log 2}=(5n)^{\log 5}$][1] asks how to solve (in some western conventions) base 10 for $(2n)^{\log 2}=(5n)^{\log 5}$. However that question's title does not seem to indicate an arbitrary base. For instance, one convention for "$\log$" is $\log c = \log_{10} c$.
Also the bas... |
How Can Solving $\left( 2n \right)^{\left( \log_b 2 \right)} = \left( 5n \right)^{\left( \log_b 5 \right)}$ be Generalized for any Base of $\log_b n$? |
> Let $f(x,y)= Ax +By +Cxy +D$. if $f(x,y)=\frac 1{xy}$ for $x,y \in \{3,4\}$,
> then what is the value of $f(5,5)$?
Here in this question I tried substituting $y=x$ thus giving :
$f(x,x) = \frac {1}{x^2}$ for $x \in \{3,4\}$
where $f(x,x)$ is a polynomial in $x$ of degree 2. So $x^2f(x,x)$ should b... |
I understand probability and statistics as a two way route. Either I begin from a random variable (r.v.) $X$ and sample from its probability distribution, or I observe a sample of values $x_1,...,x_n$ drawn from that distribution and I infer something about the distribution of the r.v. $X$. The r.v may have an expected... |
What does it really mean to take correlation between time series? |
Is the difference of these two special convex functions convex? |
The result can be derived from the theorem below on surface integrals, by setting the integrand function $f$ to unity.
I have used the following conventions below :
1. The physics convention $(r, \theta, \varphi)$ for spherical coordinates, with $\theta$ the polar angle and $\varphi$ the azimuthal angle [SPH]. ... |
**1.** Your reasoning seems valid. For a more concise argument, note that
$$ q(Y, W) = q(Y \mid W) q(W) $$
holds for any distribution $q$ for $(Y, W)$. So by choosing $q(\cdot) = p(\cdot \mid D, X)$, it follows that
\begin{align*}
p(Y, W \mid D, X)
&= p(Y \mid D, X, W) p(W \mid D, X) \\
&= p(Y \mid X, W) p(... |
**Background**
I've recently been formulating a game that incorporates elements from <a href="https://en.wikipedia.org/wiki/Egyptian_fraction">Egyptian fractions</a>, <a href="https://en.wikipedia.org/wiki/Magic_square">magic squares</a>, and <a href="https://en.wikipedia.org/wiki/Tessellation">tilings</a>. It is a ... |
I am reading a [book of E. Rio][1] and I found there a theorem (*without a proof*) about coupling. Please see below.
> **Theorem:** Let $(\xi_i)_{i \in \mathbb{Z}}$ be a sequence of random variables with values in some Polish space $X$. Assume that $(\Omega, \mathcal{T}, \mathbb{P})$ is rich enough to contain a rand... |
I was looking for a family of functions, $f_{n}:[0,1]\to\mathbb{C}$ with pointwise convergence to some function $f$ such that $\lim_{n\to\infty}\int_{0}^{1}f_ndt\neq\int_{0}^{1}fdt$.
Taking $f=1$ and $f_{n}=e^{2\pi it^{n}}$ satisfies the pointwise convergence...but I have no idea what is going on with the integra... |
Has this integral got a solution? |
In the Comments you write:
> … it's exactly a 50:50 chance that I either selected the car or the "set of all other goats and the car", isn't it? Because the car can only be in one of two sets, regardless of the number of goats in it and therefore the chance that the car is in one of two sets is 50%.
You seem to b... |
[Here][1], the Ramanujan Master Theorem is proven in the following way:
$$
\begin{split}
\int_0^\infty\sum_{n=0}^{\infty}\frac{(-1)^n}{n!} \lambda(n)x^n
& = \int_0^\infty\sum_{n=0}^{\infty}\frac{(-1)^n}{n!} E^n\lambda(0)x^n \\
& = \int_0^\infty x^{s-1}e^{-Ex}dx \;\lambda(0) = \frac{\Gamma(s)}{E^s}\lambda(0) = ... |
Let $\mathbb{F}_{11}=\{\overline{0},\overline{1},\ldots,\overline{10}\} $ be a finite field and $A=(a_{ij})_{3 \times 3}$ a matrix such that $a_{ij} \in \mathbb{F}_{11}, \ \ \forall \ 1\leq i,j\leq 3$. What is the probability that $rank(A)=1$.
I tried as follows: There is $11^9$ matrices of the described form. If $... |
I have the sphere manifold $M$ with dimension n and a function $\hat{f}_x : T_x M \rightarrow \mathbb{R}$, which is defined by $\hat{f}_x (v) = f ( exp_x(v))$, where exp is the exponential map from $T_x M$ to $M$ at a point $x \in M$. $T_x M$ is the tangent space at a point $x.$
$f(y)$ is defined by $y^T A y$ for a ... |
I have the sphere manifold $M$ with dimension n and a function $\hat{f}_x : T_x M \rightarrow \mathbb{R}$, which is defined by $\hat{f}_x (v) = f ( exp_x(v))$, where exp is the exponential map from $T_x M$ to $M$ at a point $x \in M$. $T_x M$ is the tangent space at a point $x.$
$f(y)$ is defined by $y^T A y$ for a ... |
What about $f_n(x)=n, x\in (0,1/n]$ and $f_n(x)=0, x=0 $ or $x\in (1/n,1]$?
Clearly $f_n\to 0$ but $\displaystyle\int_0^1 f_n(x) dx=1$ for all $n\geq 1$. |
So I know how to calculate percentile on a simple data set:
lets say student grades are 38, 47, 49, 58, 60, 65, 70, 79, 80, 92
Number of scores below 70 = 6
Using the percentile formula,
Percentile = (Number of Values Below “x” / Total Number of Values) × 100
Percentile of 70
= (6/1... |
I am a noob at math and I had to solve this equation for an exam:
The curve Y is the intersection of the plane x=1 and the cone `z^2=x^2+y^2`, find the curvature at a point `p(-1,2,sqrt(5))
This is how I solved it:
**1.** The intersection of the plane and the cone is giving the function `f(x,y,z)=1+y^2-z^2`
*... |
Let $k$ such that $\operatorname{char}(k)=0$ and $C\subset \mathbb{P}^2$ be the smooth projective curve defined by
$$X^4+Y^4+Z^4=0$$
I want to compute the divisor of $d\left(\frac{X}{Z}\right)$.
My approach is the following. First look at the points where $Z=1$. If we look at the affine plane $\mathbb{A}^2\cong \... |
I am reading a [book of E. Rio][1] and I found there a theorem (*without a proof*) about coupling. Please see below.
> **Theorem:** Let $(\xi_i)_{i \in \mathbb{Z}}$ be a sequence of random variables with values in some Polish space $X$. Assume that $(\Omega, \mathcal{T}, \mathbb{P})$ is rich enough to contain a rand... |
> Consider the nonlinear wave problem
$$
\begin{cases}
u_t + (x+u)u_x = 0\\
u(x,0) = x
\end{cases}
$$
>
> a) Write down Monge's equations in the for $du/d\tau$...,etc, and solve them.
>
> b) Plot the characteristics in the tx plane
So, I honestly don't know how to do this. I know the process when we have ... |
I have a simple and stupid question if I have a convex quadratic optimization problem with polyhedral constraints as follows:
$$
\begin{aligned}
\inf_{x \in \mathbb{R}^{n}} & x^{\top} Ax + b^{\top}x\\
& \text{s.t. } Hx \le h.
\end{aligned}
$$
If I introduce the Lagrange function as below:
$$
\inf_{x\in\math... |
I am a noob at math and I had to solve this equation for an exam:
The curve Y is the intersection of the plane $x=1$ and the cone $z^2=x^2+y^2$, find the curvature at a point $P(-1,2,\sqrt{5})$
This is how I solved it:
**1.** The intersection of the plane and the cone is giving the function $f(x,y,z)=1+y^2-z^2$
... |
|z-a| represents the distance of z from a
<br>
hence,
<br>
|z-2(1+i)| represents the distance of z from 2(1+i) it is true for all others too
<br>
so, if you see the points (2,2);(-1,5);(6,-2) {it's my way to represent complex no. on the argand plane (real part, imaginary part)}
<br>
so these points are collinea... |
In the Comments you write:
> … it's exactly a 50:50 chance that I either selected the car or the "set of all other goats and the car", isn't it? Because the car can only be in one of two sets, regardless of the number of goats in it and therefore the chance that the car is in one of two sets is 50%.
You seem to b... |
I am studying Hodge theory on complex manifolds; Höring's notes (https://math.univ-cotedazur.fr/~hoering/hodge/hodge.pdf p. 87) suggest, as an exercise (4.38) and I guess as a counterexample to Hodge decomposition when our manifold is not compact, that $ \mathbb{C}$ with standard Kähler structure doesn't admit a decomp... |
# Theorem
Let $C$ be a normalization constant and $\lambda_1,\ldots, \lambda_k$ a number sequence.
The integral over $\mathbb{R}^k$:
$$
\int_{\mathbb{R}^k} C \exp\left\{-\frac{1}{2}\sum_{i=1}^k\lambda_iy^2_i\right\} dy_1\cdots dy_k
$$
be finite only if **all the $\lambda$'s are positive**.
# Question
Can th... |
There are two sets of four 2D points.
There is a 1 to 1 correspondence between the two sets.
The points in both sets are close to the following nominal positions, only deviate with a random value in both direction: (-1.5, 0.0), (-0.5, 0.0), (0.5, 0.0), (-1.5, 0.0).
I would like to find a transformation (translatio... |
There are two sets of four 2D points.
There is a 1 to 1 correspondence between the two sets.
The points in both sets are close to the following nominal positions, only deviate with a random value in both direction: (-1.5, 0.0), (-0.5, 0.0), (0.5, 0.0), (-1.5, 0.0).
I would like to find a transformation (translatio... |
In the Comments you write:
> … it's exactly a 50:50 chance that I either selected the car or the "set of all other goats and the car", isn't it? Because the car can only be in one of two sets, regardless of the number of goats in it and therefore the chance that the car is in one of two sets is 50%.
You seem to b... |
There are two sets of four 2D points.
There is a 1 to 1 correspondence between the two sets.
The points in both sets are close to the following nominal positions, only deviate with a random value in both direction: (-1.5, 0.0), (-0.5, 0.0), (0.5, 0.0), (-1.5, 0.0).
I would like to find a transformation (translatio... |
In the Comments you write:
> … it's exactly a 50:50 chance that I either selected the car or the "set of all other goats and the car", isn't it? Because the car can only be in one of two sets, regardless of the number of goats in it and therefore the chance that the car is in one of two sets is 50%.
You seem to b... |
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