1
vote
1answer
123 views

Prove that gradient of $\operatorname{tr}(A \cdot B \cdot A^{T} \cdot C$) with respect to $A = C \cdot A \cdot B$ +$ C^{T} \cdot A \cdot B^{T}$

I've been poking through SEE's Machine learning notes and am having difficulty proving the relationship: Gradient, $\nabla$, of the trace $\operatorname{tr}(A \cdot B \cdot A^{T} \cdot C$) with ...
1
vote
2answers
107 views

Game with losing and winning a dollar

I found an interesting problem in my book: There is a game where player starts with $k\$$. In each step he wins or loses $1\$$ (both with probability $p=\frac{1}{2}$). The game ends when player ...
3
votes
2answers
136 views

Choosing convenient limits of integration on Basel problem

I have recently discovered ...
2
votes
1answer
410 views

What is the explicit equation that converts Cartesian coordinates to elliptical coordinates?

All of the solutions that I have seen so far require solving an implicit equation after substituting in whatever $x$, $y$, and $z$ are. What are the explicit equations for each elliptical coordinate ...
1
vote
1answer
126 views

Union of a family $J$ of open sets in $\mathbb R^n$ equals union of a countable subfamily $K$ in $J$.

Let $(U_j)_{j\in J}$ be a family of open subsets in $\mathbb R^n$. I'm asked to show that there exists a countable subset $K$ in $J$ such that $\bigcup_{j\in\ K}\left( U_j \right) = \bigcup_{j\in\ ...
1
vote
2answers
150 views

Are waiting times always going to be exponentially distributed?

I'm studying for CAS/SOA Exam P/1 and a question I have here is: We have a portfolio of $20$ insurance policies. The number of claims per policy in a $3$-month period has a Poisson distribution ...
-1
votes
2answers
163 views

If a continuous real function is additive, then it is linear

I have to prove the following problem Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function such that $f(x + y) = f(x) + f(y),\ \forall x,y \in \mathbb{Z}$. Then $f$ is a linear ...
1
vote
2answers
73 views

Show by example that this need not be true if we do not assume that the groups are finitely generated

Let $G, H,$ and $K$ be finitely generated abelian groups. If $G \times K \cong H \times K$, show that $G \cong H$. Show by example that this need not be true if we do not assume that the groups ...
9
votes
1answer
377 views

Involutions and Abelian Groups

Suppose that $ G $ is a finite group where at least three-fourths of the elements are involutions, i.e., $$ |I(G)| \geq \frac{3}{4} |G|. $$ (Here, $ I(G) $ denotes the set of all involutions of $ G $, ...
1
vote
1answer
2k views

Software to Plot 3D Vector Fields

I don't have programs like MAthematica or Maple that plot vector fields out of the box, the ones I use are Maxima and Scilab for simbolic/numeric, and none of them can easly plot 3d vector fields, so ...
9
votes
2answers
367 views

Origin and use of an identity of formal power series: $\det(1 - \psi T) = \exp \left(-\sum_{s=1}^{\infty} \text{Tr}(\psi^{s})T^{s}/s\right)$

The following is a historical question, but first some background: Let $\psi$ be a linear operator from a vector space to itself. The following two expressions, viewed as formal power series, can be ...
3
votes
1answer
872 views

Mixed Strategy Nash Equilibrium of Rock Paper Scissors with 3 players?

It seems like most game theory tutorials focus on 2-player games and often algorithms for finding Nash equilibria break down with 3+ players. So here is a simple question: Is ...
1
vote
1answer
68 views

derivation of an inequality

I'm reading a text and sometimes it skips steps in its derivations that I don't always understand. An example follows. Suppose we have, $f(y) \ge f(x) + \frac{1}{1-\alpha}\left[f(x + ...
3
votes
2answers
111 views

Nilpotent Lie Group that is not simply connect nor product of Lie Groups?

I have been trying to find for days an non-abelian nilpotent Lie Group that is not simply connect nor product of Lie Groups, but haven't been able to succeed. Is there an example of this, or hints to ...
7
votes
1answer
308 views

Is Hahn-Kolmogorov theorem a direct result of Carathéodory's extension theorem?

Both theorems assume the same condition and conclusion, except that Hahn-Kolmogorov theorem extends a premeasure on a field of subsets to a measure on a sigma algebra generated by the field of ...
3
votes
1answer
139 views

Does there exist a differentiable function $\Bbb R^2$ to $\Bbb R$ with certain partial derivative properties?

Does there exist a counter-example to the following claim: For a function $f: \mathbb{R}^2 \to \mathbb{R}$, if: $D_1f$ exists in some ball around the origin and is continuous at the origin (but not ...
1
vote
0answers
30 views

How to show that density?

Show $$ \overline{\operatorname{span}(v_j)}=L^2([0,1]),~~~~~\overline{\operatorname{span}(u_j)}=L^2([0,1]) $$ with $$ v_j(x)=\sqrt{2}\cos((j-1/2)\pi x),~~~~~u_j(x)=\sqrt{2}\sin((j-1/2)\pi x). $$ ...
4
votes
4answers
180 views

Proof of tangent half identity

Prove the following: $$\tan \left(\frac{x}{2}\right) = \frac{1 + \sin (x) - \cos (x)}{1 + \sin (x) + \cos (x)}$$ I was unable to find any proofs of the above formula online. Thanks!
2
votes
1answer
53 views

Expressing “uncountable” in $L_{\omega_1\omega}$

Given a countable signature $\tau$ I'm trying to find a uncountable $\tau$-Structure $\mathfrak{A}$ which does not satisfy the same infinitary logic $L_{\omega_1\omega}$-sentences as a countable ...
1
vote
3answers
84 views

Limit of a Cauchy Sequence

Let $\langle X_n \rangle$ be Cauchy with a sub sequence $\langle X_{n_k} \rangle$ such that $\lim_{n\to\infty} X_{n_k}=A$. Show $\lim_{n\to\infty} X_n=A$. My work so far: Choose $\epsilon$ such ...
3
votes
1answer
79 views

Real valuations on Dedekind domains

Let $D$ be a Dedekind domain. Let $v:D \to \mathbb{R}$ a valuation. We know that for every prime ideal $\mathfrak p$ of $D$ the localization $D_{\mathfrak p}$ is DVR. Does every valuation on $D$ ...
6
votes
5answers
258 views

Different ways to work out the normal in the Frenet frame

Given a curve $\gamma (t) \in \mathbb{R}^3$, when working out the Frenet frame my lecture notes define the unit normal $$\tau : = \frac{\gamma ' (t)}{\| \gamma'(t)\|}$$ and then the principle unit ...
2
votes
1answer
90 views

Finding the order of $N_G(H)$

I have this question: For a group $G=A_5$ and $H=$ $\langle(12)(34),(13)(24)\rangle$ , prove that $(123)\in N_G(H)$ and hence deduce the order of $N_G(H)$. $A_5$ is defined to be the alternating ...
6
votes
4answers
283 views

How to show $f'(0)$ exist and is equal to $1$?

Assume that $f$ be continuous on $\mathbb{R}$, $f'(x)$ exists for all $x\neq 0$, and $\lim_{x\rightarrow 0} f'(x)=1$. We need to show $f'(0)$ exist and is equal to $1$. $f'(0)=\lim_{x\rightarrow ...
11
votes
6answers
806 views

What is gained by computing additional digits of $\pi$? [duplicate]

Possible Duplicate: Do We Need the Digits of $\pi$? Given that at 39 digits, we have enough of $\pi$ to calculate the volume of the known universe with (literally!) atomic precision, what ...
1
vote
0answers
115 views

Modular form weight 0

Why is an entire modular form of weight 0 must be a constant? In particular, does a function defined on the upper half plane that is analytic everywhere, including i/infty, imply boundedness? Can we ...
1
vote
1answer
65 views

On functions in $L^p$

I am trying to solve the following problem. It seems easy, but for some reasons I am confused by the hint. Suppose $0<p_0<p_1\leq\infty$. I need to find an example of functions $f$ on ...
4
votes
5answers
154 views

how to show $f(1/n)$ is convergent?

Let $f:(0,\infty)\rightarrow \mathbb{R}$ be differentiable, $\lvert f'(x)\rvert<1 \forall x$. We need to show that $a_n=f(1/n)$ is convergent. Well, it just converges to $f(0)$ as ...
1
vote
3answers
2k views

Is the difference between consecutive prime numbers always an even number?

If we look at the difference between consecutive prime numbers, $p \gt 2$, it always appears to be an even number. For example, here are the seven consecutive primes starting at the $10^{10th}$ ...
2
votes
1answer
65 views

hopf bifurcation for an ode

I understand how to analyse a system of equations like $x'(t) = f(x,y)$ $y'(t) = g(x,y)$ set $x'$ and $y'$ to zero and find the fixed points etc, and find the stability. What Im am not sure of ...
5
votes
1answer
110 views

Quadratic Diophantic equation

Hello :) i want to give a answer op the following question: For which prime number $p$ can we give a solution of the diophantic equation given by $x^2-65y^2=p$. I want to solve the question without ...
0
votes
1answer
112 views

$\Bbb Q$ is lowest field of characteristic zero

Teorem: field of characteristic zero contains a subfield that is isomorphic to $ \Bbb Q$; so $\Bbb Q$ is lowest field of characteristic zero. proof: I did following: let char(K)=0. and $$ \beta :\Bbb ...
8
votes
1answer
195 views

Compute $\sum_{m>n=1}^{\infty} \frac{1}{m!n!}$

Compute the series $$\sum_{m>n=1}^{\infty} \frac{1}{m!n!}$$
3
votes
2answers
230 views

What is the free category on the underlying graph of a category?

Let $\mathcal{D}$ be a category. Ittay Weiss wrote about Free($\mathcal{D}$) in chat with me. He said Free($\mathcal{D}$) is the free category on the underlying graph of $\mathcal{D}$. Is ...
0
votes
1answer
33 views

Slight mistake in working out my semi direct product

Construct explcitly a non-commutative semi direct product $H \rtimes Q$ with $$H = C_{79} \hspace{2cm} Q = C_{13}.$$ You may assume that the least positive integer $k \geq 1$ such that ...
0
votes
0answers
54 views

Citation for biprofunctor

Does anyone know a book or article in a journal that defines biprofunctors between bicategories $\Bbb A$ and $\Bbb B$, as in nLab, i.e., as a pseudo functor $\Bbb A^{op}\times\Bbb B\to\Bbb C\rm{at}$? ...
13
votes
1answer
204 views

If $\lambda_n \sim \mu_n$, is it true that $\sum \exp(-\lambda_n x) \sim \sum \exp(-\mu_n x)$ as $x \to 0$?

If $\lambda_n,\mu_n \in \mathbb{R}$, $\lambda_n \sim \mu_n$ as $n \to +\infty$, and $\mu_n \to +\infty$ as $n \to +\infty$, is it true that $$ \sum_{n=1}^{\infty} \exp(-\lambda_n x) \sim ...
6
votes
2answers
215 views

The Matrix Equation $X^{2}=C$

Let matrix $C_{n\times‎n}$ and equation $X^{2}=C$ be given, i want to find matrix $X$. For $n=2$, $X$ is obtained by solving a system of equations; $$\left\{ \begin{array}{l} x_{11}^2 + ...
0
votes
2answers
2k views

Easiest way to determine all disconnected sets from a graph?

Suppose that I have a un-directed graph of nodes and edges, I would like to know all sets of nodes that do not connect with any other nodes in the graph. Here is a concrete example to help you ...
3
votes
2answers
433 views

Connected graph, each power of adjacency matrix has zeros

Could you help me with the following? Find a connected graph for which each of the matrices $A^{k}_{G}, \ k \ge 0$ contains $0$s. And another one: Show that a graph is connected if and only if ...
7
votes
1answer
148 views

Graph, two colors, no path length 3

I've just begun studying graph theory and I have some difficulty with this problem. Could you tell me how to go about solving it? In a graph $G$ all vertices have degrees $\le 3$. Show that we can ...
2
votes
1answer
71 views

Primitive solutions to $a^2 + 4b^2 = c^2$

I am trying to generate primitive solutions (GCD is 1 for $a, b, c$) to the equation $a^2+4b^2=c^2$. I attempted to do this by modifying the usual Pythagorean triplet $(m^2-n^2)^2 + (2mn)^2 = ...
3
votes
2answers
148 views

How to use equicontinuity?

Rudin asked: For $f\in L^{\infty}(\mathbb{R}^{1})$, define $f_{t}(x)=f(x-t)$. Assume that $$\lim_{t\rightarrow 0}|f_{t}-f|_{\infty}=0$$Prove that under these conditions there is a uniformly ...
2
votes
3answers
166 views

Prove inequality in complex numbers in an unit circle

Given $|\omega| < 1$, $\omega \neq 0$ and $|z| < 1$. Prove inequality: $$\frac{|\frac{|\omega|}{\omega}z+1|}{|1-z \bar \omega|} \le \frac{2}{1-|z|}$$ It is simple but i have problems with it. ...
13
votes
1answer
349 views

Manifold of Density Matrices

Let $\mathrm{M}_{d\times d}\left(\mathbb{C}\right)$ denote the set of all $d\times d$-matrices with complex entries. My goal is to show that the set $\mathcal{M}:= \left\{ \rho\in \mathrm{M}_{d\times ...
4
votes
2answers
10k views

Trapezoid area proof by dividing it into two triangles?

I am trying to figure out how the formula for the area of a trapezoid with exactly two parallel sides is deduced. In my textbook it says that the formula for the area of a trapezoid is deduced by ...
3
votes
1answer
115 views

Domain of bijectivity of function $f:\mathbb{C}\rightarrow\mathbb{C}$

There is a type of problems in my course in Complex analysis that I don't fully understand them. Given function $f:\mathbb{C}\rightarrow\mathbb{C}$, $f(z)=z^2$. You must specify the analytic and ...
1
vote
1answer
127 views

Which of the following improper integrals are convergent?

Which of the following improper integrals are convergent? a.$\quad\displaystyle \int_{1}^{\infty} \frac{dx}{\sqrt{x^3+ 2x + 2}}$ b. $\quad\displaystyle \int_{0}^{5}\frac{dx}{(x^2− 5x + 6)}$ ...
7
votes
3answers
373 views

Free group n contains subgroup of index 2

My problem is to show that any free group $F_{n}$ has a normal subgroup of index 2. I know that any subgroup of index 2 is normal. But how do I find a subgroup of index 2? The subgroup needs to have ...
8
votes
5answers
550 views

Successful approaches to the modelization of ''randomness''

If you pick a number $x$ randomly from $[0,100]$, we would naturally say that the probability of $x>50$ is $1/2$, right? This is because we assumed that randomly meant that the experiment was to ...

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