0
votes
0answers
2 views

Expected value of a roll of a fair die given that the number rolled is at least 4

I am trying to understand the solution to a probability problem, and I am having trouble understanding where some of the numbers are coming from. The textbook gives this definition for conditional ...
3
votes
3answers
3k views

What is the “Donkey Theorem”?

I was watching the Turkish version of Who Wants to Be a Millionaire? and they asked this question: What field is the Donkey Case (or I guess it can be translated as Donkey Theorem) related to? ...
0
votes
1answer
801 views

I can't find the Nash equilibrium of this 3x2 game.

Sorry for my English, I am French but i couldn't find help on the French website (so I am here). I have a question about this two-player game: $$ \begin{array}{c|cc} & y_1 & y_2 \\ \hline ...
0
votes
1answer
29 views

Determine probability based on observation

Suppose there is an urn with 100 balls, of two colors, say white and black. Let $p$ be the probability of drawing a white ball. You draw one ball, replacing after the draw. After 100 draws, each with ...
0
votes
2answers
100 views

What is the “taxonomy” or “hierarchy” (partial ordering) of algebraic objects used to attempt to capture geometric intuition?

What follows is a list of terms all of whose relationships to one another I have never fully succeeded in establishing, despite having spent much of 6-8 years trying to so. Feel no need to give ...
0
votes
0answers
6 views

How to evaluate GF(256) element

I wonder is there any easy way to evaluate element of GF(256) meaning that assume I would like to know what is alpha ^(32) or alpha^200 in polynomial form? given the primitive polynomial is D^8+D^4+D^...
-2
votes
1answer
27 views

separate $a$ & $b$ in $\sqrt{a^a*b^b}$

It is already known that $\sqrt{a^a*b^b}$ does not equal $\sqrt(a^a)*\sqrt(b^b) = a*b$ Is there any other method to separate $a$ and $b$?
0
votes
0answers
8 views

How may I use a 3x3 matrix to simulate a larger square matrix?

I am using a game engine where the library only provides 3x3 matrices with the multiplication and inverse operation. I could build my own matrix library to provide larger matrices, but it would be ...
0
votes
1answer
10 views

Product rule for derivatives: one is bounded and the other is differentiable

Let $S \subset \mathbb{R}$, $f: S \to \mathbb{R}$ be bounded and $g: S \to \mathbb{R}$ be differentiable at $c \in S$ such that $g^{\prime}(c) = g(c) = 0$. Show that $h = fg$ is differentiable at $c$ ...
0
votes
2answers
17 views

Prove that if $p \equiv 3 \pmod{4}$ then $x^2+y^2 \not \equiv 0 \pmod{p}$

Let $x$ and $y$ be integers not congruent to $0$ modulo $p$ where $p$ is a prime. Prove that if $p \equiv 3 \pmod{4}$ then $x^2+y^2 \not \equiv 0 \pmod{p}$. I thought about proving this ...
2
votes
2answers
24 views

Representing $\ln(x)$ as a power series centered at $2$ without computing any derivatives

I am working through a calc book and one of the problems asks the above question. However, taylor and maclaurin series have not been introduced yet. In some worked examples, they leverage old series,...
1
vote
3answers
19 views

Assume that $x_n \to 0$. Let $σ: N → N$ be a bijection. Define a new sequence $y_n := x_{σ(n)}$, Show that $y_n → 0$.

Let $(x_n)$ be a sequence. Assume that $x_n \to 0$. Let $σ: N → N$ be a bijection. Define a new sequence $y_n := x_{σ(n)}$, i.e. $\sigma$ is a permutation of the set of natural numbers. Show that $y_n ...
0
votes
0answers
7 views

How to know whether a contact form is only defined locally or globally?

As described e.g. here the following is the standard contact form on $\mathbb R^{2n+1}$: $$ \omega = dz + \sum_{k=1}^n x_k dy_k$$ Similarly, the following is the standard contact form on $S^{2n+1}$: ...
0
votes
0answers
9 views

Any hints on how to prove that $\ln{1\over 2\sin\left({90\over \pi}\right)}=\sum_{n=1}^{\infty}{(-1)^{n-1}B_{2n}\over 2n(2n)!}$?

How do you prove that $$\ln{1\over 2\sin\left({90\over \pi}\right)}=\sum_{n=1}^{\infty}{(-1)^{n-1}B_{2n}\over 2n(2n)!}?\tag1$$ where $B_{2n}$ is bernoulli number Any hints?
0
votes
0answers
20 views

Find the divisors of $5040$ in the Plato's dialogue “Theaetetus”

In the Plato's dialogue "Theaetetus", at a certain point, we have the following "problem" \begin{align*} 5040 &= 7! \\ &= 1\times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \\ &= 2 \...
2
votes
1answer
20 views

$X_n$ Poisson independent, $\mathbb{E}[X_n] = \lambda_n$. If $\sum \lambda_n = +\infty$, then $\frac{S_n}{\mathbb{E}[S_n]} \rightarrow 1$ a.s

Let $X_n$ be independent Poisson random variables with $\mathbb{E}[X_n] = \lambda_n$. Define $S_n = X_1 + \dots + X_n$. Show that if $\sum \lambda_n = +\infty$, then $\frac{S_n}{\mathbb{E}[S_n]} \...
-1
votes
0answers
12 views

Convolution form and $n$-th partial term of $\sum_{k=1}^{+\infty} kp^k$

With $p\in(0,1)$, let $\displaystyle x(k)=\sum_{k=1}^{+\infty}k$ and $\displaystyle \phi(k)=\sum_{k=1}^{+\infty}p^k$ So that $\displaystyle (x*\phi)(k)=\sum_{k=1}^{n}x(k)\phi(n-k)=\sum_{k=1}^{n} kp^{...
2
votes
3answers
34 views

Problem proving “p”, “k” are limit point.

Good night, i have problem with this exercises: $A=\left\{ 1-\frac{1}{n}:n=1,2,\ldots\right\} $ I make this: $$ \lim_{n\rightarrow\infty} \left(1-\frac{1}{n}\right)=1$$ Prove: Be $p=1$ and $r>...
13
votes
7answers
550 views

Why is the complex plane shaped like it is?

So it's always taken for granted that the real number line is perpindicular to multiples of i, but why is that? Why isn't i just at some non-90 degree angle to the real number line? Could someome ...
0
votes
0answers
10 views

Angle between Two Lines in 3D Space

Since my two lines are orthogonal (one in y-z plane and one in y-x), I KNOW there's a (probably simple) formula for the calculation I need, but somehow I haven't been able to find it. Equations of the ...
6
votes
1answer
598 views

Planar graphs with $n \geq 2$ vertices have at least two vertices whose degree is at most 5

This is a homework problem and my solution. I think I get the main ideas and understand what is going on, but I need to work on my proof writing. I don't get full credit when I think I should get a ...
1
vote
0answers
8 views

Minkowski Dimension of Special Cantor Set

As can be seen at the top of the page here (exercise 1), Terry Tao gives an exercise to find the Minkowski Dimension of the Quadnary Cantor Set, and of a special Quadnary Cantor Set. The two sets are:...
1
vote
0answers
14 views

Show that there aren't negative eigenvalues.

I've been trying to solve this Sturm-Liouville theory problem. Show that the problem: $$\left\{\begin{matrix} y''+(x+\lambda)y = 0\\ y(0)=0\\y(1)=0\end{matrix}\right.$$ doesn't have ...
227
votes
7answers
16k views

Best Sets of Lecture Notes and Articles [closed]

Let me start by apologizing if there is another thread on math.se that subsumes this. I was updating my answer to the question here during which I made the claim that "I spend a lot of time sifting ...
2
votes
2answers
31 views

Prove that the integral $\int_0^1 f(t) P(t - x)dt$ is a polynomial in $x$

So suppose $f$ is a generally complex, continuous function on $[0,1]$ and $P$ is a polynomial defined on the real numbers. Evidently $$ \int_0^1 f(t)P(t - x)\,dt $$ is a polynomial in $x$ but that ...
1
vote
1answer
17 views

If y is not an exterior point of $K$, then there exists a $x$ in $K$. Is it true?

For a vector $v = (x_1,\ldots,x_d)^t \in \mathbb{R}^d$, we let the function $f$ be $f(v)=|v|^2=v^tv=x_1^2+\cdots+x_d^2$. Is it possible to show that there exists a x $\in K$ which satisfies $f(x)>...
2
votes
1answer
730 views

Minimal polynomial and field extension

Suppose the degree of a field extension $[\mathbb{Q}(\alpha):\mathbb{Q}]=n\gt 1$ and $\alpha$ is a root of a monic polynomial $f \in \mathbb{Q}[T]$ and the degree of $f$ is $n$. Does the above imply ...
1
vote
1answer
19 views

Quasi-newton methods: SR1 and BFGS inverse update

In Numerical Optimization by Nocedal and Wright, (http://home.agh.edu.pl/~pba/pdfdoc/Numerical_Optimization.pdf) Chapter 2 on unconstrained optimization, page 25 top, the authors claim that "The ...
1
vote
1answer
12 views

$\tau_l(P,P)$ is a primitive $l$th root of unity

I'm trying to understand the Frey-Rück attack on elliptic curves, in particular the following lemma ($\tau_l$ being the Tate-Lichtenbaum pairing, $E(\mathbf{F_q})[l]$ the set of elements of $E(\mathbf{...
7
votes
4answers
101 views

Proving $\sum_{k=1}^{n}{(-1)^{k+1} {{n}\choose{k}}\frac{1}{k}=H_n}$

I've been trying to prove $$\sum_{k=1}^{n}{(-1)^{k+1} {{n}\choose{k}}\frac{1}{k}=H_n}$$ I've tried perturbation and inversion but still nothing. I've even tried expanding the sum to try and find ...
0
votes
0answers
11 views

Intuition/derivation for Cauchy's repeated integral formula?

https://en.m.wikipedia.org/wiki/Cauchy_formula_for_repeated_integration I'm referring to this formula due to Cauchy. The wiki page has a proof, but what I'm looking for is a more direct derivation or ...
1
vote
2answers
33 views

Prove the n-th Fibonacci number is less than $2^n$ for all n greater than zero using strong induction

I need to prove the n-th Fibonacci number is less than $2^n$ for all $n \geq 0$ using strong induction. I have been exposed to the idea that strong induction differs from weak induction in that the ...
1
vote
1answer
10 views

is the trace of inverse of positive, positive definite matrix decreasing?

Let $A, B$ be non-negative, and symmetric positive definite matrices. If $A\le B$, is it true that $\mbox{trace}(A^{-1}) \ge \mbox{trace}(B^{-1})$?
0
votes
2answers
25 views

How to prove main argument formula for any $z\in\mathbb C^*$

I would prove that for any complex number $z \in \mathbb C^*$ such that $z = x + \mathbb i y$ with $(x,y)\in\mathbb R^2$ and $x+\vert z\vert \neq 0$: $$ \arg z = 2\arctan\left(\dfrac{y}{x+\vert z\vert}...
6
votes
0answers
77 views
+50

Simplifying Trace of a Matrix Expression ${\rm Tr} \left( (I- D^{-1} BA^2) B (I- D^{-1} BA^2)^T \right)$

Let $B$ be a symmetric invertible matrix and let $A$ be a diagonal matrix with non-zero entries on the main diagonal. I am trying to simplify the following expression \begin{align} {\rm Tr} \left( ...
0
votes
1answer
15 views

How is a dominant rational map well-defined?

I am slightly confused by the definition of a dominant rational map in Hartshorne, specifically because of a comment he makes about the equivalence relation. In Chapter 1.4, he defines a rational map ...
2
votes
1answer
613 views

Reduce the payoff matrix using (weakly) dominated strategies

Below is the payoff matrix of a game. Use the principle of elimination of (weakly) dominated strategies to simplify the payoff matrix. What is the optimal solution of the game for the row player? ...
3
votes
5answers
128 views

Exponential equation without any sort of calcualtor

Is there a way i can solve the following equation only by using high school mathematics? $$4 \times2^x+3^x=5^x$$ i tried writing $5$ as $2+3$ but didn't get any result. After that i tried to divide ...
-1
votes
0answers
16 views

How to convert a 2D coordinate system into a toroidal coordinate system?

I have a 100x100 square with each point having a node attached to it such that a node at point 0,99 is more similar to a node at point 0,0 than 0,95 lets say. The same can be said about the other axis....
0
votes
0answers
16 views

What are pullbacks of finite-coproduct injections along arbitrary morphisms?

I am studying a definition of an extensive category: An extensive category is a category $E$ with finite coproducts such that pullbacks of finite-coproduct injections along arbitrary morphisms exist ...
4
votes
2answers
63 views

When is a 5th degree polynomial with at least 1 non-real root solvable by radicals?

Let $f(X)$ be an irreducible polynomial of degree 5 with coefficents in the field of rational numbers $\mathbb{Q}$. Assume that $f$ has at least one non-real root in the complex field $\mathbb{C}$. ...
0
votes
0answers
16 views

Derivative of projection

Let $\Omega \subset \mathbb{R}^n$ be a limited domain of class $C^\infty$ (open, connected) and $\varepsilon > 0$ small such that $$ \Omega_\varepsilon = \{y \in \overline{\Omega} : d(y, \partial \...
0
votes
1answer
9 views

For what functions $f$ is $x^{\sf T}Lf(x) \geq 0$?

Let $L = L^{\sf T} \in \mathbb{R}^{n\times n}$ be a (weighted) Laplacian matrix of a connected undirected graph. For those not familiar with Laplacians (they are positive semidefinite); for simplicity ...
-8
votes
0answers
22 views

Quadratic term answer fast please [on hold]

let k be a real number such that k≠0.if a and b are non zero complex numbers satisfying a+b=-2k and a²+b²=4k²-2k,then a quadratic equation having (a+b/a)and (a+b/b)as its roots is equal to
0
votes
1answer
17 views

If A is positive definite (but not necessarily symmetric) can you decompose it?

If A is a $2 \times 2$ matrix that is positive definite but may or may not be symmetric, does there exist another matrix B such that $A=B^TB$?
0
votes
0answers
20 views

Trace of the product of a Lie algebra and Lie group element

Take $U \in SU(n)$ and $X \in \mathfrak{su}(n)$. What can we learn about \begin{align} \text{Tr} (UX) \end{align} In particular Is there a closed form expression? When does $\text{Tr} (UX)$ vanish?...
0
votes
1answer
18 views

How to solve this equation: $v_1\partial_xu+v_2\partial_y u=0$

I have the following equation: $$ v_1\dfrac{\partial u}{\partial x}+v_2\dfrac{\partial u}{\partial y}=0 \tag{1}$$ $u(x,y)$ is the unknown function (a scalar-valued function), $v_1$ and $v_2$ are two ...
1
vote
0answers
30 views

Integral ${\large\int}_0^{\pi/2}\frac{x\,\log\tan x}{\sin x}\,dx$

Could you please help me to find closed form expressions for the following definite integrals: $$I_1=\int_0^{\pi/2}\frac{x\,\log\tan x}{\sin x}\,dx\approx0.3606065973884796896...$$ $$I_2=\int_0^{\pi/3}...
1
vote
0answers
37 views

Given some arbitrary roots of a polynomial p(x,y,z,…) with integer coefficients, is it possible to tell if p has a root in the Gaussian integers?

I'm trying to find if p(x,y,z,...)=0 has a Gaussian integer root (more specifically, I want to find if p has a Gaussian integer root where the imaginary components are even, but if that can't be done, ...

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