0
votes
1answer
9 views

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 ...
2
votes
1answer
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 ...
0
votes
1answer
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 $a, b$ ...
0
votes
0answers
8 views

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 ...
0
votes
1answer
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!
1
vote
2answers
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 ...
1
vote
0answers
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 ...
-1
votes
0answers
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 ...
1
vote
2answers
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 ...
0
votes
0answers
18 views

Show that $f_n\to 0$ in the distributional sense.

Let $f_n(x)=\sin{(nx)}$. Show that $f_n\to 0$ in the distributional sense. I know that this is true only if $\langle f_n,\phi\rangle=\int_{\mathbb{R}^n} f_n\phi\to \int_{\mathbb{R}^n} f\phi=\langle ...
0
votes
1answer
16 views

Find an orthonormal basis for $\mathbb{R^4}$

I have been trying to solve this problem (Fraleigh - Linear Algebra (3rd Ed.)): 6.2.17. Find an orthonormal basis for $\mathbb{R^4}$ that contains an orthonormal basis for the subspace $sp([1, 0, 1, ...
0
votes
0answers
7 views

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 ...
0
votes
1answer
13 views

Variance of turning point - Time Series

I'm trying figure out why the variance for the turning point test is $V[T] = \frac{16n-29}{90}$. Given a sequence of points $\{y_i\}_{i=1}^n$, we define a turning point at time $i$ where $1 < i ...
0
votes
0answers
16 views

How else can I tell I can do this with $5$ but not $2$ or $3$ in $\textbf{Z}[\sqrt{30}]$?

In $\textbf{Z}[\sqrt{30}]$, the number $5$ splits, since, for example, $N(5 + \sqrt{30}) = -5$. But the ideal $\langle 5 \rangle$ is a ramifying ideal, since it is equal to $\langle 5, \sqrt{30} ...
1
vote
2answers
42 views

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 ...
1
vote
0answers
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 ...
3
votes
0answers
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. ...
0
votes
0answers
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 ...
0
votes
0answers
10 views

Isometries of p-adic vector spaces?

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 ...
1
vote
0answers
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
0
votes
0answers
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 ...
4
votes
4answers
75 views

What is the most useful/intuitive way to define $\Bbb{C}$, the complex set?

I was wondering whether there is a more common or 'best' way to define/think about the set of all complex number in complex analysis? The first way I can think of is: $\Bbb{C} \triangleq \Bbb{R} ...
4
votes
4answers
168 views

Provide examples or explain why it is impossible

a) A continuous function defined on an open interval with range equal to a closed interval. My example: $f(x)=\frac{1}{2}\sin(4\pi x)+\frac{1}{2}$ on $(0,1)$ to $[0,1]$. Note: I am not considering ...
1
vote
0answers
9 views

Ellipsoid Axis at density contour, why choose biggest eigen value for axis?

I've been trying to figure out how to find the density contour for a multivariate normal density function with an arbitrary number of dimensions. I've found a lot of examples for 3Dimensions and for ...
6
votes
0answers
46 views

Prove that $a^{ab}+b^{bc}+c^{cd}+d^{da} \geq \pi$

If $a,b,c,d >0$, and $a+b+c+d=4$, prove that $$a^{ab}+b^{bc}+c^{cd}+d^{da} \geq \pi.$$ I don't think Jensen's inequality will help here, but I think first determining where equality holds ...
0
votes
0answers
13 views

Looking for examples of Discrete / Continuous complementary approaches

Among many fascinating sides of mathematics, there is one that I find very rewarding, especially in teaching at undergraduate level: the parallels that can be drawn between a "Continuous world" and a ...
2
votes
1answer
34 views

Frechet derivative in a Hilbert space

Let $\mathcal{H}$ be a Hilbert space and $A$ a self-adjoint operator. With $(\, ,\, )$ denoting the inner product and $\psi\in \mathcal{H}$, I want to formally show that the Frechet derivative of the ...
0
votes
2answers
50 views

Is my hypothesis correct?

$$\left| \left|(a^2) - 25\right|-b\right| + b = 0$$ You have to prove that $b<0$ and $b=0$ at the same time I have no problem to prove that $b$ can be $0$ the thing that I need help with is ...
0
votes
0answers
11 views

First Order Logic Tableau Multiple Universal Identifiers

I've been looking into tableau lately and I have come across multiple Universal Identifiers which I am not used to. How do I approach these to validate/invalidate with these identifiers and provide an ...
2
votes
2answers
27 views

Finding the limit of the ratio of a sequence.

I don't have any experience with formal mathematics (I took calculus in college, but that's about it), and I've come across a problem in my job that I don't quite know how to tackle. I have a ...
1
vote
1answer
36 views

Help justifying that $\mathbb Q(\sqrt[3]{2})$ is not a splitting field over $\mathbb Q$.

In a question related to introductory Galois theory, I was asked to given an example of a tower of fields $F \subset K \subset E$ such that $E$ is a splitting field for some polynomial $f(x) \in ...
1
vote
2answers
67 views

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 ...
0
votes
0answers
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 ...
-2
votes
1answer
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 ...
2
votes
1answer
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 ...
1
vote
2answers
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 ...
0
votes
2answers
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 ...
1
vote
0answers
28 views

Easy geometric sum with binomial coefficient

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} = ...
1
vote
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.) ...
1
vote
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). ...
3
votes
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 ...
1
vote
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 ...
1
vote
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 ...
2
votes
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 $f:\mathbb{R}\to[0,\infty)$ ...
-4
votes
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).
0
votes
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 ...
2
votes
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 ...
-2
votes
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 ...
0
votes
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) ...
0
votes
0answers
13 views

Modular Representations of unitriangular matrices

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, ...

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