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I need to find how distant the center of an orthogonal rectangle is from the center of a circle, given a specific angle.
The dimensions of the rectangle are proportional to the circle radius, so they use a 0-1 range, where 1 is the radius of the circle. The objective is to get the position of the center of that rect... |
I recently came across [this][1] paper by Friedgut that shows low sensitivity boolean functions are close to juntas. This was an unintuitive result to me, as I thought I could easily imagine a function on the boolean hypercube that is a linear combination of many low-degree monomials, and therefore has low sensitivity,... |
I have looked everywhere and I can't find a clear definition of hamiltonian isotopy. I have this definitions in my lecture notes but they are rather confusing
[![enter image description here][1]][1]
Acording to the last definition a Hamiltonian isotopy $\phi_t$ (the one they given in the definition of Hamiltonian... |
Is a hamiltonian isotopy real-valued of manifold-valued? |
I'm reading about holomorphic functions in section 1.2 of *Complex Analysis* by Stein and Shakarchi, and I am pretty confused about the derivative notation that the authors employ. In this section the authors derive the Cauchy-Riemann equations for a complex-valued function $f: \Omega \to \mathbb{R}$, where $\Omega$ is... |
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<h1>Question About Finding Inverse Function</h1>
<p>I am trying to find the inverse function for the gi... |
It is common for books on physics to contain something along these lines: "We have $X$ parameters and $Y$ conditions, hence, the number of free parameters is $X-Y$" or "we have $X$ components of a field, and the local gauge invariance group is $Y$-dimensional, so we can fix arbitrary $Y$ parameters". For instance, cosi... |
How does one strictly define "the number of components that can be fixed by the choice of gauge"? |
I have looked everywhere and I can't find a clear definition of hamiltonian isotopy. I have these definitions in my lecture notes but they are rather confusing
[![enter image description here][1]][1]
Acording to the last definition a Hamiltonian isotopy $\phi_t$ (the one they given in the definition of Hamiltonia... |
Winther’s suggestion in the comments is a good one. Consider the following product of $m$ power series:
$$\prod_{i=1}^m\sum_{k_i\ge 0}a_{i,k_i}x^{k_i}$$
For $n\ge 0$ what is the coefficient of $x^n$? The product is
$$(a_{1,0}+a_{1,1}x+a_{1,2}x^2+\ldots)\cdot(a_{2,0}+a_{2,1}x+a_{2,2}x^2+\ldots)\cdot\ldots\cdot(... |
I have looked everywhere and I can't find a clear definition of hamiltonian isotopy. I have these definitions in my lecture notes but they are rather confusing
[![enter image description here][1]][1]
Acording to the last definition a Hamiltonian isotopy $\phi_t$ (the one they given in the definition of Hamiltonia... |
I am trying to find the inverse function for the given function f(x) = 3x - |x| + |x - 2|. I have already found that the inverse function can be expressed as the piecewise function f^{-1}(x) = \begin{cases}
\frac{x - 2}{3} & \text{if } x < 2 \\
x - 2 & \text{if } 2 \leq x < 4 \\
\frac{x + 2}{3} & \text{if } x \g... |
When introducing $p$-adics, Kurt Hensel produced an incorrect proof of the transcendence of $e$: see https://mathoverflow.net/q/416296 for more details. The problem is that the proof relies on the "universality" of series, that is, the assumption that the series
$$\sum_{n=1}^\infty\frac{p^n}{n!},$$
which converges in... |
If A is a normal matrix, then $AA^H = A^HA$. I have observed that normal matrices tend to have good conditioning. I was wondering if this was always the case? Are there any bounds on what the condition number could be? |
I have looked everywhere and I can't find a clear definition of hamiltonian isotopy. I have these definitions in my lecture notes but they are rather confusing
[![enter image description here][1]][1]
Acording to the last definition a Hamiltonian isotopy $\phi_t$ (the one they given in the definition of Hamiltonia... |
Consider the $p$-adic field $\mathbb{Q}_p$ and $n\geq 1$.
The absolute value is $|x|=p^{-v_p(x)}$ for $x\in\mathbb{Q}_p$, where $v_p$ is the $p$-adic valuation.
Endow $V=\mathbb{Q}_p^n$ with the sup-norm:
For $v\in V$, $\|v\|=\max_{i=1}^{n} |v_i|$.
Let $A\in M_{n\times n}(\mathbb{Z}_p)$.
> Does the condition... |
I'm working in "Operator Theoretic aspects of Ergodic theory" by Eisner, Farkas Haase and Nagel. The question is as follows
> Consider the Hilbert space $H := \ell^2(\mathbb{N})$ and a sequence $(a_n)_{n\in\mathbb{N}}$ in $\mathbb{C}$ with $\sup_{n\in\mathbb{N}}\lvert a_n \rvert\leq 1$. Define $T: H\to H$ by $T(x_n)_... |
Seems like you have the $\gets$ direction OK. I'm not going to be much help with a formal proof, since I haven't done that since HS. But here's the idea: If $\left\langle x,y \right\rangle = \left\langle z,w \right\rangle$, then $\left\{ \left\{ x \right\},\left\{ x,y \right\} \right\}=\left\{ \left\{ z \right\},\left\... |
Given an integer $d > 0$, let $h: \mathbb{R}^d \rightarrow \mathbb{R}^d$ be a Lipschitz function on $\mathbb{R}^d$. Let $\Phi: \mathbb{R}^d \times [0, \infty) \rightarrow \mathbb{R}^d$, $(x, t) \mapsto \Phi_t(x)$ be the semiflow generated by the ODE $\dot{x}(t) = h(x(t))$. Given a continuous random process $\bar{x}(\cd... |
Say we have a point charge $+q$ at the origin from which it emanates an electric field
$$\mathbf E=\frac{kq}{r^2}\hat{\mathbf r}\ .$$
Now suppose we want to calculate the flux of this electric field through a surface defined by the intersection of a cone $z^2=x^2+y^2$ and the plane $z=z_0>0$, namely,
$$
\mathcal S:... |
Basically the title. As the plane minus $n$ points has as fundamental group the free product of $n$ copies of $\mathbb{Z}$, is there something like a countable free product? In that case, is it related to the mentioned fundamental group? |
Let $M$ be a compact and connected smooth manifold with a symplectic form $\omega$.
$Ham(M, \omega)$ denotes the space of hamiltonian symplectomorphisms of $(M,\omega)$.
I have the following statement in my lecture notes:
> Using Darboux’s theorem one can show that the action of $Ham(M, \omega)$ on M is trans... |
(Sorry for any misspelling or gramatical error, I´m not a native english speaker)
I am trying to solve this problem:
Vector $u=\begin{bmatrix}a &a &a &b \end{bmatrix}´$ can be written as $u=v+w$ where $v$ and $w$ are not null vector and are orthogonal and $$|(s-e_{1})´v|=||s-e_{1}||\cdot ||v||$$
with $s=\begin... |
This seems to have been mostly answered in comments,
which I have tried to organize here.
> Like when I plug in a value in the indefinite integral, we get the area from 0 to the plugged in value.
If we forget that every indefinite integral includes a $+C$ term, or if we suppose that $C=0$, we *sometimes* get the... |
This seems to have been mostly answered in comments,
which I have tried to organize here.
> Like when I plug in a value in the indefinite integral, we get the area from 0 to the plugged in value.
If we forget that every indefinite integral includes a $+C$ term, or if we suppose that $C=0$, we *sometimes* get the... |
Any good reasources on zermelo fraenkel set theory? |
(Sorry for any misspelling or gramatical error, I´m not a native english speaker)
I am trying to solve this problem:
Vector $u=\begin{bmatrix}a &a &a &b \end{bmatrix}´$ can be written as $u=v+w$ where $v$ and $w$ are not null vector and are orthogonal and $$|(s-e_{1})´v|=||s-e_{1}||\cdot ||v||$$
with $s=\begin... |
Any good resources on zermelo-Frankel-Sey theory stuff like papers, links and other educational resources would be useful Also, can someone help me understand in simple terms what exactly it is? |
(Sorry for any misspelling or gramatical error, I´m not a native english speaker)
I am trying to solve this problem:
Vector $u=\begin{bmatrix}a &a &a &b \end{bmatrix}´$ can be written as $u=v+w$ where $v$ and $w$ are not null vectors and are orthogonal and $$|(s-e_{1})´v|=||s-e_{1}||\cdot ||v||$$
with $s=\begi... |
Example #124 of Steen/Seebach's *Counterexamples in Topology* and [S121 of $\pi$-base](https://topology.pi-base.org/spaces/S000121) is "Bernstein's Connected Set":
> Let $\{C_\alpha\ : \alpha<\mathfrak c\}$ be a well-ordering of the set of
all closed, connected subsets of $\mathbb{R}^2$.
We shall define $A_\alpha$... |
I am reading a real analysis book which states that the function $f:\mathbb{R}^2\to\mathbb{R}$ defined by
$$f(x,y)=\begin{cases}
\frac{4(x^2-y)(2y-x^2)}{y^2} & \text{if } y>0\\
0 & \text{if }y\leq 0.
\end{cases}$$
is continuous on $\mathbb{R}^2\setminus\{(0,0)\}$, is not continuous at $(0,0)$ and if we fix... |
According to [this](https://bartoszmilewski.com/2017/09/06/monads-monoids-and-categories/) blog and [this](https://graphicallinearalgebra.net/2017/04/16/a-monoid-is-a-category-a-category-is-a-monad-a-monad-is-a-monoid/) blog, a 2-cell in Span(C), i.e. a morphism between spans, is a morphism between the two apices such ... |
What are 2-cells (and their horizontal compositions) in the bicategory of spans? |
Say we have a point charge $+q$ at the origin from which it emanates an electric field
$$\mathbf E=\frac{kq}{r^2}\hat{\mathbf r}\ .$$
Now suppose we want to calculate the flux of this electric field through a surface defined by the intersection of a cone $z^2=x^2+y^2$ and the plane $z=z_0>0$, namely,
$$
\mathcal S:... |
Why doesn't $RCA_0$ prove $\Sigma^0_1$-comprehension? |
I recently came across [this][1] paper by Friedgut that shows low sensitivity boolean functions are close to juntas. This was an unintuitive result to me, as I thought I could easily imagine a function on the boolean hypercube that is a linear combination of many low-degree monomials, and therefore has low sensitivity,... |
I am having difficulty of finding the ideals of $\mathbb{Z}_7$ . My preliminary attempt is that (d) is an ideal iff d is a divisor of 7. Hence, the ideals I currently have are (1) and (7). Is this correct? |
[This page][1] states "The logarithmic spiral can be constructed from equally spaced rays by starting at a point along one ray, and drawing the perpendicular to a neighboring ray... As the number of rays approaches infinity, the sequence of segments approaches the smooth logarithmic spiral"
My question is, if you ch... |
From an interview book, where the answer is not so clear I believe.
You keep generating $\mathcal U_{[0,1]}$ iid random variables until their sum exceeds 1, then compute the expected value of the last random variable, i.e. the one responsible for letting the sum of rvs overflow 1.
My idea (not working):
The $i$-... |
Source : Ayres, *Outline of Modern Algebra*
There is a stage in Ayres' construction of the set $I$ of integers that I do not understand. More precisely, I recognize the truth of the claim he makes, but I do not see the ***function or necessity*** of the claim regarding the final goal.
See image below $§4.5$
A... |
#### Background
I am following this MIT OCW course on mathematics for computer science.
In one of the recitations they come to the below result:
#### Official solution
###### Task:
A planar graph is a graph that can be drawn without any edges crossing. Also, any planar graph has a node of degree at most 5. Now... |
Let $f(z)=\lambda(-\frac{1}{1+z})=\frac{1}{1-\lambda(z)}$, where $\lambda(z)$ is the modular lambda function, $\lambda(z)=16q-128q^2+704q^3+\ldots$, where $q=e^{\pi i z}$ (https://en.wikipedia.org/wiki/Modular_lambda_function).
What is the growth rate of the values of the higher order derivatives of $f(z)$ for fixed... |
I have an assignment to find the number of operations (only multiplications and divisions) needed to solve $N \times N$ linear equation while using Jordan Elimination method. I thought about it in two different ways but I'm not sure if my work is right.
Suppose that that the augumented matrix where I need to apply t... |
Let $f(z)=\lambda(-\frac{1}{1+z})=\frac{1}{1-\lambda(z)}$, where $\lambda(z)$ is the modular lambda function, $\lambda(z)=16q-128q^2+704q^3+\ldots$, where $q=e^{\pi i z}$ (https://en.wikipedia.org/wiki/Modular_lambda_function).
What is the growth rate of the values of the higher order derivatives of $f(z)$ for a fix... |
I have the sum
$$
\sum_{n=1}^\infty \dfrac{\cos (n \theta)}{n^5} = \dfrac{\text{Li}_5 (e^{i\theta}) + \text{Li}_5 (e^{-i\theta})}{2},
$$
where $0\leq\theta < 2 \pi$ is an angle and $\text{Li}_5(z)$ is a polylogarithm function. I want to know the dominant behaviour of my function. I mostly care for the regime $\thet... |
In a [recent answer](https://math.stackexchange.com/a/4842538/170231) I showed
$$I = \int_0^\tfrac\pi2 \frac{dx}{\left(\sqrt{\sin x}+\sqrt{\cos x}\right)^2} = \sqrt2 \left[F\left(\frac\pi4,\sqrt2\right) + E\left(\frac\pi4,\sqrt2\right)\right] - 2$$
where $F(\phi,k)/E(\phi,k)$ are the incomplete elliptic integrals... |
From an interview book, where the answer is not so clear I believe.
You keep generating $\mathcal U_{[0,1]}$ iid random variables until their sum exceeds 1, then compute the expected value of the last random variable, i.e. the one responsible for letting the sum of rvs overflow 1.
My idea (not working):
The $i$-... |
Let $K$ be a complete discretely valued field of characteristic $p$, then $K \cong k((x))$, where $k$ is the residue field of $K$. This may be called a one-dimensional local field known as the field of formal Laurent power series.
I am interested if there are suitable generalisations of $K$. For example, [here](http... |
Is the $K((x_1, \cdots, x_n))$ a local field? |
How to know the number of digits in the decimals place in division results (Elliptic curves over $\mathbb F_p$)? |
$$\sum_{n=0}^{\infty} x^{n q} \prod_{k=1}^{t} \left(\sum _{m=0}^n \frac{1}{(x^{a_k})^m}\right)$$
For $t=1$
$$ \sum_{n=0}^{\infty} x^{n q} \left(\sum _{m=0}^n \frac{1}{x^{a_1 m}}\right) = \sum_{n=0}^{\infty} x^{n q} \left( \frac{\frac{1}{x^{(n+1)a_1}}-1}{\frac{1}{x^{a_1}}-1}\right) $$ $$ = \frac{1}{ \left(\fra... |
I am used to seeing examples in calculus of variations with functionals acting on curves in manifolds, i.e. $$\mathcal{L}(\gamma) = \int_0^1 L(t, \gamma(t), \dot{\gamma}(t) ) dt$$ for $\gamma : [0,1] \to N$ a smooth curve in a smooth manifold $N$, and $L: \mathbb{R} \times N \times TN \to \mathbb{R}$ (I'm not fully sur... |
In a [recent answer](https://math.stackexchange.com/a/4842538/170231) I showed
$$I = \int_0^\tfrac\pi2 \frac{dx}{\left(\sqrt{\sin x}+\sqrt{\cos x}\right)^2} = \sqrt2 \left[F\left(\frac\pi4,\sqrt2\right) + E\left(\frac\pi4,\sqrt2\right)\right] - 2$$
where $F(\phi,k)/E(\phi,k)$ are the incomplete elliptic integrals... |
Given an map invertible measurable $T$ on a space $(X,\mu)$, take a linear cocycle $\mathcal{A}(x,n)$.
The top Lyapanov exponent at $x$ is defined as
$$\chi(x)=\limsup_{n \to \infty}\frac{1}{n}\log{\|\mathcal{A}(x,n)\|}$$
To my understanding $\chi(x)=\chi(T(x))$. I.E the Lyapanov exponent is invariant under the... |
Is this Proof Involving Arithmetic Cardinality Correct? |
My company recently implemented software that uses an MCMC method. In that program, a handful of randomly generated nuisance parameters are used to calculate *expected* values in a function modeling experimental data, which are then compared to an *observed value from experimental data*. A likelihood is calculated by u... |
Is $(B_t^2 - t^2)_{t\geq 0}$ a local martingale? |
Given an map invertible measurable $T$ on a space $(X,\mu)$, take a linear cocycle $\mathcal{A}(x,n)$.
The top Lyapunov exponent at $x$ is defined as
$$\chi(x)=\limsup_{n \to \infty}\frac{1}{n}\log{\|\mathcal{A}(x,n)\|}$$
To my understanding $\chi(x)=\chi(T(x))$. I.E the Lyapunov exponent is invariant under the... |
I recently came across [this][1] paper by Friedgut that shows low sensitivity boolean functions are close to juntas. This was an unintuitive result to me, as I thought I could easily imagine a function on the boolean hypercube that is a linear combination of many low-degree monomials, and therefore has low sensitivity,... |
Is there always an automorphism distinct from identity in a simple module? |
Is this Proof Involving Cardinal Arithmetic Correct? |
Given polynomials of the form $$(1+x+x^2+x^3+\cdots+x^k)^n $$ We can calcualte the coefficients by writing it in the form $$(1-x^{k+1})^n \over (1-x)^n$$ and using the power series $(1-x)^{-n}$, as has been explained nicely on SE [here](https://math.stackexchange.com/questions/3656079/coefficient-of-xi-in-xx2-xkn?) and... |
The idea of using n-tuples to represent functions is introduced in Terence Tao's Analysis I ex 3.5.2:
> Suppose we define an ordered $n$-tuple to be a surjective function $x
> : \{i \in N : 1 \leq i \leq n\} \to X$ whose codomain is some
> arbitrary set $X$ (so different ordered n-tuples are allowed to have
> dif... |
$n$-tuples as functions - why do they need to be surjective? |
### Problem
> Let $a>0$ and $B$ be a standard $\mathbb{R}$-valued Brownian motion. Define the stopping time $S_a:=\inf\{t\geq 0\ \vert \left\lvert B_t\right\rvert = a\}$. Compute $\mathbb{E}\left[S_a^2\right]$.
### Showing that $\mathbb{E}\left[S_a\right] = a^2$
I am following the solution for a [similiar proble... |
I suspect that you have come to a conclusion about indefinite integrals by extrapolating from a sample of observations that was too small.
There's another fundamental issue here: when you learned about the indefinite integral of a function, what were you told it is?
**An indefinite integral is not a summation of ... |
I am new to the study of linear algebra and there may be errors in the description of my question. My question is that:
Are there any of the following methods for generating nilpotent matrices $M^{n\times n}$ from multivariable polynomials.
$M^{n\times n}\leftarrow\begin{pmatrix}f_{1,1}(\overrightarrow{z_1}):=\su... |
### Problem
> Let $a>0$ and $B$ be a standard $\mathbb{R}$-valued Brownian motion. Define the stopping time $S_a:=\inf\{t\geq 0\ \vert \left\lvert B_t\right\rvert = a\}$. Compute $\mathbb{E}\left[S_a^2\right]$.
### Showing that $\mathbb{E}\left[S_a\right] = a^2$
I am following the solution for a [similiar prob... |
I need to find the inverse Laplace transform of $sF''(s)$ where $F(s) = \int_0^\infty f(t)e^{-st}dt$. This is not one of the cases in my book. What can I do to compute this?
Thank you |
What is the inverse Laplace transform of $sF''(s)$? |
Let $\mathcal{E}$ be an edge measure, and function $f$ be edgewise linear, i.e., it is continuous and its restriction to edge interior is a linear function. Then, at the bottom of the page 7 of [this handout][1], the author claims that, for an edge $e=(u,v)$, we have
$$\int_{e}f\mathrm{d}\mathcal{E} = [f(u)+f(v)]\fr... |
I play a game where I have a $25%$ chance of winning \$ $3$ and a $75%$ chance of losing \$ $1$. Currently, I have \$ $5000$. I will stop playing once I either earn \$ $20000$ or lose all of my \$ $5000$. When I stop, what is the probability of having lost all of my \$ $5000$?
I had an interview question like this. ... |
What's my probability of going broke? |
I want to prove that for a graph with $n$ vertices, the size of the largest set of cycles that don't share edges with each other is at least $\frac{|E(G)| - |V(G)|}{2\lfloor \log_2(n+1)\rfloor}$.
It is trivial to prove this for all forests and trees. I can also observe that if we can prove this inequality for all gr... |
I play a game where I have a $25\%$ chance of winning $\$3$ and a $75\%$ chance of losing $\$1$. Currently, I have $\$5000$. I will stop playing once I either earn $\$20000$ or lose all of my $\$5000$. When I stop, what is the probability of having lost all of my $\$5000$?
I had an interview question like this. How ... |
I think what is confusing you is that you are "starting out" with a planar graph of $n+1$ nodes.
But note that you are not yet saying anything about its colorability. All you're trying to show here is that $P(n) \to P(n+1)$, or:
"Any planar graph with $n$ nodes is $6$-colorable" $\to$ "Any planar graph with $n+1$... |
### Problem
> Let $a>0$ and $B$ be a standard $\mathbb{R}$-valued Brownian motion. Define the stopping time $S_a:=\inf\{t\geq 0\ \vert \left\lvert B_t\right\rvert = a\}$. Compute $\mathbb{E}\left[S_a^2\right]$.
### Attempt to show that $\mathbb{E}\left[S_a\right] = a^2$
I am following the solution for a [simil... |
I play a game where I have a $25$% chance of winning \$ $3$ and a $75$% chance of losing \$ $1$. Currently, I have \$ $5000$. I will stop playing once I either earn \$ $20000$ or lose all of my \$ $5000$. When I stop, what is the probability of having lost all of my \$ $5000$?
The winning is not $1$, so I think I ca... |
I need to find the inverse Laplace transform of $sF''(s)$ where $F(s) = \int_0^\infty f(t)e^{-st}dt$. This is not one of the cases in my book. What can I do to compute this?
My hunch is that the answer is $t^2f'(t)$ seen in the beginning of this paper.
https://link.springer.com/content/pdf/10.1007/PL00001481.pdf... |
### Problem
> Let $a>0$ and $B$ be a standard $\mathbb{R}$-valued Brownian motion. Define the stopping time $S_a:=\inf\{t\geq 0\ \vert \left\lvert B_t\right\rvert = a\}$. Compute $\mathbb{E}\left[S_a^2\right]$.
### Flawed attempt to show that $\mathbb{E}\left[S_a\right] = a^2$
I am following the solution for a... |
I’m currently learning about the history of the development of the Lebesgue integral in Thomas Hawkins’s book “Lebesgue’s Theory fo Integration; It’s Origins and Development”
Hawkins is stressing how much early confusion about the Riemann integral came from conflating three different notions of “small” subsets of re... |
The Sudakov's Minoration Inequality states the following:
Let $(X_t)_{t \in T}$ be a mean zero Gaussian process. Then for any $\epsilon \geq 0$, we have
$$
\mathbb{E}\sup_{t \in T} X_t \geq c\epsilon \sqrt{\log N(T, d, \epsilon)},
$$
where $d(t,s) := \| X_t - X_s \|_{L^2}$. Here we define $N(T, d, \epsilon)$ as ... |
### Problem
> Let $a>0$ and $B$ be a standard $\mathbb{R}$-valued Brownian motion. Define the stopping time $S_a:=\inf\{t\geq 0\ \vert \left\lvert B_t\right\rvert = a\}$. Compute $\mathbb{E}\left[S_a^2\right]$.
### Flawed attempt to show that $\mathbb{E}\left[S_a\right] = a^2$
I am following the solution for a... |
My current understanding is that sorts are ranges of quantification.
[PA][1] has only one sort $N$, so it is single-sorted.
If I modify PA so that quantification over $\mathrm{Prop}$ is possible, is it two-sorted? Or, single-sorted second-order theory?
[1]: https://ncatlab.org/nlab/show/Peano+arithmetic |
Terminology: What is sort? |
The side lengths of a triangle are $x,x+4,$ and $2x-4$. The area of the triangle is $32\sqrt{2}$. What is the perimeter of the triangle?
My attempt:
According to Heron's formula
$$A=\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}$$
where
$$s=\frac{a+b+c}{2}.$$
In this scenario, $$s=\frac{x+x+4+2x-4}{2}... |
If P1<P2...<Pn are n different prime numbers, and the sum of any three numbers is still a prime number, what is the maximum value of n?
I only can find 3 numbers 3、5、11 that the sum is 19 which is still prime... |
If P1<P2...<Pn are n different prime numbers, and the sum of any three numbers is still a prime number, what is the maximum value of n? |
Given a rational function of the form $$ \frac{x^k}{(x^n-\lambda_1)(x^m-\lambda_2)}$$ with $k< n+m$, I know we can prove that there are unique polynomials $p(x),q(x)$ with
$$
\frac{x^k}{(x^n-\lambda_1)(x^m-\lambda_2)} = \frac{p(x)}{x^n-\lambda_1}+\frac{q(x)}{x^m-\lambda_2}
$$
where $\deg p(x) <n$, $\deg q(x)<1$, b... |
Let $\mathcal{E}$ be an edge measure, and function $f$ be edgewise linear, i.e., it is continuous and its restriction to edge interior is a linear function. Then, at the bottom of the page 7 of [this handout][1], the author claims that, for an edge $e=(u,v)$, we have
$$\int_{e}f\mathrm{d}\mathcal{E} = [f(u)+f(v)]\fr... |
My current understanding is that sorts are ranges of quantification.
> many-sorted logic (MSL) allows quantification over a variety of domains (called sorts) - [Stanford Encyclopedia of Philosophy][1]
[PA][2] has only one sort $N$, so it is single-sorted.
If I modify PA so that quantification over $\mathrm{Pro... |
I’m currently learning about the history of the development of the Lebesgue integral in Thomas Hawkins’s book “Lebesgue’s Theory of Integration; It’s Origins and Development”
Hawkins is stressing how much early confusion about the Riemann integral came from conflating three different notions of “small” subsets of re... |
My current understanding is that sorts are ranges of quantification.
> many-sorted logic (MSL) allows quantification over a variety of domains (called sorts) - [Stanford Encyclopedia of Philosophy][1]
[First-order Peano arithmetic(PA)][2] has only one sort $N$, so it is single-sorted.
If I modify PA so that qu... |
Let $E$ be an elliptic curve over $\mathbb{Q}_p$ with good reduction at $p$, i.e., there exists an elliptic curve $\mathcal{E}$ over $\mathbb{Z}_p$ whose generic fiber is isomorphic to $E$. I have seen on page 76, just above Theorem 5.38, of a course [note][1] by Schraen that the action of $G_{\mathbb{Q}_p}$ on $E[N]... |
I am confused about two Lemmas in the book *Associative Algebras* by R.Pierce (pages 43-44).
[![enter image description here][1]][1]
[![enter image description here][2]][2]
[![enter image description here][3]][3]
In **Lemma d**, since each right ideal of a can be considered an algebra over $A$ then if $N$ is a mi... |
I got a question below with answer from the professor, although I think the answer may be wrong, but is there something I missed?
1. Determine the form of a particular solution to the ODE below. Do not solve for the coefficient.
$y'' + 4y' + 5y = e^{2t} + t^2 \sin(t) - e^{2t} \cos(t)$
I know the solution for t... |
The tensor product of a space with itself once is $V^{\otimes1}$, but what is $V^{\otimes0}$? Since it is an empty tensor product, it is - _a fortiori_ - an empty product. So I'm looking for a "$1$" of some sort, just not sure what that would mean in this context.
>"If I take the tensor product of a vector space wi... |
What is the empty tensor product of vector spaces? |
Let $\mathcal{E}$ be an edge measure, and function $f$ be edgewise linear, i.e., it is continuous and its restriction to edge interior is a linear function. Then, at the bottom of the page 7 of [this handout][1], the author claims that, for an edge $e=(u,v)$, we have
$$\int_{e}f\mathrm{d}\mathcal{E} = [f(u)+f(v)]\fr... |
I got a question below with answer from the professor, although I think the answer may be wrong, but is there something I missed?
1. Determine the form of a particular solution to the ODE below. Do not solve for the coefficient.
$y'' + 4y' + 5y = e^{2t} + t^2 \sin(t) - e^{2t} \cos(t)$
I know the solution for t... |
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