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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... |
Let $D$ be a set of lines in the real projective plane such that for all $d\in D$, $\bigcap (D\setminus \left\{d\right\})=\emptyset$ and let $Dot(D)$ be the set of all dots that belong to at least two members of $D$.
$D$ is *complete* if for any pair of dot in $Dot(D)$ the line that contain both dots of the pair is... |
I want to use the example of a circle being divided increasing by 1 each time ( 1/1 , 1/2 , 1/3 , 1/4 , etc.). In order it would take this amount of lines: 0, 1, 3, 2, 5, 3, 7, and so on. Is it that an odd division would require an inverse amount of lines (i.e. dividing by 9 correlates to 9 lines) while even division w... |
Is there a relationship with how many lines are necessary to divide a circle? |
Recently for a bonus homework assignment in my algebra class, I was asked to review the literature and write up a proof that $\pi$ is transcendental. Essentially every source I found ("The Transcendence of $\pi$" by Steve Mayer for example) presents the classic proof of Lindemann, which heavily relies on symmetric func... |
> **[Durrett Exercise 4.2.10 (i)]**
> Let $(X_n,\mathcal{F}_n)$ be a supermartingale. Let $N_0=-1$ and for $j\geq 1$, let
> \begin{equation}
\begin{aligned}N_{2j-1}&=\inf\{m>N_{2j-2}:X_m\leq a\},\\N_{2j}&=\inf\{m>N_{2j-1}:X_m\geq b\}.\end{aligned}
\end{equation}
> Let $Y_n=1$ for $0\leq n<N_1$ and for $j\geq 1$
>... |
$\frac{1}{\sqrt{n}}\lVert x\rVert_1\le\lVert x\rVert_2\le \lVert x\rVert_1$
My idea is to show first that $\lVert \vec x\rVert_\infty \le \lVert \vec x\rVert_2 \le \sqrt{n}\lVert \vec x\rVert_\infty$, and then I could use $\lVert \vec x\rVert_\infty \le \lVert \vec x\rVert_1 \le n\lVert \vec x\rVert\infty$.
Or ... |
What are the possible applications in maths and physics of vector fields along smooth maps? |
Let $f,g:\mathbb{R} \to (0,\infty)$ be nonnegative functions on $\mathbb{R}$. If $$
\lim_{|x|\to \infty} \frac{g(x)}{f(x)} = 0.
$$
Can we conclude anything about the decay property of their respective Fourier transforms $\hat f,\hat g$ defined by
$$
\hat f(y) = \int_{\mathbb{R}} e^{-ixy} f(x)dx,
$$
similar for $... |
Let $D$ be a set of lines in the real projective plane such that for all $d\in D$, $\bigcap (D\setminus \left\{d\right\})=\emptyset$ and let $Dot(D)$ be the set of all dots that belong to at least two members of $D$.
$D$ is *complete* if for any pair of dot in $Dot(D)$ the line that contain both dots of the pair is... |
I think you are confused by this fact;
**How does multiplication correspond to area?**
1: This is simple. When we multiply two numbers, $a$ and $b$, we can think of it as a rectangle with side $a$ and side $b$. This is because of the rectangle area formula $A = lb$ corresponds to direct multiplication. The import... |
Let $T$ be a regular tetrahedron, and $\ell$ a line from the midpoint of one edge $E_1$, through the center $C$, to the midpoint of another edge $E_2$.
How do I prove that the reflection of each point on the tetrahedron across this line $\ell$ is also on the tetrahedron? I can see this visually (especially when I ro... |
In a tetrahedron, reflect each point over the line from an edge's midpoint through its center. Is it the tetrahedron itself? |
You have the right idea when it comes to solving the equation in closed form, but for finding the largest interval over which the ODE has a solution you need to write it as the IVP
$$y'(t)=F(t,y(t)):=t\sin(t)+y(t)/x \\ y(t_0)=y_0\tag{1}$$
Since $F$ is clearly continuous in its first argument and Lipschitz continu... |
It's easy to show (using the [Lindemann-Weierstrass theorem][1]) that, for $x\ne 0$, at least one of $x$ and $\sin(x)$ must be transcendental.
But what about $x\sin(x)$?
After all, the product of two transcendental numbers could be algebraic. Hence: Can the product $x\sin(x)$ be non-zero and algebraic? Might ... |
Let $p$ be a negative prime number such that $p≡5mod8$.
Let $K=\Bbb{Q}(\sqrt{p})$ and its ideal class group be $Cl_K$.
I want to prove $Cl_K[2]:=\{a\in Cl_K\mid 2a=0\}$ is trivial.
In this situation, $p$ and $2$ is unramifed in $\Bbb{Q}(\sqrt{p})/\Bbb{Q}$ because one can check corresponding local field exte... |
**Given:**
1) $\ \ a$ is even
2) $\ \ \color{#90f}{6a+11b=5}$
3) $\ \ \color{#0af}{a\equiv b\pmod{\! 5}}$
**Q: Calculate $\gcd(2a+4b,2a+8b)$**
____
**My try:**
We know there is some $i$ such that $a=2i$, plus from 3 we know there is some $j$ such that: $a-b=5j$ which means $b=a-5j=2i-5j$. From 2, we ... |
Let $f,g:\mathbb{R} \to (0,\infty)$ be nonnegative functions on $\mathbb{R}$. If $$
\lim_{|x|\to \infty} \frac{g(x)}{f(x)} = 0.
$$
Can we conclude anything about the decay property of their respective Fourier transforms $\hat f,\hat g$? They are defined by
$$
\hat f(y) = \int_{\mathbb{R}} e^{-ixy} f(x)dx,
$$
sim... |
I've had several problems in my homework asking me to find the coefficients for different quadric surfaces using only 2 given points, and a picture to possibly find more points from. For example, I have an image of this hyperbolic paraboloid:
[][1]
I know it has the basic form of $(\frac{x}{a})^2 - (\frac{y}{b})^2 =... |
How can I find the coefficients of a quadric surface, knowing only its basic form and several points it contains? |
I'm looking for some help with **Exercise 5.11.** in Bacon's *A Philosophical Introduction to Higher-Order Logics*.
1. *Construct an explicit definition of the finite Church numerals,* Num$_{\sigma}$, *in higher-order logic*.
As per the text: "the finite Church numerals are the smallest collection of entities of... |
Let $p$ be a negative prime number such that $p \equiv 5\mod 8$.
Let $K = \mathbb{Q}(\sqrt{p})$ and denote its ideal class group by $Cl_K$.
I aim to prove that $Cl_K[2] := \{a \in Cl_K \mid 2a = 0\}$ is trivial.
In this context, $p$ and $2$ are unramified in $\mathbb{Q}(\sqrt{p})/\mathbb{Q}$ because one can veri... |
I've had several problems in my homework asking me to find the coefficients for different quadric surfaces using only 2 given points, and a picture to possibly find more points from. For example, I have an image of this elliptic paraboloid
EDIT: I apparently need more reputation points to post images. Pretend you're l... |
I've had several problems in my homework asking me to find the coefficients for different quadric surfaces using only 2 given points, and a picture to possibly find more points from. For example, I have an image of this elliptic paraboloid:
EDIT: I apparently need more reputation points to post images. Pretend you'r... |
I am going through modular forms and I came to know about that, the coefficients of a modular forms f over $\mathbb{Q}$ define a finite extension $K_f/Q$. Can anyone point me to the proof of this fact? I already know the coefficients are algebraic over $\mathbb{Q}$ from Milne's Notes, Proposition $5.27$. Is the same tr... |
Let $p$ be a negative prime number such that $p \equiv 5\pmod 8$.
Let $K = \mathbb{Q}(\sqrt{p})$ and denote its ideal class group by $Cl_K$.
I aim to prove that $Cl_K[2] := \{a \in Cl_K \mid 2a = 0\}$ is trivial.
In this context, $p$ and $2$ are unramified in $\mathbb{Q}(\sqrt{p})/\mathbb{Q}$ because one can ver... |
> **[Durrett Exercise 4.2.10 (i)]**
> Let $(X_n,\mathcal{F}_n)$ be a supermartingale. Let $N_0=-1$ and for $j\geq 1$, let
> \begin{equation}
\begin{aligned}N_{2j-1}&=\inf\{m>N_{2j-2}:X_m\leq a\},\\N_{2j}&=\inf\{m>N_{2j-1}:X_m\geq b\}.\end{aligned}
\end{equation}
> Let $Y_n=1$ for $0\leq n<N_1$ and for $j\geq 1$
>... |
Suppose we have a series of RVs with $X_i \sim D$. In our instance, let's set $D$ to be Bernoulli, i.e. $X_i \sim Bern(p)$.
For some $W > 0$, we can now also create a series of RVs $Y_i = \sum_{k = 1}^W X_{i - k + 1}$, i.e. where $Y_i$ looks at a sliding window of size $W$ of the $X_i$ series.
What techniques ca... |
For ||3v|| = 1, I isolated v and got: ||v|| = 1/3
I used ||u+v||^2 = (u+v).(u+v) and got to
||u+v||^2 = ||u||^2 + 2(u.v) + ||v||^2
(1) = (1) + 2(u.v) + (1/9)
u.v = -1/18
Is my answer/process correct? |
Suppose ∥u∥=1=∥3v∥ . If ∥u+v∥=1 then u.v=? |
Consider a [two dimensional random walk][1] with step size $1$ and each step in a random direction, with the angle $\theta$ uniformly distrbuted in $[0,2\pi)$. The walk starts on the perimeter of a disk of radius $r$.
[![enter image description here][2]][2]
>What is probability that the walk will ever return to t... |
**Question:** What is a simple yet precise definition for "=" as a binary relation?
**My try:**
I find two definitions for "equality relation" which seems to be contradictory.
The first one I learnt in school is that [equality is what is called an equivalence relation][1], that is, it satisfies three axioms: Re... |
Is there a bounded random variable whose characteristic function decays exponentially in the tails?
The c.f. of uniform distribution $U(-1,1)$ is $\frac{\sin t}{t}$. I can concolute more $U(-1,1)$ together to get $\frac{\sin^{m}t}{t^{m}}$ to make the tail of it more thinner.
No matter how many terms I convolve, ... |
Is the centre of a dense sub algebra $A$ of the Von Neumann algebra $M$ is trivial ($Z_A \subset A$) then shall we conclude $Z_M(M)$ is trivial hence $M$ is factor?
Remember that we see $Z_A(A) \subset A$ within that subalgebra not in whole $M$. |
Is centre of a Von Neumann algebra trivial? |
Suppose I have some positive definite matrices $A_1, A_2, \dots A_k \in \mathbb{R}^{n \times n}$ and some values $v_1 \leq v_2\dots \leq v_k \in \mathbb{R}$. Consider the matrix $A = (\sum_{k=1}A_k v_k)(\sum_{k=1} A_k)^{-1}$. For $n=1$ (the scalar case), $A$ becomes a weighted mean of the $v_i$'s, and thus necessarily ... |
I read Poincaré's paper called [Analysis Situs](https://www.maths.ed.ac.uk/~v1ranick/papers/poincare2009.pdf). And here's the thing about chain complexes.
> **(Page 104 in [this file](https://www.maths.ed.ac.uk/~v1ranick/papers/poincare2009.pdf))**
> That being given, let ${ε}^q_{i,j}$ be a number which is equ... |
For $\|3v\| = 1$, I isolated $v$ and got: $\|v\| = 1/3$.
I used $\|u+v\|^2 = (u+v)\cdot(u+v)$ and got to
$$\|u+v\|^2 = \|u\|^2 + 2(u \cdot v) + \|v\|^2$$
$$(1) = (1) + 2(u \cdot v) + (1/9)$$
$$u \cdot v = -1/18$$
Is my answer/process correct? |
Suppose $\|u\|=1=\|3v\|$ . If $\|u+v\|=1$ then $u \cdot v=?$ |
Let $X: L^2(\mathbb{R}) \rightarrow L^2(\mathbb{R})$ be a multiplication operator, i.e. $f(x) \mapsto xf(x)$ for $f \in L^2$. Multiplication operators are known to be self-adjoint on some dense subset of $L^2$, and so by Stone's theorem on one-parameter unitary groups $X$ is the infinitesimal generator of some one-para... |
Let the *multiplicative closure* operator $h$ be defined on $\mathcal{P}(\mathbb{N})$ by setting $hA$ equal to the smallest set containing $A$ that is closed under multiplication.
In L. F. Meyers' solution to Problem E $1941$ on pp. $408$-$9$ of the April $1968$ *American Mathematical Monthly* (https://www.jstor.org/s... |
How many natural numbers are needed to generate 14 distinct subsets under complement and multiplicative closure? |
It can be shown that taking $y = \frac{1}{x}$ and rotating about the origin can produce (part of) a hyperbola. Is there such a relation in $3$D? Would it be possible to take the equation $xyz = 1$ and rotate it such that it forms a hyperboloid?
I tried figuring this out with a similar approach as one would in $2$D: ... |
I have a simple logic equivalence expression. I'm thinking there must be a name for this principal of OR inversion. Can anyone help put a name to this idea? Link to expression calc below.
(p∨q)↔¬(¬p∧¬q)
https://www.emathhelp.net/en/calculators/discrete-mathematics/truth-table-calculator/?f=%28p%E2%88%A8q%29%3D... |
I just met a result in Lemma5.1 of [ref1][1] which seems interesting and counterintuitive, it said that on 2-dimension torus $\Omega$, let $G$ be the Green's function to $-\Delta$ on $\Omega$, satisfying $\int_{\Omega} G(x, y) d y=0$ for all $x$. We have
$$
G(x, y)=\frac{1}{2 \pi} \ln \frac{1}{|x-y|}+\gamma(x, y),
$... |
I will propose a counterexample to $$((p \land q) \Rightarrow z) \Rightarrow ((p \Rightarrow z) \lor (q \Rightarrow z)).$$ Let's assume that $p$ is "$n$ is divisible by $2$", $q$ is "$n$ is divisible by $5$" and $z$ is "$n$ divisible by $10$".
Then, while it's true that ($n$ is divisible by $2$ and $n$ is divisible b... |
Use the property that $\gcd(m,n) = \gcd(m \mod n , n)$.
$$ (2a + 8b) \mod (2a+4b) = 4b \mod (2a+4b)$$
So, we have:
$$ \gcd(2a+ 4b, 2a+8b) = \gcd(2a+4b,4b)$$
Repeating the process, we have $$ \gcd(2a+4b,4b) = \gcd(2a,4b)= 4\gcd( \frac{a}{2},b)$$
Can you finish now? |
Suppose I have some positive definite matrices $A_1, A_2, \dots A_k \in \mathbb{R}^{n \times n}$ and some values $s_1 \leq s_2\dots \leq s_k \in \mathbb{R}$. Consider the matrix $A = (\sum_{k=1}A_k s_k)(\sum_{k=1} A_k)^{-1}$. For $n=1$ (the scalar case), $A$ becomes a weighted mean of the $s_i$'s, and thus necessarily ... |
>Let $S_n = \sum_{k=1}^n\frac{1}{\sqrt{n^2+k^2}}$. Calculate the following limit
$$\lim_{n \to \infty} n\left(n\Big(\ln(1+\sqrt{2})-S_n\Big)-\frac{1}{2\sqrt{2}\,(1+\sqrt{2})}\right).$$
My intuition says that every from of $n$ times a parentheses in a form of $0 \cdot \infty$ otherwise the problem would be trivial.
... |
Let $\mathcal{M}$ be a model category with class of weak equivalences $W$ and $\mathcal{J}$ a small category. The category of diagrams $\operatorname{Fun}(\mathcal{J}, \mathcal{M})$ inherits a class of weak equivalences $W_{\mathcal{J}}$, those natural transformations which are objectwise a weak equivalences. If $\math... |
According to the [nLab page of locally connected spaces][1] (Theorem 4.1), the category of locally connected spaces is a coreflective subcategory of the category of topological spaces. The coreflector (right adjoint to the canonical inclusion) assigns to a topological space the coarsest topology finer than the original... |
> Let $f: \mathbb{R}^n \to \mathbb{R}$ be a $C^2$ function with $L$-Lipschitz continuous gradients such that for some $L > 0$:
>
> \begin{gather*}
{\lVert \nabla f(x) - \nabla f(y) \rVert} \le L {\lVert x - y \rVert}, \quad \forall x,y \in \mathbb{R}^n \tag{1} \\
\end{gather*}
>
> which implies that:
>
> \b... |
I will first present a speedy proof of this, and then an explanation as to why this works that I hope is more clear than the others that have been given.
Assume that $((p \land q) \implies z) \implies ((p \implies z) \lor (q \implies z))$ is false. Then it must be the case that If $p \land q \implies z$ is true, it ... |
I read Poincaré's paper called [Analysis Situs](https://www.maths.ed.ac.uk/~v1ranick/papers/poincare2009.pdf). And here's the thing about chain complexes.
> **(Page 104 in [this file](https://www.maths.ed.ac.uk/~v1ranick/papers/poincare2009.pdf))**
> That being given, let ${ε}^q_{i,j}$ be a number which is equ... |
I read Poincaré's paper called [Analysis Situs](https://www.maths.ed.ac.uk/~v1ranick/papers/poincare2009.pdf). And here's the thing about chain complex.
> **(Page 104 in [this file](https://www.maths.ed.ac.uk/~v1ranick/papers/poincare2009.pdf))**
> That being given, let ${ε}^q_{i,j}$ be a number which is equal t... |
I know how to use the RK 4 method for a first order differential equation of the form:
$$y' = f(x, y(x))$$
$$y_{k+1} = y_k + (G_1 + 2G_2 + 2G_3 + G_4)*dx/6$$
where $G_1 + 2G_2 + 2G_3 + G_4$ are calculated as:
$$G_1 = f(t_k, x_k)$$
$$G_2 = f(t_k + 0.5*dx, x_k + G_1*dx)$$
$$G_3 = f(t_k + 0.5*dx, x_k + G_2*d... |
How to find out the Runge-Kutta 4 constants for numerically evaluating an nth order IVP? |
Consider a circle of radius $r_s$ that is tangent to two curves $r(\theta)$ and $R(\theta)$ at points $E_1, E_2$ respectively, defined in polar coordinates.
Knowing the function $r(\theta)$, find the function $R(\theta)$, so that when rotating both function in opposite directions by some angle $d\phi$ around the cent... |
Let $R_1$, $R_2 \in \mathbb R^{N \times M}$ with $M > N$ be matrices of rank $N-1$. When can we find a symmetric matrix $P\in \mathbb R^{M\times M}$ such that the matrix equation
$$ R_1 P = R_2, $$
is satisfied? When does a solution exist? I was thinking that an approach could be rewriting the equation by vecot... |
I am currently studying Chapter 3 in **Introduction to Graph Theory** by Richard J. Trudeau. In this chapter, the author proves the utility graph is nonplanar by using the Jordan Curve Theorem and the corollary:
> If $C$ is a continuous simple closed curve in a plane and two points of $C$ are joined by a continuous cu... |
> Find a Mobius transformation sending the region $D$ between $|z-1|=1$ and $|z|=1$ to ${0<Arg(z)<\frac{2\pi}{3}}$ such that 1 is mapped to $i$.
My idea:
The two circles intersect at $x+iy= \frac{1}{2} \pm \frac{\sqrt{3}}{2}i$, while the two rays of the wedge intersect at $0$ and $\infty$.
I therefore thought t... |
Let $F:[0, \infty)\to [0, \infty)$ be a continuous function. It is given that
$$
\int_0^{\infty}f(x) dx<\infty.$$ Determine whether$$
\int_0^{\infty}(f(x))^2 dx<\infty.$$
I think it holds. I write
$$[0, \infty)=A\cup B=(x\in [0, \infty): f(x)>1) \cup (x\in [0, \infty): f(x)\leq 1). $$
Clearly $A\cap B=\emptys... |
I am currently studying Chapter 3 in **Introduction to Graph Theory** by Richard J. Trudeau. In this chapter, the author proves the utility graph is nonplanar by using the Jordan Curve Theorem and the corollary:
> If $C$ is a continuous simple closed curve in a plane and two points of $C$ are joined by a continuous cu... |
I am trying to understand Discrete Fourier Transform (DFT), after only having experience with the continuous transformation. The natural idea was to try to understand DFT on the simplest function, being an exponential, which should transform into an exponential too. To check this in practice, I've drafted up a Python s... |
Imaginary components in Discrete Fourier Transform of Gaussian? |
Another partial answer based on numerical evidence. It is easy to note that your numbers can be written as
$$
\frac{a_n}{4^n}
$$
where
$$
a_n=
1,5,22,91,370,1491,5986,23986,96028,384277,1537438,6150406,\dots\tag{*}
$$
After playing a bit with the number one finds:
$$
a_1=1;\quad a_n=4a_{n-1}+\xi_n,\tag1
$... |
I will denote a pair of neighbour numbers from the sequence $(a_n)$ as $x,y$ going in that order. As you already stated, once we get a pair of coprime $x,y$, the sequence is done: it strictly increases after and hence is unbounded. Indeed, $\frac{x+y}{\gcd(x,y)}= x+y > y$ and new pair numbers $y, x+y$ are also coprime.... |
I am trying to prove that if $G$ is a group of order $1001$, then $G$ has normal subgroups of order $7$, $11$, and $13$, and thus $G$ is cyclic. I know that by Cauchy's Theorem, there exist elements $p$, $q$, and $r$ of orders $7$, $11$, and $13$ respectively, as $1001=7\cdot 11\cdot 13$, and we know that they are cycl... |
The intuition on stretching and rotating is good. But like all intuition, one needs to question it and take it with a grain of salt. For example: does it mean that you can always write a matrix as a product of a rotation(/reflection) matrix and a stretching matrix? The answer (for square matrices) is indeed yes, and th... |
Let $p$ be a negative prime number such that $p \equiv 5\pmod 8$.
Let $K = \mathbb{Q}(\sqrt{p})$ and denote its ideal class group by $Cl_K$.
I aim to prove that $Cl_K[2] := \{a \in Cl_K \mid 2a = 0\}$ is trivial.
In this context, $p$ and $2$ are unramified in $\mathbb{Q}(\sqrt{p})/\mathbb{Q}$ because one can ver... |
On a hyperbolic surface, is any compact set necessarily contained in a larger geodesically convex compact set?
The question came out when I was studying the Pick Theorem in the book *Dynamics in One Complex Variable* by John Milnor(Page22).
At the end of the proof, it seems that the answer of my question is obvio... |
Why I get the eigen value as $0$ for the minimum variance? |
If $A$ is a positive definite matrix, can we derive a lower bound for the term $x^T A y$, where $x, y$ are two vectors of the following form:
$$(x-z)^T A (x-y) \geq \alpha (x-z)^T (x-y), $$
where $\alpha$ is a positive scalar? |
Question:
>Find the smallest $n$ for which there exists a Markov chain with $n$ states such that it is periodic simultaneously with period $4$ and $6$? Put "does not exist" if such a Markov chain does not exist.
My answer is $n = 12$, because we can create a chain as below.
[![Markov chain][1]][1]
[1]: ht... |
On a hyperbolic surface, is any compact set necessarily contained in a larger geodesically convex compact set?
The question came out when I was studying the Pick Theorem in the book *Dynamics in One Complex Variable* by John Milnor(Page22).
At the end of the proof, it seems that the answer of my question is obvio... |
The intuition on stretching and rotation is good. But like all intuition, one needs to question it and take it with a grain of salt. For example: does it mean that you can always write a matrix as a product of a pure rotation(/reflection) matrix and a pure stretching matrix? The answer (for square matrices) is indeed y... |
Question:
>Find the smallest $n$ for which there exists a Markov chain with $n$ states such that it is periodic simultaneously with period $4$ and $6$? Put "does not exist" if such a Markov chain does not exist.
My answer is $n = 12$, because we can create a chain as below (cycle that can be traversed in only one d... |
Recently, I have tried to resolve the following problem.
Let periodic $f$ be defined by the following Fourier series,
\begin{align}
f(x) = \sum_{k=1}^\infty \frac{1}{k^{3/2}} e^ikx.
\end{align}
Prove that $f$ belongs to Nikol'skii space $H^1_2(-\pi,\pi)$ but
\begin{align}
\frac{1}{2\pi}\int_{-\pi}^\pi \vert f(... |
The intuition on stretching and rotation is ok, although it depends on what you mean. All functions rotate and stretch vectors, that is not so interesting; the point and the special feature of linear transformations is that they rotate and stretch vectors in a sort of "uniform" and "rigid" way.
Like all intuition, o... |
In Nakahara (Theorem 7.4, Pg 291) we have for an $r$-form on an $m$-dimensional manifold $$ * *\omega = (-1)^{r(m-r)}\omega$$ when Riemannian and $$ * *\omega = (-1)^{1+r(m-r)}\omega$$ when Lorentzian. So, for $m = 0$, it is possible to have $$ * * = -1$$ for a $0$-form in $1D$ Lorentizan manifold. Hence, my question. ... |
Is there a 1D Lorentzian manifold? |
On a hyperbolic surface, is a compact set $K$ necessarily contained in a larger compact set $K’$ so that any two points of $K$ can be joined by a minimising geodesic within $K’$?
The question came out when I was studying the Pick Theorem in the book *Dynamics in One Complex Variable* by John Milnor(Page22).
At th... |
While doing a mathematical exercise(stein Complex Analysis chapter2,exercise 3),
I managed to reduce the problem to the following one:
>$$\int_{0}^{\omega}Re^{-R\cos\theta}d\theta \rightarrow 0 \quad (\text{as } R \rightarrow \infty)$$ where $0\le \omega <\frac{\pi}{2}$.
I can prove this without much diffic... |
I want to differentiate the given function:
\begin{equation}
f(x)= \begin{cases}1 \ \ \ \ \ x\in[0,1] \\
0 \ \ \ \ \ \ x\not\in[0,1]
\end{cases}
\end{equation}
However, the simple $f'(x)=0$ seems a little bit too simple, since this is similar to the Dirac delta function, just stretched ove... |
First of all: "stretching and rotating" is not a definition, is **intuition**. The intuition on stretching and rotation you have is ok, although it depends a little on what you mean. All functions (linear and non-linear) rotate and stretch vectors, that is not so interesting; the point and the special feature of linear... |
Not sure how to go about this:
$1)$ For $ 0\le \theta \le\frac{\pi}{2}$, show that $\sin \theta \ge \frac{2}{\pi} \theta$
$2)$ By using Part 1, or by any other method, show that if $\lambda \le 1,$ then
$$\lim_{R \to \infty} R^{\lambda} \int_{0}^{\frac{\pi}{2}}e^{-R\sin\theta}d\theta=0$$
The other method ... |
Question:
>Find the smallest $n$ for which there exists a Markov chain with $n$ states such that it is periodic simultaneously with period $4$ and $6$? Put "does not exist" if such a Markov chain does not exist.
The definition of periodicity I use:
>State $a$ is periodic if there exists an integer $d>1$ (called... |
I want to differentiate the given function:
\begin{equation}
f(x)= \begin{cases}1 \ \ \ \ \ x\in[0,1] \\
0 \ \ \ \ \ \ x\not\in[0,1]
\end{cases}
\end{equation}
However, the simple $f'(x)=0$ seems a little bit too simple, since this is similar to the Dirac delta function, just stretched ove... |
First of all: "stretching and rotating" is not a definition, is **intuition**. The intuition on stretching and rotation you have is ok, although it depends a little on what you mean. All functions (linear and non-linear) rotate and stretch vectors, that is not really the interesting thing; the point and the special fea... |
[here][1] in my answer I got real part for [polylogarithm function][3] at $1+i$ for natural $n$
$$ \Re\left(\text{Li}_n(1+i)\right)=\left(\frac{-1}{4}\right)^{n+1}A_n-B_n $$
where
$$ B_n=\sum_{k=0}^{\lfloor\frac{n}{2}\rfloor} \frac{2\eta(2k)}{(n-2k)!}Re\left[\left(\frac{\ln2}{2}-\frac{3\pi}{4}i\right)^{n-2k}\ri... |
how to find closed form for $\int_0^1 \frac{x}{x^2+1} \left(\ln(1-x) \right)^{n-1}dx$? |
[here][1] in my answer I got real part for [polylogarithm function][3] at $1+i$ for natural $n$
$$ \Re\left(\text{Li}_n(1+i)\right)=\left(\frac{-1}{4}\right)^{n+1}A_n-B_n $$
where
$$ B_n=\sum_{k=0}^{\lfloor\frac{n}{2}\rfloor} \frac{2\eta(2k)}{(n-2k)!}\operatorname{Re}\left[\left(\frac{\ln2}{2}-\frac{3\pi}{4}i\r... |
I want to differentiate the given function:
\begin{equation}
f(x)= \begin{cases}1 \ \ \ \ \ x\in[0,1] \\
0 \ \ \ \ \ \ x\not\in[0,1]
\end{cases}
\end{equation}
However, the simple $f'(x)=0$ seems a little bit too simple, since this is similar to the Dirac delta function, just stretched ove... |
First of all: "stretching and rotating" is not a definition, is **intuition**. The intuition on stretching and rotation you have is ok, although it depends a little on what you mean. All functions (linear and non-linear) rotate and stretch vectors, that is not really the interesting thing; the point and the special fea... |
Too long for another comment.
Your metric can be written as
\begin{align}
g_{11}&=g_{22}=\frac{4}{(1+x^2+y^2)^2}\,, &g_{12}&=g_{21}=0\,.
\end{align}
I verified with a symbolic calculator that in this case
the Ricci tensor $R_{ij}$ is indeed equal to $g_{ij}$
and the scalar curvature is two like we get on the ... |
First of all: you can see matrices as linear transformations, that is definitely correct. "Stretching and rotating" is not a definition, is **intuition**. The intuition on stretching and rotation you have is ok, although it depends a little on what you mean. All functions (linear and non-linear) rotate and stretch vect... |
On a hyperbolic surface, is a compact set $K$ necessarily contained in a larger compact set $K’$ so that any two points of $K$ can be joined by a minimising geodesic within $K’$?
The question came out when I was studying the Pick Theorem in the book *Dynamics in One Complex Variable* by John Milnor(Page22).
At th... |
On a hyperbolic surface, is a compact set $K$ necessarily contained in a larger compact set $K’$ so that any two points of $K$ can be joined by a minimising geodesic within $K’$?
The question came out when I was studying the Pick Theorem in the book *Dynamics in One Complex Variable* by John Milnor (Page22).
At t... |
First of all: "stretching and rotating" is not a definition, is **intuition**. The intuition on stretching and rotation you have is ok, although it depends a little on what you mean. **All functions** rotate and stretch vectors, that is not really the interesting thing; the point and the special feature of linear trans... |
I have a simple logic equivalence expression. I'm thinking there must be a name for this principal of OR inversion. Can anyone help put a name to this idea? Link to expression calc below.
$$(p\lor q)\iff \neg(\neg p\land\neg q)$$
https://www.emathhelp.net/en/calculators/discrete-mathematics/truth-table-calculat... |
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