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If you are using natural deduction axioms/rules (e.g. as per Van Dalen), I think the folowing is a proof: (A stands for assumption, Best viewed on a PC; is there a good way of presenting tables in Latex?).
$Level | Line | Formula \quad\quad\quad\quad\quad \quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\q... |
Let $\Gamma \le \operatorname{PSL}(2,\mathbb{R})$ be a discrete subgroup and $\Sigma:=\mathbb{D}^2/\Gamma$ be the quotient. Then $\Sigma$ is a hyperbolic surface whose universal cover is $\mathbb{D}^2$ and the fundamental group $\pi_1(\Sigma,x)$ ($x\in\Sigma$) is isomorphic to $\Gamma$. Fix a point $\tilde{x}\in\mathbb... |
*Conceptually* $\, \dfrac{m}{18} \!=\! a \!=\! \dfrac{n}{25}\,$ [so](https://math.stackexchange.com/a/825871/242) its least denominator divides *coprimes* $18,25$ so is $1,\,$ so $\,a\in\Bbb Z$
**Remark** $\ $ This is an additive analog of this better-known (multiplicative) group result
$$ a^{\large 18} = 1 = a^... |
Let $c$ be a real consant, $X$ a real random variable and $H(c)=\mathbb E[(c-X)\chi_{X\leq c}]$. Show that $$H'(c)=P[X\leq c].$$ I already proved this if $X$ has a density function. But what if $X$ has no density? I thought about the weak derivative and started with $$\int \mathbb E[(c-X)\chi_{X\leq c}] \phi'(c)dc$$ fo... |
Short answer: these are two different definitions for two different things that are both called "condition number" (and there are many more), of course they don't agree.
Long answer :
The "condition number" of a matrix (or more generally any function) is an indicator for how much the output changes if you change ... |
I am stucked on this question and have no idea. Can somebody tell me how to solve this summation?
$\Sigma ^n_{k=1} 3^kk^2$
|
Suppose $b\in L^\infty(\mathbb{R})$ and $u\colon \mathbb{R}\times \mathbb{R}\to \mathbb{R}$, wher $(t,x)\mapsto u(t,x)$ is a solution to $$
\partial_t^2 u = \partial_x^2 u +b(x)\partial_x u,\quad (x,t)\in \mathbb{R}\times \mathbb{R}
$$ with initial conditions $u(0,\cdot)=u_0$ and $\partial_t u(0,\cdot)=u_1$.<br>
Do ... |
This answer proves that there are infinitely many $(p,q)$ satisfying the given conditions.
*Proof* :
Let $f(x):=ax^3+bx^2+cx+d$ with $a\gt 0$.
Then, we have
$$\begin{align}f'(x)&=3ax^2+2bx+c
\\\\f''(x)&=6ax+2b
\\\\g'(x)&=\pi e^{\sin(\pi x)}\cos(\pi x)
\\\\g''(x)&=\pi^2 e^{\sin(\pi x)}\bigg(\cos^2(\pi x)-\... |
To summarize the setup, we have an object with a reference frame attached to it, that specifies the object's orientation in space with respect to a world coordinate system (the coordinate system chosen for the room). In this world coordinate system we have the coordinates of $3$ points $A,B, C$. The relationship betw... |
I recently took it upon myself to investigate the knight's tour problem for a math assessment.\
I decided to investigate how the problem differs with a general knight $(m, n)$ that moves $m$ squares along one axis and $n$ squares along the other. I feel like I have dug myself into a hole as I have only been able to pr... |
For fixed real numbers $a,b,c,d$ define the map $f : \mathbb{R}^2 \to \mathbb{R}^2$ by
$$f : (x, y) \longmapsto (\phi(ax) + b \cdot \phi(ay),\hspace{1em}
\phi(cx) + d \cdot \phi(cy)),
\qquad \phi(z) := \sin(z).$$
Given a point $(x_0, y_0) \in \mathbb{R^2}$ we consider its orbits under t... |
Your **idea can work** as follows.
For $i\in \{2,3,\dots \}$, we can show that for $\color{blue}{t\in (0,1)}$:
$$\color{blue}{F_{X_{N}|N=i}(t)=1-\left ( 1+ \frac{1}{i-1} \right ) (1-t) +\frac{1}{i-1} (1-t)^{i}}, \tag{1}.$$
This gives
$$\color{blue}{\mathbb E[X_{N}|N=i]}=\int_0^1 \left [1- F_{X_{N}|N=i}(t)... |
Can someone recommend a Calculus book that emphasizes and clarifies the intuition behind the theorems to the maximum, while at the same time is fairly rigorous?
During my bachelor's in computer science, we used Stewart. I didn't like it as it was far from rigorous. Besides, I don't just want a lot of numerical examp... |
What do we count here?
$$
\frac{dy}{dx} = {2\choose1}x \
{2\choose1} = \frac{dy/dx}/{x} \
$$
Expanding $(x+dx)^n$ We discard the infinitesimal bits, the second term of the expansion, nC1•x, seems to remain as the derivative. I wonder if there is a combinatorics connection, if it can be seen on the graphs o... |
What do we count here?
$$
\frac{dy}{dx} = {2\choose1}x, \quad
{2\choose1} = \frac{dy/dx}{x}
$$
Expanding $(x+dx)^n$ We discard the infinitesimal bits, the second term of the expansion, $\binom{n}{1}x$, seems to remain as the derivative. I wonder if there is a combinatorics connection, if it can be seen on the... |
What do we count here?
$\frac{dy}{dx} = {2\choose1}x$
${2\choose1} = \frac{dy/dx}{x}$
Expanding $(x+dx)^n$ We discard the infinitesimal bits, the second term of the expansion, $\binom{n}{1}x$, seems to remain as the derivative. I wonder if there is a combinatorics connection, if it can be seen on the graphs ... |
What do we count here?
$$\frac{dy}{dx} = {2\choose1}x$$
$${2\choose1} = \frac{dy/dx}{x}$$
Expanding $(x+dx)^n$ We discard the infinitesimal bits, the second term of the expansion, $\binom{n}{1}x$, seems to remain as the derivative. I wonder if there is a combinatorics connection, if it can be seen on the gra... |
"Let $\epsilon > 0$ be given ...." is $\epsilon < 1$ ok? |
In limit definitions, is it a requirement that $\epsilon$ is arbitrary, or can I choose to work with small $\epsilon$, say below $1$? |
In [Dual Number](https://en.wikipedia.org/wiki/Dual_number#Modern_definition) it said that "It may also be defined as the exterior algebra of a one-dimensional vector space with $\varepsilon$ as its basis element." But I can't find the detailed rigorous expresion anywhere.
I learned in [Grassmann Number](https://en.w... |
What's the definition of dual number at perspect of exterior algebra? |
Introduction:
Find a such family of polynomial then we can always use Newton's method provided we have a procedure to start.
We introduce a rational function where $b_i$ are real numbers:
$f(x)=x\left(b_0(1+x)+\frac{b_1}{x+1}+\frac{b_2}{1+\frac{1}{x+1}}+\cdots+\frac{b_{n}}{1+\frac{1}{1+\frac{1}{1+\frac... |
If we consider this generating function:
$$G_{r}(x)=\sum_{n=0}^{\infty}r!{n \brace r} \frac{x^{n}}{n!}$$
is it true that $G_{r}(x)=G_{1}(x)\cdot G_{r-1}(x)$ ?
How to prove it? |
Your **idea can work** as follows.
For $i\in \{2,3,\dots \}$, we can show that for $\color{blue}{t\in (0,1)}$:
$$\color{blue}{F_{X_{N}|N=i}(t)=1-\left ( 1+ \frac{1}{i-1} \right ) (1-t) +\frac{1}{i-1} (1-t)^{i}} \tag{1}.$$
This gives
$$\color{blue}{\mathbb E[X_{N}|N=i]}=\int_0^1 \left [1- F_{X_{N}|N=i}(t) ... |
What's the name of this curve? |
My initial approach involves transforming the given congruences into linear equations:
a ≡ 7 (mod 20) to a = 20k + 7
b ≡ −3 (mod 4) to b = 4m - 3
then I plug that into the congruence of c getting:
c ≡ 40k + 14 + 4m - 3 (mod 12)
c ≡ 4k + m + 11 (mod 12)
What should I do next? |
I am reading the Evan's PDE, theorem 1 ( p.23 ) and some questions arise. I found assoicated question but I think that I don't get satisfactory answer yet.
Now assume $f\in C^{2}_C(\mathbb{R}^n)$.
> Theorem 1. ( Solving Poisson's euqation ) Define $u$ by $(8)$ ;
$$ u(x)=\int_{\mathbb{R}^n} \Phi(x-y)f(y)dy = ... |
Consider three, rather generic, equations in three variables $x,y,z$:
$$
\begin{cases}
y z+x=a\\
z x+y=b\\
x y+z=c
\end{cases}
$$
Now, assuming that there is a solution to this system, is it possible to find an expression in terms of $a,b,c$ only for the sum and/or product of the solutions $x,y,z$?
By this I... |
A bivariate function $f:\mathbb{R}^2 \to \mathbb{R} $ is called real analytic if it can be expanded into a power series about $x-x_0$ and $y-y_0$ within a neighborhood of $(x_0,y_0)$,i.e.
$$f(x,y)=\sum_{i,j=0}^{+\infty}a_{ij}(x-x_0)^i(y-y_0)^j,a_{ij}\in \mathbb{R}$$
A complex function $f:\mathbb{C}\to \mathbb{C}$ is ... |
What's the relationship between these concepts about power series expansion? |
I'm new here, nice to meet you all. I'm doing a PHD in mathematics, and I need your help.
My question is this:
Given a measure-preserving system X, And a bounded function f\in\L^\infty(X), the projection of f on the invariant factor of the system is given by
E_n foT^n.
From here, of course, we can also derive the... |
I'm trying to understand exercise 6.7 in this article:
https://www.gameludere.it/2019/05/19/funzioni-generatrici-esponenziali-e-applicazioni/#1_Le_funzioni_generatrici_esponenziali
But there is a step I don't understand.
That is, if we consider this generating function:
$$G_{r}(x)=\sum_{n=0}^{\infty}r!{n \brace r... |
I'm trying to understand exercise 6.7 in this article:
https://www.gameludere.it/2019/05/19/funzioni-generatrici-esponenziali-e-applicazioni/#1_Le_funzioni_generatrici_esponenziali
But there is a step I don't understand.
That is, if we consider this generating function:
$$G_{r}(x)=\sum_{n=0}^{\infty}r!{n \brace... |
Let G be a locally compact group and $\mathcal{H}$ a Hilbert space. Then there is a bijection between strongly continuous unitary representations of G and non-degenerate *-representations of $L^1(G)$ on $\mathcal{H}$ (see [Group algebra of locally compact group](https://en.wikipedia.org/wiki/Group_algebra_of_a_locally_... |
I will use my answer to https://math.stackexchange.com/questions/2094608/newton-method-to-solve-nonlinear-system as a guide and you can see another example using this method.
The regular Newton-Raphson method is initialized with a starting point $x_0$ and then you iterate $\tag 1x_{n+1}=x_n-\dfrac{f(x_n)}{f'(x_n)}$ ... |
I will use my answer to https://math.stackexchange.com/questions/2094608/newton-method-to-solve-nonlinear-system as a guide and you can see another example using this method.
The regular Newton-Raphson method is initialized with a starting point $x_0$ and then you iterate $\tag 1x_{n+1}=x_n-\dfrac{f(x_n)}{f'(x_n)}$ ... |
Can a Gaussian process have a degenerate sub-collection? |
In the following proof, (source https://jmp.sh/X5KwZ1Zt)
[![enter image description here][1]][1]
[![enter image description here][2]][2]
Using statement (i) as stated I only get
$X_{F^t}+D\varphi_F^tX_{G^t}\circ (\varphi^t_F)^{-1}=X_{F^t}+X_{{G^t}\circ (\varphi^t_F)^{-1}}$ but why is this equal to $ X_{F^t +G^... |
Why is this identity of hamiltonian vector fields true: $ X_{F^t}+X_{{G^t}\circ (\varphi^t_F)^{-1}}=X_{F^t +G^t\circ (\varphi^t_F)^{-1}}$? |
The following is from [Rasmussen and Williams (2006)][1]:
> A simple example of a Gaussian process can be obtained from our Bayesian linear regression model $f(x)=\phi(x)^Tw$ with prior $w\sim\mathcal N(0,\Sigma_p)$. We have for the mean and covariance
$$
\begin{aligned}
\mathbb E[f(x)]&=\phi(x)^T\mathbb E[w]=0\\... |
I have come across an integral that involves a hypergeometric function, which can be expressed as follows:
$$I = \int_0^1 x^{1/2}(1-x)^{\epsilon-1} {_{2}F_1}(\frac{1}{2}+\epsilon,1+\epsilon;\frac{3}{2};x) dx.$$
Here, $\epsilon$ is a small complex quantity where $|\epsilon|\ll1$.
I found an integral formula i... |
How can I make fast spectral radius normalization of sums of outer products? |
I think the lemma 2.3 in this [paper](https://arxiv.org/pdf/1401.7158.pdf) will help a lot in my homework, but I am confused about some steps of the proof. The lemma and its proof are as follows:(The proof in the paper begins at page 9.)
Lemma: Let $h \not\equiv 0$ be a holomorphic function on the disc $\Delta r$ in... |
$I(x, y, dx, dy) = \int^{x+dx}_{x}\int^{y+dy}_{y} e^{-\frac{1}{2}(ax^2+by^2+2cxy)}$, how can I estimate it with as little computation as possible?
Besides, I wonder if it is always hard to calculate or estimate the integration of $f(|ax^2+by^2+2cxy|)$ where $f(\cdot)$ is a monotonically decreasing function. |
Since the bins are identical, the question essentially reduces to - in how many ways can $2n$ distinct balls be divided into $n$ pairs?
Let's line up the balls in a row. The first two balls make one pair, the next two another pair, and so on. Consider any one arrangement. There are $n$ pairs and they can be arranged... |
I will use my answer to https://math.stackexchange.com/questions/2094608/newton-method-to-solve-nonlinear-system as a guide and you can see another example using this method.
The regular Newton-Raphson method is initialized with a starting point $x_0$ and then you iterate $\tag 1x_{n+1}=x_n-\dfrac{f(x_n)}{f'(x_n)}$ ... |
X and Y have the joint mass function:
$$f(x,y) = \frac{C}{(x+y+1)(x+y)(x+y-1)}$$ x, y =1,2,3...
find $V = X-Y$
if $z>=0$
$$P(X-Y = z) = P(X=k+z, Y=k)$$
$$=C\sum_{k=1}^{\infty}\frac{1}{(x+y+1)(x+y)(x+y-1)}$$
$$=\frac{1}{2}C \{\sum_{k=1}^{\infty}\frac{1}{(2k +z-1)(2k+z) - (2k+z)(2k+z+1)} \}$$
$$=\fr... |
My question originates from a short proof in Strang's LA book that shows when
$A\pmb{x}=\pmb{b}$
every vecetor $\pmb{x_r}$ in the rowspace of $A$ projects uniquely to a single vector $\pmb{b_c}$ in the column space of $A$. The proof is discussed [in this thread](https://math.stackexchange.com/questions/3585142/ho... |
What is the reason for endowing the positive semidefinite cone with an affine invariant metric instead of a Euclidean metric? |
X and Y have the joint mass function:
$$f(x,y) = \frac{C}{(x+y+1)(x+y)(x+y-1)}$$ x, y =1,2,3...
find $V = X-Y$
if $z>=0$
$$P(X-Y = z) = P(X=k+z, Y=k) (1)$$
$$=C\sum_{k=1}^{\infty}\frac{1}{(x+y+1)(x+y)(x+y-1)}(2)$$
$$=\frac{1}{2}C \{\sum_{k=1}^{\infty}\frac{1}{(2k +z-1)(2k+z) - (2k+z)(2k+z+1)} \}(3)$$... |
If have a point on an arc and other points related to it. I now want to move the point on the arc to a given target and rotate everything so that the arc-based-point has a vertical angle as the new position (like the if the arc-midpoint would now be just below (on y-axis) the target.
Example: I have an arc-midpoint ... |
Let $p\colon \mathbb{S}^2 \to X$, where $\mathbb{S}^2 = \{x \in \mathbb{R}^3 : \|x\| = 1\}$, be a covering projection. Is it possible to characterize the space $X$ up to a homeomorphism?
It is certainly clear that $X$ can be a 2-sphere, a disk, or a projective plane. Are there any other possibilities for $X$? |
Let $f_- (n) = \Pi_{i=0}^n ( \sin(i) - \frac{5}{4}) $
And let
$ f_+(m) = \Pi_{i=0}^m ( \sin(i) + \frac{5}{4} ) $
It appears that
$$\sup f_- (n) \inf f_+ (m) = \frac{5}{4} $$
Why is that so ?
Notice
$$\int_0^{2 \pi} \ln(\sin(x) + \frac{5}{4}) dx = Re \int_0^{2 \pi} \ln (\sin(x) - \frac{5}{4}) dx... |
B is a set where $x_1,x_2$ belong to $\mathbb{R}^2$ and $x_1^2 + x_2^2\leq 1\wedge x_1\geq 0.$
A function $f:B \rightarrow \mathbb{R}$ is given by
\begin{equation*}
f(x_1 ,x_2 )=x_1^2 x_2^2 +3x_1 x_2 +x_2 -4.
\end{equation*}
State the image of $f$.
From what I understand, this equation represents a unit... |
How do I find the image of a multivariable function? |
Let $p\colon \mathbb{S}^2 \to X$, where $\mathbb{S}^2 = \{x \in \mathbb{R}^3 : \|x\| = 1\}$, be a covering projection. Is it possible to characterize the space $X$ up to a homeomorphism?
It is certainly clear that $X$ can be a 2-sphere or a projective plane. Are there any other possibilities for $X$? |
**Short answer.**
Prosaically, the algebra of dual numbers $(\Bbb{D},+,\cdot)$ may be reinterpreted as the exterior algebra $\Lambda(V)$ of a one-dimensional *real* vector space $V$, in the sense that the "usual" multiplication is replaced by the exterior product, which is anticommutative for 1-vectors, as you recal... |
I am not sure. Can we interchange the integral with sum over $j$ here?
$\int_{0}^{+\infty}\Im\left(\sum_{j=0}^{\infty} \dfrac{i^{j}\mathbb{E}[X^{j}] }{j!} s^{j-1}e^{-ixs}\right)ds = \Im\left(\sum_{j=0}^{\infty} \dfrac{i^{j}\mathbb{E}[X^{j}] }{j!}\int_{0}^{+\infty} s^{j-1} e^{-ixs} ds\right) $
For example, if $x=0$,... |
Let $f$ be integrable function. I wanted to prove that $\lim_{s \to \infty}\int_0^{2pi}f(t)sin(st)dt=0$.
Firstly I wanted to do it for every simple function. I started with intervals. When $f$ is a characteristic function of interval, result follow easily.
Then my textbook says that from this result follow for ev... |
**Short answer.**
Prosaically, the algebra of dual numbers $(\Bbb{D},+,\cdot)$ may be reinterpreted as the exterior algebra $\Lambda(V)$ of a one-dimensional *real* vector space $V$, in the sense that the "usual" multiplication is replaced by the exterior product, which is anticommutative for 1-vectors, as you recal... |
Let $f$ be integrable function. I wanted to prove that $\lim_{s \to \infty}\int_0^{2\pi}f(t)sin(st)dt=0$.
Firstly I wanted to do it for every simple function. I started with intervals. When $f$ is a characteristic function of interval, the result follows easily.
Then my textbook says that from this result follow ... |
I am doing some self-study and I have this task:
Show that the signature of the following mapping, the reflection about the hyperplane $a^{\perp}$, given by
$S_a(v) = v - 2\frac{<v,a>}{<a,a>}a$
is (n-1,1,0), where a is an arbitrary nonzero vector.
I would know how to find the eigenvalues of this mapping if ... |
Assume that $X\sim Binomial(n,p)$.
Also, assume that $Y\sim Binomial(n+X,p)$. We are interested in computing the expected value of $Y$, i.e., $\mathop{\mathbb{E}}[Y]$.
**Question:** Why is the following solution incorrect?
We have:
$$\mathop{\mathbb{E}}[Y] = (n+X)p$$
We take another expectation from both si... |
> But how slowly it converge to ?
Are you not able to run numerical calculations yourself?
Let $x_N = 2/\prod_{n=1}^N (1 - 1/(4n^2))$. We have
$$
x_{100} = 3.133\ldots, \
x_{10^3} = 3.1408\ldots, \
x_{10^4} = 3.1415141\ldots, \
$$
$$
x_{10^5} = 3.1415847\ldots, \
x_{10^6} = 3.1415918\ldots, \
x_{10^... |
I'm currently learning more about PDE's and have the following exercise: Consider the PDE
$$
-\Delta u+ \ln u=f\quad in\ \Omega,\
u=u_d\quad on\ \partial\Omega
$$
where $f\in L^2(\Omega)$ is nonnegative and $u_d\in W^{1,2}(\Omega)$ with $u_d\geq \varepsilon>0$ a.e. in $\Omega$. I'm now trying to prove that there... |
I've been reading the algebraic topology book my Massey and I've been trying to do the next exercise, from chapter 4, about the van-Kamper theorem:
> *Construct for every integer n>2 a space such that it's fundamental group is cyclic and of order n*
From what I have seen we can do this exercise fairly easy with c... |
What is wrong with my induction proof? |
I'm currently grappling with a problem related to proving a statement using mathematical induction, for every integer n≥1, the expression 4^{2n} - 1 is divisible by 15.
So far, I understand the basic principle behind mathematical induction, which involves proving the statement for a base case (typically n=1) and the... |
I've been reading the algebraic topology book by Massey and I've been trying to do the next exercise, from chapter 4, about the van Kampen's theorem:
> *Construct for every integer $n>2$ a space such that it's fundamental group is cyclic and of order $n$*
From what I have seen we can do this exercise fairly easy ... |
Motivation: In $\langle\mathbb{Z}/7\mathbb{Z},\times\rangle,\ \langle 3\rangle = (3,2,6,4,5,1,3).$
> Given a $k-$tuple of distinct integers, $q_1, q_2, \ldots, q_k,$ (all
> nonzero) does $\exists$ integers $a,m,n$ such that $a m^i \equiv q_{r}
\pmod n\ \forall\ i\in \{1,2,\ldots, k\}\ ?$
Maybe there is a count... |
Given a tuple of $k$ distinct integers, is there a generator list in a $\mathbb{Z}/n\mathbb{Z}$ that matches the tuple? |
I am reading the Evan's PDE, theorem 1 ( p.23 ) and some questions arise. I found assoicated question but I think that I don't get satisfactory answer yet.
Now assume $f\in C^{2}_C(\mathbb{R}^n)$.
> Theorem 1. ( Solving Poisson's euqation ) Define $u$ by $(8)$ ;
$$ u(x)=\int_{\mathbb{R}^n} \Phi(x-y)f(y)dy = ... |
I would like to know an example showing that second-order mathematical induction is stronger than first-order mathematical induction.
For a condition $P(x)$ that can be written in the language of Peano arithmetic, we can construct the set $\{x\in\mathbb{N};P(x)\}$.
Which subset of $\mathbb{N}$ cannot be construct... |
I want to show that if the ring homomorphism $f:R \rightarrow S$ is onto, $I,J$ two ideals of $R$ and $\ker f\subseteq I$ then $f(I\cap J)=f(I)\cap f(J)$.
To prove $f(I)\cap f(J)\subseteq f(I\cap J)$, I started with $f(x)\in f(I)\cap f(J)$, thus $f(x)\in f(I)$ and $f(x)\in f(J)$, so $x\in I\cap J$ and $f(x)\in f(I\c... |
I'm currently learning more about PDE's and have the following exercise: Consider the PDE
$$
-\Delta u+ \ln u=f\quad in\ \Omega,\
u=u_d\quad on\ \partial\Omega
$$
where $f\in L^2(\Omega)$ is nonnegative and $u_d\in W^{1,2}(\Omega)$ with $u_d\geq \varepsilon>0$ a.e. in $\Omega$. I'm now trying to prove that there... |
Motivation: In $\langle\mathbb{Z}/7\mathbb{Z},\times\rangle,\ \langle 3\rangle = (3,2,6,4,5,1,3).$
> Given a $k-$tuple of distinct integers, $q_1, q_2, \ldots, q_k,$ (all
> nonzero) does $\exists$ integers $a,m,n$ such that $a m^i \equiv q_{r}
\pmod n\ \forall\ i\in \{1,2,\ldots, k\}\ ?$
Maybe there is a count... |
I've been reading the algebraic topology book by Massey and I've been trying to do the next exercise, from chapter 4, about the van Kampen's theorem:
> *Construct for every integer $n>2$ a space such that it's fundamental group is cyclic and of order $n$*
From what I have seen we can do this exercise fairly easy ... |
Let $F$ be an algebraic number field and $K,L$ its finite extensions contained in an algebraic closure of $F$. Denote the relative different by $\mathfrak d$. I want to prove that $\mathfrak d_{KL/L}$ divides $\mathfrak d_{K/F}$.
Since
$$\mathfrak d_{KL/F}=\mathfrak d_{KL/L}\mathfrak d_{L/F}=\mathfrak d_{KL/K}\ma... |
How to prove that $\mathfrak d_{KL/L}$ divides $\mathfrak d_{K/F}$? |
I am reading an interesting paper [One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational](https://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=rm&paperid=427&option_lang=eng) by V.V. Zudilin. We fix odd numbers $q$ and $r$, $q\geq r+4$ and a tuple $\eta_0,\eta_1,...,\eta_q$ of positive integer parameters satis... |
I have a question regarding interchanging the derivative and the integral, namely
$$\frac{d}{d\zeta}f^{1}(\zeta)=\frac{d}{d\zeta}\int_{-\infty}^{+\infty}f(x)e^{-2x\pi \zeta i}dx.$$
Where $f^{1}(\zeta)$ is fourier transform of $f$ ,
It is given that $xf(x)\in L^{1}(R)$.
I know that for DMT we need the integrand... |
I'm currently grappling with a problem related to proving a statement using mathematical induction, for every integer n≥1, the expression $4^{2n} - 1$ is divisible by $15$.
So far, I understand the basic principle behind mathematical induction, which involves proving the statement for a base case (typically $n=1$) a... |
Motivation: In $\langle\mathbb{Z}/7\mathbb{Z},\times\rangle,\ \langle 3\rangle = (3,2,6,4,5,1).$
> Given a $k-$tuple of distinct integers, $q_1, q_2, \ldots, q_k,$ (all
> nonzero) does $\exists$ integers $a,m,n$ such that $a m^i \equiv q_{r}
\pmod n\ \forall\ i\in \{1,2,\ldots, k\}\ ?$
Maybe there is a counter... |
Hopefully this is not a fail. But, before I explain the math, I think it will be easier if I give a background of what I'm trying to do...
*Imagine 2 line segments randomly defined in a 2D room. The line segments are 2D... Both are given a speed and omega (angular velocity). The point of the math, is to find the exa... |
How to find the first intersection point of 2 systems of symbolic equations? |
I'm learning the basics of Calculus. I'm good at differentiating (basic level). But when it comes to integration, it seems tricky as it takes time for me to think of a function for which the given one is the derivative. People seem to use some tricks when integrating two or more functions in multiplication by assigning... |
Any other ways to execute Integration rather than viewing it as the inverse of Differentiation? |
Suppose that your matrix is $A=[a_{ij}]_{i,j=0}^n$, and that $x_i:=a_{ii}$.
Then the determinant is some polynomial
$$
\Delta=\sum_{i_1<i_2<\dots<i_k}\alpha_{i_1 i_2 \dots i_k}\ x_{i_1}x_{i_2}\dots x_{i_k}
$$
where the sum is running over all the $2^n$ subsets of $\{1,2,\dots,n\}$.
How can we determine the co... |
I understood the question to be: why must $Y$ be Hausdorff for this theorem to hold? Put another way, why does the theorem fail if we merely assume $Y$ is compact and not both compact and Hausdorff?
Given any compact metrizable space $X$ with at least two points, let $Y$ have the same underlying set with the indisc... |
Motivation: In $\langle\mathbb{Z}/7\mathbb{Z},\times\rangle,\ \langle 3\rangle = (3,2,6,4,5,1).$
> Given a $k-$tuple of distinct integers, $q_1, q_2, \ldots, q_k,$ (all
> nonzero) does $\exists$ integers $a,m,n$ such that $a m^i \equiv q_{i}
\pmod n\ \forall\ i\in \{1,2,\ldots, k\}\ ?$
Maybe there is a counter... |
Motivation: In $\langle\mathbb{Z}/7\mathbb{Z},\times\rangle,\ \langle 3\rangle = (3,2,6,4,5,1).$
> Given a $k-$tuple of distinct integers, $q_1, q_2, \ldots, q_k,$ (all
> nonzero) does $\exists$ integers $a,m,n$ such that $a m^i \equiv q_{i}
\pmod n\ \forall\ i\in \{1,2,\ldots, k\}\ ?$
Maybe there is a counter... |
I would like to know an example showing that second-order mathematical induction is stronger than first-order mathematical induction.
For a condition $P(x)$ that can be written in the language of Peano arithmetic, we can construct the set $\{x\in\mathbb{N};P(x)\}$.
Which subset of $\mathbb{N}$ cannot be construct... |
What do we count here?
$$\frac{dy}{dx} = {2\choose1}x$$
$${2\choose1} = \frac{dy/dx}{x}$$
Expanding $(x+dx)^n$ We discard the infinitesimal bits, the second term of the expansion, $\binom{n}{1}x$, seems to remain as the derivative. I wonder if there is a combinatorics connection, if it can be seen on the gra... |
If $A \in \mathbb R^{n \times n}$ then I know that $\operatorname{span}\left(\left\{ e^{tA} v \mid t \in \mathbb R \right\}\right) = \operatorname{span}\left(\left\{ v, A v, A^2 v, \dots \right\}\right) = \operatorname{span}\left(\left\{ v, A v, A^2 v, \dots , A^{n-1} v \right\}\right)$.
I'm interested in a generali... |
Depending on what I'm allowed to assume, I'd approach this by showing that that this space is [metrizable](https://topology.pi-base.org/spaces/?q=Metrizable%2B%7E%24T_%7B3+%5Cfrac%7B1%7D%7B2%7D%7D%24). (In fact, though I won't show it, it is a copy of a "figure eight" in the plane.) I'll give a proof of metrizability, ... |
If roots of $$a(x^2+m^2)+amx+cm^2x^2=0.......(1)$$ are $u,v$, one can easily prove that $$a(u^2+v^2)+auv+cu^2v^2=0.........(2)$$
Interestingly, if (1) is $f(x,m)=0$, then (2) is $f(u,v)=0.$
The question: Why is this happening? |
I'm currently grappling with a problem related to proving a statement using mathematical induction, for every integer n≥1, the expression 4^{2n} - 1 is divisible by 15.
So far, I understand the basic principle behind mathematical induction, which involves proving the statement for a base case (typically n=1) and the... |
I'm reading Siegfried Bosch's *Algebra From the Viewpoint of Galois Theory*, which gives the following definition:
A finite field extension $L/K$ is said to be ***solvable by radicals*** if $L$ admits an extension field $E$ together with a chain of field extensions :
$$ K =: E_0 \subset E_1 \subset \cdots \subset E... |
[![prop 10.15][1]][1]
I don't understand the symbol $\widehat A \mathfrak{a}$.
I saw the following post, but I still have the question.
https://math.stackexchange.com/q/760052/1186046
There is following answer.
>Suppose $R\rightarrow S$ is a ring homomorphism and $\mathfrak{a}$ is an ideal of $R$. Do you know ... |
Let $x \in [0,1]$ be an *irrational algebraic number*, and $y \in [0,1]$ algebraic. Does there exist an effective procedure for determining whether there exists some integer $k \geq 1$ such that
$$
kx \equiv y \mod 1?
$$
This can be thought of a question about circle rotations from dynamics: marking a point at $0$... |
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