1
vote
1answer
62 views

Does smooth section of a quotient space $G/H$ define an immersion?

Question 1: Let $G$ be a Lie group and $H<G$ a Lie subgroup of $G$, given the projection $p:G\to G/H$ and a smooth (local) section $s:G/H\to G$ s.t. $p\circ s=id$, then does this imply that $s$ is ...
0
votes
2answers
59 views

Confused between multiple representations of Fourier Series' formula

I have never used the formula for Fourier Series with the representation that the instructor of the above video is using that involves $k$ and $\omega$. Instead, I use $n$ and $\pi$. Now, suppose ...
1
vote
1answer
69 views

Why does this interchanging of derivative and sum work?

I'm reading a stats book and, for a geometric distrubution ($E[Y]=p \sum_{y=1}^{\infty}yq^{y-1})$ it makes the claim that since $\displaystyle \frac{d}{dq}(q^y)=yq^{y-1}$ hence $\displaystyle ...
1
vote
1answer
438 views

Function not satisfying pointwise convergence and Fourier series

Can you show an example of a function that does not satisfy pointwise convergence theorem hypotheses for Fourier series but that is still expressible as Fourier series? [Added after comment] In ...
0
votes
1answer
41 views

Question about Independent events

If we know that $A,B$ are independent events, how can we proof that $\bar{A},\bar{B}$ are independent events to? We should using the definition: $\Pr(A\cap B)=\Pr(A)\cdot \Pr(B)$ Thank you!
0
votes
1answer
175 views

Probability of $3$ people in a room of $5$ having the same birthday

What is the probability that, amongst five people in a room, three have the same birthday? I was wondering about this twist on the birthday problem. I am not a major stats guy so I want your help.
3
votes
1answer
125 views

If P=NP, then NP = coNP. Why is this so?

I read that if we assume that P = NP, then NP = coNP. I am unable to understand why this is so.
1
vote
3answers
104 views

is the limit continuous or not?

is $(x^2 + y^2) / (x^2 - y^2)$ continuous or not at $(0,0)$? I think it is not continuous at $(0,0)$. To check I just plugged in the points and got $0$. Did I do that right? and also is there another ...
3
votes
0answers
79 views

Is a complex polynomial a regular covering? What is its group of deck tranformations?

We know that a complex polynomial $P$ of degree $n$ is an $n$-sheeted covering from $$\{\mathbb{C} - P^{-1}\{\text{critical values of }P\}\} \to \{\mathbb{C} - \{\text{critical values of }P\}\}. $$ ...
4
votes
2answers
187 views

What is the need for classifying numbers like integer, whole number etc?

what are the everyday life examples where we use the classification. I feel all the math behind the scenes(in computers weather etc ) is highly abstracted. I am looking for strong answers to tell the ...
0
votes
1answer
158 views

Divisibility of the difference of powers

Consider the following theorem: For any $a, b \in \mathbb{Z}^+$, there exist $m, n \in \mathbb{Z}$ such that $m > n$ and $a\ |\ b^m - b^n$. What's the best way to prove it? I have an idea ...
1
vote
1answer
340 views

Finding Limit point of given sequence

Find the limit point of the following sequence $\langle f_n\rangle$, where $$f_n=\begin{cases} \tfrac{n+1}{n}& \text{ when }n=3m,\\ \tfrac{n+2}{2n}& \text{ when }n=3m+1,\\ \tfrac{1}{n+1}& ...
0
votes
2answers
58 views

Find open sets.

Consider the set $X = \{1,2\} \times \mathbb{Z}_+$ in the dictionary order. Then this will be an ordered set with smallest element. Denote $a_n = (1,n)$ and $b_n = (2,n)$. Then elements of $X$ will ...
1
vote
1answer
111 views

Solution to linear equations as parameterized matrices.

I want to solve the following matrix equation: $$ \begin{pmatrix} a & b \\ c & d \end{pmatrix}\begin{pmatrix} 3 & 2 \\ 4 & 2 \end{pmatrix}=\begin{pmatrix} 1 & 2 \\ 3 & 4 ...
2
votes
0answers
69 views

To find the limit of three terms mean iteration

We know that the arithmetic-geometric mean $AGM(a,b)$ of $a$ and $b$ defined as $$2a_1=a+b$$ $$b^2_1=ab$$ $$2a_n=a_{n-1}+b_{n-1}$$ $$b^2_n=a_{n-1}b_{n-1}$$ $AGM(a,b)=\lim\limits_{n\to \infty} ...
4
votes
3answers
215 views

How to compute $\int_{\mathbb{C}} \frac{|dz|^2}{|z-a_1| \cdot |z-a_2| \cdot |z-a_3|}$ explicitly?

I have these integrals : $$I_1= \int_{\mathbb{C}} \frac{|dz|^2}{|z-a_1| \cdot |z-a_2| \cdot |z-a_3|},$$ and $$I_2= \int_{\mathbb{C}} \frac{|dz|^2}{|z-a_1| \cdot |z-a_2| \cdot |z-a_3| \cdot |z-a_4|}.$$ ...
2
votes
1answer
596 views

Absolute convergence of Fourier series of a Hölder continuous function

Suppose that $f$ is $2 \pi$ periodic and Hölder continuous of order $\alpha > 1/2$. Show that the Fourier series of $f$ converges absolutely. So we know that $f(x+2 \pi t) = f(x)$ for all $t \in ...
0
votes
1answer
100 views

Is there a name for these “generalised” Bernoulli processes?

What I am looking for is if there exists an official name for a "generalized" Bernoulli process, i.e. a sequence of random independent binary variables $\{X_n\}_{n\in\mathbb N}$ defined over the same ...
2
votes
2answers
224 views

Existence of a Bijective Map

I am having a little trouble with this question. Prove that there does not exist a bijective map from $\mathbb{R}^2 \to \mathbb{R}^3$ where $f$ and $f^{-1}$ are both differentiable. Thanks for any ...
0
votes
1answer
350 views

Examples of homeomorphisms between the real numbers and the positive real numbers?

I'm interested in homeomorphisms between the real numbers, $\mathbb{R}$, and the positive real numbers, $(0,\infty)$--where both spaces have the topology induced by the metric $d(x,y)=|x-y|$. Here ...
3
votes
1answer
151 views

Question on integral closure in $\mathbb{Q}[\alpha]$

Let $\alpha$ be a root of $f(x) = x^{3} -2x +6$, $ \ \mathbb{K} = \mathbb{Q}[\alpha]$. Prove that $ O _{\mathbb{K}} = \mathbb{Z[\alpha]}$. What I've done: $f$ is irreducible, so ...
2
votes
1answer
85 views

measurability w.r.t. Borel on extended real line

Following Schilling I have shown for measurable functions $$u, v \in m \mathcal{A}/ \mathcal{\hat{B}}$$ that sums, differences, products and maxima/minima are again measurable whenever they are ...
1
vote
1answer
94 views

Partition of a set and Hall's theorem

I have been wrestling with an exercise concerning latin squares from the textbook A First Course in Discrete Mathematics by Ian Anderson. The exercise is formulated thus: A set $S$ of $mn$ ...
1
vote
1answer
72 views

Dual space of $K[X]$

Let $k[X]$ be the space of polynomials over a field $k$ (regarded as a vector space over $k$). What is the dual space of this vector space? My guess is that it is somehow generated by the derivations ...
10
votes
1answer
259 views

What does the German word “Zerlegungsautomorphismus” translate to?

I would like to know if any of our German friends can translate that word for me? Zerlegung is factorisation isn't it? So what is factorisation automorphism? This is taken from Deuring's paper “Die ...
2
votes
2answers
834 views

Covering with most possible equal size subsets having pairwise singleton intersections

Assume I have a non-empty finite set $S$ with $x=|S|$. I want to divide the set $S$ into subsets $S_1, S_2, .., S_n$ (Edit: Yes, $S = \cup S_i$, and I'm embarrassed that I forgot to include that) such ...
1
vote
0answers
74 views

Maximum of the expectation of a concave function

Let's have a function $f(x, \theta)$, and some probability distribution on $x$. Let's say I have found $\theta^* = \operatorname{argmax}(f(E[x], \theta) $, and $f$ is concave in $x$. I would like to ...
1
vote
2answers
67 views

What is the mathematical term that can differentiate two same vectors?

Say i have two vectors A and B. Mathematically they are same if they have same magnitude and direction. So, say if someone asks me to draw a "vector" of 5 magnitude with 45 degree angle with ...
6
votes
1answer
99 views

Existence of solution of ODE $y^{\prime}=f(y,t)$ where $f(y,t)$ is not defined in initial value.

Consider a differential separable equation $$y^{\prime}=f(y,t)$$ with initial solution $y(t_0)=y_0$. Suppose that $f(y_0,t_0)$ is not defined. Is there a theorem which can be used to prove the ...
5
votes
2answers
99 views

For which rationals $x$ is $3x^2-7x$ an integer?

The following exercise is from [Birkhoff and MacLane, A Survey of Modern Algebra]: For which rational numbers $x$ is $3x^2-7x$ an integer? Find necessary and sufficient conditions. I think I ...
3
votes
1answer
104 views

log transformation for dummies

I have a question which is probaly very simple to answer for most people here: We have a formula: y = -log(x) Then this happens to x: ...
3
votes
1answer
95 views

What's the idea behind the proof of saturation of internal sets via ultrapower construction?

I'm trying to understand the proof of saturation of internal sets via ultrapower construction on Robert Goldblatt's Lectures on the Hyperreals: An Introduction to Nonstandard Analysis. Though, it's ...
0
votes
1answer
85 views

Translation from Spanish - Simple differentiation exercise

Could anybody please translate this exercise into English? A friend of mine sent me the translation via Facebook but I still don't understand why I have to first do the logaritmization of both sides ...
1
vote
1answer
82 views

Show that the stabilizer is a prime subgroup

We define a subgroup $H$ as being convex if $g\in H\implies h\in H$ for all $1\leq h\leq g$. A convex subgroup $P$ is prime if for any two convex subgroups $X,Y$: $X\cap Y \subseteq P$ implies that ...
3
votes
1answer
132 views

equivalent condition for moment generating function

Consider a random variable $x$ with pdf $f(x)$, and have $x \ge 0$. The moment generating function is defined as $M(t)=\int^{\infty}_{-\infty}e^{-tx}f(x)dx$ (noted that we change the sign of $t$ ...
1
vote
3answers
131 views

Definitions of the usual order in $\mathbb{N}$

I know of basically two ways of defining the usual order in $\mathbb{N}$: By using the relation "$\in$" on $\mathbb{N}$ so that $\forall m,n\in N(m<n\longleftrightarrow m\in n)$. By saying ...
2
votes
1answer
86 views

Prove that there exists walks that each edge is in $G$ [duplicate]

For some $k \in\mathbb{N}$, let $G$ be a connected graph with $2k$ odd-degree vertices, and any number of even-degree vertices. Prove that there exists $k$ walks such that each edge in $G$ is used in ...
1
vote
1answer
123 views

equivalence between axiom of choice and Zorn's lemma in a particular case.

Define $A(x)$ and $Z(x)$ as follows. $A(x)\Leftrightarrow $for every indexed family $(S_i)_{i\in I}$ of nonempty sets s.t. $\# I =x$, there exists an indexed family $(s_i)_{i\in I}$ s.t. $s_i \in ...
0
votes
4answers
3k views

Proof of the statement “The product of 4 consecutive integers can be expressed in the form 8k for some integer k”

I am slowly diving into simple number theory and learning how to craft direct proofs. I needed to proof the statement "The product of 4 consecutive integers can be expressed in the form 8k for some ...
2
votes
0answers
75 views

Loop space and $K$-theory

How can I proove without using Yoneda's lemma that $$ \Omega^2(BU \times \mathbb{Z}) \cong BU \times \mathbb{Z} ?$$ In particular how can I define a cellular map $$ f: \Omega^2(BU \times \mathbb{Z}) ...
0
votes
2answers
57 views

Show $X_t = X_{t-}$ a.s. for a Levy process $(X_t)_{t \geq 0}$

I started reading Sato's book on Lévy processes. On page 6 it says that for $(X_t)_{t\ge 0}$ Lévy process $$X_t=X_{t-},$$ for any fixed $t>0$ almost surely. It is mentioned it follows from the ...
1
vote
1answer
95 views

Expressing pushforward of a flow in integral form

Let $\phi(t,x)$ be a flow of a vectorfield $V$ on some compact domain $\tilde{U} = U\times I \in R^n \times R$. Let X be a vector field. If one wants to write $(\phi(t,x))_{*}X)(\phi(t,x)(q)) = ...
10
votes
2answers
530 views

Distributions on manifolds

Wikipedia entry on distributions contains a seemingly innocent sentence With minor modifications, one can also define complex-valued distributions, and one can replace $\mathbb{R}^n$ by any ...
1
vote
1answer
83 views

Product of (strongly) stable ideals and lexsegment ideals

(1) Is the product of lexsegment ideals again a lexsegment ideal? (2) Is the product of (strongly) stable ideals again (strongly) stable? I know that both of them are false and I can find examples ...
0
votes
1answer
175 views

Complex form of Fourier Series

So, the last part of the university syllabus in the chapter of Fourier Series is: ...
3
votes
1answer
69 views

Prove that $T$ is a subgraph of $G$

Let $T$ be a tree with $k$ edges, and let $G$ be a graph where every vertex has degree at least $k$. Prove that $T$ is a subgraph of $G$. Can someone give me tips/help on how to solve this problem?
1
vote
1answer
206 views

Remez algorithm for best degree-0ne reduced polynomials with same endpoints

Given a function $f(x)$ on [-1,1], the Remez algorithm can find the degree (at most) $n$ polynomial $P_n(x)$ that minimizes the maximum error between $P_n(x)$ and $f(x)$ on that interval. It is an ...
1
vote
0answers
17 views

similarity between bundle shift

Let $E$ be a flat unitary bundle of rank $n$ over a domain $R$ in $\mathbb{C}$. It is known that bundle shift $T_{E}$ is similar to $T_{\mathbb{C^n}}$ (which is the bundle shift corresponding to the ...
7
votes
1answer
170 views

Fourier transform using principal value

Can anyone help me compute the Fourier transform of $ 1/|x|^{n-\alpha} $ in $\mathbb{R}^n $ where $ 0 < \alpha < n $ ? Somehow it becomes the principal value of $ 1/|x|^\alpha $ which I can't ...
2
votes
3answers
834 views

Homeomorphism from the interior of a unit disk to the punctured unit sphere

I need help constructing a homeomorphism from the interior of the unit disk, $\{(x,y)|x^2+y^2<1\}$, to the punctured unit sphere, $\{(x,y,z)| x^2+y^2+z^2 = 1\} - \{(0,0,1)\}$. I was thinking you ...

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