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I recently deleted this same question, because i thought i had solved it, but I didn't, and is proving very tricky:
Find a polynomial $u_f\in\mathbb{C}[x]$ such that, for any two $ f,g\in \mathbb{Q}[x]$, $g(x)=\alpha(x-\theta_0)(x-\theta_1)$, $\mathrm{Deg}(f,g)=(n,2)$ we have that:
$$f(g(x))=u_f(x-\theta_0)u_f(x-\... |
Can a polynomial composition be written as a product of of a polynomial over linear factors? |
>Let $\alpha$ be a root of $f(X)=X^3+X^2-2X+8$ and $\beta = \frac{4}{\alpha}$. It can be shown that $O_K=\mathbb{Z}[\alpha, \beta]$.
>
>**How does one then establish the following ring isomorphism** $$\frac{O_K}{2O_K} \cong \mathbb{F}_2 \times \mathbb{F}_2 \times\mathbb{F}_2\ ?$$
You know that $α$ is a root of $f$... |
I am currently trying to write something but I got stuck. Essentially, I need to know whether a finite subset of probability distributions is a subset of the set of all probability distributions over itself.
Let me put my problem formally. Let $X$ be an arbitrary finite set, and let $D(X)$ be the set of all probabi... |
Is a finite subset of probability distributions a subset of the set of all probability distributions over itself? |
I am currently trying to write something but I got stuck. Essentially, I need to know whether a finite set of probability distributions is a subset of the set of all probability distributions over itself.
Let me put my problem formally. Let $X$ be an arbitrary finite set, and let $D(X)$ be the set of all probabilit... |
I've been playing with a boundary value problem consisting of a fourth-order, nonlinear, inhomogeneous differential equation (containing two parameters $P$ and $Q$) and four or five boundary conditions. By "four or five", I mean the following four:
1. $f\left(1\right) = 1$
2. $f'\left(1\right) = 0$
3. $f''\lef... |
A particle of mass $m$ capable of moving along the $x$-axis, is acted upon by a force $F(x) = -\frac{k}{x^3}$. At the initial time moment $t=0$, the particle is at the point $x=x_0>0$, and its velocity is $\dot{x}(0)=0$. In what time $\tau$ will the particle reach the point $x=0$?
**My attempt:**
We basically have ... |
In what time $\tau$ will the particle reach the point $x=0$? |
Is a finite set of probability distributions a subset of the set of all probability distributions over itself? |
>Let $\alpha$ be a root of $f(X)=X^3+X^2-2X+8$ and $\beta = \frac{4}{\alpha}$. It can be shown that $O_K=\mathbb{Z}[\alpha, \beta]$.
>
>**How does one then establish the following ring isomorphism** $$\frac{O_K}{2O_K} \cong \mathbb{F}_2 \times \mathbb{F}_2 \times\mathbb{F}_2\ ?$$
You know that $α$ is a root of $f$... |
-I have the following simple question find the differentiable: $D \left \{ \phi(x;y) \right \} (h_1 ; h_2) $ of the following function: $ \phi(x;y) = \begin{pmatrix}
x \\
y
\end{pmatrix} \frac{1}{\sqrt{x^2+y^2}}$ when we are in $\mathbb{R}^2 - (0;0)$
-The answer is $ D \left \{ \phi(x;y) \right \}... |
For $a,b,c>0$ and $a+b+c=1.$ Prove$:$ $$\frac{1}{ab+2c^{2}+2c}+\frac{1}{bc+2a^{2}+2a}+\frac{1}{ca+2b^{2}+2b}\geqq \frac{1}{ab+bc+ca}$$
well i try to use AM-GM but turns out:
\frac{9}{4(a^{2}+b^{2}+c^{2}+5(ab+bc+ca)} \geqq \frac{1}{ab+bc+ca}
and i always get this : ab+bc+ca \gegg a^{2}+b^{2}+c^{2}
can someon... |
I want to check the convexity of the following multivariate function:
$$K(Q, s) = h\frac{(Q-s)^2}{2Q}+b\frac{s^2}{2Q}+\hat{b}\frac{s\lambda}{Q}+K\frac{\lambda}{Q} +c\lambda,$$
where $Q>0$, $s \ge 0$ and $h, b, \hat{b}, c, \lambda, K > 0$ and are known.
I want to know/prove that this function is convex.
Calcul... |
We define the class of hypergeometric probability distribution as $\left\{P_{D}: D=0,1,2, \cdots, N\right\}$, where
$$
P_{D}(X=x)=\frac{\left(\begin{array}{c}
D \\
x
\end{array}\right)\left(\begin{array}{c}
N-D \\
n-x
\end{array}\right)}{\left(\begin{array}{c}
N \\
n
\end{array}\right)} \quad x=0,1, \cdot... |
For $a,b,c>0$ and $a+b+c=1.$ Prove$:$ $$\frac{1}{ab+2c^{2}+2c}+\frac{1}{bc+2a^{2}+2a}+\frac{1}{ca+2b^{2}+2b}\geqq \frac{1}{ab+bc+ca}$$
well i try to use AM-GM but turns out:
$$\frac{9}{4(a^{2}+b^{2}+c^{2}+5(ab+bc+ca))} \geq\frac1{ab+bc+ca}$$
and i always get this : $$ab+bc+ca \geq a^{2}+b^{2}+c^{2}$$
can so... |
-I have the following simple question find the differentiable: $D \left \{ \phi(x;y) \right \} (h_1 ; h_2) $ of the following function: $ \phi(x;y) = \begin{pmatrix}
x \\
y
\end{pmatrix} \frac{1}{\sqrt{x^2+y^2}}$ when we are in $\mathbb{R}^2 - (0;0)$
-The answer is $ D \left \{ \phi(x;y) \right \}... |
From $\frac{dv}{dt} = v \frac{dv}{dx}$ we get
$$v \frac{dv}{dx} = \frac{-k}{mx^3} \implies v \; dv = \frac{-k}{mx^3} \; dx.$$
Now we integrate to get
$$\frac{1}{2}v^2 = \frac{k}{2mx^2} + A \implies v = \sqrt{\frac{k}{mx^2} + A}.$$
Where did you get the $\frac{dv}{v}$ from? |
* I have the following simple question find the differential: $D\phi(x;y) (h_1 ; h_2) $ of the function: $ \phi(x;y):= \frac{1}{\sqrt{x^2+y^2}}\begin{pmatrix}
x \\
y
\end{pmatrix},$ defined on $\mathbb{R}^2\setminus\{(0;0)\}$.
* The answer is $$D\phi(x;y)(h_1 ; h_2) = \frac{1}{x^2 + y^2} \begin{pmatrix}
x... |
>Let $\alpha$ be a root of $f(X)=X^3+X^2-2X+8$ and $\beta = \frac{4}{\alpha}$. It can be shown that $O_K=\mathbb{Z}[\alpha, \beta]$.
>
>**How does one then establish the following ring isomorphism** $$\frac{O_K}{2O_K} \cong \mathbb{F}_2 \times \mathbb{F}_2 \times\mathbb{F}_2\ ?$$
You know that $α$ is a root of $f$... |
When im trying to determine a fourier series if I determin the $a_0$ is 0 does it follow that $a_n$ must also be 0? |
In a fourier series if $a_0$ is equal to 0 is $a_n$ also equal to 0? |
I know well that the Jacobi-Anger expansion is
or
$e^{iz\cos (x)}=\sum_{n=-\infty}^\infty i^n J_n(z)e^{inx}$
My question is how to expand with Jacobi-Anger expansion if the subscript of the exponential is a Fourier series? For example what is the solution if the subscript is the first two terms of the Fourier ... |
I know well that the Jacobi-Anger expansion is
$e^{iz\sin (x)}=\sum_{n=-\infty}^\infty J_n(z)e^{inx}$
or
$e^{iz\cos (x)}=\sum_{n=-\infty}^\infty i^n J_n(z)e^{inx}$
My question is how to expand with Jacobi-Anger expansion if the subscript of the exponential is a Fourier series? For example what is the solut... |
Let $(X_t)_{t \in [0, \infty)}$ a $\mathbb{R}$- valued
Markov process, $s, a, u >0$,
$I(a) :=
\{[k \cdot a, (k+1) \cdot a] \ : \ k \in \mathbb{Z} \} $ the
family of $a$-integral tiles covering $\mathbb{R}$.
Let $M_u(a,s)$ the random variable counting the number of tiles
$[k \cdot a, (k+1) \cdot a]$ in $I(a)$... |
* I have the following simple question find the differential: $D\phi(x;y) (h_1 ; h_2) $ of the function: $ \phi(x;y):= \frac{1}{\sqrt{x^2+y^2}}\begin{pmatrix}
x \\
y
\end{pmatrix},$ defined on $\mathbb{R}^2\setminus\{(0;0)\}$.
* The answer is $$D\phi(x;y)(h_1 ; h_2) = \frac{1}{x^2 + y^2} \begin{pmatrix}
x... |
I am learning how to prove and disprove atomic equations. I have put together a proof where I know how I want to prove the theorem but can't quite get it to flow correctly. Please critique my syntax, flow, or anything else as I am learning how to prove mathematical statements:
\begin{align*}
& \hspace{5mm} \tex... |
The normal **hyperboloids** are surfaces of degree $2$ in $\mathbb R^3$. But $xyz=1$ is a surface of degree $3$. So it is not a "hyperboloid" in that sense.
I seem to recall: Newton worked out the possible surfaces of degree $3$ in $\mathbb R^3$ ?? Or did he only do curves of degree $3$ in $\mathbb R^2$ ?? |
A particle of mass $m$ capable of moving along the $x$-axis, is acted upon by a force $F(x) = -\frac{k}{x^3}$. At the initial time moment $t=0$, the particle is at the point $x=x_0>0$, and its velocity is $\dot{x}(0)=0$. In what time $\tau$ will the particle reach the point $x=0$?
**My attempt:**
We basically have ... |
In this particular ideal case, you need to make sure that your cosines are "commensurate" with your domain. I will explain what this means.
By using `np.linspace` with `endpoint=True` (which is its default value) you are sampling the boundary value twice. This is because the FFT models functions as being periodic. F... |
As a student of numerical analysis, I understand that a finite difference approximation (FDM) of the derivative '$u_x$' of a desired solution '$u$' at the point $x_n$ in the domain is just a linear combination $\displaystyle \sum_{i=0}^Nc_i u_i,$ where, $u_i=u(x_i)$.
That is, $\displaystyle (u_x)_n=u_x(x_n)\approx c... |
How do I convert the general form of a linear equation of two variables to slope intercept form?
> More precisely, beginning with the form $ax + by + c = 0$
I have tried rearranging this for $y$, however, I am having difficulties with this. My idea is the following: use the substitution $c = -(x + y)$.
Is th... |
Let $f:\Omega\to\mathbb{R}$ be a continuous function over convex domain $\Omega$.
Is the set $$ f\left(\Omega\right)=\left\{ y:y=f\left(x\right):x\in\Omega\right\} $$
Is the set $f(\Omega)$ convex ?
Assume $$ \begin{cases}
y_{1}=f\left(x_{1}\right)\\
y_{2}=f\left(x_{2}\right)
\end{cases} $$
Looking at points ... |
I think you have confused the groups $$\langle \Bbb Z_n, +\rangle\tag1$$ and $$\langle \Bbb Z_n^\ast, \cdot\rangle\tag2$$
Often when we mention a group, we leave the group operation implicit, and often it isn't confusing. This time, it was confusing.
The theorem you cited about $\gcd(n,k)=1$ is about the _additiv... |
I think you have confused the groups $$\langle \Bbb Z_n, +\rangle\tag1$$ and $$\langle \Bbb Z_n^\ast, \cdot\rangle\tag2$$
Often when we mention a group, we leave the group operation implicit, and often it isn't confusing. This time, it was confusing.
The theorem you cited about $\gcd(n,k)=1$ is about the _additiv... |
A particle of mass $m$ capable of moving along the $x$-axis, is acted upon by a force $F(x) = -\frac{k}{x^3}$. At the initial time moment $t=0$, the particle is at the point $x=x_0>0$, and its velocity is $\dot{x}(0)=0$. In what time $\tau$ will the particle reach the point $x=0$?
**My attempt:**
We basically have ... |
In [1], I encounter the following statement:
For $ f: \mathbb{R}^{d} \rightarrow \mathbb{R} $ that is twice differentiable,
$ \nabla f (x + p) - \nabla f (x) = \int_{0}^{1} \nabla^{2} f (x + t p) ~ p ~ dt $,
which is named as "Taylor's theorem".
Can someone help me prove the statement?
In particular, doe... |
Does this spiral, formed from similar right triangles arranged around a point, have a name? |
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... |
A particle of mass $m$ capable of moving along the $x$-axis, is acted upon by a force $F(x) = -\frac{k}{x^3}$. At the initial time moment $t=0$, the particle is at the point $x=x_0>0$, and its velocity is $\dot{x}(0)=0$. In what time $\tau$ will the particle reach the point $x=0$?
**My attempt:**
We basically have ... |
I have a matrix $A =\begin{pmatrix}2&-1&3&3\\ 0&0&2&2\\ 4&-2&5&5\end{pmatrix}$ and its kernel, given by the span of $(1 2 0 0)^T$ and $(0 0 1 -1)^T$
Row reducing the matrix gives $\begin{pmatrix}2&-1&0&0\\ 0&0&1&1\\ 0&0&0&0\end{pmatrix}$
I just take the first and third rows of $A$ to be a basis for the image. Ho... |
How is the kern of a matrix also the image? |
I have a matrix $A =\begin{pmatrix}2&-1&3&3\\ 0&0&2&2\\ 4&-2&5&5\end{pmatrix}$ and its kernel, given by the span of $(1, 2, 0, 0)^{\top}$ and $(0, 0, 1, -1)^{\top}$
Row reducing the matrix gives $\begin{pmatrix}2&-1&0&0\\ 0&0&1&1\\ 0&0&0&0\end{pmatrix}$
I just take the first and third rows of $A$ to be a basis f... |
How is the kernel of a matrix also the image? |
I know that the general word problem is undecidable, but is there an effective enumeration of presentations all groups generated by $n$ elements in which each isomorphism class of a group appears exactly once?
I have no idea of how to tackle this problem, and I'm motivated by the fact that isomorphism classes of pat... |
Is there an enumeration of finitely generated groups? |
I have a line integral over a curve $\gamma$
$$\gamma(t) = <1, 1 - t>$$
$$\int_{\gamma(0)}^\gamma(1) f(x,y) ds$$
Now, because $\gamma'(t) = <0, -1>$, $\lVert \gamma'(t) \rVert = 1$
$$I_\gamma = \int_{0}^1 f(1,1-t) \lVert \gamma'(t) \rVert dt = \int_{0}^1 f(1,1-t) dt $$
Now suppose, that the path was in f... |
Line Integral over curve gamma, how would the sign change if the orientation was reversed? |
I have a line integral over a curve $\gamma$
$$\gamma(t) = <1, 1 - t>$$
$$\int_{\gamma(0)}^{\gamma(1)} f(x,y) ds$$
Now, because $\gamma'(t) = <0, -1>$, $\lVert \gamma'(t) \rVert = 1$
$$I_\gamma = \int_{0}^1 f(1,1-t) \lVert \gamma'(t) \rVert dt = \int_{0}^1 f(1,1-t) dt $$
Now suppose, that the path was in... |
Line Integral over curve $\gamma$, how would the sign change if the orientation was reversed? |
A particle of mass $m$ capable of moving along the $x$-axis, is acted upon by a force $F(x) = -\frac{k}{x^3}$. At the initial time moment $t=0$, the particle is at the point $x=x_0>0$, and its velocity is $\dot{x}(0)=0$. In what time $\tau$ will the particle reach the point $x=0$?
**My attempt:**
We basically have ... |
I have a line integral over a curve $\gamma$
$$\gamma(t) = \langle1, 1 - t\rangle$$
$$\int_{\gamma(0)}^{\gamma(1)} f(x,y) ds$$
Now, because $\gamma'(t) = \langle0, -1\rangle$, $\lVert \gamma'(t) \rVert = 1$
$$I_\gamma = \int_{0}^1 f(1,1-t) \lVert \gamma'(t) \rVert dt = \int_{0}^1 f(1,1-t) dt $$
Now suppo... |
Is *parallel* really the opposite of *anti-parallel*? While
- *anti-parallel* indicates that two vectors are directed along the same line **and are pointing in opposite directions.**
then it seems to me that
- *parallel* just indicates that two vectors are directed along the same line.
This might not be th... |
What is the opposite of antiparallel? |
A particle of mass $m$ capable of moving along the $x$-axis, is acted upon by a force $F(x) = -\frac{k}{x^3}$. At the initial time moment $t=0$, the particle is at the point $x=x_0>0$, and its velocity is $\dot{x}(0)=0$. In what time $\tau$ will the particle reach the point $x=0$?
**My attempt:**
We basically have ... |
*Too long for a comment:*
>**What are some interesting cases of $\pi$ appearing in situations that are not geometric ?**
None! :-) You did well to add “do not seem” in the title! ;-)
___
All $\zeta(2k)$ are *bounded sums of squares*, are they not ? And the equation of the circle, $x^2+y^2=$ $=r^2$, also repres... |
It's a well-known formula that $$
e = \lim\limits_{n\rightarrow\infty} (1+1/n)^n
$$
and I'm wondering if there exists a similar formula for $\pi$. All the formulas I know for $\pi$ involve series or products or special functions (such as inverse trig functions). Or one can use Stirling's approximation if factorials ... |
I recently deleted this same question, because i thought i had solved it, but I didn't, and its proving very tricky:
Find a polynomial $u_f\in\mathbb{C}[x]$ such that, for any two $ f,g\in \mathbb{Q}[x]$, $g(x)=\alpha(x-\theta_0)(x-\theta_1)$, $\mathrm{Deg}(f,g)=(n,2)$ we have that:
$$f(g(x))=u_f(x-\theta_0)u_f(x-... |
I am trying to build a formula with the data that I have below. The formula will be embedded in a Arduino sketch.
Column A is the time stamp and Column B is my voltage value. Column C is a threshold value, which I am pretty sure is not relevant to my question here.
Column E give me the slope between two points.
... |
Fitting / drawing a closed curve of a given length within given dimensions.
I'm trying to fit / draw a given length of a closed curve within given dimensions.
**Example:** Curve **length=941mm**, **width=3mm**, and **height=5mm** and it must fit within the dimensions **x=250mm** **y=210mm**
I was thinking of ... |
Is *parallel* really the opposite of *anti-parallel*? While
- *anti-parallel* indicates that two vectors are directed along the same line **and are pointing in opposite directions**
then it seems to me that
- *parallel* just indicates that two vectors are directed along the same line. But **not** necessary poi... |
I recently deleted this same question, because i thought i had solved it, but I didn't, and its proving very tricky:
Find a polynomial $u_f\in\mathbb{C}[x]$ such that, for any two given $ f,g\in \mathbb{Q}[x]$, $g(x)=\alpha(x-\theta_0)(x-\theta_1)$, $\mathrm{Deg}(f,g)=(n,2)$ we have that:
$$f(g(x))=u_f(x-\theta_0)... |
The time taken is given by:
$$
\tau = -\int_{x_0}^0\frac{dx}{\sqrt{\frac{k}{mx^2}-\frac{k}{mx_0^2}}} \\
\implies \tau = -\sqrt{\frac{m}{k}}x_0\int_{x_0}^0\frac{xdx}{\sqrt{x_0^2-x^2}}. \\
$$
We have that:
$$
\frac{d}{dx}(\sqrt{a^2-x^2})=-\frac{x}{\sqrt{a^2-x^2}}.
$$
Can you do the rest? |
I would like to add to the idea of Dániel, since I was not quite convinced at first:
"How do we know that such a proper ear exists?"
However, I could convince myself by the following line of argumentation:
Let $G$ be a 2-connected graph of order at least 3 (so $G \not \simeq K_3$) and consider an ear-decomposition... |
Can a polynomial composition be written as a product of a polynomial over linear factors? |
Fitting / drawing a closed curve of a given length within given dimensions.
I'm trying to fit / draw a given length of a closed curve within given dimensions.
**Example:** Curve **length=941mm**, **width=3mm**, and **height=5mm** and it must fit within the dimensions **x=250mm** **y=210mm**
I was thinking of ... |
**Abel's Test:** Let $f_n(x)$ be a non-increasing sequence of functions such that $0 \le f_n(x) \le M$ for all $x \in [a,b]$. If $\sum a_n$ converges then $\sum a_nf_n(x)$ converges uniformly in $[a,b]$.
**Abel's Test corollary:** Let $f_n(x)$ be a monotonic functions such that $0 \le f_n(x) \le M$ for all $x \in... |
>Let $\alpha$ be a root of $f(X)=X^3+X^2-2X+8$ and $\beta = \frac{4}{\alpha}$. It can be shown that $O_K=\mathbb{Z}[\alpha, \beta]$.
>
>**How does one then establish the following ring isomorphism** $$\frac{O_K}{2O_K} \cong \mathbb{F}_2 \times \mathbb{F}_2 \times\mathbb{F}_2\ ?$$
You know that $α$ is a root of $f$... |
Consider the subgroup $\Gamma < \operatorname{SL}(2,\mathbb{R})$ generated by the element
$$
\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}
$$
and all elements of $\operatorname{SO}(2)$. Is $\Gamma = \operatorname{SL}(2,\mathbb{R})$?
Some calculations show that
$$
\begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} \i... |
Is SL(2,R) generated by SO(2) and a single upper triangular element? |
If factorial are accepted:
By [this](https://en.wikipedia.org/wiki/List_of_formulae_involving_%CF%80#Iterative_algorithms) article if you consider the sequence
$$
a_{n+1}=\left(1+\frac{1}{2n+1}\right) a_n
$$
You have
$$
\pi= \lim_n \frac{a_n}{n}
$$
The sequence admits a [closed form](https://www.wolframalpha... |
what math concept would this be called? |
Community wiki as it does not satisfy the constrains of the OP:
By [this](https://en.wikipedia.org/wiki/List_of_formulae_involving_%CF%80#Iterative_algorithms) article if you consider the sequence
$$
a_{n+1}=\left(1+\frac{1}{2n+1}\right) a_n
$$
You have
$$
\pi= \lim_n \frac{a_n}{n}
$$
The sequence admits a [... |
I'm trying to find sufficient additional conditions to derive Coulomb equation for the electric field generated by a steady point charge in free space from Maxwell equations in said conditions.
I know that a way to do this is assuming that the solution of Maxwell equations must have spherical symmetry due to the di... |
>Let $\alpha$ be a root of $f(X)=X^3+X^2-2X+8$ and $\beta = \frac{4}{\alpha}$. It can be shown that $O_K=\mathbb{Z}[\alpha, \beta]$.
>
>**How does one then establish the following ring isomorphism** $$\frac{O_K}{2O_K} \cong \mathbb{F}_2 \times \mathbb{F}_2 \times\mathbb{F}_2\ ?$$
You know that $α$ is a root of $f$... |
I am trying to solve the following problem
Consider the ensemble consisting of the qubit states $|0⟩⟨0|$ and $|1⟩⟨1|$ occuring with probabilities 2/3 and 1/3, respectively.
Compute the quantum state $\rho$ corresponding to this ensemble. Is $ρ $ pure or mixed?
My try:
I think there is nothing to compute. The ... |
Suppose $P$ and $Q$ are combinatorially equivalent non-self-intersecting polyhedra in $\mathbb{R}^3$, with $f$ a map from edges of $P$ to edges of $Q$ under said combinatorial equivalence. Let $\theta(e)$ be the dihedral angle of edge $e$.
If it is the case that for all $e\in D$, $\theta(e)\le \theta(f(e))$, must w... |
Is it possible for the dihedral angles of a polyhedron to all grow simultaneously? |
* I have the following simple question find the differential: $D\phi(x;y) (h_1 ; h_2) $ of the function: $ \phi(x;y):= \frac{1}{\sqrt{x^2+y^2}}\begin{pmatrix}
x \\
y
\end{pmatrix},$ defined on $\mathbb{R}^2\setminus\{(0;0)\}$.
* The answer is $$D\phi(x;y)(h_1 ; h_2) = \frac{1}{x^2 + y^2} \begin{pmatrix}
... |
It is an easy exercise to show that all finite groups with at least three elements have at least one non-trivial automorphism; in other words, there are - up to isomorphism - only finitely many finite groups $G$ such that $\text{Aut}(G)=1$ (to be exact, just two: $1$ and $C_2$).
Is an analogous statement true for al... |
* I have the following simple question find the differential: $D\phi(x;y) (h_1 ; h_2) $ of the function: $ \phi(x;y):= \frac{1}{\sqrt{x^2+y^2}}\begin{pmatrix}
x \\
y
\end{pmatrix},$ defined on $\mathbb{R}^2\setminus\{(0;0)\}$.
* The answer is $$D\phi(x;y)(h_1 ; h_2) = \frac{1}{x^2 + y^2} \begin{pmatrix}
... |
The thing to keep in mind with [disk integration](https://en.wikipedia.org/wiki/Disc_integration) is the fact that the variable of integration represents the *thickness* of each disk.
You are trying to integrate the volume of a half-sphere, using:
$$
\int_0^r \pi x^2 dx
$$
with the idea being that a bunch of cir... |
Often times the notation we use for mathematical concepts is based in part on properties of that concept. For example, the notation $|A| = |B|$ in set theory means there’s a bijection between $A$ and $B$, and the notation is warranted because the “there is a bijection between these two sets” relation is an equivalence ... |
Has there ever been a substantial change in notation resulting from a landmark theorem? |
Im currently trying to derive d'Alemberts formula for wave equation in dimension $n=1$ from the Riemann function, and don't end up where I want.
I want to show that $$u(t,x)=\frac{1}{2}\big( g(x-ct)+g(x+ct)\big) + \frac{1}{2c} \int_{x-ct}^{x+ct} h(y) \ dy$$ from the general solution that is $$u(t,x)=h(x) * R_t(x)+ \... |
$\renewcommand{\al}{\alpha}
\newcommand{\bra}[1]{\left<#1\right>}
\newcommand{\brc}[1]{\left(#1\right)}
\newcommand{\Aut}[2]{\operatorname{Aut}_{#1}\brc{#2}}
\renewcommand{\C}{\mathbb{C}}
\newcommand{\si}{\sigma}$
Let $L$ be a complex semisimple finite dimensional Lie algebra with a finite dimensional faithful re... |
[![\[enter image description here\]][1]][1]
[1]: https://i.stack.imgur.com/aUDMk.png
Hi, Can you explain to me how it is possible to calculate a derivative at the zero point **without calculating the limit at zero?**
here is translation of this question:
*Show that the zero function is differentiable on... |
I'm trying to find general sufficient additional conditions to derive Coulomb equation for the electric field generated by a steady point charge in free space from Maxwell equations in said conditions.
I know that a way to do this is assuming that the solution of Maxwell equations must have spherical symmetry due t... |
In a certain exercise I am asked to find the probability of the event in which a dize is rolled seven times and the sum of the results is $14$, given the fact that the dize is regular (this is, the probability of each of the six possible results is the same: $\frac{1}{6}$.
\
I think I know the approach: let $X_i$ den... |
For example, given that a linear function is defined by its satisfying the properties $f(x+y)=f(x)+f(y)$ and $f(ax)=af(x)$, can I reduce this to $f(ax+by)=af(x)+bf(y)$, or maybe even $f\left(\frac ab x+y\right)=\frac abf(x)+f(y)$ and put $c=\frac ab$? |
Can I combine axioms to have less properties to verify? |
For example, given that a linear function is defined by its satisfying the properties $f(x+y)=f(x)+f(y)$ and $f(ax)=af(x)$, would it be okay to only check $f(ax+by)=af(x)+bf(y)$, or maybe even $f\left(\frac ab x+y\right)=\frac abf(x)+f(y)$ and put $c=\frac ab$? |
Can I combine axioms to have less properties to verify independently? |
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/10.54... |
A light bulb is automatically turned off for 21 secs. Next, it lights up automatically for 15 secs. Once again, it is automatically turned off for the following 21 secs. The process keeps repeating.
After 200 full mins, how many times has the bulb light up after turning off?
My approach was the following:
21 +... |
How many times is the light bulb turned on? |
This problem appears in Girls in Math at Yale, a high school competition which I am prepping for:
"Marie repeatedly flips a fair coin and stops after she gets tails for the second time. What is the expected number of times Marie flips the coin?"
I deduced that for there to be n flips, we have to add n(n-1)/2^n to... |
How to calculate the sum of an infinite series, such as n(n-1)/2^n? |
This problem appears in Girls in Math at Yale, a high school competition which I am prepping for:
"Marie repeatedly flips a fair coin and stops after she gets tails for the second time. What is the expected number of times Marie flips the coin?"
I deduced that for there to be n flips, we have to add $\dfrac{n(n-1... |
Is $SL(2,\Bbb R)$ generated by $SO(2)$ and a single upper triangular element? |
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