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[![enter image description here][1]][1]
I was looking over this proof in Atkinson’s Numerical Analysis 2nd Edition and I am having a hard time seeing where 2.3.5 is coming from. What is Atkinson trying to show with 2.3.5 and the line below? Thank you!
[1]: https://i.stack.imgur.com/WranV.jpg |
Where does 2.3.5 come from and how to prove it? |
Given two alphabets $A=\{a_1, a_2, ..., a_n\}$ and $B=\{b_1, b_2, b_3, ..., b_m\}$, what is the maximum average compression ratio possible to achieve by bijectively encoding strings of B using strings of A?
I guess it is trivially $\frac{|A|}{|B|}$, but I wouldn't be sure how to prove that.
Then, given a new set ... |
What's the maximum possible compression ratio for different languages? |
Let $H$ be a commutative and cocommutative Hopf algebra over an algebraically closed field $k$. I've read that if $H$ is _grouplike_ in the sense that it has no nonzero primitive elements, then $H$ is isomorphic to the group ring on its grouplike elements,but no reference or proof was given.
What is a proof of this ... |
Grouplike Hopf algebras are group rings? |
Let’s suppose I have two spheres $S_1$ and $S_2$. Let's suppose they have a non-trivial intersection, meaning its not a point. Then their intersection forms a circle.
Assume I have the centers of each sphere $P_1$ and $P_2$. Assume their radii are $r_1$ and $r_2$ respectively.
Consider this diagram:
[![ente... |
Let’s suppose I have two spheres $S_1$ and $S_2$. Let's suppose they have a non-trivial intersection, meaning its not a point. Then their intersection forms a circle.
Assume I have the centers of each sphere $P_1$ and $P_2$. Assume their radii are $r_1$ and $r_2$ respectively.
Consider this diagram:
[![ente... |
I am working with time series data, and applying FFT to calculate the power spectrum. The data is something like this:
$y = \sum_{k=1}^{5} \cos(10 k x)$
All the peaks in the FFT output should have identical magnitudes, similar to the pattern shown in the graph below:
[![expected FFT output][1]][1]
However,... |
Given two alphabets $A=\{a_1, a_2, ..., a_n\}$ and $B=\{b_1, b_2, b_3, ..., b_m\}$, what is the maximum average compression ratio possible to achieve by bijectively encoding strings of B using strings of A?
I guess it is trivially $\frac{|A|}{|B|}$, but I wouldn't be sure how to prove that.
Then, given a new set ... |
Let’s suppose I have two spheres $S_1$ and $S_2$. Let's suppose they have a non-trivial intersection, meaning its not a point. Then their intersection forms a circle.
Assume I have the centers of each sphere $P_1$ and $P_2$. Assume their radii are $r_1$ and $r_2$ respectively.
Consider this diagram:
[![ente... |
I have a function of the form $f(u) = ke^{-au}\int_{0}^{\infty}x^{z-1}(1-x)^{-z-1}e^{-bxu}dx$ where $a,b,u>0$ are all real positive numbers and $k$ is a positive normalization factor. I need to calculate the Laplace transform of $f(u)$. We have
$$F(s) = \int_{0}^{\infty} f(u)e^{-su}du = k\int_{0}^{\infty} e^{-au} \... |
I have a sequence:
1, 3, 7, 9
You can think of it as the 4 corners of a square:
[![enter image description here][1]][1]
I want to turn this sequence 1 step to the right so it becomes
[![enter image description here][2]][2]
And also turn it left:
[![enter image description here][3]][3]
What is th... |
Let $R$ be an integral domain with positive characteristic and $A$ a proper ideal of $R$. Show that the characteristic of $R/A$ is the same as the characteristic of $R$.
Attempt: Let $n$ be the characteristic of $R$ and $m$ be the characteristic of $R/A$. For any $a+A \in R/A$, $n(1+A) = n\cdot 1 +A = A.$ Hence, $m ... |
Let $R$ be an integral domain with positive characteristic and $A$ a proper ideal of $R$. Show that the characteristic of $R/A$ is the same as the characteristic of $R$.
**Attempt**
Let $n$ be the characteristic of $R$ and $m$ be the characteristic of $R/A$. For any $a+A \in R/A$, $n(1+A) = n\cdot 1 +A = A.$ Hen... |
I have a function of the form $f(u) = ke^{-au}\int_{0}^{\infty}x^{z-1}(1+x)^{-z-1}e^{-bxu}dx$ where $a,b,u,z>0$ are all real positive numbers and $k$ is a positive normalization factor. I need to calculate the Laplace transform of $f(u)$. We have
$$F(s) = \int_{0}^{\infty} f(u)e^{-su}du = k\int_{0}^{\infty} e^{-au}... |
I am taking a view on the chapter on set theory in Hinman and discovered this statement as Exercise 6.1.27.
May I please ask for a proof of it? The claim is a bit interesting, it says ``for every set $x$, the collection of sets that is not in it is empty''. What does it reveal? Is there any interesting application ... |
Let $X$ be a compact metric space and $F\subset C(X)$ be a compact subset. Show that $F$ is equicontinuous.
*Proof:* Let $f\in F$ be an arbitrary function. I want to show that $\forall \epsilon>0, \exists\,\delta>0$ such that, if $|x-y|<\delta$ then $|f(x)-f(y)|< \epsilon \ $ for all $f\in F$ and $\forall x,y\in X$.... |
I have heard that the holomorphic sectional curvature of the complex projective space (w.r.t the Fubini Study metric) is 2. Can anyone give me a self-contained reference to this fact, please?
Thanks in advance! |
Question 1: An account with an initial amount B earns compound interest at an annual effective interest rate $i$. The interest in the third year is $426$ and the discount in the seventh year is $812$. Find $i$.
The work for question 1 is simple:
$$\dfrac{B(1+i)^7}{B(1+i)^3} = \dfrac{812}{426}$$
This leads you... |
I want to show that for $u(t,x)$ which is a polynomial in $t$ and $x$ such that $$\frac{\partial u}{\partial t} + \frac{1}{2}\frac{\partial^2 u}{\partial x^2}=0$$
we have $u(t,B_t)$ is a martingale where $B_t$ stands for the standard Brownian motion.
Durrett's "Probability theory and examples" shows that $ E_x u(t,... |
I know this platform is for higher-level math. This is just a high school level question but I don't know any more suitable platform to ask. Apologise in advance.
The question is to differentiate the expression and multiply by $x$, do this twice and set $x=1$.
$$\dfrac{x^{n+1}-1}{x-1}$$
After differentiating twi... |
I know this platform is for higher-level math. This is just a high school level question but I don't know any more suitable platform to ask. Apologise in advance.
The question is to differentiate the expression and multiply by $x$, do this twice and set $x=1$.
$$\dfrac{x^{n+1}-1}{x-1}$$
After differentiating twi... |
I have a sequence:
1, 3, 7, 9
You can think of it as the 4 corners of a square:
[![enter image description here][1]][1]
I want to turn this sequence 1 step to the right so it becomes
[![enter image description here][2]][2]
And also turn it left:
[![enter image description here][3]][3]
What is th... |
After finding the exact value of the [integral](https://www.quora.com/How-do-I-evaluate-the-double-integral-displaystyle-int_-0-1-int_-0-1-left-frac-ln-1-xy-1-xy-right-2-dx-dy/answer/Lai-Johnny?ch=17&oid=1477743739641358&share=6743e386&srid=CCOmo&target_type=answer), $$\displaystyle \int_0^1 \int_0^1\left(\frac{\ln (1... |
Is there other method to deal with $\int_0^1 \int_0^1\frac{\ln^m (1+x y)}{(1+x y)^n}d x d y,$ where $m\ge 3?$ |
I have a sequence:
1, 3, 7, 9
You can think of it as the 4 corners of a square:
[![enter image description here][1]][1]
I want to turn this sequence 1 step to the right so it becomes
[![enter image description here][2]][2]
And also turn it left:
[![enter image description here][3]][3]
What is th... |
Suppose we partition the powers of $2$ into two sets in the following manner:
$$A = \{1\}, \quad B = \{2,4,8,16,\dots\}$$
Now suppose that, for each **odd** positive integer $t$, we select either $tA$ or $tB$, and form the union $S$ of all sets selected in this way. For example, if we always choose $tA$, then the res... |
Can we bound the lower and upper asymptotic density of these subsets of $\mathbb N$? |
Let $0 < c < 2/(n-1)$. Let $C_1,...,C_n$ be circles of radius $1$ center at $p_1,...,p_n$ respectively, such that $|p_{k+1}-p_k| = c, \ k=1,...,n-1$.
$\textbf{Question}$: For what configuration of $p_1,...,p_n$ is the area of $C_1 \cap \cdots \cap C_n$ minimized? (note $c$ is fixed)
Comment: I've been trying to ... |
Let $0 < c < 2/(n-1)$. Let $C_1,...,C_n$ be circles of radius $1$ center at $p_1,...,p_n$ respectively, such that $|p_{k+1}-p_k| = c, \ k=1,...,n-1$.
$\textbf{Question}$: For what configuration of $p_1,...,p_n$ is the area of $C_1 \cap \cdots \cap C_n$ minimized? (note $c$ is fixed)
Comment: I've been trying to ... |
Yes, in fact $X^*\cap X=\emptyset$. Lets write down explicitly:
$$X^*=Cauchy(X)/\sim$$
where $Cauchy(X)$ denotes the set of all Cauchy sequences on $X$ and $\sim$ is some relation. In other words $X^*\subseteq \mathcal{P}(Cauchy(X))$ as sets, where $\mathcal{P}(A)$ denotes the power set of $A$. Now what is a sequ... |
Yes, in fact $X^*\cap X=\emptyset$. Lets write down explicitly:
$$X^*=Cauchy(X)/\sim$$
where $Cauchy(X)$ denotes the set of all Cauchy sequences on $X$ and $\sim$ is some relation. In other words $X^*\subseteq \mathcal{P}(Cauchy(X))$ as sets, where $\mathcal{P}(A)$ denotes the power set of $A$. Now what is a sequ... |
Given sequence:
$$
\begin{cases}
x_{n+1}(2\cos(\frac{\pi}{m})-x_n)=1,\forall n\geq 2\\
x_1=x\in\mathbb R,m\in\mathbb N,m\geq 2
\end{cases}
$$
Minimize $$A=\frac{1}{2}\sum_{k=1}^m (x_{k+1}-x_{k})^2$$
$A$ is actually the area bounded by the zig-zag loop between $y=\frac{1}{2\cos(\frac{\pi}{m})-x}$ and $y=x$. I tr... |
Let $0 < c < 2/(n-1)$. For $p_1,...,p_n \in \mathbb{R}^2$, consider the property
$$
|p_{k+1}-p_k| = c, \ k=1,...,n-1 \tag 1
$$
where $| \cdot |$ is the Euclidean distance. Denote $C_k$ for the unit circle centered at $p_k, \ k=1,...,n$.
$\textbf{Question}$: For what geometric configuration of $p_1,...,p_n$ sa... |
Let $0 < c < 2/(n-1)$. For $p_1,...,p_n \in \mathbb{R}^2$, consider the property
$$
|p_{k+1}-p_k| = c, \ k=1,...,n-1 \tag 1
$$
where $| \cdot |$ is the Euclidean distance. Denote $C_k$ for the unit circle centered at $p_k, \ k=1,...,n$.
$\textbf{Question}$: For what geometric configuration of $p_1,...,p_n \in... |
(SOLVED) Monty Hall: the number of unknowns decreases but probabilities stay the same? |
I'm trying to see how two ideas are related:
In (Discrete) Exterior Calculus, we defined k-vectors as volumes, and that k-forms are like "measurement tools", that allow us to measure k-vectors.
However, in Differential geometry, for a real differentiable $n$-manifold $M$, and $p \in M$, we first defined the tang... |
Yes, in fact $X^*\cap X=\emptyset$. Lets write down explicitly:
$$X^*=Cauchy(X)/\sim$$
where $Cauchy(X)$ denotes the set of all Cauchy sequences on $X$ and $\sim$ is some relation. In other words $X^*\subseteq \mathcal{P}(Cauchy(X))$ as sets, where $\mathcal{P}(A)$ denotes the power set of $A$. Now what is a sequ... |
Here is Prob. 23, Chap. 4, in the book _Principles of Mathematical Analysis_ by Walter Rudin, 3rd edition:
> A real-valued function $f$ defined in $(a, b)$ is said to be _convex_ if
$$ f \big( \lambda x + (1- \lambda) y \big) \leq \lambda f(x) + (1-\lambda) f(y)$$
whenever $a < x < b$, $a < y < b$, $0 < \lambda ... |
Consider the sigma fields on $\mathbb{N}$: $F_n=\sigma(A_n)$, where $A_n=\{\{1\},\{2\},\ldots,\{n\}\}$ for $n\ge1$. Define $\mathcal{F}:=\bigcup^\infty_{n=1}F_n$. Prove that $\mathcal{F}$ is a field but not a $\sigma$-field.
---
For the $\sigma$ field part, I show that $\mathcal{F}$ is a countably infinite set ... |
Let $0 < c < 2/(n-1)$. For $p_1,...,p_n \in \mathbb{R}^2$, consider the property
$$
|p_{k+1}-p_k| = c, \ k=1,...,n-1 \tag 1
$$
where $| \cdot |$ is the Euclidean distance. Denote $C_k$ for the unit circle centered at $p_k, \ k=1,...,n$.
$\textbf{Question}$: For what geometric configuration of $p_1,...,p_n \in... |
There are two Gaussian matrices, $A\sim N(0,\sigma_A^2\mathbf{I})$ and $B\sim N(0,\sigma_B^2\mathbf{I})$, respectively.
Their dimensions are $A\in \mathbb{R}^{L\times M}$ and $B\in \mathbb{R}^{N\times M}$, respectively.
I want to find the expectation of $|AB^H|^2$.
So, how can I express the closed form of $... |
I’m trying to calculate the position of a traveler moving towards its destination a given distance.
I have a starting point A and a destination point B both with known coordinates. How can I calculate the new position of A after it travelled distance C towards point B ?[![enter image description here][1]][1]
If ... |
How can I calculate position of a point after it moved towards another point? |
How is $x^{x^{x^{x^{x^{...}}}}}$ defined? It is the limit $\lim_{n\to\infty} a_n$ of the sequence
\begin{align*}a_0 & = 1 \\ a_1 & = x \\ a_2 & = x^x \\ a_3 & = x^{x^x} \\
a_4 & = x^{x^{x^x}} \\
& \ldots
\end{align*}
Starting from $a_2$, these are powers whose exponent is not necessarily an integer. Such kinds of ... |
Assume I have an alphabet $\{A,B,C...\}$ with a total of $K$ symbols. For words of same length, I define the distance $d$ between them as the number of positions in which they have differing symbols (reduces to Hamming distance for $K=2$).
Starting from a fixed word $w$, I am interested in finding sets of words such... |
Let $E=E(k), K=K(k)$ be the complete elliptic integrals of the second and the first kind respectively. It is well-known that $\frac12\int_0^1K\,dk=G$ is the Catalan constant. We can also find, by switching integral trick that
\begin{align}
\int_0^1E\,dk
&=\int_0^{\frac\pi 2}\int_0^1\sqrt{1-k^2\sin^2\theta}d\theta\\... |
I am taking a view on the chapter on set theory in Hinman and discovered this statement as Exercise 6.1.27.
May I please ask for a proof of it? The claim is a bit interesting, it says ``for every set $x$, the intersection of the collection of sets that is not in it is empty''. What does it reveal? Is there any inte... |
How would I prove that the Riemann curvature tensor $R: \scr X(M)^3 → \scr X(M)$,
$R(X, Y )Z := \nabla X\nabla Y Z − \nabla Y \nabla XZ − \nabla [X,Y ]Z$, is indeed a tensor ?
I thought I could use this:
[![enter image description here][1]][1]
(pag 41 in https://radbouduniversitypress.nl/site/books/m/10.5419... |
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))$, i... |
Your first problem is equivalent to wonder when
$\langle x,y\rangle>0$ implies that $\lambda_1x_1y_1+\cdots+\lambda_nx_ny_n>0$ where $\lambda_i\geq 0$ for all $i.$ To see this, just diagonalize $A$ in an orthonormal basis. You are now facing with a problem in $R^{2n}$ with the quadratic forms $(x,y)\mapsto \langle x,... |
**Short answer**
$x^{x^{x^{x^{x^{...}}}}}$ is not defined for $x < 0$.
**Explanation**
How is $x^{x^{x^{x^{x^{...}}}}}$ defined? It is the limit $\lim_{n\to\infty} a_n$ of the sequence
\begin{align*}a_0 & = 1 \\ a_1 & = x \\ a_2 & = x^x \\ a_3 & = x^{x^x} \\
a_4 & = x^{x^{x^x}} \\
& \ldots
\end{align*}
St... |
I was trying to prove that $\varinjlim_{U \ni x} A(U) \cong A_{\mathfrak{p}}$ where $U$ are basic open sets containing $x = \mathfrak{p} \in \operatorname{Spec} A$, and $A(U) = A_f$ where $U = X_f$. $A$ is a commutative ring.
I know that the restriction maps from $A(X_f)$ to $A(X_g)$ are given by $a/f^k \mapsto au^k... |
I have an expression of the form
\begin{equation}
i\operatorname{tr}(ABC - B^\dagger AC),
\end{equation}
where $A$ and $C$ are Hermitian matrices, but $B$ is not. $B-B^\dagger := i D$, where $D$ is also a Hermitian matrix. It can be seen that the above is real by cyclic permutation of the second term inside the tra... |
**Short answer**
$x^{x^{x^{x^{x^{...}}}}}$ is not defined for $x < 0$.
**Explanation**
How is $x^{x^{x^{x^{x^{...}}}}}$ defined? It is the limit $\lim_{n\to\infty} a_n$ of the sequence
\begin{align*}a_0 & = 1 \\ a_1 & = x \\ a_2 & = x^x \\ a_3 & = x^{x^x} \\
a_4 & = x^{x^{x^x}} \\
& \ldots
\end{align*}
St... |
What is $\textbf{I}$? a matrix with all coefficients equal to $1$? What is $B^H$? Hermitian conjugate? So we are with complex Gaussian variables? What is $|AB^H|? $| A determinant? So we are in the case $L=N$? |
Let $m$ and $n$ be two integers and $m <= n$. There are a matrix $A$ of $m$-by-$m$ with $A(i,j) = 1/(2n+2j-2i)!$ and a vector $r$ of $m$ entries with $r(i) = 1/(2n+2i)!$.
Is there a formula for the inner product of $r$ and the first column of the inverse of $A$?
Or, can it be shown that the inner product aforem... |
I know that $a|x-y|=|ax-ay|$
Question: If $(X,d)$ is a metric space (arbitrary) : $a d(x,y)=d(a x, a y)$? |
$a d(x,y)=d(a x, a y)$? |
**Short answer**
$x^{x^{x^{x^{x^{...}}}}}$ is not defined for $x < 0$.
**Explanation**
How is $x^{x^{x^{x^{x^{...}}}}}$ defined? It is the limit $\lim_{n\to\infty} a_n$ of the sequence
\begin{align*}a_0 & = 1 \\ a_1 & = x \\ a_2 & = x^x \\ a_3 & = x^{x^x} \\
a_4 & = x^{x^{x^x}} \\
& \ldots
\end{align*}
St... |
Given that $\Delta\,ABC$ is an isosceles right triangle with $AC = BC$ and $\angle{ACB}= 90^\circ$. D is a point on AC and E is on the extension of BD such that $AE \perp BE$. If $AE = \frac{1}{2}BD$, prove that BD bisects $\angle{ABC}$.
[![enter image description here][1]][1]
[1]: https://i.stack.imgur... |
I have the below algorithm that appears to find a Hamiltonian path in undirected graphs (if one exists). I've tested with every graph I could find or come up with and it appears to work. I've included an example of it working on a decision problem graph.
**I'm trying to find a counter example graph and am hoping som... |
Let $m$ and $n$ be two integers and $m <= n$. There are a matrix $A$ of $m$-by-$m$ with $A(i,j) = 1/(2n+2j-2i)!$ and a vector $r$ of $m$ entries with $r(i) = 1/(2n+2i)!$.
Is there a formula for the inner product of $r$ and the first column of the inverse of $A$?
Or, can it be shown that the inner product aforem... |
Let $(M, g)$ be a Riemannian manifold. The length of an admissible
curve $\gamma : [a, b] \to M$ is defined by $L(γ) = \int_a^b ∥\dot \gamma (t)∥dt$.
1)Compute the Euler–Langrange equations for the variation of the length functional.
2)I want to show that these equations are the geodesic equation if and only if the... |
There are two complex Gaussian distributed matrices, $\mathbf{A}\in \mathbb{C}^{L\times M}$ and $\mathbf{B}\in \mathbb{C}^{N\times M}$.
The elements of $\mathbf{A}$ and $\mathbf{B}$ are followed i.i.d. complex Gaussian distribution with mean 0 and variances $\sigma_A^2$ and $\sigma_B^2$, respectively, i.e., $a_{ij}... |
How would I prove that the Riemann curvature tensor $R: \scr X(M)^3 → \scr X(M)$,
$R(X, Y )Z := \nabla X\nabla Y Z − \nabla Y \nabla XZ − \nabla [X,Y ]Z$, is indeed a tensor ?
I thought I could use this:
[![enter image description here][1]][1]
(pag 41 in https://radbouduniversitypress.nl/site/books/m/10.5419... |
**Short answer**
$x^{x^{x^{x^{x^{...}}}}}$ is not defined for $x < 0$.
**Explanation**
How is $x^{x^{x^{x^{x^{...}}}}}$ defined? It is the limit $\lim_{n\to\infty} a_n$ of the sequence
\begin{align*}a_0 & = 1 \\ a_1 & = x \\ a_2 & = x^x \\ a_3 & = x^{x^x} \\
a_4 & = x^{x^{x^x}} \\
& \ldots
\end{align*}
St... |
My idea to calculate the volume of the hemisphere is to sum up the area of circles of all radii up to the radius of the hemisphere we are interested in: $$\int_0^r \pi x^2 dx$$
This gives $\frac{1}{3}\pi r^3 \neq \frac{2}{3}\pi r^3$, so deviating by a factor of $2$ from the known formula for the volume of a hemisphere... |
What's wrong with this derivation of the volume of a hemisphere? |
Here: https://en.wikipedia.org/wiki/Imaginary_hyperelliptic_curve#The_divisor_and_the_Jacobian it says: "Actually, Bézout's theorem states that a straight line and a hyperelliptic curve of genus 2 intersect in 5 points. So, a straight line through two point lying on
{\displaystyle C} does not have a unique third inte... |
I'm trying to follow the proof of Vizing's Theorem given here https://math.uchicago.edu/~may/REU2015/REUPapers/Green.pdf
Unfortunately, I soon ran into trouble with this line:
> If $c_k$ is absent at v, then we can reassign colours $c_i$ to $\{v,
> w_i\}$ for $i ∈ [k]$ and we are done.
The problem is that e... |
I asked about this and the answer is true if we assume AF. As a more interesting question, does it still hold without AF? According to the suggestion in the comment, I should post it as another question.
Original question: https://math.stackexchange.com/questions/4866146/prove-that-under-zf-forall-x-bigcap-zz-notin-... |
Does $(\bigcap \{z:z\notin x\} = \emptyset)$ hold for every set $x$ in ZF$-$AF? |
I'm trying to prove that class number of the quadratic field with discriminant $17$ is $1$. Let $K =Q[\sqrt{17}]$ be the quadratic field and $\mathfrak{o}_{K}=\mathbb{Z}[\frac{1+\sqrt{17}}{2}]$ be the ring of integers. The Minkowski's bound tells us that the one representative of each ideal class should have norm 1 or ... |
In the Riemannian context you always have a metric to identify vector fields with 1-forms. Proposition 2.7 says you that a tensor of type $(k,l)$ is nothing but a $\mathcal C^\infty(M)$-multilinear map
$$
\mathfrak X^k(M) \times \Omega(M)^l \to \mathcal C^\infty(M)
$$
By the universal property of tensor product, an... |
How many trees with 10 nodes are there, given that each vertex must have a degree of either 1, 2, or 5?
I initially approached the problem by attempting to construct each tree individually, focusing first on maximizing the number of edges with vertices of degree 5, then proceeding similarly for vertices of degree 2.... |
I'm following [*A primer for unstable motivic homotopy theory*][1] written by Antieau, Elmanto as my first intro to motivic homotopy theory. Their treatment of the singular construction on p. 30 confuses me. Let me introduce you to their setup:
We write $\mathbf{sPre}(\mathbf{Sm}_S) = \operatorname{Fun}(\mathbf{Sm}_... |
Let $M$ be a simply connected smooth manifold ($M=R^3$ is enough for my purposes).
Then, the following Proposition 1 is obvious.
>**Proposition 1**
>For all piecewise smooth loop, $c[0,1]\to M$, there is $H$ such
> that;<br>
>  (1) $H:I\to M$ is a continuous mapping, Here $I:=[0,1]\times[0,1]... |
I have a sequence:
1, 3, 7, 9
You can think of it as the 4 corners of a square:
[![enter image description here][1]][1]
I want to turn this sequence 1 step to the right so it becomes
[![enter image description here][2]][2]
And also turn it left:
[![enter image description here][3]][3]
What is th... |
I am currently reading "[Restrictions on harmonic maps of surfaces][1]" by J. Eells and J. C. Wood and have trouble understanding the proof of the lemma on p. 265. The paper states:
> We have the complexified tangent bundle $T^c(X) = T(X) \otimes_\mathbb{R} \mathbb{C}$ and the decomposition $T^c(X) = T^{1,0}(X) \opl... |
I am wondering if it's true in general that:
> $ ( A \text{ is an antichain in } P)^{ M} \Leftrightarrow ( A \text{ is an antichain in } P)$ whenever $M$ is a transitive ZFC model.
I recall that 'being a relation' is absolute. Is this enough to say what I want? Is being a maximal antichain also absolute for tran... |
In proposition 2.7 the map $\tau$ has values in $C^\infty (M)$.
But the curvature tensor is a map $\mathfrak{X}(M)^3 \to \mathfrak{X}(M)$ as you have defined it. Therefore you can not apply proposition 2.7 in the way you want to.
To apply proposition 2.7 we first need to transform the Riemann "Tensor" (note that ... |
**Given any equation, can we definitively determine in finite time whether or not this equation is solvable by non-numerical methods to give a symbolic answer?**
It is trivial to show that there exist equations which is provable to have a symbolic answer (linear equations, quadratic equations, etc). Some equations m... |
Given any equation, can we definitively determine in finite time whether or not this equation can be solved to give a symbolic answer? |
Let be $x_{1},x_{2},x_{3}$ i.i.d. random variables following a normal distribution with $\mu=0$ and $\sigma=1$. I'm intrigued by the following random variable, which is a ratio of a cubic form and a quadratic form:
$$\frac{1}{\sqrt 2}\frac{x_1^2 x_2 + x_1 x_2^2 + x_1^2 x_3 - 6x_1 x_2 x_3 + x_2^2 x_3 + x_1 x_3^2 + x_... |
Suppose you have a hemispherical bowl that is open at the top. You tilt the bowl by an angle $\theta$ in any direction. Then you fill it with water till the water surface reaches the lowest point on the rim of the bowl. If the radius of the bowl is $R$, what is the volume $V$ of water as a function of $\theta$?
*... |
I know that a representation of the Dirac delta function is $\sum_{n=-\infty}^\infty e^{inx}=2 \pi \delta(x)$.
I am trying to figure out if the series with positive $n$ only $\sum_{n=0}^\infty e^{inx}$ has a simple limit as a distribution, but I cannot find any sources. The real part should be $\pi \delta(x)+\frac{1}{... |
What is the limit (as a distribution) of this Fourier series, similar to that of a Dirac Delta? |
In proposition 2.7 the map $\tau$ has values in $C^\infty (M)$.
But the curvature tensor is a map $\mathfrak{X}(M)^3 \to \mathfrak{X}(M)$ as you have defined it. Therefore you can not apply proposition 2.7 in the way you want to.
To apply proposition 2.7 we first need to transform the Riemann "Tensor" (note that ... |
When Shapiro says that the polynomials "span" a set $B$ "in" a topology, of course he means that the *closure* of the set of polynomials in *that* topology is $B$. I make this somewhat pedantic point only to emphasize that there is nothing counterintuitive about potentially getting different things when one computes th... |
In proposition 2.7 the map $\tau$ has values in $C^\infty (M)$.
But the curvature tensor is a map $\mathfrak{X}(M)^3 \to \mathfrak{X}(M)$ as you have defined it. Therefore you can not apply proposition 2.7 in the way you want to.
To apply proposition 2.7 we first need to transform the Riemann "Tensor" (note that ... |
I begin by an example showing that the solutions of
$$AL+L^T=0\tag{1}$$
can depend on parameters.
Let us take
$$A=\pmatrix{1&0&0\\0&1&1\\0&0&-1}$$
then the general solution of (1) is :
$$L=\pmatrix{0&a&0\\-a&b&-2b\\0&-2b&4b}$$
How such solutions can be obtained ? By the following method :
Begi... |
# 1D basis functions
In 1D space, I'm looking for a set of [basis functions](https://en.wikipedia.org/wiki/Basis_function) like below. They would be equal to one at certain points and zero at others.
[![Schematic 1D][1]][1]
## 2D schematics
In 2D space, the basis function could be something below, taken fro... |
A problem in _Complex Numbers_ (Andreescu, Andrica)
> Let us solve the equation $z^3 = 18 + 26i$, where $z = x + yi$ and $x$, $y$ are
integers.<br><br>
We can write $(x + yi)^3 = (x + yi)^2 \times (x + yi)$ = $(x^2 − y^2 + 2xyi)(x + yi)
= (x^3 − 3xy^2) + (3x^2y − y^3)i = 18 + 26i$
<br><br>
Using the definition ... |
Why is this particular substitution made? ($y=tx$) |
The following result holds true:
> For a real function $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ we have that $\inf_{x,y} F(x,y)\leq \inf_x (\inf_y F(x,y)) $.
My question is on whether or not it is possible to provide an example where we have strict inequality? If so, I would be interested in seeing an example. I... |
Suppose you have a hemispherical bowl that is open at the top. You tilt the bowl by an angle $\theta$ in any direction. Then you fill it with water till the water surface reaches the lowest point on the rim of the bowl. If the radius of the bowl is $R$, what is the volume $V$ of water as a function of $\theta$?
*... |
Let $A(a_1, a_2, a_3), B(b_1, b_2, b_3), C(c_1, c_2, c_3), D(d_1, d_2, d_3), E(e_1, e_2, e_3), F(f_1, f_2, f_3)$ in the 3d system of coordinates.
What is the conditiones to have the plane $(ABC) $ and $(DEF) $ perpendicular? |
What is the conditiones to have the plane $(ABC) $ and $(DEF) $ perpendicular? |
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