0
votes
1answer
11 views

Help with proof about functions and subsets

Problem: let $f: A \rightarrow B$. Prove that $f$ is injective if and only if for all $D \subset A$ we have that $f^{-1}(f(D)) = D$. Proof: => Suppose $f$ is injective. Let $x \in f^{-1}(f(D))$. ...
1
vote
0answers
5 views

Does strictly positive density function on the real line with infinite expected value exist?

The problem is as stated in the title. I am looking for an example or a disproof, whether there exists a continuous density function on the whole real line with infinite expected value. Once again: ...
0
votes
0answers
9 views

Reversed Cayley transformation for any unitary matrix

It is well known that if $Q$ is a unitary matrix such that $I+Q$ is invertible (where $I$ is the identity matrix), then $$ A:=(I-Q)(I+Q)^{-1} $$ is skew-Hermitian ($A=-A^*=-\bar{A}^T$). This fact is ...
0
votes
0answers
14 views

Prove that $a+ib$ is prime in $\Bbb Z[i]$, of $a^2+b^2$ is prime in $\Bbb Z$.

Prove that $a+ib$ is prime in $\Bbb Z[i]$, of $a^2+b^2$ is prime in $\Bbb Z$. My Try: We can easily show that $\Bbb Z[i]$ is a FD but how can we show that $\Bbb Z[i]$ is a UFD. Because if we can show ...
0
votes
1answer
11 views

Proving that a matrix product is singular

I just played around in mathematica and found out that it seems like if $A$ is an $m\times n$ matrix and B is an $n\times m$ matrix, with $m>n$, then $AB$ is singular. How does one go about proving ...
1
vote
0answers
18 views

Is there a way to define the concept of manifolds so it looks more like “generalised affine spaces”?

What I have in mind is along the lines of this: Let $M$ a topological space, $V$ a normed vector space, and $$ \boxminus \colon M\times M \to V, $$ $$ \boxplus \colon M\times V \to M. $$ Then ...
1
vote
0answers
16 views

Stupid question on linear subspaces formed by functions

The set of differentiable real-valued functions on (0,3) such that f'(2)=b is a subspace of $\mathbb{R}\leftarrow(0,3)$ if and only if b=0 ($\mathbb{R}\leftarrow(0,3)$ denotes the set of functions ...
1
vote
0answers
7 views

Complex projective manifolds and smooth projective varieties

Look at the following theorem: The following two categories are equivalent: The category of smooth projective varieties over $\mathbb C$. (Where a variety is understood as in Hartshorne's ...
-1
votes
0answers
13 views

Math Combination

I have 3 bags with 3 balls each Bag 1 - white, blue , green Bag 2 - yellow, orange, purple Bag 3 - black, red, brown How do we calculate all possible 3 balls combinations(1 ball from each bag) ?
4
votes
6answers
27 views

How does one prove that if $\cos(t) = \cos(t')$ and $\sin(t) = \sin(t')$ then $t = t' + 2k\pi$?

How does one prove that if $\cos(t) = \cos(t')$ and $\sin(t) = \sin(t')$ then $t = t' + 2k\pi$ ? I've tried proving the above statement, which I think is valid. I know $\sin(t)$ is injective on ...
2
votes
0answers
21 views

Principle of explosion: Other arguments?

I've come across a proof-theoretic argument for explosion on Wikipedia, which is as follows: $A \ \ \wedge\sim A$ $A$ $ \sim A$ $ A \lor B$ $B$ $(A \ \ \lor \sim A) \implies B$ I've thought of ...
1
vote
0answers
8 views

Simple composition factor

We know that if $e$ is a primitive idempotent in a semiperfect ring then $Re/Je$ is a simple left $R$-module, where $J$ is the Jacobson radical of $R$. Now, suppose that $e$ is an idempotent in an ...
1
vote
1answer
8 views

Proof via strong induction of a string output

I'm still new to the whole proof thing (first class of discrete mathematics and analysis right now). I could do general induction problems, but the fact that 'n' is the output here along with the ...
3
votes
0answers
21 views

Rank in row echelon form

$$A= \begin{bmatrix} a & 1 & a & 0 & 0 & 0 \\ 0 & b & 1 & b & 0 & 0 \\ 0 & 0 & c & 1 & c & 0 \\ 0 & 0 & 0 & d & 1 ...
1
vote
0answers
8 views

Is my proof that the sheaf of smooth functions is soft correct?

As mentioned in the title, I proved the following statement and need help from someone who checks it for correctness: Let $X$ be a paracompact smooth manifold. Then $U \mapsto C^\infty(U)$ where $U$ ...
1
vote
1answer
11 views

Split n balls to k boxes

I have $n$ different balls $(1,2,..., n)$ and $k$ different boxes $(1,2,...,k)$. I want to put all balls to boxes, but if ball i has smaller nuber than j (i < j) than ith ball must be put to box ...
1
vote
0answers
9 views

(representation theoretic) meaning of sum over even rows of a Young tableau

Think of a Young tableau $R$ as composed by $d$ rows with number of elements $\mu_i:=\mu_i^R$ $\mu_1 \geq \mu_2 \geq \cdots \geq \mu_d > \mu_{d+1}=0$ (and $\mu_i =0\, \forall i >d$) and define ...
1
vote
0answers
10 views

Sigma-additive measures on finite discrete spaces

Let $\mathcal{A}$ be a finite algebra of sets, and $\mu$ a $\sigma$-additive function defined on $\mathcal{A}$. Then, the Hahn-Kolmogorov theorem tells us, amongst other things, that $\mu$ can be ...
2
votes
0answers
10 views

Poincare duality isomorphism problem in the book “characteristic classes”

This is the problem from the book, "characteristic classes" written by J.W. Milnor. [Problem 11-C] Let $M = M^n$ and $A = A^p$ be compact oriented manifolds with smooth embedding $i : M \rightarrow ...
1
vote
0answers
24 views

What is the step in this proof “because $\omega$ is closed”?

I am working through this proof of the Poincare lemma here but I don't understand one step. First, there is the following equation $$ {\partial \over \partial x^j} f(x) = \int_0^1 \left (t ...
1
vote
0answers
5 views

Inclusion of Sobolev spaces with fractional order

Let $W^{k,p}\mathbb(R^n)$ the usual Sobolev space. We know that if $k>l$ and $1\leq p<q<\infty$, $(k-l)p<n$ and $$\frac{1}{q}=\frac{1}{p}-\frac{k-l}{n}$$ then ...
1
vote
0answers
12 views

Updated conditional probability - Elo rating

Consider a tournament where all teams play against each other, and suppose we have the probability for win-draw-loss for every match (e.g. with the prediction from Elo rating). Is there a feasible ...
1
vote
1answer
17 views

probability of two successive random numbers has the same starting number

Question/problem(subtask b): What is the probability of two successive random numbers has the same starting number? What we do know is that a random number generator randomizes numbers of 6-digits ...
1
vote
0answers
2 views

what is the difference between renormalized solution and entropy solutions for nonlinear elliptic PDE?

I want to know the difference between renormalized solution and entropy solutions for nonlinear elliptic PDE when right hand side of the elliptic operator is $L^1.$ Also how does right hand side with ...
1
vote
0answers
5 views

Saturation of the Babenko–Beckner inequality

The Babenko-Beckner inequality $|| \mathcal F f ||_q \geq C(q,p)||f||_p$ is a well-established theorem. It relates the $q$-Norm of a Fourier transform $\mathcal F f$ of a function $f$ to its ...
1
vote
0answers
10 views

Understanding Tensor product of modules

This is follow up question Understanding the Details of the Construction of the Tensor Product If $M$ and $N$ are 2 A-modules, we define $Z = A^{MxN}$ as module generated by $MxN$. Then $D$ is ...
1
vote
0answers
10 views

Boundedness and Schrödinger operators

I'm working on Schrödinger operators, currently struggling with the proof of $\nabla(\Delta + i)^{-1/2}$ bounded. The domain is $H^2(\mathbb R^3)$. So, of course $(\Delta + i)^{-1}$ is bounded, and ...
-2
votes
1answer
21 views

Find diff. function $f:\mathbb{R}^2 \to \mathbb{R}$ such that $f(x,y) \neq 0, f(0,0) = 0$ and $|R(0,x)|/|x|^n \to 0 \forall n$. [on hold]

Give an example of a differentiable function $f:\mathbb{R}^2 \to \mathbb{R}$ such that $f(x,y) \neq 0, f(0,0) = 0$ and $|R(0,x)|/|x|^n \to 0 \ \forall n$. Here $R(x,y)$ denotes the error term while ...
0
votes
0answers
7 views

Numerically stable way to compute the conditional covariance matrix

The Wikipedia article on multivariate normal distribution contains the well-known fact about the conditional "sub-distribution": If $μ$ and $Σ$ are partitioned as follows: $$ \boldsymbol\mu = ...
1
vote
0answers
11 views

Weak solution of elliptic equation depends continuously on parameter

Suppose I have a weak formulation of the form: find $u \in H^1_0(\Omega)$ such that $$\int_\Omega b(p)(\nabla u \nabla v + \lambda uv)=0$$ holds for all $v \in H^1_0(\Omega)$ where $b:[a,b] \to ...
0
votes
3answers
53 views

Parametrization of the ellipse $\frac {x^2} {p^2} + \frac {y^2} {q^2} = 1$.

Consider the ellipse ($p > q > 0$) $$\frac {x^2} {p^2} + \frac {y^2} {q^2} = 1$$. I want to prove that $$\mu(t) = (p \cos(t), q \sin(t))$$ is a parametrization of the ellipse. I see that $(x/p, ...
0
votes
0answers
7 views

Decomposition of 4x4 or larger affine transformation matrix to individual variables per degree of freedom.

There are a couple of problems and solutions where affine matrices are decomposed into their seperate tranformations. However they are all for the 2D case and I`m finding it difficult to generalise it ...
0
votes
1answer
17 views

To show module homomorphism being injective

Suppose $R$ is a field, $M$ is a right $R$ module, and $f : R_R \rightarrow M$ is a non-zero homomorphism. Show that $f$ is injective. My work: The homomorphism is uniquely determined by $f(1)$. If ...
1
vote
1answer
21 views

Maximum value of the area of triangle

If two of the medians of a triangle have lengths x and y, what is the maximum value for the area?
0
votes
1answer
11 views

Generating Borel algebra - proof

In the second paragraph of the proof it says: "To prove that $\mathcal{B}(\mathbb{R})$ is also generated by the other classes of intervals, it suffices to prove that any interval $]a,b[$ is contained ...
1
vote
0answers
18 views

Essay about PDEs

I've taken an introductory course in PDEs and I have to write an essay of 4-8 pages on a topic in partial differential equations. The topics we touched on are: First order linear partial ...
0
votes
0answers
12 views

Strict total ordering

I'm not able to understand how the below relation is example of "strict total order". Consider a set $X = 2^Y$ where $Y = \{1,2,3,4,5,6,7,8,9\}$. The expected order of $X$ is for all $x, y$ ...
0
votes
2answers
38 views

Matrices that commute with all matrices [duplicate]

Let $Z_n$ be the set of all $n \times n$ matrices that commute with all $n \times n $ matrices. Show that $$Z_n = \{\lambda I_n \ | \ \lambda \in \mathbb R\}$$ ($I_n$ is the $n \times n$ identity ...
0
votes
1answer
11 views

Prove: if the complementary graph is connected, then graph isn't necessarily unconnected.

I have such a question. There is a theorem related to graphs that says, that if a graph is disconnected then it's complementary graph is connected. But how can I prove that the inverse is not true, ...
0
votes
0answers
5 views

Tangent line to the unitary group $U_1$

I have been working through a small project and the last part has me completely stumped. I have just shown that the matrix $\exp\tau X$ is unitary for all $\tau\in\mathbb{R}$ iff $X$ is ...
0
votes
0answers
24 views

Prove that the probability of getting at least $k$ heads when using coins that skew more to heads cannot be worse that the alternative

Let $0 \le p < p' \le 1$. Let $X_i$ be the Bernoulli random variable that takes the value of $1$ with probability of $p$ and zero otherwise. Similarly, let $X'_i$ be the Bernoulli random variable ...
0
votes
2answers
22 views

Why $(10^ab+c)^{4d+1}-c \mid 10$?

I came across the following equation: $$x=(10^ab+c)^{4d+1}-c$$ Why is $x$ a multiple of $10$ for any natural number values for $a$, $b$, $c$ and $d$? The only progress I made was that $a$ could be ...
0
votes
0answers
3 views

Convert position and velocity vector to TLE for propagation

I have a set of position and velocity vector for a satellite at time t0. I want to propagate position and velocity of the satellite at time t (t>t0) I converted these vectors to Keplerian Elements ...
1
vote
3answers
22 views

How many ways there are?

I cant solve the following problem. In how many ways we can divide 6 balls between 3 children if every children must receive at least 1 ball. I don't understand the problem. Is it permutations or ...
0
votes
1answer
24 views

Sum of exponential square series

I have a infinite sum which I wonder if it will converge to a simpler function $f(r) = \Sigma_n r^{n^2} , r<1$, I also interested in case $r$ is a complex number on unity circle $r = ...
1
vote
0answers
13 views

Flatness of integral closure over an integral domain

The problem 1.2.10 from Qing Liu's book "Algebraic Geometry and Arithmetic Curves" is the following: Let $A$ be an integral domain, and $B$ its integral closure in the field of fractions ...
0
votes
0answers
11 views

Time derivative of a function involving absolute value

I have a functional looking like this $$ L[u] = \int \int_{\Omega} k \left| \nabla u \right|^2$$ , for which I want to take the time derivative $\dot{L}$. I am not sure that I handled the time ...
1
vote
0answers
4 views

Pre-dual of distributions with support in a closed subset

Usually, in order to define the space of distributions $\mathcal{D}'(\Omega)$ on an open subset $\Omega \subseteq \mathbb{R}^n$, one considers the space $\mathcal{D}(\Omega)$ of $C^\infty$-test ...
1
vote
2answers
35 views

Matrix notation of vectors?

My linear algebra book says that a vector is a one-column matrix. However, in the context of what we are studying (linear equations) it would make more sense if a vector was of the form of the ...
0
votes
0answers
6 views

Assessing the “Quality ” of a problem solved by using lagrangian multipliers

I have an ill-defined question. I work in machine learning and am trying to learn the parameters of a model, such that my problem amounts to constrained optimization. That is, I have some training ...

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