0
votes
0answers
4 views

Online Video Lectures for Graduate Level Mathematics?

I am wondering if there is a nice compilation of good video lectures in graduate level mathematics? I mean a website, or a forum, or maybe something like course-era (which is mostly undergraduate ...
0
votes
0answers
15 views

Proof by induction

I've proved the base case where $n=1$ and made the assumption that $n=k$ is true, but I'm stuck on the $n=k+1$ part. I just cannot seem to get the algebra to work in my favor. Here is the original:...
0
votes
0answers
7 views

B of Eilenberg-MacLane space

Let $K(m, \mathbb{Z})$ be the Eilenberg-MacLane space. I've read that $BK(m, \mathbb{Z}) = K(m+1, \mathbb{Z})$, but I'm trying to understand this. I'm familiar with the bar construction $BG$ for a ...
0
votes
1answer
16 views

Show that $(1-i)^{7}=8(1+i)$

Homework Problem: Show that $(1-i)^{7}=8(1+i)$ by converting to polar form, taking the seventh power, and then converting back to standard form. Here is what I have: $r=\sqrt{2}$ and $\theta=\arctan{(...
0
votes
0answers
9 views

What is probability that a team reaches final if we know the probabilities of all opponents in the semi-final?

Our Discrete Math professor asked us a question as the Euros are going on. Given the following info, what is the probability that Portugal will make it to the final? Win Probabilities in quarter ...
1
vote
0answers
8 views

How can I compute completions of rings?

I want to learn about how to compute the completions of local rings. For example, I want to be able to compute the completions of \begin{align*} \left(\frac{\mathbb{C}[x,y]}{(y^2 - x)}\right)_{(x,y)} ...
-2
votes
0answers
14 views

how to solve this manually with using scientific calculator— (2.0012) raise to the power 107.

How to solve this manually with using scientific calculator-- (2.0012) raise to the power 107.
0
votes
0answers
7 views

How to integrate discrete data by Gaussian quadrature method

I'm trying to numerically integrate discrete data by Gaussian quadrature method. The file attached "test.mat" is a discrete data set taken from a finite-element mode solver and has the following ...
1
vote
0answers
8 views

Let $f \colon \mathbb{R}^n \to \mathbb{R}^n$ of class $C^{1}$ , $f(x)=0 \forall |x| \geq r>0$ so $\displaystyle \int_{B[0,k]}$ det$Jf(x)=0$

Let $f \colon \mathbb{R}^n \to \mathbb{R}^n$ of class $C^{1}$, exists $r>0$ such that $f(x)=0$ for any $|x| \geq r$. Prove that exists $k>0$ such that: $\displaystyle \int_{B[0,k]}$ det$Jf(x)=0$...
2
votes
0answers
33 views

Project Euler's, Problem #565

Project Euler's, Problem #565 states: Let $\sigma(n)$ be the sum of the divisors of $n$. E.g. the divisors of $4$ are $1, 2$ and $4$, so $\sigma(4)=7$. The numbers $n$ not exceeding $20$ ...
1
vote
1answer
18 views

Does the following fractional series converge?

Does $\sum_{n=1}^{\infty}(\frac{5^{2n+1}}{n^n})$ converge?. So far I believe it does not since numerator seems to me to grow faster but I was not able to write the proof, so any hint is very ...
0
votes
1answer
18 views

Smallest number $n$ for which $p\mid n!+1$ and $n\nmid p-1$

My question is that: What is the smallest positive integer $n$ such that $n!+1$ is divisible by $p$ and $p-1$ is not divisible by $n$ and give some examples for $n$ This is my question, I try to ...
1
vote
0answers
11 views

Fourier series: Where is the source of resonance in the original input signal?

I understand that Fourier series approximate the input signal well and series converge to the original function. But, the system "ODE", such as "$x''+Ax'+Bx=f(t)$", where $f(t)$ is a periodic function ...
2
votes
0answers
12 views

Seeking help with solving a Volterra integral equation of the second kind

I have the following equation \begin{equation} F(\theta) + (c)^{\frac{1}{c}-1}\sqrt{\frac{c}{2\pi}}\int\limits_0^\theta \frac{\theta^{\frac{1}{c}} - \tau^{\frac{1}{c}}}{(\theta - \tau)^{3/2}}\exp{\...
0
votes
0answers
5 views

Solve the closed form solution for argmax of $ x^Ty - x^T\ln(x)$

Let $y \in \mathbb{R}^n, \ln(x) = \begin{bmatrix} \ln(x_1) \\ \vdots \\ \ln(x_n) \end{bmatrix} \in \mathbb{R}^n$ Show that $$x^* = \text{argmax}_{x \in \mathcal{D}} \quad x^Ty - x^T\ln(x)$$ ...
0
votes
0answers
7 views

Can I calculate a fractional sum with functional equations and/or infinite series?

I was wondering if I can calculate fractional sums (non-integer sums) with functional equations in the following manner $$f(x):=\sum_{n=1}^xg(n)$$ $$f(x)=g(x)+f(x-1)\tag1$$ $(1)$ most certainly ...
0
votes
1answer
27 views

A group is simple if an only if its homomorphic images are the trivial group and G itself(up to isomorphism)

I need to prove the following: Let $G$ be a group. Then it's simple if and only if there is only surjective homomorphism $G \to G'$ for $G' = \{ e \}$ or $G' \cong G$. Not sure how to approach ...
4
votes
0answers
20 views

is the number of isomorphism classes of quotients of a finite dimensional commutative ring over a field finite?

If $A$ is a finite dimensional unital and commutative algebra over some infinite field $k$, what is the number of isomophism classes of rings of the form $A/I$ where $I$ is a proper ideal of $A$? Is ...
1
vote
4answers
30 views

Switch algebraic sign

I can't believe that I seriously ask this question as it is so simple. Given this $-x^3+4x$ I'd like to factor out -x, so I did $-x(x^2-4)$ which equals $-x(x^2-2^2)$ equals $-x(x-2)(x+2)$ ...
-5
votes
3answers
35 views

Please sketch the graph of this function. [on hold]

Please sketch the graph of this function. y=2x/x-1 Showing of solution is very much appreciated. THANK YOU
0
votes
2answers
23 views

Linear approximation with error

You want to tile a square floor. You were told that the measurements are 15 feet by 15 feet. You buy 225 square feet of tile, but after using up all of your tile you discover that there are still 5 ...
0
votes
1answer
36 views

How to prove $(1-x) y \ge 0$ is a convex set?

$x \epsilon [0,1], y> 0 $ Let $(1-\underline{x}) \underline{y} \geq 0 $ and $(1-\bar{x}) \bar{y} \geq 0 $ Let $t \epsilon [0,1]$ $[1- (t\underline{x}+ (1-t)\bar{x})] (t\underline{y}+ (1-t)\bar{y}...
0
votes
1answer
36 views

Checkers Board Problem

Here we consider a checkerboard expanded to size 12 × 12 instead of the ordinary 8 × 8 checkerboard. a) How many squares on this board contain more than a third of the total number of dark small ...
0
votes
0answers
11 views

Something is wrong with this argument (Lorentz and Rosenthal-Woo sequence spaces)

Fix once and for all $1<p<\infty$. Throughout, $w=(w_n)_{n=1}^\infty$ will denote a sequence of positive real weights satisfying \begin{equation}1=w_1\geq w_2\geq w_3\geq\cdots>0\;\;\;\text{ ...
1
vote
0answers
18 views

Ito's Formula applied to a weird equation…

I was just wondering if someone could explain how to solve this problem. I have an equation $X(t, S_{t}) = s\partial_{s}u - u$ where we have $u(t,S_{t})$. Now, according to the paper, evaluating $dX$ ...
1
vote
2answers
63 views

How to understand this integral result?

I was reading this page on Wikipedia: Birthday Attack I can understand up until how to approximate the minimal number of attempts for a given probability $$n(p; H) \approx \sqrt{2H \log \frac 1{1-p}...
1
vote
1answer
30 views

Finding the average distance between bounces of an object randomly bouncing on the inside of a sphere

I have a sphere or radius r that has a small point like object moving at a constant velocity inside it. Each time it hits the side it bounces in a random direction but always away from the tangent ...
0
votes
0answers
16 views

Martingale system expected winnings

Suppose we have a starting amount of money and gamble them using the martingale system by doubling the starting bet on a loss and resetting to out starting bet on a win. How can we calculate the ...
-2
votes
1answer
18 views

How to convert liters to quarts by linear approximation?

One liter is 1.0567 quarts. How do you use linear approximation to estimate how many liters are in one quart?
0
votes
1answer
20 views

How to prove $Acos(\omega t-\phi)$ = $acos(\omega t)$ + $bsin(\omega t)$ using $e^{i\theta}$?

I want to show that $Acos(\omega t-\phi)$ = $acos(\omega t)$ + $bsin(\omega t)$ First I verified for myself through the angle addition proof that: $$ cos(\omega t+ \phi) = cos(\omega t)cos(\phi) - ...
0
votes
0answers
9 views

Feasible Region of QCQP and Semidefinite Programs

I am trying to visualize the feasible region of a Quadratically Constrained Quadratic Program (QCQP) which is expected to be non convex (actually is a set of ellipses in $\mathbb{R}^2$) and the ...
0
votes
0answers
21 views

Limit of a probability vector.

There will be a 200 rep bounty for this question. Let $U_1,...,Un$ be iid uniform on (0,1). Set $L_n=\max_{i\leq n} U_i$. Also $S(n)= \inf\{i\leq n| U_i = L_n \}$ the time were the highest value is ...
1
vote
0answers
8 views

Resolvent Inequality

Let $H$ be a Hermitian matrix and $h$ some vector of the same length. The resolvent of $H$ at $z\in\mathbb C$ shall be denoted by $$G(z):=(H-z\cdot1)^{-1}.$$ Is it true that $$\frac{(\Im z)\left(1+\...
1
vote
1answer
12 views

how to generate a Bernoulli sequence of 1 and -1 with autocorrelation 0.3

Please provide some hint about how to generate a Bernoulli sequence of 1 and -1 with autocorrelation 0.3 .
1
vote
0answers
29 views

On comparing two different notions of compactly generated space

I have encountered, in different circumstances, the following two slightly different categories: The full category of $\mathsf{Top}$ consisting of all objects that are: a) topological spaces ...
0
votes
0answers
17 views

smooth manifold, canonical embedding

Let $X$ be a smooth manifold and $U\subseteq X$ open. Define a canonical smooth structure on $U$, for which the embedding $U\to X$ is smooth. Hello, I want to solve this task. My try was as follows:...
0
votes
1answer
41 views

Numbers not expressible as a sum of an arithmetic progression

For every integer $d \geq 1$, let $M_d$ be the set of all positive integers that cannot be written as a sum of an arithmetic progression with difference $d$, having at least two terms and consisting ...
2
votes
0answers
9 views

Incompatible first and second fundamental forms

Say the first and second fundamental forms of a surface (a and b) in 2D are incompatible (i.e. they do not satisfy the Codazzi-Mainardi equations), then the "surface" cannot be embedded in 3D. Is this ...
0
votes
0answers
13 views

$0$-cohomology of a presheaf and the associated sheaf

Let $F$ be a presheaf of abelian groups on $X$ (topological space for example, nice enough) and $\tilde{F}$ the associated sheaf. Then if $f\in H^0(X,\tilde{F})$, we can find a cover $\mathfrak{U}=(...
5
votes
0answers
30 views

Generalizing Ramanujan's cube roots of cubic roots identities

(This extends this post.) Define the function, $$\sqrt[3]{G(t)} = \sqrt[3]{t+x_1}+\sqrt[3]{t+x_2}+\sqrt[3]{t+x_3}\tag1$$ where the $x_i$ are roots of the cubic, $$x^3+ax^2+bx+c=0\tag2$$ While $G(t)...
0
votes
0answers
15 views

Proof of propositional logic theorem using Induction on Formulas

How to prove the following theorem using induction on formulas? Let V and V' be two valuations of L. Let $\alpha$ be a formula such that V(p) = V'(p), for all atomic formula p that is subformula ...
1
vote
1answer
29 views

Soft Question: Good strategies for writing a technical book or textbook

Most people would agree that doing technical work, whether it be pure or applied, and learning the background knowledge necessary to do this work comprises most of the literature and curriculum in ...
1
vote
1answer
26 views

Maximize the number of non zero elements of a product of binary matrices.

I want to find two binary matrices $A$ of size $N \times M$ and $B$ of size $M \times N$ such that: $AB=C$ is a strictly lower-triangular matrix ($j \geq i \implies C_{ij}=0$) The number of ...
0
votes
0answers
13 views

representations of SL_n(R) or SL_n(C)

Could someone point me to a reference for the finite dimensional representation theory of $G=SL_n(F)$ where $F = \mathbb{R}$ or $\mathbb{C}$? In particular, I want to know what this "highest weight" ...
1
vote
1answer
40 views

Automorphisms action on $\mathbb C$

If we know that the $$Aut \ \mathbb C=\{z\mapsto az+b:a\in \mathbb C^*,b\in \mathbb C\}$$ What are the subgroups of $Aut \ \mathbb C$ that act withiut fixed points and properly discontinuously on $\...
0
votes
0answers
14 views

“Solving” for a sequence given an (expected) expression for the summation

Consider the "equation" \begin{equation} \frac{1}{a_n}\sum_{k=1}^n ka_k = \mathcal{O}\left(\frac{n^2}{\log n}\right).\tag{1}\label{eq:conjec} \end{equation} Does there exist some monotonically ...
0
votes
0answers
28 views

For what value of $c$ is $f$ periodic?

Let $f(x)=a\sin(cx)+b\cos(cx)$, where $a$, $b$ and $c$ are constants. Since $\sin$ and $\cos$ have a period of $2\pi$, if $c\in\mathbb{Z}$ then $f$ has a period of $2\pi$. How to prove the converse? ...
0
votes
0answers
5 views

Structures in Non-linear Sigma Model

I debated whether this belongs here, or on Physics SE, but all my questions correspond to the algebraic/complex geometry, not physics, so hopefully it's okay here. The non-linear sigma model ...

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