0
votes
1answer
262 views

Prove a space has a countable dense subset

I'm doing this exercise in Munkres book and got no clue about the solution. Hope some one can help me solve this. Show that the product space $R^{I}$, where $I = [0,1]$, has a countable dense ...
4
votes
2answers
280 views

Polynomials $P(x)$ satisfying $P(2x-x^2) = (P(x))^2$

I am looking (to answer a question here) for all polynomials $P(x)$ satisfying the functional equation given in the title. It is not hard to notice (given that one instinctively wants to complete the ...
1
vote
1answer
75 views

number of generators of MASA

Let $\mathcal{H}$ be an infinite-dimensional Hilbert Space. Do the maximal abelian self-adjoint subalgebras of $\mathcal{B}(\mathcal{H})$ always have infinitely many generators as an algebra ? (The ...
1
vote
0answers
49 views

Are all the square roots of non-square numbers surds? [duplicate]

Quite self explanatory really, basically, are $\sqrt5 ,\sqrt3$ and $\sqrt7 $ and surds? (So basically, every square root of any non-square number)
0
votes
1answer
28 views

Order embedding and graph embedding of Hasse digraphs

If there is an order embedding from order A to order B, is there a graph embedding between their Hasse digraphs? What if we replace 'embedding' by 'order preserving' and 'homomorphism' etc?
2
votes
0answers
67 views

An inner product on a space of linear maps

Let $V$ and $H$ be two complex Hilbert spaces. We suppose $V$ to be finite-dimensional. I'd like to understand the structure of Hilbert space on the space of linear mappings $\mathrm{Hom}(V,H)$. ...
1
vote
1answer
222 views

Principle of recursion for inductively defined relations

If we consider a relation $R$ and then its symmetric-reflexive-transitive closure --say $R^*$--, is there a recursion principle associated with $R^*$? It seems to me that such unique function is not ...
2
votes
3answers
168 views

How fast is the water draining out after 5 min?

The volume $V$, in liters, of water in a water tank after $t$ min it starts draining, is given by $$V(t)=260(60−t)^2$$ How fast is the water draining out after 5 min? Do I calculate the ...
0
votes
1answer
73 views

conjugate prime ideals of integral extensions and relevance of the characteristic of the ground field

This question refers to the proof of theorem 9.3, p. 66 in Matsumura's Commutative Ring Theory: "if $A$ is an integrally closed domain, $K$ its field of fractions and $L/K$ a normal field extension, ...
2
votes
0answers
63 views

What are co-products for directed graphs?

I define a digraph as a set $V$ (vertexes) and a relation $E$ (edges) on $V$. Morphisms of digraph are functions which preserve $E$. So we have a category. What are co-products in this category? (I ...
2
votes
0answers
315 views

Is there a particular meaning to the sum of Fourier coefficients $a_{n^2}$?

The formula $$\sum_{n=-\infty}^{+\infty} e^{-in^2x}$$ does not converge in any function space but it is perfectly valid in $\mathcal{D}'(\mathbb{R})$. When applied on a test function $\psi(x) = ...
1
vote
1answer
105 views

Learning as a combinatorialist

Sorry if this isn't appropriate for stackexchange, but I just have been curious. For a combinatorialist, is more effort spent learning specific tools in combinatorics or learning other areas of math ...
2
votes
0answers
74 views

Smoothness does not depend on the choice of atlases

Here is a part of a lecture note: I need some help to solve the exercise. I want to show that if $\psi\circ f\circ\phi^{-1}$ is differentiable and $\alpha, \psi$ and $\beta,\phi$ are in the same ...
2
votes
3answers
434 views

Given $\sin z=5$. Find $e^{iz}$ (Complex Trigonometric Function)

Given $sin$ $z=5$. Find $e^{iz}$. Here is what I have done: \begin{align} \sin z &= \frac{e^{iz}-e^{-iz}}{2i}=\frac{e^{i(x+iy)}-e^{-i(x+iy)}}{2i}\\ &=\frac{e^{-y}(\cos x+i\sin x)-e^{y}(\cos ...
4
votes
1answer
221 views

What is Galois theory for schemes?

I have heard about "Galois theory for schemes" in this note. I haven't read it yet and I know Galois theory and a litle bit about schemes (as what it is, some properties like seperated, irreducible, ...
0
votes
4answers
279 views

If $A\subseteq B\subseteq C$ and that A is dense in B s odo B is dense in C how do I prove that A is dense in C?

If $A\subseteq B\subseteq C$ and that $A$ is dense in $B$ s odo $B$ is dense in $C$ how do I prove that $A$ is dense in $C$? Please provide me with a headstart, greatly appreciated!
1
vote
1answer
84 views

an issue with expectation

in book's Bernt.Øks SDE i read that book and i have some serious issues :( page 21 Example 7.4.2 ) Consider n-dimensional Brownian motion $W=(W_1, \ldots ,W_n)$ starting at $a=(a_1,\ldots,a_n) \in ...
0
votes
1answer
46 views

A more rigorous proof that if $\forall n, \int_{[0,1]} f^n = C$, $f(x) = \chi_{\{f = 1\}}(x)$

Consider the following problem: Let $f \geq 0$ be measurable on $[0,1]$. If there is a constant $C$ such that for all $n \in \mathbb{N}$, $\int_{[0,1]} f^n = C$, then show $f(x) = \chi_{\{ f = 1 ...
0
votes
1answer
40 views

Show that every sequence in $l_2 (ℝ)$ is the limit of a sequence in $l_0(ℝ)$.

Consider the real linear space of sequences of real numbers $$l_0(ℝ) := \{(a_n)_{n∈ℕ} : a_n ∈ℝ ,a_n≠0 \text{ for finite many $n$} \} $$ and $$l_2(ℝ) := \{(a_n)_{n∈ℕ} : a_n ∈ ℝ, \sum_{n=1}^{∞} {a_n}^2 ...
0
votes
0answers
48 views

Prove that $S$ clusters at $p$ iff for each $r>0$ there is a point $q\in M_r(p)\cap S$ such that $q\ne p$

Prove that $S$ clusters at $p$ iff for each $r>0$ there is a point $q\in M_r(p)\cap S$ such that $q\ne p$ Can someone please give me a hint to start?
1
vote
0answers
61 views

Does existence of weak spatial derivative imply existence of classical time derivative in this situation?

Let $f(x_1,...,x_n,t)$ be a function, where $(x_1,...,x_n) \in \mathbb{R}^n$ and $t \in [0,T].$ Denote by $f_{x_i}$ the weak (partial) derivative of $f$ wrt. $x_i.$ Is it possible for ...
0
votes
1answer
138 views

For which intervals $[a,b]$ in $\mathbb{R}$ is the intersection $[a,b]\cap \mathbb{Q}$ a clopen subset of the metric space $\mathbb{Q}$?

For which intervals $[a,b]$ in $\mathbb{R}$ is the intersection $[a,b]\cap \mathbb{Q}$ a clopen subset of the metric space $\mathbb{Q}$? Can someone give me a hint on how to start?
2
votes
1answer
147 views

Derive power series for $\frac{1}{1-x^2}$ from $\frac{1}{1-x}$?

The series $\sum_{k=0}^{\infty} x^k$ is known and equals $\frac{1}{1-x}$ for $|x|<1$. Can I use this fact and derive the power series for $\frac{1}{1-x^2}$ from it, eg using ...
1
vote
2answers
49 views

characterizing open subgroups of profinite groups

I am studying Brian Osserman's notes on infinite Galois theory and i am a little bit confused in his proof of Lemma 2.2. In particular, we have a profinite group $G$ being the inverse limit of ...
15
votes
3answers
455 views

Proving that $\int_a^b\frac{dx}{\sqrt{(x-a)(b-x)}}=\pi$

How do I prove that $$\int_a^b\frac{dx}{\sqrt{(x-a)(b-x)}}=\pi?$$ I'm just wondering if LHS even equal to the RHS in the first place? Thanks for the help!
2
votes
1answer
43 views

Does $F$ have a local minimum at $0$?

I have tried to do this exercise. I'm always dubious about the use of the chain rule, is my solution correct? Let $F \colon \mathbb{R}\mapsto\mathbb{R}$ with $F(x,y):=xye^x+ye^y-e^x+1$ and let $C$ be ...
2
votes
1answer
200 views

The derivative of the norm, radial harmonic functions

Let $f=f_{n,k}$ be a $C^{2}(\mathbb{R}^{n})$ function with compact support such that if $R\leq |x|\leq 2^{k}R$ $$f(x):=\begin{cases} -\log|x| & \text{when } n=2 , \\ |x|^{2-n} & \text{when } ...
0
votes
2answers
38 views

Determining if two things are equal when they contain a floor function

I have two equations: $ 50.8 \lfloor \frac{w - 1}{50.8} \rfloor $ and $ 25.4 \lfloor \frac{w - 1}{25.4} \rfloor $ I need to determine if they are equal, how can I do this? What I think the the ...
1
vote
2answers
525 views

Decomposition of a group into disjoint double cosets

This is taken from Lang's Algebra, exercises on groups. Let $G$ be a group and $H,H'$ be subgroups of $G$. A double coset of $H,H'$ is a subset of the form $HxH'$, $x\in G$. The first question asks ...
1
vote
3answers
282 views

A sequence $(a_n)$ converges to $L$, $|L| < 1$. Prove that $(a_n)^n$ converges to $0$

So $a_n$ converges to $L$, and the absolute value of $L$ is less than $1$. How do we go about proving that the sequence $(a_n)^n$ converges to 0? I tried a couple of basic methods but I don't seem to ...
3
votes
1answer
185 views

A Functional Differential Equation

I was having a play with some trig. identities and noticed the following: $\cos{x}=\frac{\sin{2x}}{2\sin{x}}$ Now, $\cos{x} = \frac{d}{dx}\sin{x}$ so I made the following analogous differential ...
2
votes
0answers
75 views

differentiable function in R but non continuous derivative for any point in R

Need your help with the next Question (I thought about her and could not find an answer) Is there differentiable function for all R , so that the derivative is non continuous for any point in R ? ...
0
votes
1answer
57 views

Integration by parts: Is this correct?

It's been a while since I have done integration by parts and I just wanted to see if I was correct or not, I have $$f(x)=\frac{x}{\theta^2}\exp\left(\frac{x^2}{2\theta^2}\right)$$ where $\theta$ is a ...
0
votes
2answers
1k views

Deformation retract and homotopy equivalence

If $A\subset X$ is a deformation retract of $X$. Are $X$ and $A$ homotopy equivalent?
2
votes
1answer
100 views

To show that a linear operator is '''cyclic'.

I am stuck at this problem for quite a long time now. Problem. Let $F_p$ denote the field of $p$ elements, where $p$ is prime. Let $n$ be a positive integer. Let $V$ be the vector space $(F_p)^n$ ...
0
votes
1answer
1k views

Finding the angles for a certain value of sine, cosine or tangens

I want to solve the following task: Which angle between 0° and 360° has a cosine (or sine, or tangens) of 0.5? Same task, but for an angle between 540° and 720°? I want to solve it without ...
3
votes
3answers
2k views

Strong Induction Proof: Fibonacci number even if and only if 3 divides index

The Fibonacci sequence is defined recursively by $F_1 = 1, F_2 = 1, \; \& \; F_n = F_{n−1} + F_{n−2} \; \text{ for } n ≥ 3.$ Prove that $2 \mid F_n \iff 3 \mid n.$ Proof by Strong Induction : ...
0
votes
2answers
99 views

Discrepancies in the solution of an irrational inequality.

$\sqrt { -6x+10 } + \sqrt {-x+2} \gt \sqrt {4x+5}$ In the equality above I've tried to solve several times, several problems were discovered...solving it the regular way doesn't yield the correct ...
3
votes
1answer
370 views

Scalar potential and vector potential

Let $D$ be a simply connected open subset of $\mathbb{R}^3$. It is well-known that if ${\bf v}(x,y,z)=[P(x,y,z), Q(x,y,z), R(x,y,z)]$ is a $C^1$ function on $D$ such that $$\mbox{curl }\bf{v}=\bf{0}$$ ...
2
votes
0answers
68 views

Stable fiber products of commutative rings

Let $R_1 \to T$ and $R_2 \to T$ be homomorphisms of commutative rings. Consider the fiber product $R=R_1 \times_T R_2$. Let $R \to R'$ be a homomorphism of commutative rings, and define $R'_i$ to be ...
0
votes
2answers
1k views

Find $dy/dx$ in terms of $x$ and $y$ when $x^2y^3=7x−3y$

Find $dy/dx$ in terms of $x$ and $y$ when $$x^2y^3=7x−3y$$ Not sure how to start here, would be nice with some pointers
0
votes
1answer
198 views

Lattices in the complex plane

Consider the ring $R=\mathbb{Z}[\sqrt{-2}]$. It is a lattice in the complex plane: the set of points with integer coordinates with respect to the basis: $1,\sqrt{2}i$. Each mesh of the lattice is a ...
3
votes
5answers
2k views

The Fourier series of $\sin^3 t$ in trigonometric form

I'm trying to calculate the Fourier series of $\sin^3t$ in trigonometric form. In previous excercises I have been able to use trigonometric identities to be able to calculate the coefficents, but here ...
1
vote
1answer
244 views

Ellipse arcs. Draw a tangent line in the end point or make arc longer?

I read this article: link It describes how to draw ellipse arcs at all from svg. Each ellipse is described with the following params (and I know them): x1, y1, x2, y2 - arc from point (x1, y1) to ...
0
votes
1answer
48 views

Help with getting the right direction on a boolean algebra question

Need some help getting in the right direction for answering the following question: Prove the following property and interpret this in $\mathcal P \left ({V} \right)$: if $x+ \bar y=$ 1, then ...
1
vote
2answers
107 views

Definition Of 'Inner Derivation'

I need a useful definition of an inner derivation of modules or even better a good reference to read about.
1
vote
5answers
94 views

Need help with a limit

I'm trying to determine $$\lim_{x \to 0} {x^2 \over \cos (3x) - 1}$$ My guess is that using the fact that $\lim_{x \to 0} {\sin x \over x} = 1$ or perhaps $\lim_{x \to 0} {\tan x \over x} = 1$ could ...
1
vote
1answer
59 views

Normal numbers in base 2 are meager (of the first category).

What is a simple proof that the set of normal numbers in base 2 in $[0,1]$ is of the first category (meager)? Definition 1. A set is called meager or of the first category if it is the countable ...
2
votes
1answer
419 views

Show that the intersection of any two distinct Sylow $2$-subgroups of $G$ has order $8$

Suppose that $G$ is a group of order $48.$ Show that the intersection of any two distinct Sylow $2$-subgroups of G has order $8.$ Let $H,K$ be two distinct Sylow $2$-subgroups of $G.$ Then ...
0
votes
1answer
2k views

Flowerpot falls off a windowsill

A flowerpot falls off a windowsill and falls past the window below. You may ignore air resistance. It takes time $0.420sec(s)$ to pass this window, which is of height $1.90meters(m)$. How far is the ...

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