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I am reading "Analysis on Manifolds" by James R. Munkres.
>Definition. Let $k\leq n$. Let $A$ be open in $\mathbb{R}^k$, and let $\alpha:A\to\mathbb{R}^n$ be a map of class $C^r (r\geq 1)$. The set $Y=\alpha(A)$, together with the map $\alpha$, constitute what is called parametrized-manifold, of dimension $k$. We de... |
Why did the author need to extend $\alpha$ to a neighborhood of $Q$? ("Analysis on Manifolds" by James R. Munkres.) |
I am self-studying Boyd's convex optimization textbook, and I am befuddled by Example 9.1 shown here in screenshot. Specifically, I am confused how he gets that the supremum of q^T z over the ellipsoid is equal to that first term on the right-hand side.
I tried to formulate it as a constrained optimization problem o... |
Using quantum field theory arguments, in https://arxiv.org/pdf/2401.05733.pdf, we found that there is a parametric representation of the (slightly generalised) Euler-Beta function:
\begin{equation}\label{gen-beta}
\frac{\Gamma\left(\alpha-s_{1}\right)\Gamma\left(\alpha-s_{2}\right)}{\Gamma\left(\beta-s_{1}-s_{2}\... |
Using quantum field theory arguments, in https://arxiv.org/pdf/2401.05733.pdf, we found that there is a parametric representation of the (slightly generalised) Euler-Beta function:
\begin{equation}\label{gen-beta}
\frac{\Gamma\left(\alpha-s_{1}\right)\Gamma\left(\alpha-s_{2}\right)}{\Gamma\left(\beta-s_{1}-s_{2}\... |
Given independent random samples $(X_1,...,X_m)$ and $(Y_1,...,Y_n)$, respectively, from the following distributions of $X$ and $Y: X \sim \lambda_1 \exp (-\lambda_1 x) I(x >0)$, and $Y \sim \lambda_2 \exp (-\lambda_2 y) I(y >0)$. Consider the problem of testing the null hypothesis $H_0:\lambda_1=\lambda_2 $ against th... |
How to design this two sample exponential distribution test? |
I'm currently looking at the following proof in my textbook:
A nontrivial vector space $V$ over an infinite field $F$ is not the union of a finite number of proper subspaces.
Suppose that $V = S_{1} \cup \cdots \cup S_{n}$, where we may assume that
$$S_{1} \nsubseteq S_{2} \cup \cdots \cup S_{n}$$
Let $w \... |
I have a function f(x) defined below as,
$f(x) = \sum_{n=0}^N c_n(g_n(x + n) + g_n(x - n))$
I have no knowledge of what $c_n$ is, but I do have access to f(x) numerically. Is there a way to apply an operator that would cancel all terms except for one associated with a particular n? My goal is to extract $g_n(x)$.... |
When can $\,x\bmod m\,$ be computed from $\,x\bmod n?$ |
I have the following statement.
Suppose $(a_n)$ is a sequence of complex numbers satisfying the following properties.
i) $a_{nm} = a_na_m$ for all positive integers $n, m$.
ii)
\begin{align}
\sum_{n \geq 1} |a_n|^2 < \infty.
\end{align}
Then $|a_n| \leq 1/n$ for all positive integers $n$.
I have 3... |
>__Conjecture.__ For all sufficiently large $n$ (something like $n \geq 3$); either $\exists k \geq 0$ such that $\gcd(2^k - 1, \frac{p_n\#}{6}) = 1$, _or_ $\exists k \geq 0$ such that $\gcd(3^k - 1, \frac{p_n\#}{6}) = 1$, or both.
Where $p_n\# := p_{n}p_{n-1} \cdots p_1$ is primorial.
__Question.__
How can ... |
Using quantum field theory arguments, in https://arxiv.org/pdf/2401.05733.pdf, we found that there is a parametric representation of the (slightly generalised) Euler-Beta function:
\begin{equation}\label{gen-beta}
\frac{\Gamma\left(\alpha-s_{1}\right)\Gamma\left(\alpha-s_{2}\right)}{\Gamma\left(\beta-s_{1}-s_{2}\... |
I have the following statement.
Suppose $(a_n)$ is a sequence of complex numbers satisfying the following properties.
i) $a_{nm} = a_na_m$ for all positive integers $n, m$.
ii)
\begin{align}
\sum_{n \geq 1} |a_n|^2 < \infty.
\end{align}
Then $|a_n| \leq n^{-1/2}$ for all positive integers $n$.
I h... |
Why Are There Finitely Many Regular Polytope Compounds? |
I am self-studying Boyd's convex optimization textbook, and I am befuddled by Example 9.1 shown here in screenshot. Specifically, I am confused how he gets that the supremum of $q^{\top} z$ over the ellipsoid is equal to that first term on the right-hand side.
I tried to formulate it as a constrained optimization pr... |
I've recently been studying about the General relativity and Einstein field equation. When I reading of the derivation of the field equation, I encounterd a method called Einstein-Hilbert action. This action is given as
\begin{align}
S_{EH}[g_{ij}]\,=\,\intop_M \sqrt{-g}R\,dx_3dx_2dx_1dt
\end{align}
where $S_{EH}[... |
$n=10x+y. 13|n \iff 13|(x+4y)?$
$(40x+4y)-(x+4y)=39x$
$(x+4y)+39x=4(10x+y)$
If a prime divides two numbers, it divides their sum and their difference.
IF $13|(x+4y)$ and $13|39x$, so $13|4(10x+y)$. Since $13\not|4 \implies 13|10x+y$
By similar reasoning, one can prove the other direction .
|
>__Conjecture.__ For all sufficiently large $n$ (something like $n \geq 3$); either $\exists k \geq 0$ such that $\gcd(2^k - 1, \frac{p_n\#}{6}) = 1$, _or_ $\exists k \geq 0$ such that $\gcd(3^k - 1, \frac{p_n\#}{6}) = 1$, or both.
Where $p_n\# := p_{n}p_{n-1} \cdots p_1$ is primorial.
__Question.__
How can ... |
I've recently been studying about the General relativity and Einstein field equation. When I reading of the derivation of the field equation, I encounterd a method called Einstein-Hilbert action. This action is given as
\begin{align}
S_{EH}[g_{ij}]\,=\,\intop_M \sqrt{-g}R\,dx_3dx_2dx_1dt
\end{align}
where $S_{EH}[... |
__Context & Interest.__
See this MSE post about a [twin-prime related topology][1]. Basically $0$ is a easily seen to be a generic point in this topology. Every generic point is clearly dense as a singleton subset, meaning its closure is the whole space $\Bbb{Z}/p_n\#$. In an approach toward potentially proving the... |
I was doing a homework for measure theory, and had a question involving two functions being equal almost everywhere. I've solved the problem, but am not sure about this other example I thought of. Take $f,g : \{1,2\} \rightarrow \mathbb{R}$ to be two functions such that $f(1) = f(2) = 0$, and $g(1) = 0, g(2) = 1$. Then... |
Given a manifold $M$, a differential $k$-form $\omega$ assigns to each point $p$ $\epsilon$ $M$ a $k$-covector $\omega_p$ $\epsilon$ $\bigwedge^k \left(T_p^*M\right)$, where $\bigwedge^k \left(T_p^*M\right)$ is the space of alternating $k$-tensors on the tangent space $T_pM$.
If $\omega = \sum f_i dx^i$ denotes a 1... |
Let $(X, \mathcal{A}, \mu)$ be a finite measure space and $u$ be a measurable function. Show that
$$
\sum_{n = 1}^\infty \mu\{\vert u \vert \ge n\} \le \int \vert u \vert d\mu \le \sum_{n = 0}^\infty \mu\{\vert u \vert \ge n\}
$$
In the proof, I am able to show that
$$
\sum_{n = 1}^\infty 1\{\vert u \ver... |
I was recently reading a paper (Lumley, R. R., & Zervos, M. (2001). A Model for Investments in the Natural Resource Industry with Switching Costs.) that contains the following statement:
Elements of the Sobolev space $W^{2,p}_{loc}((0,\infty))$, $p\geq 1$, have sufficient smoothness to be able to apply Ito's formula... |
> Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a lipschitz
> continuous function that satisfies $ \langle f(x)-f(y), x-y\rangle
\geq \alpha\|x-y\|_2^2 \text { for all } x, y \in \mathbb{R}^n $ and
> fixed $\alpha>0$.
>
> I am trying to find a range $\omega>0$ such that the fixpoint
> iteration $$ x=\Phi... |
In information theory, it is well known that the capacity of a channel can be achieved asymptotically using random coding method (Section 7 of [Cover & Thomas' textbook][1]).
However, in coding theory, it is only shown that we can get a code that achieves Gilbert-Varshamov bound (I will show how this is related to c... |
In information theory, it is well known that the capacity of a channel can be achieved asymptotically using random coding method (Section 7 of [Cover & Thomas' textbook][1]).
However, in coding theory, it is only shown that we can get a code that achieves Gilbert-Varshamov bound (I will show how this is related to c... |
This originates from a computer science problem: suppose you have a variable $i$ with init value $0$. Then you create $n$ threads that each thread $i\rightarrow i+1$ for $m$ times. What's the possible value of $i$ when every thread finish.
One would expected that $i=mn$, but the tricky part is, there is a phenomena ... |
**Full exercise.**
Suppose $V$ is finite-dimensional and $U$ is a subspace of $V$. Then
$$
\text{dim}\ U^0 = \text{dim}\ V - \text{dim}\ U.
$$
**My Question.**
My strategy is exactly how the first proof in this post shows goes about showing the above result: https://math.stackexchange.com/a/2475382/64575... |
Have I shown these linear functionals span the annihilator? |
**Full exercise.**
Suppose $V$ is finite-dimensional and $U$ is a subspace of $V$. Then
$$
\text{dim}\ U^0 = \text{dim}\ V - \text{dim}\ U.
$$
**My Question.**
My strategy is exactly how the first proof in this post goes about showing the above result: https://math.stackexchange.com/a/2475382/645756
... |
There is a questions that how to draw the estimators (ratio estimator) if we know the prior for the numerator or the ratio estimator.
Assume that we have three coefficients: $a_1,a_2$ and $a_3$ . And we beleive that all the three coefficients follow the same exponential distributions say $a_1,a_2,a_3\to Exp(\lambda)... |
Let $u$ be a tempered distribution and $\psi$ a Schwartz function.
I have seen two definitions for $\psi*u:$
1. $\psi*u:$ defined as a function of $\mathbb{R}^d,$ that is, for $x \in \mathbb{R}^d,\psi*u(x)=u(\psi(\cdot-x)).$
2. $\psi*u$ defined as a tempered distribution, that is, for $\phi \in \mathcal{S},(\psi*... |
**Full exercise.**
Suppose $V$ is finite-dimensional and $U$ is a subspace of $V$. Then
$$
\text{dim}\ U^0 = \text{dim}\ V - \text{dim}\ U.
$$
**My Question.**
My strategy is exactly how the first proof in this post goes about showing the above result: https://math.stackexchange.com/a/2475382/645756
... |
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... |
**I am trying to understand why the Quadratic Variation property of Brownian Motion is true.**
That is, if $W$ is a Wiener Process (https://en.wikipedia.org/wiki/Wiener_process), then a very small change in the time results in an equally small change in position squared.
Mathematically, we can express the Quadr... |
Why is the Quadratic Variation true in Brownian Motion? |
**I am trying to understand why the Quadratic Variation property of Brownian Motion is true.**
That is, if $W$ is a Wiener Process (https://en.wikipedia.org/wiki/Wiener_process), then a very small change in the time results in an equally small change in position squared.
Mathematically, we can express the Quadr... |
In information theory, it is well known that the capacity of a channel can be achieved asymptotically using random coding method (Section 7 of [Cover & Thomas' textbook][1]).
However, in coding theory, it is only shown that we can get a code that achieves Gilbert-Varshamov bound (I will show how this is related to c... |
**I am trying to understand why the Quadratic Variation property of Brownian Motion is true.**
That is, if $W$ is a Wiener Process (https://en.wikipedia.org/wiki/Wiener_process), then a very small change in the time results in an equally small change in position squared.
Mathematically, we can express the Quadr... |
**I am trying to understand why the Quadratic Variation property of Brownian Motion is true.**
That is, if $W$ is a Wiener Process (https://en.wikipedia.org/wiki/Wiener_process), then a very small change in the time results in an equally small change in position squared.
Mathematically, we can express the Quadr... |
Are there simple examples of non-invertible elements (i.e. zero divisors) in a real Clifford algebra $C(V)$ other than $0$?
Notation: $V$ has a positive definite metric $q$, and $uv+vu=-2q(u, v)$. Therefore, if $V$ is
$1$-dimensional, $C(V)$ is isomorphic to ${\mathbb C}$; if $V$ is $2$-dimensional, $C(V)$ is is... |
Concrete example of non-invertible element in a Clifford algebra? |
**I am trying to understand why the Quadratic Variation property of Brownian Motion is true.**
That is, if $W$ is a Wiener Process (https://en.wikipedia.org/wiki/Wiener_process), then a very small change in the time results in an equally small change in position squared.
Mathematically, we can express the Quadr... |
**I am trying to understand why the Quadratic Variation property of Brownian Motion is true.**
That is, if $W$ is a Wiener Process (https://en.wikipedia.org/wiki/Wiener_process), then a very small change in the time results in an equally small change in position squared.
Mathematically, we can express the Quadr... |
**I am trying to understand why the Quadratic Variation property of Brownian Motion is true.**
That is, if $W$ is a Wiener Process (https://en.wikipedia.org/wiki/Wiener_process), then a very small change in the time results in an equally small change in position squared.
Mathematically, we can express the Quadr... |
In information theory, it is well known that the capacity of a channel can be achieved asymptotically using random coding method (Section 7 of [Cover & Thomas' textbook][1]).
However, in coding theory, it is only shown that we can get a code that achieves Gilbert-Varshamov bound (I will show how this is related to c... |
**I am trying to understand why the Quadratic Variation property of Brownian Motion is true.**
That is, if $W$ is a Wiener Process (https://en.wikipedia.org/wiki/Wiener_process), then a very small change in the time results in an equally small change in position squared.
Mathematically, we can express the Quadr... |
In the following articles: <https://proceedings.mlr.press/v202/yuan23b/yuan23b.pdf>(section 7.4), <https://arxiv.org/pdf/2207.02917.pdf>(Theorem 4), The Reproducing Kernel Hilbert Spaces(RKHS) are considered as an application of the Yoneda Lemma, that interprets K(x,y) = <K(x,-),K(y,-)> as the Yoneda embedding. However... |
[The question is here ][1]I graphed the functions, and it is a semi-circle that lies on the x-axis plus two lines Then I calculated it by area and it is half square minus half semi-circle which is 8-2pi and the other two triangles are 1 and 2 so the total area is 11-2pi, but if I calculate the definite integrals in the... |
**I am trying to understand why the Quadratic Variation property of Brownian Motion is true.**
That is, if $W$ is a Wiener Process (https://en.wikipedia.org/wiki/Wiener_process), then a very small change in the time results in an equally small change in position squared.
Mathematically, we can express the Quadr... |
Given the function f(x)=2-sqrt(4-x^2) x∈[-2,2] and |x-5|-1 x>2, on the interval [-2,6],and use it to calculate the definite integral
int(f(x),x=-2..6)in terms of area.
I graphed the functions, and it is a semi-circle that lies on the x-axis plus two lines Then I calculated it by area and it is half square minus hal... |
The title says it already. For a Boolean ring B which is also a field, I have to prove that the relation $B \cong \mathbb Z /2\mathbb Z$ holds. Any help? I have to prove this as a part of an exercise and in the previous part of the exercise, I have already shown that $x+x=0$ for any $x\in B$ and that $B$ is commutative... |
**I am trying to understand why the Quadratic Variation property of Brownian Motion is true.**
That is, if $W$ is a Wiener Process (https://en.wikipedia.org/wiki/Wiener_process), then a very small change in the time results in an equally small change in position squared.
Mathematically, we can express the Quadr... |
In the following articles: <https://proceedings.mlr.press/v202/yuan23b/yuan23b.pdf>(section 7.4), <https://arxiv.org/pdf/2207.02917.pdf>(Theorem 4), The Reproducing Kernel Hilbert Spaces(RKHS) are considered as an application of the Yoneda Lemma, that interprets $K(x,y) = \langle K(x,-),K(y,-)\rangle_H$ as the Yoneda e... |
(a) Graph the function \( f(x) =
\begin{cases}
2 - \sqrt{4 - x^2}, & -2 \leq x \leq 2 \\
|x - 5| - 1, & x > 2
\end{cases} \)
on the interval \[-2, 6] \) and use it to calculate the definite integral f(x) in terms of area.
I graphed the functions, and it is a semi-circle that lies on the x-axis plus two lines... |
(a) Graph the function \( f(x) =
\begin{cases}
2 - \sqrt{4 - x^2}, & -2 \leq x \leq 2 \\
|x - 5| - 1, & x > 2
\end{cases} \)
on the interval \[-2, 6] \) and use it to calculate the definite integral $$ \int_{-2}^{6} f(x) \,dx $$ in terms of area.
I graphed the functions, and it is a semi-circle that lies on ... |
(a) Graph the function f(x) =
\begin{cases}
2 - \sqrt{4 - x^2}, & -2 \leq x \leq 2 \\
|x - 5| - 1, & x > 2
\end{cases} on the interval \[-2, 6] \) and use it to calculate the definite integral $$ \int_{-2}^{6} f(x) \,dx $$ in terms of area.
I graphed the functions, and it is a semi-circle that lies on the x-a... |
Stimulated emission in laser can be described by the classical oscillator theory (A.E. Siegman *Lasers* chapter 2 University Science Books 1986).
According to Siegman, the overall susceptibility is
$$\tilde{\chi}={Ne^2\over m\epsilon}{1\over \omega_a^2-\omega^2+j\omega\Delta\omega_a}$$
The exact form of the ima... |
(a) Graph the function
$$
f(x) =
\begin{cases}
2 - \sqrt{4 - x^2}, & -2 \leq x \leq 2 \\
|x - 5| - 1, & x > 2
\end{cases}
$$
on the interval [-2, 6] and use it to calculate the definite integral
$$
\int_{-2}^{6} f(x) \, dx
$$
in terms of area.
I graphed the functions, and it is a semi-circle that l... |
So, a visual of that function is below: positive area in blue, negative in red:
[![enter image description here][1]][1]
> Should I count the area below the x-axis as negative in this question?
Yes. The integral gives the *signed* area. You are being told to calculate the integral, using the area as the motivat... |
Stimulated emission in laser can be described by the classical oscillator theory (A.E. Siegman *Lasers* chapter 2 University Science Books 1986).
According to Siegman, the overall susceptibility is
$$\tilde{\chi}={Ne^2\over m\epsilon}{1\over \omega_a^2-\omega^2+j\omega\Delta\omega_a}$$
The exact form of the ima... |
I just came to know that it is possible to define a _generalized base_ as an infinite sequence of natural numbers $\mathbf b=(b_1,b_2,\dots)$ where $b_i\ge 2$ for all $i$. With this definition, any $m\in \mathbb N$ can be uniquely expressed as
$$m=\overline{x_n\dots x_1}^\mathbf b := x_1 + x_2\cdot b_1 + x_3\cdot b_1b... |
Let ∈ℕ₀ and ∈ℚ₊ such that ≥2 and 100 ∈ ℕ₊. We would like to compute ⌈/⌉. At our disposal we have only operations on integers with downrounding: addition, subtraction, integer multiplication, integer division div (= ⌊…/…⌋), taking the remainder after division rem. If were an integer, we'd compute (+-1) div = ⌊(+-1)/⌋.... |
How to reduce division with uprounding to arithmetic operations with downrounding if the denominator is not necessarily an integer? |
This answer is partial. It concerns mainly natural solutions of the equation
The brief search showed several related papers, which I listed below as the references. Unfortunately, I had no time to study them all, so my survey below is incomplete.
It seems that all of these references concern natural solutions of ... |
Consider the following:
Let $(x_n)_{n \in \mathbb{N}}$ be a sequence of real valued functions $[0,1] \rightarrow \mathbb{R}$. And $\mathcal{P}:={p_t:t \in [0,1]}$ a family of seminorms such that $p_t(x)=0$ for any function $x$ with $x(t)=0$. Given that, we can talk about the pointwise convergence of a sequence of... |
This answer is partial. It concerns mainly natural solutions of the equation
The brief search showed several related papers, which I listed below as the references. Unfortunately, I had no time to study them all, so my survey below is incomplete.
It seems that all of these references concern natural solutions of ... |
I am looking into repeated operations, and it seems really hard to *precisely* define multiplication.
Of course, for integer $b$ and real number $a$, we use the grade school definition we all know:
$$ab = \underbrace{a + a + a + \cdots + a}_{b\text{ times}}$$
but what about for real numbers $a$ and $b$?
Fo... |
Stimulated emission in laser can be described by the classical oscillator theory (A.E. Siegman *Lasers* chapter 2 University Science Books 1986).
According to Siegman, the overall susceptibility is
$$\tilde{\chi}={Ne^2\over m\epsilon}{1\over \omega_a^2-\omega^2+j\omega\Delta\omega_a}$$
The exact form of the ima... |
>> Given $E(X)=100,\sigma(X)=10$ and $P(X\le 10k)\ge 4/5$, how do we find a lower bound on $k$ using the Chebyshev's inequality
Let $X$ be any r.v. with finite mean, $μ$, and finite variance. Then $∀a > 0,$
$$
P\bigg(|X-\mu|\ge a\bigg)\le\frac{\sigma^2(X)}{a^2}
$$
which is the Chebyshev’s inequality.
$$
P(... |
This answer is partial. It concerns mainly natural solutions of the equation
The brief search showed several related papers, which I listed below as the references. Unfortunately, I had no time to study them all, so my survey below is incomplete.
It seems that all of these references concern natural solutions of ... |
This answer is partial. It concerns mainly natural solutions of the equation
The brief search showed several related papers, which I listed below as the references. Unfortunately, I had no time to study them all, so my survey below is incomplete.
It seems that all of these references concern natural solutions of ... |
Let ∈ℕ₀ be an unknown variable and ∈ℚ₊ be a known constant such that ≥2 and 100 ∈ ℕ₊. (So we can pre-compute anything regarding , e.g., represent 100 as a product of powers of primes.) We would like to compute $\left\lceil\frac{}{}\right\rceil$. At our disposal we have only operations on integers with downrounding: add... |
Let $X$ be a separated scheme of finite type over an algebraically closed field $k$ of characteristic $p>0$, possibly not equidimensional. Assume that $X$ has only two irreducible components $X_1$ and $X_2$, and that each $X_i$ is a nonsingular complete variety. Assume further that $X_1$ and $X_2$ meet transversally at... |
Suppose I have two sets and they have the same cardinality which I believe means that there exists a bijection between them, kind of like an isomorphism but I don't know if I can call it that since all sets may not be groups.
Eg, let's consider the set $(0,1)$ and the set of irrationals between $(0,1)$, since they bo... |
Let ∈ℕ₀ be an unknown variable and ∈ℚ₊ be a known constant such that ≥2 and 100 ∈ ℕ₊. (So we can pre-compute anything regarding , e.g., represent 100 as a product of powers of primes.) We would like to compute $\left\lceil\frac{}{}\right\rceil$. At our disposal we have only operations on integers with downrounding: add... |
Consider the question
>Let $f:\mathbb N \to \mathbb R$ satisfy $f(mr)<f(r)$ for all $m,r\in \mathbb N$. Is $f(k)$ decreasing for all $k>k_0$ for some $k_0$?
The answer is clearly no as we have the following counterexample
$$f(n)=\begin{cases}
1/n & \text{when $n$ is even}\\
2/n & \text{when $n$ is odd}
\end{cas... |
Let $f$ satisfy $f(mr)<f(r)$ for all $m,r\in \mathbb N$. Is $f(k)$ decreasing for all large $k$? |
How to reduce division with uprounding to integer-arithmetic operations with downrounding if the denominator is not necessarily an integer? |
Let ∈ℕ₀ be an unknown variable and ∈ℚ₊ be a known constant such that ≥2 and 100 ∈ ℕ₊. (So we can pre-compute anything regarding , e.g., represent 100 as a product of powers of primes.) We would like to compute $\left\lceil\frac{}{}\right\rceil$. At our disposal we have only operations on integers with downrounding: add... |
Let $a\in\mathbb R^n$ be a tuple of length $n\in\mathbb Z_{>0}$. Let $X=\{a_i:1\le i\le n\}$ be the set of elements of $a$ for $x\in X$ let
$$i(x)=\min\{j:a_j=x\}$$
be the first occurence of $x$ in $a$. Now, for $x,y\in X$ let $x<y$ if $i(x)<i(y)$.
This order is known as *order of first occurrence*.
Let $x:\{1,\d... |
I'm studying Stochastic calculus and applications, Cohen. He claims that
> As $M \in \mathcal{H}^{2,d}$, we know $\lim_k \sum_{j=1}^k \mathbb{E}[(\Delta M_{T_n})^2] < \infty$.
Note that $T_n$ is all jump times of $M$, which is either totally inaccessible or predictable time.
But how this be true? I think it is t... |
I'm studying Stochastic calculus and applications, Cohen. He claims that
> As $M \in \mathcal{H}^{2,d}$, we know $\lim_k \sum_{j=1}^k \mathbb{E}[(\Delta M_{T_n})^2] < \infty$.
Note that $T_n$ is all jump times of $M$, which is either totally inaccessible or predictable time.
But how this be true? I think it is t... |
What does "differentiate with respect to $g(x)$" actually mean intuitively? |
I'm studying Stochastic calculus and applications, Cohen. He claims that
> As $M \in \mathcal{H}^{2,d}$, we know $\lim_k \sum_{j=1}^k \mathbb{E}[(\Delta M_{T_n})^2] < \infty$.
Note that $T_n$ is all jump times of $M$, which is either totally inaccessible or predictable time.
But how this be true? I think it is t... |
Brainstorm: If $a+c \le w \le b+d$ then $w$ is "some proportion" between $a+c$ and $b+d$. Pick $x,y$ so that the are the same proportion between $a$ and $b$ and $c$ and $d$ respectively. How?
Let $r = \frac {w- (a+c)}{(b+d)-(a+c)}$. Thats the "some proportion". Note: $0 \le r \le 1$.
Let $x = a + r\times (b-a... |
It is obvious that $d+1$ points uniquely determine a circumscribed hypersphere in $d$ -dimensional space. If the number of points $n \le d$, the circumscribed hypersphere is not unique, but the smallest one exists and must have its center lying inside the space spanned by these $n$ points.
For example, when the hype... |
Consider the construction of a Kalman filter for estimating the attitude of an aircraft with IMU and a system dynamics in state space representation like this $\dot{x} = Ax + Bu$
There are many papers focusing on this, some have treated angular velocity $\omega$ which can be measured by the IMU as an control input $... |
How is the sensor measurement treated in a Kalman filter? |
Zorich first define differentiable as
[![enter image description here][1]][1]
Thus, Zorich only requires a point to be a limit point that is in the domain. Many of the zorich's later theorem used this general definition. On the other hand, when proving Taylor's Theorem, zorich state the following:
[![enter ima... |
I have a process/model the depends on a single parameter $\lambda$ that generates $n+1$ outcomes. From $N$ events I can estimate the probabilities $\hat{p}_k$ from $N_k/N$ using a MLE for a multinomial distribution (MD). From my model I can generate probabilities $p_k$.
How do I construct the best estimate of $\... |
Consider the question
>Let $f:\mathbb N \to \mathbb R$ satisfy $f(mr)<f(r)$ for all $m,r\in \mathbb N$, $m>1$. Is $f(k)$ decreasing for all $k>k_0$ for some $k_0$?
The answer is clearly no as we have the following counterexample
$$f(n)=\begin{cases}
1/n & \text{when $n$ is even}\\
2/n & \text{when $n$ is odd}
\... |
Now we have several independent and identically distributed random variables following the uniform distribution on the interval [0, 1].They are denoted as $x_1, x_2, x_3, ..., x_m$ and $y_1, y_2, ..., y_n$ respectively.Let
$$
D_{i,j} = \begin{cases}
x_i-y_j, & x_i\ge y_j \\\\
1+x_i-y_j, & x_i<y_j
\end{cases}
$... |
I recently learned about hypercomplex systems that are taken over the reals, i.e. the dual numbers for which $j^2=1$, $j≠1$, and the dual numbers for which $ε^2=0$, $ε≠0$. These number systems, along with the complex numbers, have some nice properties, namely they represent all the possibilities for 2-dimensional unit... |
At the end of pg 65 of Milne's ANT notes (https://www.jmilne.org/math/CourseNotes/ANT.pdf), he claimed $X^4+X^3+X^2+X+1\equiv (X+4)^4$ (mod 5) to be obvious. I understand that LHS is Eisenstein at 5 by substituting $X=Y-1$, and that if $f$ is Eisenstein at $p$, then $p$ ramifies totally in $\mathbb{Q}[\alpha]$ for $\al... |
To denote a triangle we write $\triangle ABC$.
How about a circular sector with $O$ at the center of the circle and with the arc $AB$.
Is there a standard way such as $\lhd OAB$ where $\lhd$ has a round end? |
In this [other question](https://math.stackexchange.com/q/4838311/909869) I found an approximation as follows: $$\ln(x+1) \approx \dfrac{\ln(x)}{1-x^{-\frac{1}{\ln(2)}}}$$
By differentiating both sides, since just one logarithmic terms remains you could find the following approximation:
$$\ln(x)\approx \dfrac{\ln(2... |
How the inequality given below $$\left(p+\frac{2-p^2}{r}\right)^2<2$$ will eventually be simplified to the inequality $$2pr+2-p^2<r^2$$
I got the reference from this quora [Link](https://www.quora.com/How-do-you-explain-in-simple-easy-steps-the-reason-why-Walter-Rudin-chose-q-p-p-2-2-p-2-in-his-book-Principles-of-Mat... |
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