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0
votes
1answer
17 views

Is the set of all multilinear forms on a vector space $V$ over a field $F$ itself a vector space? [closed]

If so, what is vector addition on this space? Is it the usual pointwise addition of functions?
2
votes
3answers
40 views

Diophantine's Problem

Find all integer solutions for $$x^3+1=y^2.$$ Attempt: By guessing, I found five pairs of integer solutions for the equation: $(2, \pm 3)$, $(0, 1)$, $(-1, 0)$ and $(0, -1)$, but really I don't know ...
0
votes
1answer
20 views

Let $f:\Omega\to\textbf{R}^{m}$ be a function. Then $f$ is measurable if and only if $f^{-1}(B)$ is measurable for every open box $B$.

Let $\Omega$ be a measurable subset of $\textbf{R}^{n}$, and let $f:\Omega\to\textbf{R}^{m}$ be a function. Then $f$ is measurable if and only if $f^{-1}(B)$ is measurable for every open box $B$. My ...
1
vote
0answers
15 views

Decomposition group does not depend on the prime

Suppose you are working with an abelian Galois group $G=G(L/K)$ of the Galois extension $L/K$. You know the Decomposition group is: $D=D(Q,P) = \{ \sigma \in G : \sigma(Q) = Q \}$ where $Q$ (in $L$...
0
votes
0answers
11 views

Clarifying the residue theorem to compute a real trigonometric integral where denominator vanishes

I was asked to solve this exercise in class where we are dealing with the residue theorem and its use for computing real trigonometric integrals on the real axis. We are asked to compute: $$ I = \...
0
votes
0answers
10 views

English version of Gelfand's Normierte Ringe

Where can I find an English version of Gelfand's Normierte Ringe? Below is in German: http://www.mathnet.ru/links/5fca09985bb55cb0e9f6eeb1aecb896c/sm6046.pdf
1
vote
1answer
24 views

Proof Explanation: If $m \in n$, $\exists p \in \omega$ for which $m + p^+ = n$

Synopsis In Exercise 4.23 of Enderton's Elements of Set Theory, we are asked to show that if $m \in n$, $\exists p \in \omega$ for which $m + p^+ = n$. This seems like an obvious statement, but I ...
0
votes
1answer
14 views

Toeplitz matrix, definition not well understood

I was reading about Toeplitz matrix and found the following: If the i,j element of A is denoted Ai,j, then we have Ai,j = A i+1,j+1 = a i-j So I understood that ...
1
vote
4answers
40 views

I'm stuck trying to factor $x^2-4$ to $(x-2)(x+2)$

I am trying to understand each step in order to get from $x^2-4$ to $(x-2)(x+2)$ I start from here and got this far... $x^2-4 =$ $x*x-4 =$ $x*x+x-x-4 =$ $x*x+x-2+2-x-4 =$ $x*x+x-2+2-(x+4) =$ After ...
1
vote
0answers
6 views

Symmetry of (the Partial Derivative of) a Multivariate Logistic Function

Fix $N\in \mathbb{N}$. Let \begin{equation} H(\mathbb{w}) := \frac{1}{1+e^{-\sum_1^Nw_i}},\quad (\mathbb{w}\in\mathbb{R}^N). \end{equation} I am interested in the symmetries of its partial derivative ...
0
votes
1answer
29 views

Stuck on matrix derivative

I am stuck with this (probably simple) derivatives: $$ \frac{\partial}{\partial X}Tr((A\odot(B^{T}XB))C)\;\;and \;\;\frac{\partial}{\partial X}Tr((A\odot(B^{T}XX^{T}B))C) $$ where $A,B,C$ are ...
0
votes
1answer
14 views

Find the volume of an enclosed surface in spherical coordinates where one variable is equal another

I have an enclosed surface in spherical coordinates, where the distance $\rho$ from the origin to an arbitrary point equal to the polar angle $\phi$: $$\rho = \phi$$ I need to find the volume of ...
2
votes
1answer
35 views

Cohomological criterion for non-triviality of negative part of graded module

Let $R$ be a graded ring and $M$ a graded module. Then for sufficently large $n$, we have $$H^0(\operatorname{Proj}(R), \widetilde{M}(n))\cong M_n.$$ Hence if I want to show that $M_{>0}$ is non-...
0
votes
1answer
25 views

Closed form of $\sum _{i=1}^n\:\frac{\left(m-i\right)!}{\left(n-i\right)!}$

I am interested in a closed form of the following sum: $$ S_n := \sum _{i=1}^n\:\frac{\left(m-i\right)!}{\left(n-i\right)!}\;\; n,m \in \mathbb{N},\ n < m$$ Amongst other strategies I have also ...
1
vote
0answers
12 views

Tamely ramified extensions.

I have a question about a result on tamely ramified extensions in Neukirch's Algebraic Number Theory. Proposition 7.7 in chapter II section 7. The question I have is about the proof which starts by ...
0
votes
1answer
8 views

How to get rid of the increasing index in the denominator in this power series?

So we're supposed to calculate the limit of $\frac{x\sin(x)}{1-\cos(x)}$ as $x\to 0$ using power series. Using the series expansions for $\sin(x)$/$\cos(x)$ and multiplying in the $x$ I get $$\frac{\...
0
votes
0answers
12 views

A Schmidt decomposition proof

Consider the pure state $\boldsymbol{\eta} \in \mathcal{H}_{AB}$. It exists an orthonormal set $\{\alpha_1, \alpha_2 \dots \alpha_i\} \subset \mathcal{H}_A$ and $\{\beta_1, \beta_2 \dots \beta_i\} \...
2
votes
1answer
17 views

How to compute generators in biquadratic extension

I'm working in this setting: $K = \mathbb{Q}[\sqrt{2},\sqrt{3}]$, and let $K_1= \mathbb{Q}[\sqrt{2}],K_2= \mathbb{Q}[\sqrt{3}],K_3= \mathbb{Q}[\sqrt{6}]$ the three subfields. I know that $2 \in \...
0
votes
1answer
16 views

Find a ring homomorphism $\theta$ s.t. Ker $\theta = \mathbb{Z}_6 \times \{[0]\}$.

Find a ring homomorphism $\theta: \mathbb{Z}_6 \times \mathbb{Z}_{14} \to \mathbb{Z}_6 \times \mathbb{Z}_{14}$ for which Ker $\theta = \mathbb{Z}_6 \times \{[0]\}$. Attempt: I know that Ker $\theta =...
1
vote
2answers
32 views

Limit of $\frac{e^{xy}}{x+1}$ as it goes to $0$

I have to do the following limit: $$\lim_{(x,y)\rightarrow (0,0)}\frac{e^{xy}}{x+1}$$ I did that if $x = 0 \Rightarrow \lim \rightarrow 1$ but if $y = 0 \Rightarrow \lim \rightarrow$ undefined. By ...
0
votes
0answers
9 views

Question on the use of the Markov Kernel for conditional probability

We define a Markov kernel Let $(\Omega_{1},\mathcal{A}_{1})$ and $(\Omega_{2},\mathcal{A}_{2})$ be some measurable spaces. A map $K$ where $K : \Omega_{1}\times \mathcal{A}_{2}\to [0,\...
0
votes
0answers
15 views

Extracting a smaller Markov chain from a larger Markov chain

I am not very familiar with Markov chains, hence the probably ill titled questions. If we have 5 random variables $X, Y, Z, W$ and they form a Markov chain such that $$X \leftrightarrow Y \...
1
vote
0answers
8 views

Infinitesimal generator of the standard Brownian motion

As explained in this Wikipedia page, the infinitesimal generator of the standard Brownian motion is $\frac{1}{2}\Delta$ and for the Brownian motion it has an extra $\partial_t f$ term. Can anybody ...
0
votes
0answers
22 views

Derivative $ dy/dx$ of a complex $f(x)\log{f(x)}$ function with respect to $x$?

I am trying to take the derivative of the function. The fundamental way seems tedious for me. I apologize for that. I was trying to find a closed-form expression of the derivative $dy/dx$ for such a ...
1
vote
0answers
22 views

Which topic studies the symmetry/antisymmetry of products?

Suppose we have a ring $R$ and $u, v, w, a_0, ..., a_n$ are elements of the ring I have encountered some work which takes the following concepts for granted: 1) formation of symmetric/antisymmetric ...
5
votes
2answers
63 views

Grothendieck group “commutes” with direct sum

The Grothendieck completion group of a commutative monoid $M$ is the unique (up to isomorphism) pair $\langle \mathcal{G}(M), i_M\rangle$, where $\mathcal{G}(M)$ is an abelian group and $i_M\colon M\...
0
votes
1answer
27 views

SIR Model Specifics

I read on Wikipedia (under "Compartmental models in epidemiology") that the differential equations for the SIR Model was the following, $$S'(t)=-\frac{\beta}{N}I(t)S(t)$$ $$I'(t)=\frac{\beta}{N}I(t)S(...
0
votes
0answers
26 views

Integral for homework (It's supposed to solved with integration by parts) [closed]

There is this integral in my homework I solved it using some trigonometry and u substitution bu it's supposed to be solved by integration by parts. If anyone could help it would be fantastic. Problem:...
0
votes
0answers
11 views

How do I solve the distributional equation T.x =1?

I am having a bit of trouble solving distributional equations. An example that I am currently working on is to show that the distributional equation $T.x = 1$ has a solution if and only if $T=p.v.(\...
0
votes
0answers
8 views

$x_0+M\subset K$ for some linear subspace $M$ of a Hilbert space implies $x_0$ is orthogonal to $M$, where $K$ is a closed convex set

Suppose $K$ is a closed convex set in a Hilbert space $H$. If $x_0+M\subset K$ for some linear subspace $M$ of $H$, prove that $\langle x_0,y\rangle=0$ for all $y\in M$; in other words, $x_0$ is ...
1
vote
1answer
17 views

A type of set used in convergence in measure theory

This is not a specific problem, but a general question. Often when we're showing convergence of functions (particularly pointwise) or even of sets in certain cases, a set of the following form ...
0
votes
2answers
22 views

Limit of $\frac{1}{r}\ln\left(1+r\sum\limits_{i=1}^n p_i \ln(x_i)+ \omicron(r)\right)$

Let be $n \in \mathbb{N}$ arbitrary but fixed, $\sum\limits_{i=1}^n p_i =1$ and $\forall ~ 1\leq i \leq n$ we assume: $x_i \in \mathbb{R}$. What is the limit of $\lim\limits_{r\to 0}~\frac{1}{r}\ln\...
0
votes
0answers
18 views

The intuition behind using the Euler-Lagrange equation to solve PDEs

I was looking at some videos on using the Euler-Lagrange equation and the Calculus of Variations. I understand that the Euler-Lagrange equation can be used to solve some PDEs analytically, or simplify ...
0
votes
0answers
9 views

Which is the correct way of showing $D \otimes \bar{D}$ is a representation?

To show that $D \otimes D$ is a representation, I know that I must show that*: $$\tag{1} D \otimes D (U_1 U_2) \phi = D \otimes D (U_1) (D \otimes D(U_2) \phi)$$ I understand that this can be ...
0
votes
1answer
47 views

What can you say about $T$ if dim$(V) =$ Rank$(T - \lambda I)$?

I stumbled across this condition and I wanted to know what you could say about this: Let $T:V \to V$ be a linear transformation, with $V$ having a finite dimension. What can you say about $T$ if ...
0
votes
1answer
25 views

Is this proof that relative entropy is never negative correct?

I wish to prove that relative entropy(Kullback-Liebler divergence) is always non-negative. I.e. that $$I^{KL}(F;G)=E_F\left[\log\frac{f(X)}{g(X)}\right]\geq0$$ where F,G are two different probability ...
0
votes
1answer
14 views

Calculate the curvature $k(t)$, for the curve $r(t)=\langle 1t^{-1},-5,3t \rangle$

I have that $k(t)=\frac{\mid r'(t)\times r''(t) \mid}{\mid r'(t)\mid^3}$. So first, $r'(t)=\langle -\frac{1}{t^2},0,3 \rangle$. $r''(t)=\langle \frac{2}{t^3},0,0 \rangle$. $\mid r'(t)\mid = \sqrt{...
4
votes
1answer
51 views

How should I self-study set theory/cardinality?

So, I am an absolute beginner in mathematics; only being knolwdegable on some basic ideas in the subject. My interest in maths started only recently, while reading about set theory and cardinality (...
0
votes
1answer
20 views

Need to prove (or disprove) this equation regarding Complex numbers

i am trying to solve this question: given $a,z \in \mathbb{C}$ $|\bar{a}\cdot z-1|^2 -|z-a|^2=(1-|a|^2)(1-|z|^2)$ i tried to assign several expamles for the given a,c complex numbers and got truth ...
1
vote
2answers
30 views

“Modern analysis” books in languages other than English or Russian

I confess that I am not quite sure about the usefulness of this question, but after getting to know of a few textbooks in German and Portuguese, I've certainly gotten curious: Do you know of ...
1
vote
1answer
18 views

How to quantify asymptotic growth?

Specifically, my research question is to find operator $A: (\mathbb{R}^+\rightarrow\mathbb{R}^+)\rightarrow\mathbb{S}$, where $\mathbb{S}$ is some totally ordered set, such that for $f, g: \mathbb{R}^+...
0
votes
0answers
5 views

Create a table of values and an associated graph that are increasing, with a point of inflection.

X Y 3 .. 4 9 5 14 6 17 7 19 8 .. The question is how I can find the missing value. As I did the problem, I got 1 and 21 but they are the wrong answers in this case.
1
vote
2answers
12 views

Estimation of the Kolmogorov test power [Rstudio].

Let $x$ be a 20-element sample from the T-Student distribution with 4 degree of freedom. I want to do a simulation to estimate the power of the Kolmogorov test for $H_0=F\sim N(0,1)$ if $\alpha=0.01$. ...
1
vote
0answers
13 views

Representations of elements in polynomial rings

I am reading a paper by Banaschewski (Krull implies Zorn) and I am trying to prove everything I find in order to understand it fully. There's a few facts he states before getting into the actual proof,...
1
vote
1answer
20 views

Finding a fixed polynomial under the multiplicative inversion automorphism

Can anyone find a polynomial $f ∈ ℚ\left(X+\frac{1}{1-X} + \frac{X-1}{X}\right) ⊆ ℚ(X)$ that is fixed under the automorphism $(X ↦ \frac{1}{X})$? $f = X+\frac{1}{X}$ would be nice, but I don't know ...
0
votes
1answer
25 views

Limit $\lim_{x\to\infty} x(\arctan(a^2x)-\arctan(ax))$

I have a limit $\lim_{x\to\infty} x(\arctan(a^2x)-\arctan(ax))$ and I know the solution $\frac{a-1}{a^2}$, but I dont have any Idea, how to calculate this limit or at least how to start. Any idea?
1
vote
0answers
26 views

How many functions does it take to solve polynomial equations?

It is well known that polynomial equations of degree $\ge 5$ cannot be solved by just using addition, multiplication, division and radicals (Abel–Ruffini theorem). So what, who cares? Let's just add ...
2
votes
1answer
21 views

'Locally' Convex Function

I have a continously differentiable function $f:\mathbb{R}^{n}\rightarrow\mathbb{R}$ which I am trying to prove is globally convex. Computing the Hessian directly is very difficult as it is a somewhat ...
3
votes
0answers
29 views

Apply Pigeonhole Principle to pick numbers such that at least two of them have a digit in common

How many integers from 100 through 999 must you pick in order to be sure that at least two of them have a digit in common? (For example, 256 and 530 have the common digit 5.) In the worst case I ...
1
vote
2answers
80 views

Proof Without contradiction $x^2 = 2$ has no rational solutions

I am in the process of learning real analysis and I was wondering if there was a way to prove no $x \in \mathbb{Q}$ satisfies $x^2 = 2$ without a proof by contradiction.

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