instruction stringlengths 12 30k |
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I have angle between two vectors:
pseudocode for func angleBetween:
dotProduct = v1.x * v2.x + v1.y * v2.y;
determinant = v1.x * v2.y - v1.y * v2.x;
angle = atan2(determinant, dotProduct);
How to compare
$$\text{angleBetween}(v_1,v_2) < \text{angleBetween}(v_3,v_4)$$
using only deter... |
How to compare angles without function atan2? |
We want to show:
$$*(e_{i_1} \wedge\ldots \wedge e_{i_k})=\epsilon e_{j_1} \wedge\ldots \wedge e_{j_{n-k}}$$
where $\epsilon$ is the sign of the permutation $i_1 \cdots i_k \bar i_1 \cdots \bar i_{n-k}.$
The book doesn't provide any short of proof but according to [Wikipedia][1] we look the Gram determinant ... |
I am interested in analytically continuing the differentiation operation on functions, and I am currently focusing on trigonometric functions. Any high schooler can conjecture that $\frac{d^a}{dx^a}\sin(x) = \sin(x+\pi a/2)$ applies not only for integers, but for all complex numbers; I, however, am trying to prove the ... |
Proof that $\frac{d^a}{dx^a}\sin(x) = \sin(x+\pi a/4)$ iff these two infinite series are equivalent? |
Elementary proof that If $n+1$ is prime in $\mathbb{Z}$, then $1+\sqrt{-n}$ is prime in $\mathbb{Z}[\sqrt{-n}]$? |
It is a well-known (and frankly magical) property that first-order logic is strongly semantically complete (Gödels completeness theorem).
Independence-friendly logic is just like first-order logic but it allows quantifiers to only depend on some of the variables outside its scope.
For example: $\forall w \forall ... |
Does independence-friendly logic have a completeness theorem? |
Both methods give the same result. The first method, which I used in https://math.stackexchange.com/questions/4751226/find-the-complete-integral-of-following-partial-differential-equation-z-p2?noredirect=1&lq=1, gives the following result:
$$
z=\frac{1}{2}(x+a)^2+\frac{1}{2}(y+b)^2+ab. \tag{1}
$$
Now, let's use the... |
Suppose I have an $N \times N$ matrix of the form $M_{ij} = \sum_{k=0}^{N-i}\sum_{l=0}^{N-j}\frac{1}{2^{k+l+1}}\binom{k+l}{l}$. From numerical experiments, I've noticed that $\det(M)$ appears to be a power of $\frac{1}{2}$ (specifically, $\det(M) = \frac
{1}{2^{N^2}}$), and that $M^{-1}$ has all integer entries. Is th... |
Ok, I made an attempt, but it might be wrong:
Existence of $v \in V / m_f=m_{f,v}$:
- For any $v \in V, m_{f, v}$ divides $m_f$, since if $m_f(f)=0$, then $m_f(f)(v)=0 \ \forall \ v \in V$, implying that $m_{f, v}$, which annihilates $v$, must divide $m_f$.
- The primary decomposition theorem states that $V$ ... |
While reading Donald Knuth's "Sorting and Searching" I have come across a table of inversions which lists the numbers of inversions $k$ for a number of permutations of lengths $n$ (varying from 1 to 6).
There is a zigzag line which separates $I_{n}(n-1)$ from $I_{n}(n)$ as $n$ increases from 1 to 6 down the table.
... |
In a precise sense the answer is **no**.
In terms of expressive power, IF-logic is equivalent to existential second-order logic ESO. Now while ESO is reasonably tame in several ways (e.g. compactness and Lowenheim-Skolem), as far as *complexity* goes it's quite wild: for example, we can reduce the problem of telling... |
Suppose I have an $N \times N$ matrix of the form $M_{ij} = \sum_{k=0}^{N-i}\sum_{l=0}^{N-j}\frac{1}{2^{k+l+1}}\binom{k+l}{l}$. From numerical experiments, I've noticed that $\det(M)$ appears to be a power of $\frac{1}{2}$ (specifically, $\det(M) = \frac
{1}{2^{N^2}}$. How to derive this?), and that $M^{-1}$ has all i... |
As far as I understand, given a function f(x,y) if one wants to show that it is bilinear,
It is sufficient to show that f can be expressed in a form $f(x,y) = x^tAy$ as these forms are always bilinear.
Could one provide an example of a function f(x,y) that cannot be expressed in this form? I dont really understand ... |
I am studying *Multiplicative number theory I: Classical theory by Hugh L. Montgomery, Robert C. Vaughan*. In Appendix A, Theorem A.1. states that $I = \int f dg$ exists if $f$ is continuous and $g$ is of bounded variation. Let $a=x_0 \le x_1 \le \dots \le x_N=b$ is a partition of $[a,b]$, and $S(x_n, \xi_n) = \sum_1^N... |
I was reading Ozawa's book and in Chapter 5 they discuss $C^* $-algebras of locally compact Hausforff étale groupoids. I have a question regarding the reduced $C^* $-algebra construction.
It is first defined by considering the $* $-algebra $C_c(G)$ and endow it with a $C_0(G^{(0)})$-valued inner product defined for ... |
Consider for simplicity a real-valued random variable $X$ with finite moments $m_X(n) = E[X^n]$, $n>0$. Now suppose we wish to construct a new random variable $Y = f(X)$, where $f$ is some measureable function such that the first $N$ moments are equal, e.g., $m_Y(n) = m_X(n)$ for $n \leq N$ for some fixed number $N$. I... |
Let $f : \mathbb{R} \to \mathbb{R}$ a monoton function in $[−r, r]$ for some $r>0$. Prove that the if $$\sum a_n$$ converge absolutely then $$\sum f(a_n)a_n$$ converges absolutely\
I have not idea how to approach this, but i attempted to prove that $$\sum f(a_n)a_n<\sum a_n$$ but i think think it's not so true.
It's... |
> Suppose $H = L_2(B)$ where $B$ is the unit ball in $\mathbb{R}^d$. Let $K(x,y)$ be a measurable function on $B \times B$ that satisfies $|K(x,y)| \leq A|x-y|^{-d+\alpha}$ for some $\alpha > 0$ whenever $x,y \in B$. Define $$Tf(x)=\int_B K(x,y)f(y) \, dy$$
>$\bullet$ Prove that $T$ is a bounded operator on $H$.
... |
While reading Donald Knuth's "Sorting and Searching" I have come across a table of inversions which lists the numbers of inversions $k$ for a number of permutations of lengths $n$ (varying from 1 to 6).
There is a zigzag line which separates $I_{n}(n-1)$ from $I_{n}(n)$ as $n$ increases from 1 to 6 down the table.
... |
While reading Donald Knuth's "Sorting and Searching" I have come across a table of inversions which lists the numbers of inversions $k$ for a number of permutations of lengths $n$ (varying from 1 to 6).
There is a zigzag line which separates $I_{n}(n-1)$ from $I_{n}(n)$ as $n$ increases from 1 to 6 down the table.
... |
The following equality is true for every positive integer $n$ :
$$\sum_{k=0}^n {n \choose k} = 2^n $$
It is a special case ($p = 0$) of the sequence :
$$S_{p, n}=\sum_{k=0}^n k^p {n \choose k} $$
For example :
$$S_{0,n}= 2^{n} \qquad S_{1,n}=n 2^{n-1} \qquad S_{2,n}=n(n+1)2^{n-2} \qquad S_{3,n}=n^2(n+3)2^{n-3} $$
... |
The function $f(x,y)=a(x-b)^2+c(y-d)^2$ is convex.
It is well known and I can prove it ($a>0, b>0$).
What if $F(x,y)=a(g(x,y)-b)^2+c(h(x,y)-d)^2$?
1) Is $F$ always a convex function?
Or
2) if $g$ and $h$ is convex, then is $F$ always a convex function? |
Every convex form of functions is convex? |
I have a real-valued function $f(x)=\sum_{n=x+1}^\infty \frac{1}{n^2}$ where $x \in \mathbb{N}$. I want to understand how $f(x)$ scales with respect to $x$.
One thing I tried is as follows: Since it is not hard to see that $f(x) < \frac{1}{x}$ for all $x$, using the [Big-O notation][1], $f(x) = O\big(\frac{1}{x}\bi... |
Consider an m-by-n grid of unit squares, indexed by
(i, j) with 1 ≤ i ≤ m and 1 ≤ j ≤ 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
(m+n-2) choose (m -1)?
could someone give me some intuition for this? |
if you have a grid of m by n squares why is the number of paths from (1,n) to (m,1) equal to m+n-2 choose m - 1? |
Let $M$ be a Riemannian manifold, $E \to M$ a vector bundle and $\nabla : \Gamma(E) \to \Gamma(T^*M \otimes E)$ a connection on $E$.
I'm a bit confused on the definition of flatness. The word is used a bit vaguely here and there.
Lee calls a Riemannian manifold flat if and only if the curvature tensor $R $ vanish... |
Since $1.681>\phi$, to show $e^{e^2}>1000\phi$ it is enough to show
$$e^2>3\ln10+\ln2+\ln(1-0.19095)$$
We have
$$e^2>(2.71828)^2>7.38904$$
$$6.90776>3\ln10$$
$$0.69315>\ln2.$$
And utilising the fifth degree Maclaurin polynomial of $\ln(1-x)$
$$-0.21188>\ln(1-0.19095).$$
Check: $$7.38904\stackrel{?}{>}6.90776+0.... |
Consider an $m \times n$ grid of unit squares, indexed by
$(i, j)$ with $1 \leq i \leq m$ and $1 \leq j \leq n$.
Then why is the number of possible paths from $(1,n)$ to $(m,1)$ such that you are only going down and to the right equal to $\binom{m+n-2}{m -1}$?
could someone give me some intuition for this? |
if you have a grid of $m \times n$ squares why is the number of paths from $(1,n)$ to $(m,1)$ equal to $\binom{m+n-2 }{m - 1}$? |
This is a question I ended up with while trying to make a program that would find an (at least locally-) minimal-energy configuration for a piecewise linear 1-dimensional object in $\mathbb{R}^3$ in relation to a practical knot theory type of question. If it helps at all, the setup described here is essentially asking ... |
Exists a perfect square that divided by 6 equals a prime number? |
Given a (autonomous) dynamical system, one can prove instability of a point via the Lyapunov method, by simply finding a Lyapunov function that increases in a neighbourhood of the point. This ensures a stronger property than instability. It ensures the existence of a small enough neighbourhood, such that all orbits sta... |
The functor $\Omega$ is the composite of a bunch of functors, which I will now describe. Write $\mathsf{A}$ for the diagram $2\to 3\leftarrow 1$. The inclusion $i_3\colon *\to\mathsf{A}, *\mapsto 3$ induces a left Kan extension $\mathrm{Lan}_{i_3}\colon\mathscr{C}\to\mathrm{Fun}(\mathsf{A},\mathscr{C})$ (informally sen... |
$$\int_{-\infty}^\infty\sin(xe^x)dx=\underbrace{\int_{-\infty}^0\sin(xe^x)dx}_A+ \underbrace{\int_0^\infty\sin(xe^x)dx}_B=\underbrace{\frac i2\int_0^1x^{-i x-1}-x^{ix-1}}_Adx+ \underbrace{\frac i2\int_1^\infty x^{-i x-1}-x^{ix-1}}_B$$
Integral $A$:
$A$ is reminiscent of the [sophomore’s dream integrals](https://e... |
8 people total. Seated around a circular table. You can only talk to the 2 people seated directly next to you.
Each person has a connection value with each other person. eg : #1 and #3 are friendly and have a "conversation value" of 7. #1 and #2 are not friends, and have a "conversation value" of 3.
Pict... |
8 people seated around a circular table. Each person has a connection value with each other person. How do you arrange the table to maximize? |
$$\int_{-\infty}^\infty\sin(xe^x)dx=\underbrace{\int_{-\infty}^0\sin(xe^x)dx}_A+ \underbrace{\int_0^\infty\sin(xe^x)dx}_B=\underbrace{\frac i2\int_0^1x^{-i x-1}-x^{ix-1}}_Adx+ \underbrace{\frac i2\int_1^\infty x^{-i x-1}-x^{ix-1}}_B$$
Integral $A$:
$A$ is reminiscent of the [sophomore’s dream integrals](https://e... |
I have a system of 6 equations for 8 variables. There are many solutions. I want a solution that minimizes $x_1^2 + x_2^2 + ... + x_8^2$.
Is there a formulaic method to solve this?
(Application: 8 motors in an octocopter. Thrust and torque for each motor is approximately linear with throttle/RPM. Power consum... |
Suppose we have a random sample $X=(X_1,\dots, X_n)$ which depends on $\theta$, treated as a value of the random variable $\Theta$. The posterior pdf is given by $$f_{\Theta | X}(\theta|x)=\frac{f_{X|\Theta}(x|\theta)f_\Theta(\theta)}{\int_{-\infty}^\infty f(x,\theta)d\theta}$$
where $f_\Theta$ is the prior pdf... |
Suppose we have a random sample $X=(X_1,\dots, X_n)$ which depends on $\theta$, treated as a value of the random variable $\Theta$. The posterior pdf is given by $$f_{\Theta | X}(\theta|x)=\frac{f_{X|\Theta}(x|\theta)f_\Theta(\theta)}{\int_{-\infty}^\infty f(x,\theta)d\theta}$$
where $f_\Theta$ is the prior pdf... |
$$\frac{\tan(\alpha)\tan(\beta)}{\tan(\alpha)+ \tan(\beta)}= -\cot(\alpha+ \beta) +\frac{1}{\tan(\alpha)+ \tan(\beta)}$$
$$\displaystyle I:=\int _0^{\frac{\pi}2}\int _0^{\frac{\pi}2}\frac{\tan(\alpha)\tan(\beta)}{\tan(\alpha)+ \tan(\beta)} d \alpha d \beta=\int _0^{\frac{\pi}2}\int _0^{\frac{\pi}2}-\cot(\alpha+ \b... |
Why is the formula $f_{X|Y}(x|y) = \frac{P_{Y|X}(y|x) f_X(x)}{P_Y(y)}$ true? |
Suppose we are given a finite system of linear equations $a_{i1} x_1 + \cdots + a_{in} x_n = b_i$ in the variables $x_i$, where the constants $a_{ij}$ and $b_i$ are all integers, and moreover suppose that the system of equations has a solution when viewed in $\mathbb Z/ p$ for every prime $p$ and also has a solution ov... |
Does a system of linear equations with solutions modulo all primes have an integer solution? |
$$ I = \int_{0}^{\infty} \frac{\sqrt{2} \left( e^{-2vz - 2b\sinh(z)}\cos\left(\pi\left(\frac{1}{4} + v\right)\right) + e^{2vz - 2b\sinh(z)}\sin\left(\pi\left(\frac{1}{4} + v\right)\right) \right)}{\sqrt{e^z - e^{-z}}} \, \mathrm{d}z $$
Going through Desmos, the function does converge.$f(z)$, the integrand, is holomorp... |
For $d\times d$ matrix $A$ and $x\in \mathbb{R}^d$ with IID standard normal entries, I'm interested in the the following quantity:
$$E_{A,x}[\cos(A^T x,A^{-1} x)]$$
It appears to be $\frac{1}{\sqrt{2}}$, in simulations, can this be proven?
Motivation: This justifies the use of transpose+line search to approxim... |
This is a question I ended up with while trying to make a program that would find an (at least locally-) minimal-energy configuration for a piecewise linear 1-dimensional object in $\mathbb{R}^3$ in relation to a practical knot theory type of question. If it helps at all for the purposes of context or intuition, the se... |
Let $X,Y\geq 0$ be i.i.d. random variables with a continuous distribution with PDF $f$ and CDF $F$.
My question is: Under what conditions does the following inequality hold,
\begin{align}
\mathbb{E}[|X-Y|] \leq \mathbb{E}[X].
\end{align}
If $X,Y\sim Exp(\lambda)$, then the above identity can easily be shown to ... |
When does $\mathbb{E}[|X-Y|]\leq\mathbb{E}[X]$ hold for $X,Y\geq 0$ i.i.d.? |
I have a real-valued function $f(x)=\sum_{n=x+1}^\infty \frac{1}{n^2}$ where $x \in \mathbb{N}$. I want to understand how $f(x)$ scales with respect to $x$.
One thing I tried is as follows: Since it is not hard to see that $f(x) < \frac{1}{x}$ for all $x$, using the [Big-O notation][1], $f(x) = O\big(\frac{1}{x}\bi... |
For $d\times d$ matrix $A$ and $x\in \mathbb{R}^d$ with IID standard normal entries, I'm interested in the the following quantity where $\cos(u,v)$ refers to [cosine similarity](https://en.wikipedia.org/wiki/Cosine_similarity):
$$E_{A,x}[\cos(A^T x,A^{-1} x)]$$
It appears to be $\frac{1}{\sqrt{2}}$, in simulation... |
If A=LF(r) and B is finite dimensional algebra with the dimension are l, living in a bigger von Neumann Algebra, Further A and B are * free, no, is there any formula to calculate what is LF(r)\*B? |
The problem occurs in the [Convex Optimization II lecture by Stephen Boyd][1]. There we want to choose $\alpha_i$'s to minimize $$\frac{R^2+G^2 \sum_{i=1}^k \alpha_i^2}{2\sum_{i=1}^k \alpha_i}$$
where $\alpha_i>0,R>0,G>0$. The professor argued this in the [video 20:00][2] with the statement that a quadratic over a p... |
Suppose we have an arbitrary finite collection $X = (x_i)_{i = 1} ^ n$ of the boundary sphere $\partial D$ where $D$ represents the Poincare disk. Does there exist a (discrete?) subgroup $G$ of the isometry group of $D$ whose limiting set $\Lambda(G)$ is $X$? I know $G$ is not elementary as long as $n > 2$.
I am par... |
**Problem:**
The lines $l_1$ and $l_2$ have vector equations
$$\mathbf r_1=
\begin{pmatrix}
1 \\
4 \\
3 \\
\end{pmatrix} +\lambda
\begin{pmatrix}
2 \\
1 \\
2 \\
\end{pmatrix}
$$
$$\mathbf r_2=
\begin{pmatrix}
4 \\
4 \\
5 \\
\end{pmatrix} +\mu
\begin{pmatrix}
1 \\
2 \\
2 \\
\end{pmatrix}
... |
If $C=\sum\limits_{r=1}^K A_r \otimes B_r$, for elements $\{ A_r\}_r^K $ and $\{B_r\}_r^K$ of $C^*$ algebras $\mathscr{A}$ and $\mathscr{B}$, respectively; then if $\sigma$ is a positive linear functional of $\mathscr{B}$, that is $\sigma(D^*D) \geq 0, \,\forall \, D\in \mathscr{B}$, then if one were to calculate:
$... |
It is known that the meaning of a conditional statement in fuzzy logic can vary depending on the interpretation and context. In certain fuzzy logic books, I have come across the interpretation that "if A, then B" can be understood as "A *coupled* with B" when it means $\mu(x,y) = T_{norm} (\mu_{A}(x), \mu_{B}(y))$, ins... |
I have been recently reading about properties of Legendre functions in several sources and cannot seem to find any properties of Legendre functions of the second kind with negative integer degree. For example, in the [Digital Library of Mathematical Functions][1] and in Lebedev's book on special functions, they usually... |
I know there is a there is a virtually same [question][1] and [its answer][2] on this site, but neither the questioner nor the answerer appears to be active at the moment.
Define a group action on Grassmannian
$$\begin{aligned}
\mathrm{GL}_n(\mathbb{C})\times\mathrm{Gr}_{n,k}&\longrightarrow\mathrm{Gr}_{n,k},\\
(... |
This is essentially very similar to Sangchul's solution, but I think the approach might be easier to understand/motivate.
Hints/Observations towards a solution. If you're stuck, explain what you've tried.
1. Show that $a_n$ is a decreasing sequence.
- OP mentioned this, but did not explicitly sho... |
I am so confused about the following problem (maybe I have made some silly conceptual mistakes)
The reference I used is Lee's introduction to Riemannian manifold.
[![enter image description here][1]][1]
There are two theorems in this book.
[![enter image description here][2]][2]
[![enter image description ... |
Let all untyped variables be natural numbers.
__Formula?__
Given $x \geq 1$, $0 \leq r \lt d$ there are two cases to handle: $x \lt r$ and $x \geq r$.
What is the 2-case closed form for $\sum_{i = 1}^{x} \lfloor \frac{i - r}{d}\rfloor$?
__Attempt.__
Case $x \lt r$. This implies that $1 \lt r$ so the ter... |
What is the 1-case closed form for $\sum_{i = 1}^{x} \lfloor \frac{i - r}{d}\rfloor$? |
Both integrals are examples of this integral,
which (as you observed) you can solve by partial fractions and arctan:
$$
\int_0^\infty \frac1{x^4 + x^2 \left(k^2 + \frac1{k^2}\right) + 1} \,\mathrm dx. \tag1
$$
We write
\begin{align}
\frac1{x^4 + x^2 \left(k^2 + \frac1{k^2}\right) + 1}
&= \frac1{\left(x^2 + ... |
I know there is a virtually same [question][1] and [its answer][2] on this site, but neither the questioner nor the answerer appears to be active at the moment.
Define a group action on Grassmannian
$$\begin{aligned}
\mathrm{GL}_n(\mathbb{C})\times\mathrm{Gr}_{n,k}&\longrightarrow\mathrm{Gr}_{n,k},\\
(g,W)&\longm... |
I am trying to learn more about the Inverse Gaussian Distribution and its applications (e.g. First Passage Time).
First, define a **Brownian Motion with Drift**: If $X(t) = \mu t + W(t)$ represents a Brownian motion with drift $\mu$, where $W(t)$ is a standard Brownian motion (i.e. $W(t) \sim N(0,t)$). Here, the in... |
Why is the Brownian Motion related to the Normal Distribution? |
This is a question I ended up with while trying to make a program that would find an (at least locally-) minimal-energy configuration for a piecewise linear 1-dimensional object in $\mathbb{R}^3$ in relation to a practical knot theory type of question. If it helps at all for the purposes of context or intuition, the se... |
Is this quotient space even defined? If I add some $v \in V$ to some $u \in U$, I would be adding a polynomial and a constant for the first coordinate (possible) and an ordered pair and a constant for the second coordinate, which does not seem possible.
My intuition is telling me that if we mod out $\mathbb{R}^2$ we... |
I'm wondering wether the class $C$ defined by:
$$x\in C\Leftrightarrow x\subseteq C\quad\quad (1)$$
Is unique. Its existence is ensured by the following argument:
Let $U$ be the class of all sets, and $x\in U$ a set in it. Since $U$ contains all sets, then it must contain $\{x\}$, and so:
$$x\subseteq\{x\}\in U\R... |
Is the class of all sets unique? |
Let $X$ be an $m$-dimensional random variable and let $C$ be its covariance matrix. I am following a proof that says if $X$ is centered then $P(X \in \mathrm{Im } C) = 1$ where $\mathrm{Im } C$ is the image of $C$.
In the proof they show that for any $\xi$ in the kernel of $C$ we have
$$0 = \langle C\xi, \xi\rang... |
I found an old book in uncle's house called Concrete Mathematics, apparently this is a undergraduate-level textbook and I am only in Highschool but I was still curious about the book.
In the 1.3 problem of the book, it wanted to solved recurrent relations without "clairvoyance".
According to the book, the recurr... |
I found an old book in my uncle's house called Concrete Mathematics, apparently this is a undergraduate-level textbook and I am only in Highschool but I was still curious about the book.
In the 1.3 problem of the book, it wanted to solved recurrent relations without "clairvoyance".
According to the book, the rec... |
I am reading "Analysis on Manifolds" by James R. Munkres.
I cannot understand the last part of Step 5 in the proof of change of variables theorem (Lemma 19.1)
[https://archive.org/details/MunkresJ.R.AnalysisOnManifoldsTotal/page/n177/mode/2up][1]
[![enter image description here][2]][2]
The author wrote as... |
I found an old book in my uncle's house called Concrete Mathematics, apparently this is a undergraduate-level textbook and I am only in Highschool but I was still curious about the book.
In the 1.3 problem of the book, it wanted to solved recurrent relations without "clairvoyance".
According to the book, the rec... |
Curvature is a property of a connection. When talking about connections on tangent bundles, we abuse language slightly and speak of curvature of the underlying manifold. See [here][1] for some more remarks about why we omit the distinction in language about objects ‘living on $M$’ vs ‘living on $TM$’.
Next, given an... |
Let $X$ be an $m$-dimensional random variable and let $C$ be its covariance matrix. I am following a proof that says if $X$ is centered then $P(X \in \mathrm{Im } C) = 1$ where $\mathrm{Im } C$ is the image of $C$.
In the proof they show that for any $\xi$ in the kernel of $C$ we have
$$0 = \langle C\xi, \xi\rang... |
Let $X$ be an $m$-dimensional random variable and let $C$ be its covariance matrix. I am following a proof that says if $X$ is centered then $P(X \in \mathrm{Im } C) = 1$ where $\mathrm{Im } C$ is the image of $C$.
In the proof they show that for any $\xi$ in the kernel of $C$ we have
$$0 = \langle C\xi, \xi\rang... |
If $M$ is a flat $R-$module and $I$ is an ideal of $R$, why is $I\otimes_R M$ isomorphic to $IM$? |
If a complete ordered set induces a metrizable topological space, then will that be a complete metric space? |
Why is it adding $1$ to both sides of the equation $Tn = 2T_{n-1} + 1$? |
I recently got an explanation of the Monty Hall problem and I thought I understood it but after giving it more thought, it still looks wrong.
Instead of using goats and doors, the example used a 52-card poker deck. The cards are facing down on a table and you need to pick one card without looking at it. Let's say th... |
Monty Hall: the number of unknowns decreases but probabilities stay the same??? |
This post directly follows this [question](https://math.stackexchange.com/questions/4862992/using-euler-lagrange-equations-show-that-the-following-eom-can-be-written-as-d) and is very similar in nature:
>Consider the following Lagrangian:
>$$\mathcal{L}=\frac14\left(\partial_\mu A_\nu-\partial_\nu A_\mu\right)\left(\... |
Suppose that Ax = b and Cx = b have the same general solution for every b. why is it that then A = C? |
My professor tells me that a proof by contradiction structure below is poor:
> Proof. To prove: A implies B.
So, assume A (in order to prove that B follows).
Assume not B (in order to arrive at a contradiction).
Succeed at showing that not A follows from the assumption of not
B, thus arriving at the contradictio... |
What is wrong with this proof by contradiction? |
The following is loosely related to this [question](https://math.stackexchange.com/questions/4860279/what-is-the-correct-sign-for-the-four-vector-potential-gauge-transform-a-mu-t/4860666?noredirect=1#comment10362260_4860666):
>[...], the most general renormalisable Lagrangian that is invariant under both Lorentz trans... |
The tail of $p$-series ($p > 1$) can be upper bounded by (from e.g. [here][1]):
\begin{equation}
\sum_{j=n}^\infty \frac{1}{j^p} = O(n^{1-p}).
\end{equation}
I'm interested in finding the upper bound of
\begin{equation}
\sum_{j=n}^\infty \frac{\sin(cj)}{j^p},~~ \text{ or } ~~ \sum_{j=n}^\infty \frac{\cos(cj... |
This post is a follow-up to this [previous question](https://math.stackexchange.com/questions/4861221/show-that-partial-mu-phi-ast-a-mu-phi-a-mu-phi-ast-partial-mu-phi-a-mu).
>Consider the following Lagrangian:
>$$\mathcal{L}=\frac14\left(\partial_\mu A_\nu-\partial_\nu A_\mu\right)\left(\partial^\mu A^\nu-\partial^\... |
Suppose we have a [complete ordered set][1] (i.e. a totally ordered set with the [lub property][2]) and the order topology happens to be metrizable. Does it follow that this space is a [complete metric space][3]?
[1]: https://en.wikipedia.org/wiki/Total_order#Completeness
[2]: https://en.wikipedia.org/wiki/L... |
$$\int_{0}^{1} \frac{x^{2n+2}}{x^2-2x+2} \,{\rm d}x \tag{1}$$
This integral came up as a result of another integral I was trying to evaluate. Just for recreational purposes. Is there some closed form for this, even in terms of special functions?
In general, is it possible to evaluate integrals of the form:
$$... |
I am consider the following free boundary problem: $u_t + \mathcal{L}u = 0, 0<x<s(t)$, with the boundary conditions: $u(T,x) = f_T(x)$, $u(t,s(t)) = g(t)$ and $u_x(t,s(t))=C$. Here, $\mathcal{L}$ is a linear differential operator.
I am planning to use the finite difference scheme to solve the problem. However, how... |
Seems simple question but don't know to prove rigorously.
In Eisenbud's Commutative Algebra book, in chapter 7, p.179 he wrote that "More generally, if $k$ is a field and $R:=k[x_1, \dots , x_n]/I$, then the completion of $R$ with respect to $\mathfrak{m}:=(x_1, \dots , x_n)$ ( $\mathfrak{m} \supseteq I$ ) is ... |
Completion of $(k[x_1, \dots, x_n]/I)_{\mathfrak{m}/I}$ is $k[[x_1, \dots , x_n ]] / Ik[[x_1, \dots , x_n ]]$? |
Suppose $M^n$ is a smooth projective variety. A line bundle $L$ on $M$ is nef (numerically effective) if on any complete curve $C$ in $M$, $L$ has positive degree, i.e.
$$
L\cdot C=\int_{C}R_h>0.
$$
where $R_h$ is the curvature form of any Hermitian metric $h$ on $L$.
A line bundle $L$ on $M$ is big if there exi... |
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