2
votes
2answers
63 views

How do I see that function $e^{-4x}$ is not bounded?

How do I see that function $e^{-4x}$ is not bounded? How do I proove that?
0
votes
0answers
14 views

Nonlinear differential equations

How to solve the following set of differential equations analytically $$\begin{align} a_1 \ddot{\phi} + b_1 \dot{\phi} + c_1 \phi & = d_1 \dot{\theta}\dot{\psi} \\ a_2 \ddot{\theta} + b_2 ...
2
votes
3answers
31 views

Linear equation of parallel lines [on hold]

Suppose I have two equations -x + y = 0 -x + y = -2 Suppose I don't know geometry, I don't know slopes, suppose I have just started reading algebraic equations ...
3
votes
1answer
18 views

Prove that it is sufficient to check $\lceil \log(k) \rceil$ pairs to tell if a set of integers is pairwise coprime

I am reading chapter 31 of Introduction to Algorithms (CRLS) and I encountered some difficulties while solving 31.2-9. I managed to prove the first part of a problem, but I can't prove the generalized ...
1
vote
1answer
33 views

Why is the polynomial $F \in K[Z_0, \dots, Z_n]$ on $K^{n+1}$ not a well defined function on $\mathbb{P}^n$ in general?

I'm reading Joe Harris' Algebraic Geometry and he says "A polynomial $F \in K[Z_0, \dots, Z_n]$ on the vector space $K^{n+1}$ does not define a function on $\mathbb{P}^n$" where $K$ is a ...
-1
votes
4answers
38 views

reasoning based question [on hold]

If 100 apples are to be divided among 25 people,how they can be divided so that none of them gets an even number of apples?
2
votes
3answers
66 views

Find the inverse function of $f(x) = \dfrac{x}{1-x^2}$

The function $F : (-1, 1) \to \mathbb {R}$ is defined by $F(x) = \dfrac{x}{1-x^2}$. Example 5 page 106 at Munkers' Topology says that its inverse is $$G(y) = \dfrac{2y}{1+(1+4y^2)^{\frac12}}.$$ I ...
0
votes
1answer
28 views

Direct sum Geometry

I am given a projection (an afinity that verifies $f'^{2} = f'$) and since $u = f'(u) + (u - f'(u)) \longrightarrow$ $E = ker(f' - Id)\bigoplus ker(f')$, I don't see how this follows. Thanks in ...
3
votes
1answer
51 views

Proof of Tychonoff's Theorem for an undergrad

In the midst of learning about compactness I come across Tychonoff's Theorem: Let $\{X_i : i \in \mathcal{A}\}$ be any collection of compact spaces. Then $\displaystyle\prod_{i \in ...
1
vote
1answer
34 views

Find the minimum value of expression involving real numbers

Let $n$ positive integer. Find the minimum value of expression: $$ E=max(\frac {x_1} {1+x_1},\frac {x_2} {1+x_1+x_2}, ... , \frac {x_n} {1+x_1+..+x_n})$$ where $x_1,x_2, .. , x_n$ are ...
0
votes
1answer
27 views

How can we find the elementary divisors?

We consider the polynomial ring $R=\mathbb{R}[t]$ and the $R$-module $M=\mathbb{R}^3$, where $a\cdot x$ ($a\in R,x\in M$) is defined as usual if $a\in \mathbb{R}$, and $a\cdot x=(x_1, 0, 0)$ if ...
2
votes
2answers
60 views

Is $\sin z/z$ analytic at the origin?

For $z\in\Bbb C$ let $$ f(z) = \frac{\sin z}{z} $$ Along the real line this is well behaved, and approaches $1$ as $z\to 0$. But is $f(z)$ analytic at the origin ($z=0$)? I tried explicitly checking ...
3
votes
2answers
73 views

Smallest gaps between powers of 2 and 3

I am trying to find the smallest gaps between powers of 2 and 3. Such examples are: (2, 3) (3, 4) (8, 9) (27, 32) (243, 256) (2048, 2187) (16384, 19683) (524288, 531441)... What are the next ...
0
votes
1answer
12 views

Prove similarity of matrix $A^{-1}$ to matrix $A^{*}$ which is Hermitian adjoint

Let $A \in \mathcal M_{n}(\Bbb C)$ and $A$ is similar to unitary matrix. Prove that $A^{-1}$ is similiar to $A^{*}$, where $A^{*}$ is Hermitian adjoint. $A = C^{-1}UC$, where $U$ is unitary matrix ...
6
votes
1answer
136 views

Why was I wrong about the monster-gem riddler

Every week I like to do the fivethirtyeight.com Riddler, an interesting and pleasantly challenging (at least for me) weekly math puzzle which comes out Fridays, with the answer and explanation to the ...
2
votes
2answers
33 views

Finding subgroups of $D_8$

$$D_8=\{(),(1234),(13)(24),(1432),(13),(24),(14)(23),(12)(34) \}$$ Am I right to say that to find the subgroups, we have to make sure that the identity can be generated or is in the group and the ...
1
vote
2answers
26 views

Implicit differentiation of Product of Two Functions of $y$

I am asked to find the derivative of $$e^y\cos^2y$$ with respect to $x$. I think it is $$y'e^y\cdot2y'\cos y \sin y$$ Since there is no $x$ and $y$ term, such as $xy$, the product rule does not ...
0
votes
1answer
12 views

can I get weak convergence in sobolev spaces from convergence of distributions

my question is the following. Given a sequence $(f_k)_k$ in $W^{1,q}(\Omega)$ with $q \in (1,\infty)$ and $\Omega \subseteq \mathbb{R}^n$ open and bounded. If I want to show $f_k$ converges in ...
0
votes
0answers
24 views

Notation: should Markov chains steps be noted by uppercase or lowercase letters?

I'm reading the chapter about perfect sampling of the "Monte Carlo Statistical Methods" by Robert and Casella, 2004. I've got an issue about notation, when they talk about random mappings, they say ...
0
votes
1answer
36 views

Is these angles 90 degrees?

If I have the following triangle: Where $\angle B=\angle C = O$ And $AP$ bisects $\angle A$ so essentially $\angle BAP = \angle CAP = \frac12 \angle A$ We can prove that $\angle APB = \angle APC$ but ...
2
votes
1answer
35 views

Probability Mass Function of infinitely re-rolled dice

I play a game called Shadowrun. It is a role-playing game that uses a dice pool mechanic. A player has a dice pool of $x$ six-sided, unbiased dice. Every 5 or 6 counts as a success. The more ...
1
vote
2answers
28 views

Consecutive a-smooth numbers

I am looking for two large numbers $n, n+1$ such that both are $7$-smooth numbers. The two largest pairs I found are $2400, 2401$ and $4374, 4375$. Can anyone find a larger pair if it exists? Second, ...
2
votes
4answers
54 views

prove $\sum_{k = 0}^{n} \binom{n}{k} \binom{m-n}{n-k} = \binom{m}{n}$

prove $\sum_{k = 0}^{n} \binom{n}{k} \binom{m-n}{n-k} = \binom{m}{n}$ Attempt:I was thinking of trying to prove this through induction, but I am having trouble with a base case: base case: let $n = ...
0
votes
1answer
18 views

How to reduce a matrix algebraic expression with a division?

I'm not really sure about the algebra rules here, but it feels like my expression should be reducible. Say I have a 1xN vector called w. Then say I have an NxN matrix called M. Say I have the ...
0
votes
0answers
4 views

Can you find shearlets in CNNs?

I read that you can find wavelets in the first layers of a CNN. Can you also find shearlets, and if not, why? Edit (further explanations): I read that, when you train a CNN the filters in the first ...
0
votes
0answers
20 views

Elliptic functions

I would like some references to compute such limits: Jacobi elliptic functions: $\lim_{k\to 0} dn(z/k,k)$ Second species: $\lim_{k\to 0} Z(z/k,k)$ I didn't find what I wanted in the Lawden's book. ...
1
vote
2answers
20 views

Laplacian of a function restricted to a submanifold

Let $f$ be a sufficiently smooth function on a manifold $S$. Let $M$ be a sub manifold of $S$. Can someone show how I can write $\Delta^M( f|_M)$ in terms of $\Delta^{S}f$ ? ((i.e the relation ...
0
votes
1answer
12 views

Let $x,y \in l_p$, proof that $2^k (||x||^p + ||y||^p)^{2/p} \leq 2 (||x||^2 + ||y||^2)$, where $k = 2 - 2/p$ and $1<p\leq 2$

Let $x,y \in l_p$, proof that $2^k (||x||^p + ||y||^p)^{2/p} \leq 2 (||x||^2 + ||y||^2)$, where $k = 2 - 2/p$ and $1<p\leq 2$ My attempt: It's equivalent to proof the following inequality: $$ ...
-1
votes
2answers
29 views

Count the expected value from mimimum [on hold]

Random variable $S_{N}=X_{1}+\dots+X_{N}$ has a Poisson distribution(I assume that the author mean than $N$ has a Poisson distribution). wih $\lambda=5$. Random variable $X$ takes two values ...
0
votes
1answer
27 views

Isomorphism of group algebras $k[G\times H]\cong k[G]\otimes k[H]$ and interpretation

could someone check it the following is correct? I want to show the isomorphism of $k$-modules ($k$ a Ring) as mentioned in the title. I would like to simplyfy the situation to two finite groups ...
3
votes
1answer
27 views

Infimum of distance between point and (closed) set

I'm having a little trouble with the following exercise: Let $V \subset\mathbb{R}^p$ be a non-empty, closed set and $a \in \mathbb{R}^p$. For $x, y \in \mathbb{R}^p$ we note $d(x,y) = \|x-y\|$. ...
-2
votes
2answers
38 views

$p,q$ coprime polynomials - are $p^n,q^m$ coprime? [on hold]

Suppose that $p,q \in K[x]$ are coprime (there is no polynomial that divides both) and let $n,m \in\mathbb{N}$. Are $p^n$ and $q^n$ coprime and, if so, how to prove it?
1
vote
1answer
24 views

Good introduction to cohomology of spaces?

I'm trying to read chapter $3$ of Hatcher but I find it a bit difficult to read. I really only made it through the first two chapters because I had in-class lectures to go along with the reading. Does ...
0
votes
0answers
19 views

If $S \subset X$ does a subsequence of $S$ converge in $S$ or in $X$?

Let $X$ be a metric space and let $S\subset X$ be a compact space. By definition, $S$ is compact implies that for all sequences $(x_n)$ of $S$, there exists a subsequence $(x_{n_{\alpha}})$ that ...
1
vote
1answer
23 views

A definition of the Legendre transform from Zorich

This is from exercise 8.5.5.2 from Mathematical Analysis I by Zorich. The Legendre transform of a (presumably differentiable) function $f:\mathbb R^n\to\mathbb R$ is "the transformation to the new ...
1
vote
0answers
8 views

Gradient and Laplacian of a submanifold

let $M^n$ be a smooth an $n$-dimensional sub-manifold of a $\mathbb{R}^{m}$. Denote by $\nabla^{M}$ and $\nabla^{\mathbb{R}^{m}}$ be the gradient of $M$ and $\mathbb{R}^{m}$ respectively. Similarly ...
0
votes
0answers
15 views

Image of base change of immersion

I'm revising for my exams now and struggling with the following exercise: Let $f: X \to S$ be an open or closed immersion and $g: S' \to S$ another morphism where $X,S,S'$ are schemes. Then the ...
14
votes
4answers
490 views

Gaining Mathematical Maturity [on hold]

I was redirected here by a kind fellow from math.overflow. This is not a typical math question, so I apologize if that is discourteous. I am currently a sophomore in my undergraduate mathematics ...
0
votes
0answers
4 views

Reference for $L$ functions

What is the best possible reference to study $L$ functions for a beginner in this field? I have studied up to the Chebotarev Density Theorem in Algebraic Number Theory. Depending on that, which ...
1
vote
2answers
46 views

What is the boundary of $\mathbb{Q} \times \mathbb{Q}$ in $\mathbb{R} \times \mathbb{Q}$?

Consider the space $X = \mathbb{R} \times \mathbb{Q}$ with the standard Euclidean topology, and the subset $A = \mathbb{Q} \times \mathbb{Q} \subset X$. I am trying to determine the boundary of A. ...
0
votes
0answers
7 views

Complexity of Munkres Algorithm

At the moment I'm trying to derive the complexity of the Munkres algorithm in the worst case. For this I'm working with this website and currently I have following complexities for the steps: ...
3
votes
4answers
242 views

Condition probability distributions: Two people flipping fair coins

Suppose that two people are playing a game where they each flip a fair coin 100 times. The winner of this game is the person who has flipped the most heads. What is the expected number of heads ...
1
vote
1answer
54 views

What happens when $\beta_1 + \beta_2=1$ and when $0<\beta_1 + \beta_2<1$?

I have the following example of the Scwarz-Christoffel integral formula: $$S(z)=\int_0^z w^{-\beta_1}(1-w)^{-\beta_2}dw$$ with $0<β_1 <1, 0<β_2 <1$, and $1<β_1 +β_2 <2$ and I know ...
0
votes
1answer
12 views

The signature of an inner product space does not depend on its basis

In R.W.R. Darling's "Differential Forms and Connections" an inner product is defined for a vector space $V$ as a bilinear, symmetric and nondegenerate (but not necessarily positive-definite) map from ...
2
votes
0answers
24 views

Hilbert curve: how can I find the image of an irrational point in $[0,1]$?

I have a question on the construction of the Hilbert curve: how can I find the image of an irrational point in $[0,1]$? Consider the Hilbert curve $$ f_h:[0,1]\rightarrow [0,1]^2 $$ Consider a ...
1
vote
4answers
44 views

Are these integrations the same?

For the integration of: $$v^2 + 6k^2 = -Mv \frac{dv}{dx}$$ I rearranged to get: $$\int \frac{1}{-M} dx = \frac{1}{2} \int \frac{2v}{v^2 + 6k^2} dv$$ Is this the same as the following integral that is ...
0
votes
0answers
11 views

Let $A\in M_{n \times n}(\mathbb{C})$, $P_A(x)=\prod_{j=1}^{k} (x - \lambda_j)^{n_j}$, and $g\in \mathbb{C}[X]$. Find a C.P. for $g(A)$.

Let $A\in M_{n \times n}(\mathbb{C})$. Its characteristic polynomial $P_A(x)=\prod_{j=1}^{k} (x - \lambda_j)^{n_j}$, and $g\in \mathbb{C}[X]$. Find a characteristic polynomial for $g(A)$. I believe ...
-1
votes
1answer
24 views

Proof/disprove contunuity of a map [duplicate]

I need help with proving / disproving something: Look at the map $$\Phi: (C([0,1], \mathbb R), ||\cdot||_{\infty}) \to (\mathbb R, |\cdot|); \,\,\,\,\,\,\Phi(u) := \int_0^1 u²(t) dt$$ a) ...
0
votes
1answer
32 views

Evaluating $\int R(X)sin(x) dx$ with residue theorem.

The integral I am trying to evaluate is: $$I = \int_{-\infty}^\infty \frac{x}{1+x^2}\sin x\ dx = \int_{-\infty}^\infty f(x)\ dx$$ The standard approach to this is to realise $\sin x$ as the complex ...

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