1
vote
0answers
10 views

PDE basic traffic flow problem

I am analyzing a basic example of traffic flow presented here http://people.uncw.edu/hermanr/pde1/PDEbook/PDE_Main.pdf and have a question to the last transition in the traffic flow equation ...
1
vote
2answers
24 views

If $F ⊂ \mathbb{C}$ and $α_1,\ldots,α_n$ ∈ $\mathbb{C}$ are algebraic over $F$, prove that $F(α_1, \ldots, α_n)$ is a simple extension of $F$.

The statement is true for $n = 1$: $F(α_1)/F$ is finite (since $α_1$ is algebraic over $F$) and is simple, so we need to show that it holds for $n ≥ 2$. By induction on $n$; suppose the theorem is true ...
0
votes
0answers
6 views

Solution for the inhomogeneous 3D heat equation with initial temperature distribution

Can anyone describe the general solution for the inhomogeneous 3-dimensional heat equation: $u_t = K\nabla^2u + \frac{1}{c\rho}f$, with initial condition $u(x, 0) = g(x)$, no boundary conditions. ...
0
votes
1answer
16 views

Determine Integrability without use of Riemann Integral

Determine the function is integrable or not on its interval of definition: $f(x)=\begin{cases}0 \quad \textrm{if} \quad 0\le x\le1\\ x \quad \textrm{if} \quad 1\lt x \le2\end{cases}$ So in our class ...
3
votes
1answer
37 views

Proving that $\lim_{x\to 2}\frac{x^2-5x}{x^2+2}=-1$ using the $\epsilon$-$\delta$ definition of a limit

My attempt: $$ \left|\frac{x^2-5x}{x^2+2}+1\right|<\left|\frac{x^2-5x}{x^2+2}\right|< \left|\frac{x^2-5x}{x^2}\right|<\frac{1}{x^2}|x^2-5x|,$$ using the restriction $|x-2|<2$, so $0<x&...
1
vote
1answer
24 views

Matrices representing injective homomorphisms

Let $R$ be a ring and $M$, $N$ finitely genereated free modules modules over $R$. Let $A$ be a matrix representing a homomorphism $f: M \rightarrow N$. We know that the map $f$ is injective if and ...
1
vote
2answers
23 views

Question about surjective functions

Let $S$ and $T$ be sets and let $f : S \to T$ be a function. Prove the following: If $R$ is a set and $h:R \rightarrow S$ is a function such that $f \circ h$ is surjetive then also $f$ is ...
1
vote
1answer
32 views

How can these row-equivalent matrices have the same determinant?

I am trying to prove that the geometric multiplicity of an eigenvector is bounded by the algebraic multiplicity. One particular proof of this theorem that I like is contained in the answer by Mariano ...
1
vote
0answers
26 views

Optimal Apple Eating Strategy

You hate apples. As a result, you have angered the apple king and are being punished. You will have to eat $n$ apples before the apple king is willing to let you leave. The apples are marked from $1$ ...
3
votes
0answers
24 views

Why is it some regularized sums end up like this?

Suppose we have a regularizable divergent series or product, something along the lines of $$\sum_{n=k}^\infty a_k\text{ - or - }\prod_{n=k}^\infty a_k$$ In particular, we may have $\sum_{n=1}^\infty ...
2
votes
1answer
16 views

If set $A_n$ converges to $A$ in $L^1$, then is it also true for the $r$-neighborhood?

Let $A$ be a bounded simply connected set in $\mathbb{R}^2$, and $A^r$ denote the $r$-neighborhood of $A$, that is, $A^r:=\{x: d(x,A) \le r\}$. Suppose there exists $A_n \rightarrow A$ in $L^1$, that ...
3
votes
1answer
66 views

What is the derivative of $x^i$?

What would the derivative be of $x^i$? Would it simply be $ix^{(1-i)} $? I tried running the Power rule, and I got that is that right?
3
votes
3answers
56 views

Prove that all trees are bipartite

I've been trying to prove this for a while. I can think about it intuitively, but I can't come up with a formal proof. I would appreciate some help. Here's how I'm thinking about it Let T be the ...
0
votes
0answers
8 views

Is it possibile to compute the transient response from the FFT?

I'm measuring the user acceleration using the iPhone accelerometer, but I have to discard the initial data that is taken when the user is still not moving and has to put the phone in the pockets. Like ...
0
votes
0answers
9 views

Topology on function space between smooth manifolds

I'm having trouble reconciling my intuitive notion of "functions close to $f$" with the seemingly very technical definition of the weak and strong topologies on $C^r(M,N)$ in Hirsch's Differential ...
2
votes
3answers
30 views

$\epsilon$-$\delta$ proof, $\lim\limits_{x \to a}$ $\frac{1}{x}$ = $\frac{1}{a}.$

It's been a while since I've studied $\epsilon$-$\delta$ proofs so I'm trying to get a good understanding of how to go about solving this. Working backwards using this as a reference: $\epsilon$-$\...
1
vote
1answer
12 views

Un-monochromatic Arithmetic Progressions

Prove that the set $\{1,2,3,...,2008\}$ can be colored with two colors such that any $18$ term arithmetic progression in this set is not monochromatic.
0
votes
1answer
19 views

Question about injective functions

Let $S$ and $T$ be sets and let $f : S \to T$ be a function. Prove the following: If $U$ is a set $g:T\rightarrow U$ is a function such that $g \circ f$ is injective then also $f$ is injective ...
1
vote
1answer
15 views

Proving a Legendre function using generating function

I must prove that $\int_{-1}^1 (1-2xt+t^2)^{-1/2}P_n(x)dx=\frac{2t^n}{2n+1}$. I know that the generating function is $(1-2xt+t^2)^{-1/2}=\sum_{n=0}^\infty P_n(x)t^n$. I also know that the ...
0
votes
1answer
15 views

A question about retraction mapping

Let $f:\overline{B}_r(0)\rightarrow \overline{B}_r(0)\setminus\{0\}$ be a continuous function. Suppose that $f(x)=x$, when $x\in \partial \overline{B}_r(0)$. Then we can consider a continuous ...
0
votes
1answer
15 views

Clouser of the set of a convergent sequence

Let $a_n$ be a sequence convergent to $q$ in $\bar U$ where $U$ is a bounded open set in $\mathbb R^m$. Let $B=\{a_n; n\in \mathbb N\}$. Why is $\bar B=B\cup\{q\}$ if $q$ in the boundary of $U$ and $\...
0
votes
0answers
14 views

Floor function algebraic manipulation

Well, I'm working with a matrix $D=\vert d(j,x):1\leq j\leq m,1\leq x\leq 2^{p}\vert$ where $m=\frac{n}{2}$ and $j,x,p,n$ are positive integers and $n \equiv 2 \pmod 4$. Its elements are defined by: $...
6
votes
0answers
37 views

Practicality of the Lebesgue integral

I am getting pretty frustrated with the Lebesgue integral mainly because it seems highly impractical to calculate anything non-trivial. Whenever I look for a concrete calculation all I see are ...
1
vote
1answer
30 views

I can't derive the integrating factor of this first order ODE from the Dover textbook

I'm a junior mechanical engineering student. I can't derive the integrating factor of this first order ODE $(x^ 2 - y^2 - y) dx - (x^ 2 - y^2 - x) dy = O$ The textbook provides 5 integrating ...
8
votes
0answers
60 views

Is there a probability measure on $[0,1]$ with no subsets with measure $\frac{1}{2}$?

I have a decidedly weird question. Does there exist a probability measure $(\mu, \mathcal{F})$ on $[0,1]$ such that 1) $\mu(x) = 0$ for every $x \in [0,1]$ 2) For every $r \in [0,1] \setminus \...
1
vote
1answer
29 views

Rearrangement of equations

Trying to find solutions to the equation of $$(1/x)+1=2\sin⁡x+3$$ and I have to rearrange it into the form of $x=f(x)$ as I am using the fixed point method and iterations. Struggling to find new ...
1
vote
0answers
9 views

Adjacent order statistic of transformed random variables

I have $n$ draws $X_1, X_2, \ldots,X_n$ from a random variable $X$ which is continuous and has values between zero and $\overline{x}$. Let $Y(k)$ be a linear transformation of $X$ such that $Y(k)=a(k)...
0
votes
1answer
31 views

Poisson distribution with more than one lambda.

An archaeologist has two old pieces of wood and shall decide which piece of wood is the oldest. Radioactivity from the pieces of wood are recorded by a counter. Number of registrations per unit time ...
2
votes
2answers
18 views

The lower bound of the smallest eigenvalue of a symmetric positive definite matrix

I encounter a symmetric positive definite matrix whose features are all diagonal entries are $1$. all the other entries are in $[0, 1)$, but the matrix is not diagonally dominant. Now I am ...
3
votes
2answers
20 views

Derivation of Taylor expansion with the $a$ term

If we have $$f(x) = \sum_{n=0}^\infty a_n x^n$$ The $k$th derivative is $$f^{(k)}(x) = \sum_{n=0}^{\infty} a_{n+k} \frac{(n+k)!}{n!} x^n$$ Which also means that $$f^{(k)}(0) = k! a_k$$ Implying ...
0
votes
1answer
28 views

help with finding roots using excel or matlab

I've been asked to finding the roots of the equation $x^3-6x+2=0$. They've called it the 'pivot method' but I can't seem to find the 'real' name of it. I've looked at secant, newton, bisection etc... ...
1
vote
1answer
6 views

Spanning 2-regular subgraphs in even regular graphs.

Theorem: Every regular graph of positive even degree has a spanning 2-regular subgraph. This was taken from Corollary 5.10 of ETH Zurich's notes on graph theory. The proof constructs a Eulerian tour,...
2
votes
2answers
26 views

Is orthogonality of column vectors preserved after right-multiplication by unitary matrix?

$\mathbf V$ is an $n \times (n-1)$ matrix with mutually orthogonal columns. $\mathbf Q$ is a unitary matrix of size $(n-1) \times (n-1)$. Is there a concise algebraic proof that the columns of $\...
0
votes
0answers
14 views

Recursive formula in term of original value

$$P_1=P_0G_{0,1}A_1\\ P_m=P_{m-1}G_{m-1,m}A_m+A_0\sum_{i=0}^{m-2}P_i G_{i,m}~~\text{for}~~m\geq 2$$ Is it possible to write $P_m$ in terms of only $P_0$, i.e., without other $P_j$ terms?
1
vote
2answers
39 views

How to formally prove whether this function is onto or not?$ K(x) = x^ 2$ where $x \ge 0$.

$K(x) = x^2.$ The domain and range of this function comprise of non-negative real numbers. If it were real numbers instead of "non-negative" real numbers, then it seems easy to prove it by ...
-4
votes
1answer
24 views

equations, formula, or proofs that cause one to feel “sad,” or negatively in general. [on hold]

It's extremely commonplace for mathematician's to refer to work as "elegant," "beautiful," and I've seen many compare the process of doing mathematics to painting, or playing music. I think that for ...
0
votes
0answers
7 views

A matrix decomposition problem for row/column element order

Indeed, I don't know how to classify this problem, but I try to use matrix to describe it. The problem is that there exists a function $f(x, y)$ and its exact form remains unknown. But I have some ...
1
vote
0answers
38 views

I need someone to explain this shape to me.

The hexagon fits in the circle perfectly, the circle fits in the square perfectly. but the hexagon doesn't fit in the square perfectly. Doesn't this defy this formula below? a = b b = c a = c (what ...
1
vote
4answers
36 views

Why must the determinant of the hessian of a scalar function be positive for there to be a local min/max? Intuition needed

Is there any intuition behind having the determinant of the Hessian matrix being negative corresponding to a saddle point, and positive corresponding to a max/min depending on the sign of $f_{xx}$ for ...
2
votes
1answer
24 views

Adjoint of a normal operator A is a polynomial in A

Is it true that adjoint of a normal operator A can be written as a polynomial in A?
2
votes
1answer
27 views

Complementary Text to Gunning and Rossi - Analytic functions in several complex variables

I'm currently a second year student who has a background in group theory, ring theory, galois theory, metric spaces and point set topology. I'm currently taking courses in algebraic topology, advanced ...
1
vote
0answers
21 views

Evaluation of $\exp\left(a\frac{d^2}{dx^2}\right)f(x)$

I know that \begin{align*} \exp\left(a\frac{d}{dx}\right)f(x)=f(x+a)\,, \end{align*} by comparing the Taylor expansions of both sides ($f(x)$ is an arbitrary function). However, if I have, where $f(...
0
votes
0answers
14 views

What could we say about the isometric matrix $(df_{v,w})^*$?

Let an $n$-gon $P$ and a regular $n$-gon $Q$ be isospectral. If they were isometrics, what could we say about the transpose matrix $(df_{v,w})^*$ from $g(v,w) = f^* g'(v,w) = (df_{v,w})^* g'(f(v),f(w)...
0
votes
3answers
30 views

Fermat's Little Theorem and Legendre symbol

I have two questions: Q1: Why is the order of $19$ modulo $29$ equal to $28$? We know by Fermat's Little Theorem that $a^{28} \equiv 1 \pmod{29}$, but why is $28$ the smallest here? Q2: Let $\left(\...
2
votes
1answer
11 views

If $\sigma (1 \cdots n) = (1 \cdots n) \sigma$ then $\sigma = (1 \cdots n)^i$ for certain $i$

Question In my group theory course, I am asked to show for $\sigma \in S_n$ that if $\sigma (1 \cdots n) = (1 \cdots n) \sigma$ then $\sigma = (1 \cdots n)^i$ for certain $i$. My answer Let $\sigma \...
0
votes
0answers
7 views

Applying the Hoeffding bound compenentwise to a vector

Given $v \in [0,1]^n$ and let $r$ be a random vector such that $\mathbf{E}[r_i] = v_i$ for $i$ in $\{ 1, \dots, n \}$, i.e., $r_i$ is an unbiased estimate for $v_i$. Can I now just apply a Hoeffding +...
2
votes
1answer
7 views

Of $n^2$ points of intersection, $np$ lie on curve of deg. $p < n$, then remaining $n(n - p)$ lie on a curve of deg. $n - p$

Let $C$, $C'$ be two plane curves of degree $n$. Is the following statement true or not? Suppose that of the $n^2$ points of intersection, $np$ lie on a curve of degree $p < n$, then the ...
-2
votes
1answer
36 views

Loaded coin: probability I will never get two heads [on hold]

I'm currently starting to learn probability in an intro course in college and am wondering how to solve this. Given a loaded coin that gives a 60% chance of flipping heads, and 40% chance of tails, ...
-1
votes
0answers
11 views

solve the initial value problem on the half line for the diffusion equation $U_x(t,0)=\sin t$ [on hold]

solve $U_t-U_{xx}=0$ for the half line with initial conditions: $$\quad Ux(t,0)=\sin t\\ U(0,x)=x$$
1
vote
0answers
12 views

Equivalence of two definitions of proper morphism for separated varieties

In Harthshorne (and pretty much everywhere else I know) a proper morphism is defined to be A morphism of schemes $f:X\to Y$ is said to be proper if it is sperated, of finite type and ...

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