instruction stringlengths 12 30k |
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Around 1670, following Renee DesCartes' mid-century introduction of the Cartesian Coordinate system, Isaac Newton is supposed to have identified ten different coordinate systems. I am seeking to identify these ten coordinate systems and whether those were all those known to mathematicians of that era, or what others w... |
I have to establish the inequality $|tanh(x) - P_4(x)|≤0.005$ for |x| $< \frac{1}{2}$. I have established that tanh(x)$<$ x for all x $>$ 0. I now have to establish that
x → tanh(x)-P_4(x) is growing in (0,∞). But I am not sure how to do so?
I know that $P_4 = x- \frac{1}{3}x^3$. |
Let $X$ be an equidimensional reduced scheme of finite type over an algebraically closed field $k$. Assume that $X$ has two irreducible components $X_1$ and $X_2$. Assume also that $X_1$ and $X_2$ are nonsingular varieties which meet transversally at $Y := X_1 \cap X_2$, so that $Y$ is a nonsingular variety of codimens... |
In the bisection method for optimization, we look at the first derivative and then depending on whether it is positive or negative replace the boundary point a or b. I tried to implement it in python and it worked (at least for the simple examples I replicated). But it seems that I can arbitrarily change a=c and b=c po... |
I'm trying to compute this series.
$$S=\sum_{k=4}^\infty \dfrac{9}{(k-1)^2} - \dfrac{9}{(k+2)^2}$$
I have found the answer using a "shortcut" aka the Basel sum...
$$S=\sum_{k=4}^\infty \dfrac{9}{(k-1)^2} - \dfrac{9}{(k+2)^2} = \sum_{k=4}^\infty \dfrac{9}{(k-1)^2} - \sum_{k=4}^\infty \dfrac{9}{(k+2)^2}$$
... |
Let me show how to transform the problem into an equivalent problem which I suspect might be easier to find in the literature.
Let $g$ and $l$ denote $\text{gcd}(a,b)$ and $\text{lcm}(a,b)$ respectively, then we have $gl=ab$. If $d_1$ and $d_2$ are such that $l = d_1 d_2$, $d_1 \mid a$, $d_2 \mid b$ and $\text{gcd}(... |
I have to establish the inequality $|\tanh(x) - P_4(x)| \le 0.005$ for $|x| < \frac{1}{2}$. I have established that $\tanh(x) < x$ for all $x>0$. I now have to establish that $x \to \tanh x - P_4(x)$ is growing in $(0,\infty)$. But I am not sure how to do so?
I know that $P_4 = x- \frac{1}{3}x^3$. |
What does "much less than" mean while comparing measures? |
Let me show how to transform the problem into an equivalent problem which I suspect might be easier to find in the literature.
Let $g$ and $l$ denote $\text{gcd}(a,b)$ and $\text{lcm}(a,b)$ respectively, then we have $gl=ab$. If $d_1$ and $d_2$ are such that $l = d_1 d_2$, $d_1 \mid a$, $d_2 \mid b$ and $\text{gcd}(... |
Let $E\subset\mathbb{R}^n$ be a bounded domain (open and connected) and $F\Subset E$ be a compactly contained (or even compact) subset of $E$. We set $a := dist(F,\partial E) = \inf\limits_{ y\in F} \inf\limits_{x\in \partial E}|x-y|>0$, where the distance function is taken with respect to the usual euclidean norm $|\c... |
I'm trying to compute this series.
$$S=\sum_{k=4}^\infty \dfrac{9}{(k-1)^2} - \dfrac{9}{(k+2)^2}$$
I have found the answer using a "shortcut" aka the Basel sum...
$$S=\sum_{k=4}^\infty \dfrac{9}{(k-1)^2} - \dfrac{9}{(k+2)^2} = \sum_{k=4}^\infty \dfrac{9}{(k-1)^2} - \sum_{k=4}^\infty \dfrac{9}{(k+2)^2}$$
... |
Please refer to the attached image.
I am struggling to understand how the integration by parts is performed for the weak form.
Can anyone please assist with an explanation?
The equation is not written as a PDE since it is in 1D
[Weak form][1]
[1]: https://i.stack.imgur.com/z2tgu.jpg |
The operator norm of a mapping $\mathcal{A}:\mathbb{R}^{n_1\times n_2} \to \mathbb{R}^{n_1\times n_2}$ is defined by
$$
\|\mathcal{A}\|=\sup_{\|X\|_{F} \leq 1}\|\mathcal{A}(X)\|_{F},
$$
where $X$ is a matrice, and $\|\cdot\|_{F}$ is $F$-norm defined by $\|X\|=\left<X,X\right>^{\frac{1}{2}}$.
If we have $\|\mathc... |
Let $a,b,c$ be elements in a GCD domain (or just a GCD monoid) $R$. Suppose that $\text{gcd}(a,b)=1$ and $c \ne 0$. If $R$ is a UFD, it is possible to write $c=a'b'$ with $\text{gcd}(a',a)=\text{gcd}(b',b)=\text{gcd}(a',b')=1$. Is this still true in any GCD monoid, or is there a counterexample?
This question [is equ... |
Let me show how to transform the problem into an equivalent problem which I suspect might be easier to find in the literature.
Let $g$ and $l$ denote $\text{gcd}(a,b)$ and $\text{lcm}(a,b)$ respectively, then we have $gl=ab$. If $d_1$ and $d_2$ are such that $l = d_1 d_2$, $d_1 \mid a$, $d_2 \mid b$ and $\text{gcd}(... |
Please refer to the attached image.
I am struggling to understand how the integration by parts is performed for the weak form.
Can anyone please assist with an explanation?
The equation is not written as a PDE since it is in 1D
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/z2tgu.... |
$$\iint\limits_{\Omega}{(x+y)dx dy}$$, where the area Ω is bounded by a curve $x^2+y^2=x+y$. Note: Evaluate by converting to polar system.
My Solve: $$\iint\limits_{\Omega}{(x+y)dx dy}=\int_0^{2\pi}d\theta \int_0^{\frac{1}{\sqrt 2}} r \,(rsin\theta + rcos\theta)dr=$$
$$=\int_0^{2\pi}(sin\theta+cos\theta)d\theta*\... |
> Evaluate $\displaystyle\iint\limits_{\Omega}{(x+y)dx dy}$ where the area $\Omega$ is bounded by a curve $x^2+y^2=x+y$. Note: Evaluate by converting to polar system.
My Solve: $$\iint\limits_{\Omega}{(x+y)dx dy}=\int_0^{2\pi}d\theta \int_0^{\frac{1}{\sqrt 2}} r \,(r\sin\theta + r\cos\theta)dr=$$
$$=\int_0^{2\pi}... |
I stumbled upon what seems like an interesting bifurcation. I wonder if anyone has an explanation for this. Here are the details:
Let $a_n\in\mathbb{R}$, $n \in \mathbb{N}$ be a sequence.
Define a new sequence for $n\in\mathbb{N}$:
$$
\sigma_n = \begin{cases}
a_1 & \text{if } n = 1 \\\\
\sigma_{n-1}+... |
In 3D, we tend to define the angle between two planes as the minimum angle of rotation required to turn one plane into the other. Rarely does this make sense in 4D. Let's see why.
In 4D we must get more specific about what a rotation is. In 3D, there is no distinction between a planar rotation, that is, a rotation w... |
Let $\mathbb{F}$ be a field and $A\in M_n(\mathbb{F})$.
Let the minimal polynomial of $A$ be $\prod_{i=1}^m q_i(x)^{\ell_i}$, where each $q_i(x) \in \mathbb{F}[x]$ is monic irreducible, each $\ell_i>0$, and gcd$(q_i(x),q_j(x))=1$ for $i \neq j$. If $s=\deg q_1(x)^{\ell_1}$, does there exist a $v \in \mathbb{F}^n$ such... |
In 3D, we tend to define the angle between two planes as the minimum angle of rotation required to turn one plane into the other. Rarely does this make sense in 4D. Let's see why.
In 4D we must get more specific about what a rotation is. In 3D, there is no distinction between a planar rotation, that is, a rotation w... |
I am confused about elliptic curves. How can we replace this
$$E: ay^2=bx^3+c$$
such that $a,b,c\in\mathbb Z, a,b\not=0$ to be its Weierstrass form? Can we just transform them like this? Substitute $y$ by $y/a^2b$ and $x$ by $x/ab$ yields
$$E: y^2=x^3+a^3b^2c.$$
Is easy way like this justified so that the above is... |
I saw a problem where it asked: What question can you ask a knight (who always tells the truth) such that he cannot respond? For this, I came up with the question "Is the answer to this question No?" -- the knight can't say Yes (because he would be lying) and he also can't say no. Similarly for a knave, "Is the answer ... |
i've done lebesgue vitali theorem and i'm looking for some examples of Riemann integrable function that is discontinous in a dense set. I'm thinking about caratheristic function but i cannot prove continuity. Any idea? |
I have a continuous and nondecreasing function $F:\mathbb{R}\to\mathbb{R}$ satisfying following conditions:
1. $F(x) = 0$ for any $x\in (-\infty, 1]$,
2. $\lim\limits_{x\to +\infty} F(x) = 1$,
2. for any $x\in \mathbb{R}$ we have
$$F(x) \;=\; 0.8 F(2x - 1) + 0.2 F(x/2 - 1).$$
Is there any solution to this "fun... |
For my question (2), instead of using Munroe's approach to establish the limit, we can use problem 19.7 in Aliprantis and Burkinshaw 2nd edition, or Theorem 2.30 in Folland:
Assume that $\{f_n\}$ is Cauchy in measure, if and only if there exists a measurable function $f$ such that $\lim_{n}f_n=f$ in measure, i.e. for ... |
Let $(X,\mathcal{A},\mu)$ be $\sigma$-finite measure space.
Claim: $L^\infty$ is complete.
Ideas of the proof:
- (i) Assume Cauchy in norm - let $\{f_n\}$ be a Cauchy sequence which converges in $L^\infty$ norm. That is, let $\epsilon>0$, and assume $\left\Vert f_n-f_m \right\Vert_\infty<\epsilon.$
Prove: $
... |
Assume a random vector $$X=(X_1,\ldots,X_p) \sim N(0,\Sigma)$$ with $\Sigma$ positive definite matrix. Is there literature regarding the covariance (formulas) of the form
$E[X_j^kX_{j'}^l]$? with $k,l \in \{1,\ldots, q\}$ for some $q\in \mathbb{N}$.
Additional Note: More concretely i am concerned whether a covaria... |
I was writing a simple program to help visualize inequalities based on 2 variables. The test inequality that I was using was this: $$\sin\left(0.1(x^2+y^2)\right)\geq0$$
Regions that satisfy the inequality are coloured in white.
Here's the result, for a bounding box of $(-25,-25),(25,25)$:
[![enter image descrip... |
I wrote up the following proof for proving $L^\infty$ is complete. Please check if I made any mistakes, thank you!
There is a common/standard treatment for establishing the limit of a Cauchy sequence in $L^\infty$-norm using the approach suggested by Munroe, Aliprantis and Burkinshaw, Wheeden and Zygmund, Folland, a... |
> I think that is can be assumed that: $(_)=^−(−1)^$
I believe you meant the number of ways in which the desired event can happen. You need to divide by $n^i$ for it to be the probability.
> Now, I don't know how calculate $(A_i ∧ A_j)$.
If you consider the complement of this, we have three possibilities. (Wit... |
I have the following system $$\ddot{x}+w^2 x=0,$$ with the following initial conditions: $\dot{x}(0)=0$ and $x(0)=x_o$, the solution reads: $$x(t)=x_o cos(t).$$ Now I want to include the fluctuations due to thermal fluctuations (in frictionless system), so the equation of motion is modified as $$\ddot{x}+w^2 x=\lambda... |
how to determine amplitude and frequency of Oscillator with random fluctuations? |
It seems that
$$\left(\frac{{\pi}^n}{nn!}\right)_{n\ge 1}$$
is a decreasing sequence. How can I prove it?
I noticed that replacing $\pi$ by $2\pi$ makes the sequence *increasing* for a bit, and removing the $n$ in front of $n!$ makes the sequence increasing for a bit as well. The problem in hand seems quite non-tr... |
I'm reading the following proof of the matrix-tree theorem:
[https://www.sciencedirect.com/science/article/pii/0012365X9200059Z][1]
in particular, his Theorem 3.1.
I was not able to validate the last part:
[enter image description here][2]
**How do I show the identity $(-1)^{\nu(D)-|W|}=\epsilon(gf)$?**
... |
[![enter image description here][1]][1]
The solution to this definite integration problem involves considering $t=1/y$ to make a change of the integration bounds. However after solving in terms of $y$ they directly substitute back into $t$ disregarding the earlier substitution.
Shouldn't it be substituted back as... |
I'm reading the following proof of the matrix-tree theorem:
[https://www.sciencedirect.com/science/article/pii/0012365X9200059Z][1]
in particular, his Theorem 3.1.
I was not able to validate the last part:
[![enter image description here][2]][2]
**How do I show the identity $(-1)^{\nu(D)-|W|}=\epsilon(gf)$... |
I'm reading the following proof of the matrix-tree theorem:
[https://www.sciencedirect.com/science/article/pii/0012365X9200059Z][1]
in particular, his Theorem 3.1.
I was not able to validate the last part:
[The last lines about equation (3.5)][2]
**How do I show the identity $(-1)^{\nu(D)-|W|}=\epsilon(gf)$?... |
How can one prove that $ \lim\limits _{n\rightarrow \infty}\int \limits_{0}^{\infty}\frac{\sin (x^{n})}{x^{n}}dx=1 $? |
I have a basic question in Galois theory.
>For any given natural number $n$ is there a Galois extension of $\mathbb Q$ of degree $n$?
I want to show that there are splitting fields of polynomials in $\mathbb Q[X]$ of arbitrary degrees over $\mathbb Q$.
A hint would be appreciated- many thanks.
|
Question: Suppose $[L:K]=4$, $\operatorname{char}K \neq 2$ and $L$ is algebraically closed. Show that there is an intermediate field $M$ such that $[L:M]=2$ and $X^2 + 1$ splits over $M$. Show that this leads to a contradiction.
I have successfully found such $M$. Would somebody please give me some hints to the last... |
We are going to find the definite integral $${I=\int\limits_{0}^{1}\frac{\ln x\sin^{-1}x\cos^{-1}x}{x}dx}$$
Ah, this integral. The more I looked into it the more it looked like it might have a closed form. Arriving at it just feels so satisfying!
<hr>
Firstly, by the substitution $x=\sin \theta$, we obtain
$$\b... |
I am taking a course in Analytic Number Theory. The main object of study is arithmetical functions. Moreover, if we look at the arithmetical functions which do not vanish at $1$, then they form a group under Dirichlet multiplication, $*$. An important subgroup of this group is the group of multiplicative functions. My ... |
**EDITED**
***
If anyone is interested in this question,
now there are [OEIS draft][1] and [repository][2] with detailed data.
***
Let $n,b\in\mathbb{N}$ and $P_{n_b}$ is polynomial whose coefficients *are digits* of $n$ in base $b$.
Let $Q_{1}(x) \cdot Q_{2}(x) \cdots Q_{m}(x)$ - polynomial factorization (o... |
I'm studying Galois Theory and I stumbled upon this question:
> Let $E/K$ be a Galois extension of degree $75$. Show that there exists an intermediate subfield $K \subsetneq F \subsetneq E$ such that $F/K$ is also a Galois extension.
I started like this: let $G = \text{Gal}(E/K)$. Because $E$ is Galois over $K$,... |
Prove, if $\delta>0$, then $\lim_{n\to\infty}\int_{\delta<|s|<a}K_n(s)ds=0$ if
$$K_n(s)=
\begin{cases}
&n, |s|<1/2n\\
&0, |s|>1/2n
\end{cases}
$$
This shouldn't be too difficult, but I'm struggling somewhat. If I just set up the problem, then we have
$$\int_{-1/2n}^{-\delta}n ds+\int_{\delta}^{1/2n}n ds=1-2n\de... |
In urn I there are $3$ white balls and $2$ black balls. Urn II contains $2$ white balls and $3$ black balls.
- We draw a ball from urn I and either throw it back or put it into urn II - each of these two possibilities has probability $\frac{1}{2}$.
- We then do the same for urn II.
What is the probability that... |
The problem is as follows:
> Senior students tend to stay up all night and therefore are not able
> to wake up on time in morning. Not only this but their dependence on
> tuitions further leads to absenteeism in school. Of the students in
> class XII, it is known that 30% of the students have 100% attendance.
> ... |
I am reading a paper where authors denote $\bar{G_k} = I - G_k(G_k^TG_k)^{-1}G_k^T$. Now, I am trying to verify it and reduce the second term to see what it is. I am doing the following, please tell me what I am doing wrong.
$$= G_k(G_k^TG_k)^{-1}G_k^T$$
Simplifying the inverse term, using $(AB)^{-1} = B^{-1}A^{-... |
Suppose there is a continuous dynamic system of order $n+1$ given by
$$\begin{align}
\dot{x}_1 &= Ax_1 + F(t)x_1 + G_1(t)x_2 \\
\dot{x}_2 &= kx_2 + G_2(t)x_1
\end{align}$$
where $x_1\in\mathbb{R}^n$ is a vector and $x_2\in\mathbb{R}$ is a scalar. Moreover, we know that matrices $A, F(t), G_1(t), G_2(t)$ are bound... |
This question is a follow-on of this [answer](https://math.stackexchange.com/a/4855444/305862) of mine.
This answer provides a numerical solution ; the initial question was about possible closed form expressions.
Q1 : Is there a reference in the existing literature about these orthogonal curves ?
Q2 : These ort... |
I am trying to find the general solution for $u(x,y)$ of this problem with the method of Lagrange, but I could not finish my attempt. The Lagrange system we had is $$\frac{dx}{y+u}=\frac{dy}{y}=\frac{du}{x-y}.$$
Working on this system:
\begin{align}
& \frac{dy}{dx}=\frac{y}{y+u}, \\
& \frac{du}{dx}=\frac{x-y}... |
Say I have a (smooth) function $f : \mathbb{R}^n \to \mathbb{R}$, and a critical point $x$ (*ie*, $f'(x) = 0$). I call this point **degenerate** if $\det \text{Hess}_x f = 0$ (so, equivalently, if the kernel of the Hessian at $x$ is non-trivial). An example is $x = 0$ for $f : \mathbb{R} \to \mathbb{R} : x \mapsto x^3$... |
Birth-death : Always more than 1 bifurcation? |
[![A spherical triangle][1]][1]
[1]: https://i.stack.imgur.com/rxTH7.png
In the sphere above, the shaded area defined by the points A, B, C clearly makes a triangle. My question is, can the complement of this area, that is everything on the sphere that is white, also be considered a triangle? It is continuous... |
Is the complement of a triangle on the sphere also a triangle? |
I have iid random variables $Z_1,Z_2,...$ that are uniformly distributed on the set $\{1,2,...,n\}$. I define $X(n)=\min\{k:Z_k=Z_j\text{ for some }j<k\}$ and $Y(n)=\min\{k:Z_k=Z_j,Z_i=Z_l\text{ for some }j<k \text{ and }i<l<k\}$ so X(n) is the first repeat of a number and Y(n) is the second repeat (not necessarily a r... |
Pick **any** absolutely integrable function $\eta(x)$ such that $\int_{-\infty}^\infty dx \ \eta(x)=1 $. We use $\eta$ to 'construct' a Dirac delta (a so called nascent delta or [approximation to the identity](https://en.wikipedia.org/wiki/Dirac_delta_function?useskin=vector#nascent_delta_function) )
$$\tag{1}
\f... |
Given $f\in L^1_{loc}(\mathbb{R}^n)$, we consider the maximal function
$$
Mf(x) := \sup_{r>0} \frac{1}{|B(x,r)|} \int_{B(x,r)} |f(y)|dy.
$$
It is known that $M : W^{1,p}(\mathbb{R}^n)\to W^{1,p}(\mathbb{R}^n)$, for $1<p<\infty$, is a bounded operator : for any $f\in W^{1,p}(\mathbb{R}^n)$, it holds
$$
\|Mf\|_{W^{... |
X, Y, Z: discrete random variables
Probability of X, Y, Z have distinct values is 1.
Let
$a=P(X>Y)$
$b=P(Y>Z)$
$c=P(Z>X)$
(i) Find min{a,b,c}.
(Note: in an election, it's possible for more than half of the voters to prefer candidate A to candidate B, more than half B to C, and more than half C to A)
... |
> Can we clear out $a,b,c$ in four equations: $ax+by+cz=0, cx+ay+bz=0, a+b+c=1, a^2+b^2+c^2=1$?
In my search of characteristic cone of $u_{xy}+u_{yz}+u_{zx}=0$, I need to clear out $a,b,c$ in the above formula to get an equation depending only on $x,y,z$, which should be a polynomial of $x,y,z$ of homgeneity $2$ (s... |
Can we clear out $a,b,c$ in four equations: $ax+by+cz=0, cx+ay+bz=0, a+b+c=1, a^2+b^2+c^2=1$? |
[enter image description here][1]
[1]: https://i.stack.imgur.com/RKfcw.png
I'm trying to solve this problem. I don't know what the problem means.
First, I can show that a given function has the inverse at a given point c. Is it not enough? Or I have to solve the system of equations with two variables? |
In quantum mechanics, we construct all the representations of the $SU(2)$ algebra $$[J_j,J_k]=i\varepsilon_{jk\ell}J_\ell$$ in the $|j,m\rangle$ basis. The allowed values of $j$ turn out to be $$j=0,\frac{1}{2},1,\frac{3}{2},\ldots$$ which are all said to be irreducible.
Is there a theorem that we can invoke to con... |
Why are $j=0,\frac{1}{2},1,\frac{3}{2},\ldots$ representations of $SU(2)$ algebra irreducible? |
> Can we clear out $a,b,c$ in four equations: $ax+by+cz=0, cx+ay+bz=0, a+b+c=1, a^2+b^2+c^2=1$?
In my search of characteristic cone of $u_{xy}+u_{yz}+u_{zx}=0$, I need to clear out $a,b,c$ in the above formula to get an equation depending only on $x,y,z$, which should be a polynomial of $x,y,z$ of homgeneity $2$ (s... |
> Let $U=\{(x,y)\in\mathbb{R}^2\colon x\neq 0\}$. For each $(x,y)\in U$, let
$$u(x,y)=\frac{x^4+y^4}{x}\quad\text{ and }\quad v(x,y)=\sin(x)+\cos(y).$$
Let $f=(u,v)$ on $U$ and $c=(\pi/2, \pi/2).$ Can we solve for $x,y$ in terms of $u,v$ near $c$? Find $\partial x/\partial u$ near $f(c).$
I'm trying to solve this ... |
Can we solve $u=(x^4+y^4)/x,v=\sin x+\cos y$ for $(x,y)$ near $(\pi/2,\pi/2)$? |
Let $G$ a connected unimodular Lie group, endowed with Haar measure $X={X_1,\cdots,X_k}$ a Hörmander system of left-invariant vector fields. The sublaplacian $\Delta = - \sum_{i=1}^k X_i^2$ generates a symmetric Markovian semigroup $H_t$ and $H_t$ admits a right convolution kernel $h_t$ such that
$$H_tf(x) = \int_G h_... |
An implication is the compound statement of the form “if p , then q .”
It is denoted $p \rightarrow p$.
If $p$ is false, the implication $p \rightarrow q$ is always true.
So the implication $0 =1 \rightarrow 3=6$ is true.
Indeed, by assuming $0 =1$, one can show $1=3$ as follows:
$0=1 \Rightarrow 0+1=1+1... |
I watched wikipedia page [Moduli space][1], and the definition of $n$-marked moduli space as follow
> One can also enrich the problem by considering the moduli stack of genus $g$ nodal curves with $n$ marked points. Such marked curves are said to be stable if the subgroup of curve automorphisms which fix the marked ... |
An implication is the compound statement of the form “if p , then q .”
It is denoted $p \rightarrow q$.
If $p$ is false, the implication $p \rightarrow q$ is always true.
So the implication $0 =1 \rightarrow 3=6$ is true.
Indeed, by assuming $0 =1$, one can show $1=3$ as follows:
$0=1 \Rightarrow 0+1=1+1... |
The following definition is taken from Wikipedia:
In mathematics, a $\textit{contraction mapping}$, or $\textit{contraction}$ or $\textit{contractor}$, on a metric space $ (M, d) $ is a function $ f $ from $ M $ to itself, with the property that there is some real number $ 0 \leq k < 1 $ such that for all $ x $ and ... |
I just started learning about set theory and the countability of sets. And there is something my brain does not quite understand about infinities. It might be a dumb question, but here it goes:
You can prove that the integers are countable and have the same cardinality as the natural numbers by finding a bijection b... |
Why doesent the integers have twice as large cardinality as the natural numbers? |
I have the following equation
$$a=a_0+Mx,$$
where $a$, $a_0$, and $x$ are $N\times 1$ complex vectors, and $M$ is a (non-invertible) $N\times N$ complex matrix.
For a given $a_0$ and $M$, I want to find a vector $x$ such that the absolute value of each element of $a$ is equal to 1.
Is there a nice algebraic... |
We know that $H^\infty + C(\mathbb{T})$ is the closed subalgebra of $L^\infty(\mathbb{T})$ containing $H^\infty$.
How to show that
$H^\infty + C(\mathbb{T})$ = clos$[\cup_{n\geq 0} \chi_{-n} H^\infty$], where $\chi_n (e^{it})=e^{int}$ for $n\in \mathbb{Z}$ ?
Also we have $H^\infty + C(\mathbb{T})$ = clos$[H^\infty... |
An implication is the compound statement of the form “if p , then q .”
It is denoted $p \rightarrow q$.
If $p$ is false, the implication $p \rightarrow q$ is always true.
So the implication $0 =1 \rightarrow 3=6$ is true.
Indeed, by assuming $0 =1$, one can SHOW $3=6$ as follows:
$0=1 \Rightarrow 0+1=1+1... |
In a nutshell: how can we integrate rough paths using Lebesgue integration?
---
**(The real case)** Let $\gamma:[a,b]\to\mathbb{R}^n$ be a path and let $f:\mathbb{R}^n\to\mathbb{R}$ be a function. If $\gamma$ is smooth, we often define
$$\int_\gamma f := \int_a^b(f\circ\gamma)|\gamma'|.$$
The integral on the ri... |
I was reading the definition of CW-complex in terms of pushouts given by Lück's *Algebraische Topologie: Homologie und Mannigfaltigkeiten* (Chapter 3). It is stated (though not proven) that such a topological space is always Hausdorff. I guess this follows readily by induction if we can prove that the adjunction space ... |
> Why is $\{(\Delta_1,...,\Delta_k) \in B_1 : k\geq 1, B\in
\mathcal{B}(\mathbb{R}^k)\}$ a $\cap$-stable generator of $\mathcal{F}_u$?
Suppose $(W_t)_t$ is a standard Brownian motion and $0=s_0<s_1<\cdots s_k\leq u$. Define $\Delta_1=W_{s_1}-W_{s_0}$, $\Delta_2=W_{s_2}-W_{s_1}$ and so on up to $\Delta_k=W_{s_k}-W_{... |
Why is $\{(\Delta_1,...,\Delta_k) \in B_1 : k\geq 1, B\in \mathcal{B}(\mathbb{R}^k)\}$ a $\cap$-stable generator of $\mathcal{F}_u$? |
**Is there a non-combinatorial way of proving the given expression?**
$$\sum_{r=0}^{n} (-1)^r\binom{n}{r}(n-r)^{r}(n-r-1)^{n-r}=\sum_{k=0}^{n}(-1)^k \frac{n!}{k!}$$
The given expression represents the no. of onto functions $f$ from $A$$\to$$A$ such that $f(i)$$\neq$$i$ and $n(A)$=n, which I found using the ... |
In a nutshell: how can we integrate rough paths using Lebesgue integration?
What I mean by a rough path is a continuous map $[a,b]\to\mathbb{R}^n$, or $[a,b]\to\mathbb{C}$ in the complex case.
---
**(The real case)** Let $\gamma:[a,b]\to\mathbb{R}^n$ be a path and let $f:\mathbb{R}^n\to\mathbb{R}$ be a functio... |
For a group $G$, let $G'$ denote the commutator of the group $G$, and if $H \leq G$ the left cosets will be denoted as $gH$. Now, I understand the fact that $[SL_2(\mathbb{Z}):GL_2(\mathbb{Z})'] = 2.$ My question is, does there exist an element of order $4$ in $GL_2(\mathbb{Z})/GL_2(\mathbb{Z})'$? What I have worked ou... |
I am interested to solve the following PDE .
$$\partial_t f(t,x,y)=-y\partial_xf(t,x,y)+(x+y)\partial_y f(t,x,y)+\alpha f(t,x,y)$$
with $f:\mathbb{R}^+\times\mathbb{R}\times \mathbb{R}\to\mathbb{R}$.
This reminds me kinetic fokker planck without the diffusion term in $y$ and under harmonic potential in both $x$ an... |
I'm reading these [notes][1], and in the proof of this **Proposition 2**, which states:
> Let $d\ge 2$ and $S\ge 1$ be integers. Let also $(\alpha_1,\dots,\alpha_d)\in\mathbb{N}^d \setminus\{ 0 \}$. Let $(P_n)_{n\ge0}$ be the $d$-color Pòlya urn random process having $S\cdot Id$ as replacement matrix and $(\alpha_... |
Convergence of Pòlya urns: waht does it mean "expanding real polynomials in a basis"? |
While reading [this](https://github.com/python/cpython/blob/10fbcd6c5dc25bfe14e02fd93ef93498a393860c/Objects/dictobject.c#L289) I found that a recurrence $j = (5*j + 1)\,mod\,2^k$ generates $j$ recurrently in range $[0, 2^k)$ starting from any non-negative $j$ and positive $k$.
I've not found any sources with a proo... |
I think Gouvea's book *$p$-Adic Numbers: An Introduction* explains this idea more clearly in his section titled "Interpolation", and the account below is similar to that. It shows what happens when the OP's third question "why do this fix an $s_0$ business at all?" is taken seriously, but instead of using exponents in... |
In urn I there are $3$ white balls and $2$ black balls. Urn II contains $2$ white balls and $3$ black balls.
- We draw a ball from urn I and either throw it back or put it into urn II - each of these two possibilities has probability $\frac{1}{2}$.
- We then do the same for urn II.
What is the probability that... |
In urn I there are $3$ white balls and $2$ black balls. Urn II contains $2$ white balls and $3$ black balls.
- We draw a ball from urn I and either throw it back or put it into urn II - each of these two possibilities has probability $\frac{1}{2}$.
- We then do the same for urn II.
What is the probability that... |
In urn I there are $3$ white balls and $2$ black balls. Urn II contains $2$ white balls and $3$ black balls.
- We draw a ball from urn I and either throw it back or put it into urn II - each of these two possibilities has probability $\frac{1}{2}$.
- We then do the same for urn II.
What is the probability that... |
I'm working on visualizing the reflections of a fundamental domain in the Poincaré disk along each side of it using SageMath. The figures below show the reflections of a polygon (a 4-gon and a 3-gon, the innermost ones around the origin), where some of the reflected polygons overlap. Do these look correct to you?
[!... |
In my answer to a recent
[MSE question](https://math.stackexchange.com/q/4875434/)
I mention the elliptic curve
[LMFDB 1728.n4](https://www.lmfdb.org/EllipticCurve/Q/1728a1)
with integer points $(-1,\pm1)$. The elliptic divisibility
sequence associated with these points is defined by initial values
and recursion
... |
The problem is trivial - it *gives* you that if a student does not have perfect attendance, the probability that they make an A is $10\% = \frac{1}{10}$.
If you want to use the definition of conditional probability here, you can first calculate the probability that a student does not have perfect attendance, which i... |
I have seen the bounds $\left(\frac{n}{k}\right)^k \leq {n \choose k} \leq \left( \frac{en}{k}\right)^k$ for integers $n \geq k >0$ for the binomial coefficient. I can prove the upper bound in this inequality, but I'm missing some detail for the lower bound. Does anyone know some simple way to prove this lower bound?
... |
I'm working on visualizing the reflections of a polygon in the Poincaré disk along each side of it using SageMath. The figures below show the reflections of a polygon (a 4-gon and a 3-gon, the innermost ones around the origin), where some of the reflected polygons overlap. Do these look correct to you?
[![enter imag... |
Suppose that we are given a smooth, closed, connected homology sphere $M^n \subset \mathbb{R}^{n+1}$ with $n \geq 6$. I want to kill some elements in the fundamental group of $M$ by surgeries. As far as I understand it correctly, this is done by the following procedure:
1) Choosing a representative $S^1 \longrightar... |
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