instruction stringlengths 12 30k |
|---|
> is it true that, if two spaces satisfy the same topological properties written in an appropriate (formal) language, then they are homeomorphic?
since first order theories with infinite models always have elementarily equivalent but non-isomorphic models, one has to look for something else. hen:
if you're willin... |
First I will say that the exact same question has been uploaded to the site already few years ago but it seems no final answer were given there. Let
$$X = \{a,b,c\}, \quad Y = \{1,2,3,4,5,6,7\}.$$
I need to count how many injective functions $f: X\to Y$ there are such that
$$f(a)\ne 1,2,\quad f(b)\ne 2 ,\quad f... |
In the framework of optimal control we have introduced in class the HJB equation to solve optimal control problem of the form
$$
\inf_{u\in\mathcal{U}}\left[\int_{0}^{T}L(t,x_u(t),u(t))dt + h(x_u(T));\quad x’(t)=f(t,x(t),u(t)), x(0)=x_0\right]
$$
Where $x(t)\in\mathbb{R}^{d}$, $u(t)$ is a control, $L$ a cost f... |
I have found an answer in the shape of an essentially brute-force Python 3 program I threw together. I will not accept my own answer because I'm hoping there are better takes on this, like a proper recursion or Lindenmayer system to quickly construct this solution. I am not entirely sure at this point, whether the resu... |
I've been reading through the Polytope-Wiki entry on [helices][1]. To my understanding, an $n$-gonal helix is a blend of a planar $n$-gon $\{n\}$ with the regular linear apeirogon $\{\infty\}$. The blend of the apeirogon with a line segment produces the only two dimensional "helix", the zig-zag. Finally, $\{\infty\}$ i... |
$\textbf{Hint}$
Following lulu's advice to use friendly nomenclature for symbols, and realising that the red balls are irrelevant,
Let $w$ = # of white balls, $b$ = # of black balls, $n = w+b$, then
P(black) $=p = \dfrac{b}{n},\;\;$ P(white) $=1-p = q$
The earliest the $k_{th}$ black could come would be ... |
I have a question regarding boundary conditions. In my problem I’m given a system of partial differential equations, and some boundary conditions. The first question asks me to prove that the given boundary conditions are sufficient for solving the problem. How can I formally prove it?
Thanks in advance! |
I have found an answer in the shape of an essentially brute-force Python 3 program I threw together. I will not accept my own answer because I'm hoping there are better takes on this, like a proper recursion or Lindenmayer system to quickly construct this solution. I am not entirely sure at this point, whether the resu... |
I think the short answer to the question
> *Can it really be that the same abstract polytope can have such
> qualitatively different realisations?*
Is "yes".
Even a square (as an abstract graph) has some qualitatively quite different embeddings in the plane. Many more in space. |
It is clear that it is true that f(x,y)-f(x',y)=g(x,x'), but is the function g necessarily a separable additive function? Thank you! |
Is it true that if f(x,y)-f(x',y) is independent of y for all y, then f(x,y)-f(x',y)=g(x)-g(x') for some g? |
It is clear that it is true that $f(x,y)-f(x',y)=g(x,x'),$ but is the function $g$ necessarily a separable additive function? Thank you! |
Is it true that if $f(x,y)-f(x',y)$ is independent of $y$ for all y, then $f(x,y)-f(x',y)=g(x)-g(x')$ for some $g?$ |
> Let $f : \mathbb{R}\rightarrow \mathbb{R}$ be a continuous function such that $\left | f(x) - f(y) \right | \geq \left |x - y\right |$ for all $x, y \in \mathbb{R}$. Prove that the range of $f(x)$ is all of $\mathbb{R}$.
----------
I found a solution [here][1] which goes like:
> If $f(x)=f(y)$ then $0\geq |x-y|$... |
suppose we have, in the ring $\mathbb{Z} [x,y,z,w,v]$, the following polynomials:
$f=xw-yz$,
$g=x^2z-y^3$,
$h=yw^2-z^3$,
$k=xz^2-y^2w$.
The question is to prove that $I=(f,g,h,k)$ is the radical ideal of $J=(f,g,h)$.
I tried several things to no avail: by imposing $x>y>z>w$ I tried to divide $k^2$ by ... |
[question 1][1]
[question 2][2]
[material from the note][3]
I was so confused by the vector "y", why would it have different signs for Q1 and Q2 matrices in this case? I feel like vector "a" doesn't change at all so should the sign stay the same, could this be explained to me please?
[1]: https://i.... |
I've been reading through the Polytope-Wiki entry on [helices][1]. To my understanding, an $n$-gonal helix is a blend of a planar $n$-gon $\{n\}$ with the regular linear apeirogon $\{\infty\}$. The blend of the apeirogon with a line segment produces the only two dimensional "helix", the zig-zag. Finally, $\{\infty\}$ i... |
> Let $f : \mathbb{R}\rightarrow \mathbb{R}$ be a continuous function such that $\left | f(x) - f(y) \right | \geq \left |x - y\right |$ for all $x, y \in \mathbb{R}$. Prove that the range of $f(x)$ is all of $\mathbb{R}$.
----------
I found a solution [here][1] which goes like:
> If $f(x)=f(y)$ then $0\geq |x-y|$... |
The constraints are $f(0,0)=1, f(0,k)=0\space \forall k \neq 0, f(N,k)=0 \space \forall k>N\space0\leq p\leq 1$ .
When working on a probability problem, I came across this recursion when working with random walks. For symmetric random walks, with $p=\frac{1}{2}$, I found a closed form solution of $2^{-N}*$ ${N}\choo... |
Let's say $f(x)\ge -M$ for some $M>0$ for all $x$. Since $f$ is monotone, WLOG assume it is strictly increasing. This means $x\to-\infty$, $f(x)\to-M$.
What can you say about $f(x)-f(y)$ under this asymptotic behaviour? |
The Direct Comparison test states that if we have two series $a_n$ and $b_n$ with $b_n ≥ 0$ and $a_n ≤ b_n$ for all n. If b converges so does a. If a diverges so does b.
I was given the series $\sum_{n=2}^{\infty}\frac{1}{n^n}$ and I attempted to use the direct comparison test on this, but was unable to come up with... |
can anybody prove the statement below using only basic identities?
$A$ and $B$ are disjoint languages.
$$(A \cup B)^∗AB(A \cup B)^∗ \cup B^∗A^∗ = (A \cup B)^∗$$
|
![Image for visualization][1]
I recently came across this variation of linear programming problem where given a set of $2$-D points, we would like to find 1 point which maximizes certain objective function. Let's say the objective function is to maximize $y = m(x+A) + B$ where $m$ is gradient, $A$ and $B$ are consta... |
Let $p$ be a large prime, and $S$ a set of non-negative integers less than $p$. Consider the distribution $D_{S,p}$ over complex numbers defined as $\sum_{x\in S}e^{i2\pi x y/p}$ for $y$ chosen uniformly at random from the set of non-negative integers less than $p$.
**Is it possible to show, for arbitrary sets $S$, ... |
It's not a typo.
What is a "tuple"? Normally we think of a "tuple" with index set $I$ as a function with domain $I$. So, viewed as a set of ordered pairs, a tuple $(x_i)_{i\in I}$ would be the set
$$\{ (i,x_i)\mid i\in I\}.$$
For index sets that are of the form $I=\{1,2,3,\ldots,n\}$, we often denote this as $(a_... |
let $X \sim Bin(10,p)$
We would like to test the following hypotheses:
$H_{0}: p=0.5 $ vs $H_{1} : p=0.3$
The Statistic $T(x)$ is defined as $T(x)=x$ and the critical (rejection) region is $R=\{x:T(x)\le4\}$
I want to calculate the conditional probability of the null hypothesis given $T(x)\in R $ ... |
> Let $A$ be the $\Bbb R$-algebra defined by endowing $\Bbb R^2$ with the multiplication $$(a,b)\cdot (c,d)=(ac,ad+bc).$$ Show that $A \cong \Bbb R[x]/(x^2)$.
So I'm planning to use the first isomorphism theorem. For that I need a surjective map $\varphi : \Bbb R[x]\to A$ with kernel $(x^2)$. For any $p = \sum_ka_kx... |
I want to use the Lagrange inversion theorem from *Whittaker and Watson*, p. 133, to find the power series of $\sin^{-1}$ at the origin. It should be a valid procedure because $\sin^{-1}$ is analytic at the origin. The theorem is stated as follows:
>Let $f(z)$ and $\phi (z)$ be functions of $z$ analytic on and inside ... |
I just took the derivative with respect to x as:
F'(x) = $\int_1^{3x}e^{xy} \,dy$ which should be equal to $e^{3x^{2}} -e^x\over x$
However my textbook made the calculation as:
F'(x) = $3x^{-1}*e^{3x^{2}}*3+\int_1^{3x}e^{xy} \,dy$ which equals to $2e^{3x^{2}} -e^x\over x$
I don't understand why it is like t... |
I just took the derivative with respect to x as:
F'(x) = $\int_1^{3x}e^{xy} \,dy$ which should be equal to $e^{3x^{2}} -e^x\over x$
However my textbook made the calculation as:
$F'(x) = 3x^{-1}*e^{3x^{2}}*3+\int_1^{3x}e^{xy} \,dy$ which equals to $2e^{3x^{2}} -e^x\over x$
I don't understand why it is like t... |
Let $X \in \mathbb{R}^d$ be a zero-mean multivariate Gaussian, with independent components, i.e., $X \sim \mathcal{N}(0,\Sigma)$, with $\Sigma = diag(\sigma_1^2,\ldots,\sigma_d^2)$ a diagonal matrix. Leveraging component independence and (sub-)Gaussianity, it can then be shown that, for any $0 < \lambda \leq \frac{1}{\... |
I am trying to solve the following problem taken the book Markov Chains and Mixing Times 2nd edition (Exercise 12.2):
Let $P$ be irreducible transition matrix, and suppose that A is a matrix with $0 \le A(i,j) \le P(i,j)$ and $A \ne P$. Show that any eigenvalue $\lambda$ of $A$ satisfies $|\lambda| < 1$.
Now I kn... |
[Green's theorem](https://en.wikipedia.org/wiki/Green%27s_theorem) says that:
$$
\int_C L \ dx + \int_C M \ dy = \iint_D \frac{\partial M}{\partial x} - \frac{\partial L}{\partial y} \ dx \ dy
$$
If the M and N statisfy $\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y} = 1$, then the formula can be use... |
I've been reading through the Polytope-Wiki entry on [helices][1]. To my understanding, an $n$-gonal helix is a blend of a planar $n$-gon $\{n\}$ with the regular linear apeirogon $\{\infty\}$. The blend of the apeirogon with a line segment produces the only two dimensional "helix", the zig-zag. Finally, $\{\infty\}$ i... |
Let $X \in \mathbb{R}^d$ be a zero-mean multivariate Gaussian, with independent components, i.e., $X \sim \mathcal{N}(0,\Sigma)$, with $\Sigma = diag(\sigma_1^2,\ldots,\sigma_d^2)$ a diagonal matrix. Leveraging component independence and (sub-)Gaussianity, it can then be shown that, for any $0 < \lambda \leq \frac{1}{\... |
Let me take this back to a really fundamental level for a moment.
If we want to start a Group we need a set and an operation. What the set and operation are can be decided later, but for argument's sake I will use $S={0,1,2,3,4,5...}$
In order to successfully define an operation, the set $S$ must first be **close... |
Let $p$ be a large prime, and $S$ a set of non-negative integers less than $p$. Consider the distribution $D_{S,p}$ over complex numbers defined as $\sum_{x\in S}e^{i2\pi x y/p}$ for $y$ chosen uniformly at random from the set of non-negative integers less than $p$.
**Is it possible to show, for arbitrary sets $S$, ... |
**Definition:** Let $\mathfrak{g}$ be a Lie algebra. A *universal enveloping algebra* of $\mathfrak{g}$ is a pair $(\mathfrak{U},i)$ consisting of a unital associative algebra $\mathfrak{U}$ and a Lie morphism $i: \mathfrak{g} \rightarrow \mathfrak{U}_L$ satisfying the following universal property:
for every pair $(... |
I've gotten stuck on a particular sum, to which I think I know the answer (thanks to Wolfram:Alpha), but not the method leading to it. I wonder if someone here can help me solve it.
Let $d$ be a positive, odd integer, and define $u$ to be the first nontrivial $d$th root of unity (that is, $u = \exp\left(\frac{2\pi i... |
Let's say I have a multivariable function $f(x,y)$ that is continuous across $0<x<\infty$ and $0<y<\infty$. I'm trying to solve the following integral:
$\int_{a}^{b} dx \frac{\partial f}{\partial y} \Big\vert_{y=x}$
This is probably a silly question, but can I use the Fundamental Theorem of Calculus to reduce thi... |
Let's say I have a multivariable function $f(x,y)$ that is continuous across $0<x<\infty$ and $0<y<\infty$. I'm trying to solve the following integral:
$\int_{a}^{b} dx \frac{\partial f}{\partial y} \Big\vert_{y=x}$
where $a,b>0$ and are real. This is probably a silly question, but can I use the Fundamental Theor... |
I've gotten stuck on a particular sum, to which I think I know the answer (thanks to Wolfram:Alpha), but not the method leading to it. I wonder if someone here can help me solve it.
Let $d$ be a positive, odd integer, and define $u$ to be the first nontrivial $d$th root of unity (that is, $u = \exp\left(\frac{2\pi i... |
Consider the following statements:
> Finite Ramsey Theorem ($\mathrm{FRT}$): For any $r,k,p\in\mathbb{N}$, there exists $N\in\mathbb{N}$ such that for any set $S$ of cardinality $N$ and any function $c:[S]^{p}\rightarrow [r]$ there exists a subset $H\subseteq S$ of cardinality $k$ such that $c|_{[H]^{p}}$ is constan... |
Well assuming that the [Prime Number Theorem][1] is true, when substituting $\delta(x)$, the density of primes near $x$—which I am being vague of what it means 'cause I don't have enough foundation about the topic—is asymptotic to $\frac{1}{\log(x)}$. Then, when I graphed the difference between $\log(x!)$ & $x\cdot\sum... |
I am trying to calculate the probability that a jar with 200 jelly beans and 40 possible jelly bean colors does not feature all colors at least 1 time.
I am aware of the way to solve this using inclusion-exclusion but have been wondering why the following approach does not work.
Let 40 glasses represent the 40 di... |
Assume that $X\in \mathrm{PSD}(n)$ is a symmetric, positive semi-definite matrix with real entries and $C>0$ is a positive real number. If $X$ satisfies $$
\forall \alpha\in\mathbb{Z}^n\setminus\{0\},\qquad \alpha^T X \alpha \geq C,
$$ then is it true that $X$ is positive definite? Can we give a concrete lower bound ... |
Let's say I have a multivariable function $f(x,y)$ that is continuous across $0<x<\infty$ and $0<y<\infty$. I'm trying to solve the following integral:
$\int_{a}^{b} dx \frac{\partial f}{\partial y}\vert_{y=x},$ where $a,b>0$ and are real. This is probably a silly question, but can I use the Fundamental Theorem of... |
In the square $ABCD$, the points $M$ and $N$ belong to the sides $BC$ and $CD$ respectively, such that $\angle MAN = 45^°$. Let $O$ be the point of intersection of the circle that goes through $C$, $M$ and $N$ with the segment $AC$. Is the point $O$ always the circumcenter of $\triangle MAN$?
[Illustration on Geogebra... |
How to prove if the given point is always the circumcenter of the triangle? |
>What if there are more than just plus and minus basics signs?
Various authors have attempted to construct new number systems with multiple "signs". For example, in my [prior answer](https://math.stackexchange.com/a/32114/242) you will find some links to so-called PolySign numbers that were discussed frequently on ... |
In the square $ABCD$, the points $M$ and $N$ belong to the sides $BC$ and $CD$ respectively, such that $\angle MAN = 45^°$. Let $O$ be the point of intersection of the circle that goes through $C$, $M$ and $N$ with the segment $AC$. Is the point $O$ always the circumcenter of $\triangle MAN$? [Illustration on Geogebra]... |
You may find of interest some more general results besides the mentioned Frobenius Theorem. Weierstrass (1884) and Dedekind (1885) showed that every finite dimensional commutative extension ring of $\mathbb R$ without nilpotents ($x^n = 0\,\Rightarrow\, x = 0$) is isomorphic as a ring to a direct sum of copies of $\rm... |
Problem: Let $A_1A_2\dots A_{100}$ be a regular $100$-gon with a circumcircle of diameter 1. Let M be the midpoint of the minor arc $A_1A_2$. Find $\displaystyle \sum^{100}_{i=1} |MA_i|^2.$
My attempt:
Let the points $A_k$ be of the form $\displaystyle \frac12 e^{2\pi k/100} = \frac12 \cos(\frac{2\pi k}{100})+\frac... |
I was solving a problem that asked to estimate the value of $N$ in an event whose probability was given by:
$$\mathbb{P}(N)=\frac{\binom{N-100}{90}\binom{100}{10}}{\binom{N}{100}}$$
i.e., the question asked to find a reasonable local maximum of $\mathbb{P}(N)$. As this function is too complicated, the hint given was ... |
I am trying to prove that the sequence of functions $f_n (t) = n^2 t e^{-nt}$ does not converge uniformly on $\mathbb{R}_{\geq 0}$. I was able to show that it converges point-wise to the zero function. To prove that it doesn't converge uniformly, I need to show that
$$
\exists x, \exists \epsilon, \forall N, \exists... |
i am reading the book an introduction to non abelian class field theory by Seiken Saito and Toyokazu Hiramatsu in page 5 ,and i am stuck in the following part
let
$\eta(6z)\eta(18z)=\sum_{x,y \in \mathbb{Z}} (-1)^{n+m} q^{((6m+1)^2+3(6n+1)^2)/4)}$
Where $q=e^{2\pi iz}$
and let A(p) be the number of soluti... |
i am reading the book an introduction to non abelian class field theory by Seiken Saito and Toyokazu Hiramatsu in page 5 ,and i am stuck in the following part
let
$\eta(6z)\eta(18z)=\sum_{x,y \in \mathbb{Z}} (-1)^{n+m} q^{((6m+1)^2+3(6n+1)^2)/4)}$
Where $q=e^{2\pi iz}$
and let A(p) be the number of soluti... |
Does there exist two binary operations $+$ and $*$ on $\{0,1\}$ such that $+$ and $*$ satisfy the same equations, but not the same quasi-equations? If not, are there such binary operations on a finite set of higher cardinality, and if so, what is the smallest possible cardinality? For those who don't know, a quasi-equa... |
Examples of two finite magmas which satisfy the same equations but not the same quasi-equations? |
This [definite integral](https://math.stackexchange.com/q/2286805/) has an equivalent closed form in terms of complete [elliptic integrals](https://mathworld.wolfram.com/EllipticIntegralSingularValuek1.html),
$$\begin{align*}
I &= \int_0^\tfrac\pi2 \frac{\sqrt{\sin x}}{1+\cos x} \, dx \\
& = 2 - \sqrt{\frac2\pi}\,... |
Can we reduce $\int_0^{\pi/2}\frac{\sqrt{\sin x}}{1+\cos x}\,dx$ to complete elliptic integrals? |
I originally added the following as an update to the question. I decided to post it as an asnswer since no objections have been raised so far. That way the question will be removed from the unanswered list. In any cae, here is my attempt at answering the question.
Over $\mathsf{PA}^{-}+\mathsf{I\Sigma}_{0}^{0}$ (see... |
In the square $ABCD$, the points $M$ and $N$ belong to the sides $BC$ and $CD$ respectively, such that $\angle MAN = 45^°$. Let $O$ be the point of intersection of the circle that goes through $C$, $M$ and $N$ with the segment $AC$. Is the point $O$ always the circumcenter of $\triangle MAN$?
![Illustration on Geogeb... |
I want to show that a statement with several quantifiers, e.g., "$f(a, b) <= 0$ for all $a\in [0, 3]$ and all $b\in [-\infty, -1]$", is equivalent to another statement with several quantifiers, e.g., "$g(a, b) <= 0$ for all $a \in[-10, 15]$ and $b < 0$", where $f$ and $g$ are two functions from $\mathbb{R}^2 \rightarr... |
> In the axiomatic treatment of natural numbers, can we define what a natural number is?
If you mean 'just writing down the usual Peano axioms in first order logic and working from there *with just them, nothing more*', then everything is a natural number already (well, the objects at least, not the functions)
If... |
I originally added the following as an update to the question. I decided to post it as an answer since no objections have been raised so far. That way the question will be removed from the unanswered list. In any cae, here is my attempt at answering the question.
Over $\mathsf{PA}^{-}+\mathsf{I\Sigma}_{0}^{0}$ (see ... |
I'm exploring the interaction between different types of stochastic matrices and came across a scenario that I find quite intriguing. Consider a nonnegative $(n \times n) $ matrix \($A$\) that is row-stochastic (each row sums to 1) but not column-stochastic (at least one column does not sum to 1). My question is about ... |
Can the Product of a Row-Stochastic and Non-Column-Stochastic Matrix and Another Stochastic Matrix Be Doubly Stochastic? |
Problem: Let $A_1A_2\dots A_{100}$ be a regular $100$-gon with a circumcircle of diameter 1. Let M be the midpoint of the minor arc $A_1A_2$. Find $\displaystyle \sum^{100}_{i=1} |MA_i|^2.$
My attempt:
Let the points $A_k$ be of the form $\displaystyle \frac12 e^{2\pi ki/100} = \frac12 \cos(\frac{2\pi k}{100})+\fra... |
What's the difference between zig-zags and a helixes? |
> Shuffle an ordinary deck of 52 playing cards containing four aces.
> Then turn up the cards from the top until the first ace appears. On
> the average how many cards are required to be turned before producing
> the first ace?
The question is answered [here][1] using symmetry. I have seen other such [questions][... |
Just for curiosity I want to represent $x^n$ as a sum of $P_k:= (x)(x-1)\dots(x-k+1)$.
----
Since $x=P_1,\ xP_n= P_{n+1} +nP_n$, this proves that it is possible for any $x^n$ to be represented as a sum of $P_k:= (x)(x-1)\dots(x-k+1)$ where $1\le k \le n$, $ \ n \in \mathbb{N}$. We have $$x^n = \sum\limits_... |
I am going to formulate an optimization problem for finding a feasible portfolio with the largest value of Sortino ratio. It will defined as follows:
\begin{align*}
\max_{x} \quad & STR(x) = \frac{\mu^T x - R}{\sqrt{\mathbb{E}[\min\{0, (r-\mu)^T x\}]^2}} \\
\text{s.t.} \quad & x \in X
\end{align*}
where:
- $x ... |
I originally added the following as an update to the question. I decided to post it as an answer since no objections have been raised so far. That way the question will be removed from the unanswered list. In any cae, here is my attempt at answering the question.
Over $\mathsf{PA}^{-}+\mathsf{I\Sigma}_{0}^{0}$ (see ... |
I am trying to solve the following ODE $$\begin{aligned} & y^{\prime \prime}+2 y^{\prime}+y=0 \\ & y(0)=2 \quad y^{\prime}(0)=-1 \\ & 0 \leqslant x \leqslant 1 \end{aligned}$$
Using central FD, I can write $$\frac{y_{i+1}-2 y_i+y_{i-1}}{h^2}+2 \frac{y_{i+1}-y_{i-1}}{2 h}+y_i=0$$ which I have simplied to $$(1+h) y_{i+1... |
I am trying to solve the following ODE $$\begin{aligned} & y^{\prime \prime}+2 y^{\prime}+y=0 \\ & y(0)=2 \quad y^{\prime}(0)=-1 \\ & 0 \leqslant x \leqslant 1 \end{aligned}$$
Using central FD, I can write $$\frac{y_{i+1}-2 y_i+y_{i-1}}{h^2}+2 \frac{y_{i+1}-y_{i-1}}{2 h}+y_i=0$$ which I have simplied to $$(1+h) y_{i+1... |
My understanding of the bounding scheme is rudimentary. But it is an induction principle after all, just for a restricted class of formulas. Hence, it appears to me that the [proof theoretic ordinal][1] (also called $\mathsf{\Pi_{1}^{1}}$-ordinal) of $\mathsf{B\Sigma}_{2}^{0}$ (equivalently, $\mathsf{B\Pi}_{1}^{0}$) sh... |
Let $\sigma:\mathbb{N}\longrightarrow\mathbb{N}$ be strictly increasing, and consider the power series
$$ S_{\sigma}(x)=\sum_{n=0}^{+\infty}(-1)^nx^{\sigma(n)}. $$
Can any real number in $[0,2]$ be obtained as the limit $\lim\limits_{x\rightarrow 1^-}S_{\sigma}(x)$ for some $\sigma$ ?
According to [this][1] answer... |
Spherical coordinates will help you to notice that any term containing $h_1^{p_1}h_2^{p_2}h_3^{p_3}/||\vec h||$ with $p_1+p_2+p_3>1$ will tend to $0$ in the limit $\vec h\to\vec 0$. How is this important? It means that you can skip all higher order terms, like $h_i^2$ or $h_ih_j$.
$$f(\vec a+\vec h)-f(\vec a)=(-4+h_2)... |
I am implementing a finite difference scheme for the following PDE:
$$
u_t=u_{xx}+f(u)
$$
with the nonflux boundary condition and a given initial condition. The scheme is backward in time and central in space(which should be unconditionally stable) as
$$
\frac{u^{k+1}_i-u^k_i}{\Delta t}=\frac{u^{k+1}_{i-1}-2u^{k+... |
The [proof-theoretic strength][1] of a theory is measured by the $\mathsf{\Pi}_{1}^{1}$-ordinal of the theory (indeed, there are other ordinal analyses, like the $\Pi_{2}^{0}$-ordinal of the theory).
Consider the following statements:
> Finite Ramsey Theorem ($\mathrm{FRT}$): For any $r,k,p\in\mathbb{N}$, there ... |
I'm working through Advanced Calculus - Theory and Practice by John S. Petrovic and is currently working on problem 3.4.2, which is as follows:
Find the limit and prove that the result is correct using the definition (epsilon-delta) of the limit: $\lim_{{x \to 4}} \frac{1}{{x^2 + 5x - 24}}$
This is my attempt:
... |
Is $f(x) = \frac{1}{x^2 + 5x - 24}$ continuous at $x=4$? |
The [proof-theoretic strength][1] of a theory is measured by the $\mathsf{\Pi}_{1}^{1}$-ordinal of the theory and it is called the proof-theoretic ordinal (PTO) of the theory (there are other ordinal analyses, like the $\Pi_{2}^{0}$-ordinal of the theory).
My understanding of the bounding scheme is rudimentary. But ... |
I originally added the following as an update to the question. I decided to post it as an answer since no objections have been raised so far. That way the question will be removed from the unanswered list. In any cae, here is my attempt at answering the question.
Over $\mathsf{PA}^{-}+\mathsf{I\Sigma}_{0}^{0}$ (see ... |
> **Problem:** Let $A_1A_2\dots A_{100}$ be a regular $100$-gon with a circumcircle of diameter $1$. Let $M$ be the midpoint of the minor arc $A_1A_2$. Find
$$
\sum^{100}_{i=1} |MA_i|^2.
$$
**My attempt:** Let the points $A_k$ be of the form
$$
\frac12 e^{2\pi ki/100}
= \frac12 \cos\Bigl(\frac{2\pi k}{100}\... |
I'm working through *Advanced Calculus: Theory and Practice* by John S. Petrovic and is currently working on problem 3.4.2, which is as follows:
> Find the limit and prove that the result is correct using the definition (epsilon-delta) of the limit:
$$
\lim_{x \to 4} \frac{1}{x^2 + 5x - 24}
$$
This is my atte... |
If $c,d$ are two relatively prime positive integers, then we can find integers $a,b$ such that $ad-bc=1$. But $a$ and $b$ are not unique: we can replace $a$ with $a+kc$ and $b$ with $b+kd$ for any integer $k$ and get another solution. So there is a solution $(a,b)$ of minimal distance from the origin, in the sense that... |
Problem:
Use the Laplace form to solve the following initial-value problem.
$$ y'' + 6y' + 8y = 16 $$
with
$$ y(0) = 0 $$
$$ y'(0) = 10 $$
and the independent variable is $t$.
Answer:
\begin{align*}
s^2Y(s) -sy(0) - y'(0) + 6 ( sY(s) - y(0)) + Y(s) &=\dfrac{16}{s} \\
s^2Y(s) - 10 + 6 ( sY(s) ) + Y(s)... |
> The random variables $X$ and $Y$ are said to independent if, for any
> two sets of real numbers $A$ and $B$,
>
> $$P(X \in A, Y \in B) = P(X \in A)P(Y \in B)$$
>
> It can be shown by using the three axioms of probability that the
> equation will follow if and only if, for any $a, b \in R$: $$P(X \leq
a, Y \l... |
How to increase a grid using this equation on desmos? |
I am trying to solve the following ODE $$\begin{aligned} & y^{\prime \prime}+2 y^{\prime}+y=0 \\ & y(0)=2 \quad y^{\prime}(0)=-1 \\ & 0 \leqslant x \leqslant 1 \end{aligned}$$
Using central FD, I can write $$\frac{y_{i+1}-2 y_i+y_{i-1}}{h^2}+2 \frac{y_{i+1}-y_{i-1}}{2 h}+y_i=0$$ which I have simplied to $$(1+h) y_{i+1... |
Problem:
Use the Laplace form to solve the following initial-value problem.
$$ y'' + 6y' + 8y = 16 $$
with
$$ y(0) = 0 $$
$$ y'(0) = 10 $$
and the independent variable is $t$.
Answer:
\begin{align*}
s^2Y(s) -sy(0) - y'(0) + 6 ( sY(s) - y(0)) + Y(s) &=\dfrac{16}{s} \\
s^2Y(s) - 10 + 6 ( sY(s) ) + Y(s)... |
I am trying to solve the following ODE $$\begin{aligned} & y^{\prime \prime}+2 y^{\prime}+y=0 \\ & y(0)=2 \quad y^{\prime}(0)=-1 \\ & 0 \leqslant x \leqslant 1 \end{aligned}$$
Using central FD, I can write $$\frac{y_{i+1}-2 y_i+y_{i-1}}{h^2}+2 \frac{y_{i+1}-y_{i-1}}{2 h}+y_i=0$$ which I have simplied to $$(1+h) y_{i+1... |
Find $F'(x)$ where $F(x) = \int_1^{3x} y^{-1}e^{xy} dy$.
I just took the derivative with respect to x as:
$F'(x) = \int_1^{3x}e^{xy} \,dy$ which should be equal to $e^{3x^{2}} -e^x\over x$.
However my textbook made the calculation as:
$F'(x) = 3x^{-1} e^{3x^{2}} 3+\int_1^{3x}e^{xy} \,dy$ which equals to $2e... |
The Supplement for《Measure, Integration & Real Analysis》by Sheldon Axler is in the following page:
https://measure.axler.net/SupplementMIRA.pdf
Exercise 34 on p43 of the Supplement is the following:
Prove that if $F_1$ is a closed subset of $R^n$ and $F_2$ is a closed bounded subset of
$R^n$, then $F_1 + F_2$... |
Question about the sum of a closed and a closed bounded subsets of $R^n$? |
Why $\min (s,t) =\int_0^1(t-u)_{+}^0(s-u)_{+}^0 du$?
Background of the question is:
An elementary illustration of the integral representation theorem arises from the kernel
$$
\begin{aligned}
K(s, t) & =\min (s, t) \\
& =\int_0^1(t-u)_{+}^0(s-u)_{+}^0 d u
\end{aligned}
$$
defined on $[0,1] \times[0,1]... |
Why $\min (s, t) =\int_0^1(t-u)_{+}^0(s-u)_{+}^0 d u$? |
Below i am trying to make the asked points clear, computations do not stay in focus. First of all a picture:
[![mse 43881764][1]][1]
The problem starts somehow with the middle, making the least number of words in a random language count for exposing the problem. This is not the way we think, and not the way we st... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.