0
votes
0answers
2 views

model definitions for tautology, contradiction, and connectives quantify too much, no?

Occasionally I come across a definition based on what will happen in all models, for example, that a contradiction is a statement that is false in all models, that a tautology is a statement that is ...
0
votes
0answers
2 views

Polar coordinates for vector field to find sticking flow

I am currently working on an impacting system which is basically just a spring damper and a circular enclosure. Because of the rotational symmetry of the problem I need the vector field in polar ...
0
votes
0answers
5 views

Number of equivalence classes based on a relation regarding a non-principal ultrafilter

We have an equivalence relation on $\mathbb{N}^\mathbb{N}$ given by $$f\equiv g \iff \{n\in\mathbb{N}: f(n)=g(n)\}\in\mathbb{U},$$ where $\mathbb{U}$ is a non-principal ultrafilter on $\mathbb{N}$. ...
0
votes
1answer
5 views

How small need it be to approximate integral as one area of product of initial value times length.

$$\left(\int_{t}^{t+\Delta t}a(t')dt'\right), a(t) \text{ is scalar}$$ How small need $\Delta t$ be to approximate $$\left(\int_{t}^{t+\Delta t}a(t')dt'\right)$$ as $$a(t)\Delta t$$ [ Just one ...
0
votes
0answers
6 views

The set of all maximal ideals

I am trying to prove a proposition and in my proof I somehow need to find the set of all maximal ideals of a Banach Algebra. This is my working environment: Let $A(\mathbb{R}^2)$ be the (Wiener ...
0
votes
2answers
14 views

Prisoners' Dilema

I started to learn about game theory just now. I am confusing about the prisoners' dilema, when 2 prisoners given a choice whether to keep silent or rat out the other guy. From what I red, if one rat ...
0
votes
0answers
4 views

Proof regarding little o for vector field

I'm struggling with one part of a proof of a theorem. Let $\gamma(t) : A \subseteq \mathbb{R} \rightarrow \mathbb{R}^m$, with $\gamma \in \mathscr{C}^1(A)$ (hence $\gamma$ differentiable). ...
-1
votes
0answers
19 views

Prove without differentiation that $\log_{n+1}n$ is increasing

I want to show that the sequence $\{ \log_{n+1}n \}$ is increasing without differentiation. I don't have any idea. How can I prove that?
1
vote
0answers
8 views

Questions on color theory, mostly linear algebra related

I'm reading into color theory and there were a few questions which I asked myself along the way, maybe you can put me forward to some source where I can find answers or give them directly. The human ...
2
votes
1answer
9 views

Problem with proof of $H \cap K $ is of finite index if $ H,K$ are finite index subgroups

I came across a proof earlier for a solution to a Herstein Topics in Algebra question earlier that I'm not convinced with, from AOPS site. If $G$ is a group and $H,K$ are two subgroups of finite ...
1
vote
0answers
8 views

First principle of differentiation needs to approximate a sufficiently small integral as area?

$$y(t+\Delta t) = e^{- \int_{t}^{t+\Delta t}H(t')dt'}y(t)$$ is the solution to the differential equation $$\frac{dy}{dt} = -H(t)y$$, $H(t)$ and $y$ are scalar. However, in showing that $$y(t+\Delta t)...
1
vote
2answers
12 views

Can Someone help me with my triginomitry torarion, algorithim? I think it's called an algoritham…

I've been working on some code for a game to make a hit box, this question is just about the math though. Basically I'm trying to rotate an X, Y point(i guess according to the game it's Z,X Not sure ...
0
votes
1answer
25 views

Simple formula difficult solution

I've thinking a lot about it, but is there a simple way to get $\frac{A}{C}$ from $X = \frac{A + B}{C + D}$ where it does not depend on A and C anymore? This seems so easy but it's quite hard for ...
0
votes
0answers
8 views

Calculating the coefficients of a separable 2-qubit state

Given a separable 2-qubit state φ = φ0 ⊗ φ1 with φi= ai0|0> + ai1|1> φ thus can be written as φ = b00|00> + b01|01> + b10|10> + b11|11> with bij = a0ia1j. Now let some bij be given, i.e....
1
vote
2answers
15 views

Existence of such a meromorphic function?

Is there a function $f$ that is holomorphic on $\mathbb{C}-\mathbb{Z} $ and maps into or onto $\mathbb{C}-\mathbb{R}$ ? Into or onto $\mathbb{C}-\mathbb{R}^{+}\cup\{ {0} \}$? All I have been able to ...
0
votes
2answers
21 views

Showing this set $A$ is closed, bounded and not compact?

Let $$ l^1(\mathbb{N}) = \left\{ (x_n)_n \mid \sum_{n = 0}^{\infty} | x_n | \ \text{converges} \right\}, $$ the space of all sequences whose associated series converge absolutely. On this space we ...
1
vote
1answer
32 views

Calculation of integral $ \int_{0}^{\infty} \frac{x}{e^x - 1}\mathrm{d}x$

Apparently the solution is trivial, since Landau-Lifshitz does not even lose a word on it. However, I have no idea, any suggestions? $$ \int_{0}^{\infty} \frac{x}{e^x - 1} \mathrm{d}x$$
-4
votes
2answers
18 views

simplify factorials: $\frac{(k-1)!}{(k+2)!}$

Question: simplify $$\frac{(k-1)!}{(k+2)!}$$ What I did was: $$\frac{(k - 1)!k!}{(k + 2)! \times (k + 1)!}$$ This I did following the rule $n! = n \times (n - 1)!$. can this be simplified ...
-2
votes
0answers
16 views

Rational numbers problem

I have a problem with rational numbers, how do i find the numbers behind the following equation: 0,a(b) = a/b, using the following rationale: a/b*10=a,(...
0
votes
1answer
8 views

Table of e8 representations

I want to understand the representation theory for the (complex-valued) $e8$ exceptional Lie algebra. An ideal answer to this question would contain a link to a text file (or any other format) ...
0
votes
1answer
19 views

Show $ \frac{(-1)^{n}}{n-\ln(n)}=\frac{(-1)^{n}}{n}+\mathcal{O}\left(\frac{\ln(n)}{n^{2}} \right) $

I would like to show that : $$ \dfrac{(-1)^{n}}{n-\ln(n)}=\dfrac{(-1)^{n}}{n}+\mathcal{O}\left(\dfrac{\ln(n)}{n^{2}} \right) $$ by starting from the left side and get the right side My proof: ...
0
votes
3answers
49 views

A pretty hard problem about functions.

Let there be $$f:(-1,4)→ R$$ $$\text{differentiable on} (-1,4) , f(3)=5 , f'(x)≥-1$$ $$\text{which is the maximum value of}$$$$f(0)$$
0
votes
2answers
43 views

Simplifying factorials: $\frac{(n-1)!}{(n-2)!}$

Question: simplify $$\frac{(n-1)!}{(n-2)!}$$ What I did was: $$\frac{(n - 1)!}{(n - 2)! \times (n - 3)!}$$ This I did following the rule $n! = n \times (n - 1)!$. But my answer just doesn't look ...
2
votes
0answers
21 views

Quotient ring of Gaussian integers $\mathbb{Z}[i]/(a+bi)$ when $a$ and $b$ are NOT coprime

The isomorphism $\mathbb{Z}[i]/(a+bi) \cong \Bbb Z/(a^2+b^2)\Bbb Z$ is well-known, when the integers $a$ and $b$ are coprime. But what happens when they are not coprime, say $(a,b)=d>1$? — For ...
0
votes
0answers
10 views

Can you explain Tate's algorithm by the example?

Consider the elliptic surface $\mathcal{E}$ over $\mathbb{P}_k^1$ with homogenous coordinates $t$, $s$ and a field $k$ of even characteristic: $$ \mathcal{E}\!: s^{10}(y^2z + yz^2) = t^4s^6x^3 + (t^...
1
vote
0answers
19 views

$A,B$ be countable dense subsets of $\mathbb R$ , let $A,B$ be given usual subspace topologies , then there exists a homeomorphism $f:A \to B$?

Let $A,B$ be countable dense subsets of $\mathbb R$ (with usual euclidean topology ) let $A,B$ be given usual subspace topologies , then is it true that there exists a homeomorphism $f:A \to B$ ?
2
votes
0answers
12 views

conditional probability on zero probability events and conditional Radon-Nikodym derivatives

Consider a stochastic process $\{x_t\}_{t\in T}$ adapted to some filtered probability space $(\Omega,\mathcal{F},\{\mathcal{F}\}_{t\in T},\mathbb{P})$ taking values in the state space $(\mathbb{R},\...
0
votes
0answers
13 views

Topological Join of Unit Balls

I have seen that apparently one has for spheres that $S^n*S^m=S^{n+m+1}$. Is there a similar result for unit balls? Thank you.
0
votes
2answers
23 views

Boundedness and convergence of a sequence $(x_n)$

Suppose that $x_0 = \alpha \in \mathbb{R}$ and $x_{n+1} = x_n ^2-x_n +1$. I am asked to study the boundedness of $(x_n)$ and then asked if $(x_n)$ converges. How can I show that $(x_n)$ is bounded? ...
2
votes
2answers
62 views

Is it possible / allowed to use L'Hôpitals rule for products?

In our readings, we had L'Hôpitals rule and defined it like that: $\lim_{x\rightarrow x_{0}}\frac{f'(x)}{g'(x)}$ Because we had it in our readings, we are allowed to use this to find limit of ...
1
vote
0answers
14 views

Gauss-Bonnet formula for flat surfaces with border

Let $S$ be a compact surface without border of genus $g$ with a riemannian metric $g$ which is flat except for a finite number of points where it has cone singularities (this is what is called "flat ...
0
votes
0answers
7 views

Difference between discrete maximum principle and $l^{ \infty} $ stability for parabolic PDEs

In our Finite Difference Approximations class we briefly when over the concepts, but it is not that clear to me. Can someone please explain in a simple intuitive way what is the difference between the ...
2
votes
0answers
47 views

What is the proof that $\int e^{-x^2} \cdot dx$ is not elementry.

Is there a proof that gives the evidence there is no closed form for $\int e^{-x^2} \cdot dx$? or just because they were not able to find that elementry form for a long time of trying without any ...
0
votes
0answers
11 views

Eigenvalue perturbation of singular matrix

Consider a Hermitian matrix $\mathbf{A_0} \in \mathbb{C}^{N \times N}$ with one singularity, i.e. its eigenvalues in increasing order are: \begin{equation} 0 < \lambda_2 \leq \lambda_3 \leq \ldots \...
0
votes
0answers
8 views

Canonical bundle of the Lagrangian Grassmannian

I'd like to compute the canonical bundle of the Lagrangian Grassmannian $\mathbb{LG}_n$, the set of Lagrangian subspaces of dimension $n$ of a vector space together with fixed symplectic bilinear ...
0
votes
1answer
27 views

Proof that $n = 3k + 5l$ for $n > 7$

Show that for every n greater than $7$, there are non-negative integers $k$ and $l$ such that $$n = 3k+ 5l.$$ So induction seems like a possibility. $n = 3k + 5l$ and so $n + 1 = 3k + 5l + 1$. ...
0
votes
0answers
8 views

Generalizing integral identity from characteristic function to $f\in L^1$

I read the proof, on F.J. Jones, Lebesgue integration on Euclidean space, of the fact that, if $u:[a,b]\to\mathbb{R}$ is an absolutely continuous non-decreasing function (therefore differentiable ...
1
vote
1answer
14 views

what does “a set of sets that are not members of themselves” of Russell’s Paradox mean

Russell’s Paradox begins with a statement of "Let $R$ be the set of sets that are not members of themselves", i.e. $R=\{S\mid S\notin S\}$. I'm a little bit confused with the statement, for example, ...
0
votes
0answers
9 views

expectation and variance of an implicit estimator

Suppose the following equation holds \begin{align*} p_2=\int\limits_{-\infty}^{\Phi^{-1}(p)}\int\limits_{-\infty}^{\Phi^{-1}(p)} \frac{1}{2\pi\sqrt{1-\rho^2}}\exp\bigg({-\frac{1}{2}\frac{x^2-\rho xy+...
0
votes
0answers
13 views

Subset of Coins with maximal value

Let $ n \in \mathbb{N} $ with $ n\ge 3 $ be given. Assume that you have $ k-1 $ coins of value $ 1/k $ for all $ k \in \lbrace 2,\ldots,n \rbrace $. Now you have to choose a subset of these given ...
-6
votes
1answer
39 views

Why Do We Need To Make Math Tough Unnecessarily?

If you are downvoting please atleast tell me why are you doing so. I really need to find the answer of the question i have put forward. Basically I'm asking is why should math should be connected to ...
0
votes
0answers
16 views

An Extreme Point of a closed ball of $\ell^\infty$

I am trying to prove that all "closed unit ball" of $$ c_0 = \{ \{x_n\}_{n=1}^\infty \in \ell^\infty : \lim_{n\to\infty} x_n = 0\} $$ do not have any extreme point. (Extreme Point) Let $X$ be a ...
0
votes
0answers
15 views

Find the maximum of the $k$ such $0\le x^2(3-2x)(2x^k+(3-2x)^k)\le 3$

Find $k_{\max}$,such $$0\le x^2(3-2x)(2x^k+(3-2x)^k)\le 3,0\le x\le 1$$ since $$x^2(3-2x)>0\Longrightarrow 2x^k+(3-2x)^k\ge 0$$ it is clear for $k\in R$ and other case it's not easy to solve
1
vote
1answer
11 views

Show $(-1)^{n}\ln\left[ \frac{n(n+2)}{n^2-n+1} \right]=3\frac{(-1)^{n}}{n}+\mathcal{O}\left( \frac{1}{n^2}\right) $

I would like to show that : $$(-1)^{n}\ln\left[ \dfrac{n(n+2)}{n^2-n+1} \right]=3\dfrac{(-1)^{n}}{n}+\mathcal{O}\left( \dfrac{1}{n^2}\right) $$ by starting from the left side and get the right ...
-1
votes
0answers
15 views

Calculate integral $x_1^{k-1}x_2^{l-k-1}(1-x_1-x_2)^{n-l}$ [on hold]

Calculate $\int_0^{1-x2}\ x_1^{k-1}x_2^{l-k-1}(1-x_1-x_2)^{n-l}dx_1$ If $n,l,k$ is fixed -it's easy - just expand polynomial and cause similar terms. But what I can do in this common case?
2
votes
0answers
11 views

Integration over the Haar measure of a compact Lie group preserves smoothness?

Let $G$ be a compact Lie group. Then there is a unique Haar (probability) measure on $G$. If $f \colon G \to \mathbb{R}$ is a smooth function, is the function $$ G \to \mathbb{R}, \qquad x \mapsto \...
1
vote
2answers
26 views

Does line-onto-line imply affine?

Let $n>1$ be an integer. Is every map $A : \mathbb{R}^n \to \mathbb{R}^n$ that maps lines onto lines (the image of a line is a line) affine?
0
votes
1answer
12 views

How do I interpret following equations on fibonacii numbers?

I went through an online tutorial (http://codeforces.com/blog/entry/14385) on finding n-th fibonacci number which explains a method as, You are standing at position n in Ox axis. In a step, ...
1
vote
4answers
18 views

What is the number of subspaces of a particular dimension?

If we have vector space $V$ with dimension $n$ then how many subspaces of $V$ with dimension $m<n$ are there? In my opinion the answer should be the number of ways to choose $m$ linearly ...
0
votes
2answers
18 views

$A$ be closed in $X$ , $U \subseteq A$ is open in $A$ , $V$ be open in $X$ s.t. $U \subseteq V$ , then is $U \cup (V\setminus A)$ open in $X$?

Let $X$ be a topological space , $A$ be closed in $X$ , $U \subseteq A$ is open in $A$ , $V$ be an open subset of $X$ such that $U \subseteq V$ , then is it true that $U \cup (V\setminus A)$ is open ...

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