All Questions

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For a positive integer $k$, let $B_{k}=\{\,x \in \Bbb Z \mid x \leq 2k\,\}$

I need a simple explanation on what the answer is to $B_{k}=\{\,x \in \Bbb Z \mid x \leq 2k\,\}$. Question asks: Determine $\bigcup_{k=1}^{2016} B_{k} =?$ I understand the it will go on like ...
22 views

what is the terminology of this form of equation $x^2 +1/x^2 + \sqrt{x}$

what is the terminology of this form of equation. It has only one variable, but with rational exponents, it can be positive, negative or fraction such as below: $ax^2 +b/x^2 + c\sqrt{x} =0$ I ...
32 views

Iterative method for finding real solutions to $a+b+c+d = abcd = 7.11$

I have "come up with" a method for finding $a,b,c,d \in \Bbb{R}$ such that their sum and product is equal and wanted to ask if the method is sound. First, rearrange both equations so that only $a, b$ ...
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Computing the group of deck transformations w.r.t. a polynomial

Let $p: \mathbb{C}\backslash Y' \to \mathbb{C}\backslash X'$ be a polynomial where $Y'$ is the set of branch points and $X'$ is the image of $Y'$ under $p$. If $\deg p = n$ then $p$ is an unbranched ...
23 views

Can integrals be solved using MATLAB's ode solver? [on hold]

I was wondering how I could solve a definite integral using a MATLAB ode solver? Thanks!
19 views

Derivation of Spherical Law of Cosines

I am trying to get a derivation of the spherical law of cosines. The Wikipedia page [https://en.wikipedia.org/wiki/Spherical_law_of_cosines] contains a proof that I don't understand because there are ...
19 views

Does $\phi: A \otimes \mathbb{Q} \to B \otimes \mathbb{Q}$ surj. imply that for $b \in B$, $b = n \phi(a)$ for some $n \in \mathbb{Z}$, $a \in A$?

Let $A$ and $B$ be abelian groups. Suppose that we have a morphism $\phi: A \to B$ and that $\phi \otimes_\mathbb{Z} \mathbb{Q}: A \otimes_\mathbb{Z} \mathbb{Q} \to B \otimes_\mathbb{Z} \mathbb{Q}$ is ...
22 views

True or false: Every transformation T: Cn --> Cn (n ≥ 2) has n distinct eigenvectors

True or false: 1) Every transformation T: Cn --> Cn (n ≥ 2) has n distinct eigenvectors 2) Every transformation T: Cn --> Cn (n ≥ 2) has at least 1 eigenvector 3) Every transformation T: Rn --> Rn ...
20 views

Help showing inductively defined sequence is monotone for $n \geq 2$

I'm having trouble proving the following sequence is monotone: Let $a > 0$, $(s_n)$ be a sequence defined by: $$s_{n+1} = \frac{1}{2}\left(s_n + \frac{a}{s_n}\right), \quad s_1 = a_0>0$$ I've ...
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Continuity of the Box-Cox transform at λ = 0: Why is it the log?

The Box-Cox power transform frequently used in statistical analysis takes the value (x^λ -1) /λ for λ not equal to zero, and ln(x) for λ=0. I would like to see a demonstration, that need not be a ...
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In which order should i learn the foundations of mathematics

I know from Wikipedia that those are the four pillars of the foundations of mathematics: Proof theory Aximatic Set theory Model Theory Recursion Theory and i want to learn all of them, the problem ...
10 views

Show that $\lim_{n\to \infty}\prod_{i=n}^{Bn}\frac{\arctan(i\phi)}{\arccos\left(\frac{\phi}{i}\right)}=B^{\frac{2}{\pi}}$

Inspired from Gosper's formula $$\lim_{n\to \infty}\prod_{i=n}^{2n}\frac{\pi}{2\arctan(i)}=4^{\frac{1}{\pi}}$$ (See pi formulas; maths world) Through mathematical experimental we found another ...
26 views

An equality in the proof of Proposition 3 of Section 2.7 of Pierre Samuel's Algebraic Theory of Numbers

I am reading Pierre Samuel's Algebraic Theory of Numbers. I get stuck at an equality within the proof of Proposition 3 of Section 2.7. The statement of the proposition is as follows: Proposition 3. ...
14 views

Ito formula proof

Is there a simple way to prove $$x=f(t,x_t)\\dx_{t}=\frac{\partial f}{\partial t}dt+\frac{\partial f}{\partial x}dB_t+\frac{1}{2}\frac{\partial^2 f}{\partial x^2}(dB_t)^2$$? can we prove it by ...
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In Euclidean space $\mathbb{R}^d$ there are ${d+1 \choose 2}$ isometries (translations and rotations). In the $p$-adic space $\mathbb{Q}_p^d$ what do isometries look like? How many are there? Does ...
13 views

Diophantine Equation & Probability [on hold]

I'm asking please if there is any link betwen Probability and Diophantine equation ? For exemple can we estimate the numbers of solutions of the Diophantine equation: $F(x)=G(y)$ Thinks
19 views

Simplifying trick for extremizing functionals

If I have a functional $I[y] = \int{({y^\prime}^2- 1)^2}dx$ , since $f(x) = x^2$ is an increasing function for $x > 0$, can I make the conclusion that $I[y]$ is extremized for at the same ...
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Why do authors make a point of $C^1$ functions being continuous?

I've just got a little question on why authors specify things the way they do. Is their some subtlety I'm missing or are they just being pedantic? I've encountered the function spaces $C^k[a,b]$ a ...
7 views

Proof of $\delta$-Hyperbolicity of $\mathbb H^n$ just with the hyperboloid model?

Do you know any proof of the fact that $\mathbb H^n$ is Rips-hyperbolic (i.e., geodesic triangles are $\delta$-slim for some $\delta$, also called "Gromov-hyperbolic" in some contexts), which makes no ...
26 views

Interesting relationship between cardinality and Lebesgue outer measure

If two sets $A$ and $B$ defined on bounded intervals have the same cardinality and $A \bigcap B$ is non empty and the Lebesgue outer measure of A is greater than zero. Is it then true that the ...
18 views

Trigonometry, obtuse angles and a negative length?

I've been looking for an answer as to why when $\cos x<0$ and $\tan x<0$ the angle is obtuse. I found a few identical explanations online where, a right-angle triangle is formed in the second ...
19 views

Is it true that finite intersection distributes over arbitrary unions?

I have come across the problem of showing that $$\bigcap_{i=1}^n \Big ( \bigcup_{\alpha\in A} X_\alpha^{(i)}\Big) = \bigcup_{\alpha\in A} \Big ( \bigcap_{i=1}^n X_\alpha^{(i)} \Big)$$ for some family ...
21 views

Diff. Eq. solution by inspection

Ran into an interesting problem that is bugging me! Determine by inspection a solution to this differential equation: 4y'' = y What this says to me is that we must find a function that if we ...
In the context of stochastic processes I came across the following equality, where $|s| < 1, p \in [0,1]$: $$\sum^\infty_{k=0}(s^2p(1-p))^k\begin{pmatrix} 2k \\ k \end{pmatrix} = ... 0answers 9 views Is there an equation/algorithm to find the axis of a curved helix with varying pitch and radius? I'm trying to find a way to numerically extract or estimate the axis of rotation for an arbitrarily shaped helix, which may be curved and have varying pitch and radius. (See here for an example.) ... 2answers 12 views Examples of cubic graphs in which every cycle of length divisible by 3 has a chord Cubic graphs (graphs in which every vertex has valency 3) cannot be trees, so they contain a lot of cycles. Some of these cycles have length divisible by 3 (e.g. triangles, hexagons, nonagons etc). ... 4answers 31 views Percentages, discrete or continuous? I have this question I can't figure out. A basketball player starts a game. During the first period of the game, the success rate of his shots is less than 80%. At the end of the game, his success ... 0answers 14 views Reduction of functions with Lie group symmetries If I have a function f:\mathbb{R}^n\rightarrow \mathbb{R} with a Lie group G as a symmetry, f(Ax)=f(x),\quad A\in G how might I go about obtaining a reduced function \tilde{f} on ... 0answers 19 views Options on Futures Black-Sholes I am taking the Financial Risk Management course, and the topic now is "Variations on the Black-Scholes Model". I am following Paul Wilmott's "The Mathematics of Financial Derivatives: A Student ... 2answers 29 views Let f:\mathbb{R}\to[0,\infty) measurable and f\in L^1. Show that \mu(E)<\delta \implies \int_E f < \varepsilon. I'm learning about measure theory, specifically the Lebesgue integral of nonnegative functions, and need help to understand the solution to the following problem: Let f:\mathbb{R}\to[0,\infty) ... 1answer 39 views How do I calculate the probability that Joe scores at least 3 points higher than say Bob? [on hold] I have two normal distributions of two people's test scores (mean and stand. dev). 0answers 26 views Knapsack or bin packing problem? I have i items and I should pre-packed m knapsacks with identical items where only K<n items can be packed. Also, we should have only one of each item in each sack. The time capacity for ... 1answer 43 views prove or disprove that [∫f(x)g(x)\,dx]^2 ≤ ∫f(x)^2dx ∫g(x)^2dx for all f and g over any interval I have managed to prove that if f and g have this property that f and f+g will also have the property but I have failed to prove that it is true in general. Ideally, the proof would use only ... 0answers 18 views How to show convegence of function series in Norms? I've been given different norms and function series and I shall proof the convergence of the series in the norm. But I don't know what to do - can anyone explain me which steps I have to do ? Thanks ... 0answers 12 views An inequality combining Hölder and Euler's aproach in his proof for infinitude primes By Hölder inequality$$ \left( \sum_{n=1}^N\mu(k)\log k \right)^{q_n}\leq \left(\sum_{k=1 }^N \left| \mu(k) \right|^{p_n} \right)^{\frac{q_n}{p_n}}\cdot \left(\sum_{k=1} ^N (\log k)^{q_n} \right) ...
Assume that $U_3(\mathbb{F}_p)$ is the group of unitriangular matrices with entries from the field $\mathbb{F}_p$ of $p$ elements. Do you have any idea how can we compute the indecomposable (modular, ...