1
vote
0answers
19 views

Wave equation. Perturbation methods

The problem I want to solve is, \begin{equation*} u_{tt} = u_{xx}, \, t >0, \end{equation*} \begin{equation*} u_x(0,t) = - \epsilon, \, u_x(1,t) = \epsilon, \, t > 0, \end{equation*} ...
0
votes
1answer
27 views

Union of compact sets

Let $K_1$,$K_2$,...,$K_N$ be compact subsets of the metric space $(X,d)$. Show that a) $K_1\cap K_2\,\cap\,...\cap\,K_N$ is compact. b) $K_1\cup K_2\,\cup\,...\cup\,K_N$ is compact. My own idea is ...
0
votes
1answer
16 views

Absolute convergence of an infinite complex series

Let $(z_n)$ be a sequence of complex numbers. If $\sum_{n=1}^\infty \vert z_n\vert$ converges then $\sum_{n=1}^\infty z_n$ does too. Show that this statement is true assuming we know this is true for ...
0
votes
1answer
22 views

Can a graph be a subgraph if each component is not connected?

Say you two connected components of G. Can they be "combined" as one subgraph with no edges connecting them?
0
votes
1answer
27 views

Which $y$ does verify: $y \pm \sqrt{y^2-4} > 0$? [on hold]

How should one analyze this kind of inequalities?
0
votes
0answers
17 views

Characteristics of $\vec u$ in equation of the reflection of $\vec x$ about the line $N$

The linear transformation of the reflection of $\vec x$ about line $N$ is $$\vec x = 2 \text{proj}_N(\vec x)\vec x - \vec x= 2(\vec x \cdot \vec u) \vec x- \vec x.$$ Is the unit vector, $\vec u$, ...
0
votes
2answers
31 views

Why is the infimum of the following relation 1?

The question was: What is the infimum from $K=\{x ~ |~0 \leq x \lt 1\}$ for the partial order $\{ (x,y) ~ | ~ ( \exists z \in \mathbb{R}_{\geq 0})[x-z=y] \}$ on set $\mathbb{R}$. This was a question ...
0
votes
2answers
15 views

For the ring $\mathbb{Z}[\sqrt{d}]$

For the ring $\mathbb{Z}[\sqrt{d}]$, we define the norm function $f:\mathbb{Z}[\sqrt {d}] \rightarrow \mathbb{N}\cup \{0\}$ by: $ f(a+b\sqrt d)=|a^2-db^2|$. How can I prove that $f(x)=0$ if and only ...
1
vote
1answer
11 views

How to find arbitrage for forward rate of exchange?

Assume that $S(0)$ is the current rate of exchange for foreign currency. Assume that and $K_h$ and $K_f$ are rates of return on home and foreign currency if it is invested over a period $T$. (A) ...
1
vote
0answers
17 views

Every set of pairwise disjoint sets has a choice function implies AC

Suppose that for every collection $A$ of non empty pairwise disjoint sets has a choice function. I need to prove that this implies the axiom of choice. Let $S$ be a collection of sets. For every $B$ ...
0
votes
2answers
13 views

Sign of a permutation including a trivial cycle

This may be a rather basic question, but I can't see mention of this anywhere. Suppose I have a permutation $p\in S_5$ (say). Suppose further that $p$ decomposes as $p=(1 2)(3)(4 5)$. What is the ...
-1
votes
2answers
52 views

Why does this bound of integration become pi/4? [on hold]

Perhaps it is something easy that I am not seeing but I cannot seem to understand why, in this integral, a changes to $\frac\pi4$ 3$\int_0^a$$\frac {dx}{\sqrt {a^2+x^2}}$ I was able to solve the ...
0
votes
0answers
15 views

Are these functions Lebesgue Integrable? How to show this?

I have recently learnt the comparison test, MCT, Fatou's Lemma and DCT for Lebesgue integrals, but have been struggling with the details of the proofs. 1) f is 0 a.e. so is integrable to 0 2) not ...
0
votes
0answers
11 views

Representation formula for a Hilbert space valued Brownian motion. Prove independence of the real-valued Brownian motions in the expansion.

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge 0}$ be a filtration of $\mathcal A$ $(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a separable Hilbert space $Q$ be ...
1
vote
0answers
15 views

Find finite-history recurrence from full-history recurrence?

In this question I asked for a closed-form solution to this functional differential equation: \begin{align*} f'(z) &= e^{-2} (f(z/e))^2 \\ f(0) &= 1. \end{align*} It doesn't look like there ...
0
votes
0answers
13 views

Optimal Airline Pricing under Modified Two-Class Littlewood

I will explain (and attempt to prove) the original problem before introducing the version I am having difficulty with. The original problem goes as follows: An airline owns a plane with capacity ...
1
vote
3answers
65 views

How many numbers between $1$ and $9999$ have sum of their digits equal to $8$? $16$?

How many numbers between $1$ and $9999$ have sum of their digits equal to $8$? $16$? Can someone tell me if I got the right answers? I solved both cases and I've got $148$ for $8$ and $633$ for $16$. ...
1
vote
1answer
46 views

How do I solve $\int_ {-\infty}^\infty xe^{-m(x-a)^2}$?

Evaluate $$\int_{-\infty}^\infty xe^{-m(x-a)^2}$$ where $m$ and $a$ are constants I can solve this if the exponential is simple $e^{x^2}$ by substitution, but this one doesn't work that way as ...
0
votes
0answers
18 views

Euler exponential continued fraction to compute the trigonometric functions and the golden ratio

Using the Euler continued fraction for the exponent, which is convergent everywhere on the complex plane: $$e^{-z}=1-\cfrac{z}{1+z-\cfrac{z}{2+z-\cfrac{2z}{3+z-\cfrac{3z}{4+z-\cdots}}}}$$ We can ...
1
vote
1answer
15 views

Proof verification: If $M$ is maximal, $R/M$ is a field.

Let $M$ be maximal, we want to show that $R/M$ is a field. To do this we wish to show that for all $a\in R$, where $a+M\not\equiv 0+M$, $a+M \in R/M$ is a unit. So we have already excluded $a\in M$, ...
0
votes
1answer
7 views

polygon angle names

Regular polygons can be divided up into triangles of equal sizes. For example, a pentagon would have 5 triangles. Each would have one angle with a value of 72 degrees and two angles of 54 degrees. ...
0
votes
1answer
12 views
1
vote
1answer
12 views

Proof improvement for $(a^2+b^2)(c^2+d^2)(e^2+f^2)(g^2+h^2) = A^2 + B^2$ given $(a+ib)(c+id)(e+if)(g+ih) = A + iB$

If $(a+ib)(c+id)(e+if)(g+ih) = A + iB$, prove that $(a^2+b^2)(c^2+d^2)(e^2+f^2)(g^2+h^2) = A^2 + B^2$ My approach is pretty straightforward: $$(a+ib)(c+id)(e+if)(g+ih)$$ ...
-3
votes
0answers
26 views

prove $\bar{f^{-1}} \cdot \bar{f}=\bar{f(1)}$ [on hold]

prove $\overline{f^{-1}} \cdot \overline{f}=\overline{f(1)}$, where $f$ is a path between $a$ and $b$ in a topological space such that $f:I\rightarrow X$ and $f(0)=a,f(1)=b$ and $\bar{f}$= All paths ...
0
votes
2answers
30 views

Find 2x2 matrix such that its inverse equals its transpose

Find some matrix $B\in GL_2 (\mathbb{R})$ such that $B^{-1} = B^T$ and $B \neq I$ What I tried: I tried to create a simultaneous equation i.e. if B = $\begin{bmatrix} a&b\\c & ...
0
votes
0answers
16 views

How to visualize complex domains

I was hoping if someone can help me visualize complex domains. I know how simplex ones like $|z|<1$ or $\text{Re}z < 1$ look like but for the more complicated ones such as $$\text{Im } z < ...
-4
votes
0answers
20 views

A ball is thrown vertically upwards from ground level and its height ${h}$ meters after ${t}$ seconds is given by the function ${h= 12t-5t^2}$.

At what ${times}$ is the ball at a height of ${7}$ m from the ground At what ${times}$ will the ball return to the ground The ${domain}$ of the above function The greatest ${height}$ reached by the ...
0
votes
1answer
7 views

Find unit vector perpendicular to x-z,x-y, and y-z plane

I'm guessing that the unit vector perpendicular to the x-z plane is $\begin{bmatrix}1\\0\\1\end{bmatrix}$ I'm guessing that the unit vector perpendicular to the x-y plane is ...
0
votes
0answers
18 views

Average number of steps to return to the origin of a random walk on a 2-d lattice.

Suppose I have a random walker on a 2-d square lattice with periodic boundary conditions with equal probability of going in any of the four directions. The walk ends when the walker reaches the point ...
0
votes
0answers
10 views

If $e \in E$ where $E$ is a component of a permutation group, then $e$ has order $2$ or $3$ if $|\mbox{fix}(g)| \le 3$ for all nontrivial $g \in G$

Suppose $G$ is a transitive permutation group such that all four-point stabilizers are trivial, and $G$ has some nontrivial three-point stabilizer. Said differently this means that every nontrivial ...
0
votes
1answer
17 views

Prove that f has a primitive on the annulus

Suppose $f$ is holomorphic on the annulus $D = \{1/2 < |z| < 2\}$. Show that $f$ has a primitive on $D$ if and only if $$\int_{|z|=1}f(z)dz=0$$ Any hints on the main idea
1
vote
1answer
22 views

Distance in metric space, triangle inequality problem

Let $(X, d)$ be a metric space. Let $t\in (0,1]$. Show that $d^t: X\times X\to\mathbb{R}$ $$d^t (x,y) := d(x,y)^t, \forall x,y\in X$$ is also a distance function. Problematic bit is the triangle ...
1
vote
0answers
12 views

Can someone provide some examples to illustrate the difference between Pointwise equicontinuity and Uniform equicontinuity?

I don't know what is with the subject of pointwise and uniform equicontinuity, pretty much all the material you can find online are either: Proofs i.e. pointwise equicontinuity is uniform ...
1
vote
1answer
18 views

can someone explain how bonds work and how to calculate the value of a bond?

I am taking financial mathematics and in class I learned a formula to calculate the value of a bond, $B=\frac{F}{1+r}$ if one payment or if coumpund $B=\frac{F}{(1+r)^n}$ where $B$ is the value of the ...
-1
votes
0answers
10 views

doing a math project my slopes don't make sense

I'm doing a project using slopes. I have a track and have released a marble at 3 different intervals on the track. I have 3 rises and 3 runs. They go from larger numbers (rise 58, 55, 49 to run 62, ...
1
vote
1answer
14 views

Does this use of the delta function make sense?

NOTE: Please do not provide any sort of a solution to what I am trying to do, as this ia an assessed question. Just let me know whether or not it is a valid use and explain please. I am trying to ...
1
vote
1answer
9 views

Show that sample variance is unbiased and a consistent estimator

I am having some trouble to prove that the sample variance is a consistent estimator. I have already proved that sample variance is unbiased. I understand that for point estimates T=Tn to be ...
1
vote
2answers
22 views

Missing step to prove an inequality for bounded analytic function.

The exercise is as follows Exercise: Let $f : \overline{B(0,R)} \to \mathbb{C}$ be a holomorphism with $|f(z)| \leq M$, for some $M > 0$. Show that $$\left| \frac{ f(z) - f(0)}{M^2 - ...
1
vote
1answer
35 views

Limit of $\frac{\tan(x)-x}{x^3}$ as $x$ approaches $0$ without L'Hospital's Rule

I am trying to find the limit of $\frac{\tan(x)-x}{x^3}$ as $x$ approaches $0$. I know that this can be found by using L'Hospital's Rule 3 times. Is there a way to solve this problem without using ...
1
vote
0answers
16 views

Random walks in $\mathbb{Z}^2$

Consider a random walk on the integer lattice in the plane. If a “particle” making a random walk arrives at a lattice point $p = (k_1,k_2)$ at the time $t$, then one of the four neighbors $(k_1±1, k_2 ...
0
votes
0answers
15 views

Classification of the irreducible group representations of the dihedral groups

Let $D_n$ be the dihedral group of order $2n$. Show that all irreducible representations have vector space dimension $1$ or $2$, and describe them up to isomorphism. Any hints how to even start?
0
votes
0answers
5 views

“L-normal” finite strings?

Consider finite strings on a finite alphabet of size $b\ (\ge 2)$. Let's call such a finite string "$L$-normal" just if, for each $l\in\{1,2,...,L\}$, all $b^l$ possible length-$l$ substrings occur in ...
0
votes
2answers
15 views

Prove constant times invertible matrix is also invertible

Let $B\in GL_n(\mathbb{R})$ and $\beta \in \mathbb{R}$ with $\beta \neq 0$. Show $\beta B \in GL_n(\mathbb{R})$ What I tried: I know it intuitively makes sense that this would be the case, but I ...
-1
votes
0answers
22 views

NAUTY algorithm

NAUTY is a Graph Isomorphism(GI) software developed by Brendan McKay to test isomorphism of Graphs. It provides a practical solution to the Graph Isomorphism problem. It is a program for isomorphism ...
0
votes
1answer
30 views

Calculate the limit $\lim_{x \to + \infty}\int_{\mathbb{R}} \frac{t^2}{1+t^2}e^{-(x-t)^2}dt$

I am really confused with x approaching $+ \infty$. How can I solve this limit: $$\lim_{x \to + \infty}\int_{\mathbb{R}} \frac{t^2}{1+t^2}e^{-(x-t)^2}dt$$
3
votes
1answer
25 views

What causes the equating-the-coefficients method not to work?

When searching here to find methods for solving quartic polynomials, I came across a question where one of the solutions (at the very end) mentions that the equating coefficients method can fail. ...
0
votes
0answers
15 views

In how many ways can we select $r$ doughnuts from a box of a dozen doughnuts that has $2$ apple fritters, $3$ sprinkled, $3$ jelly and $4$ glazed?”

In how many ways can we select $r$ doughnuts from a box of a dozen doughnuts that has $2$ apple fritters, $3$ sprinkled, $3$ jelly and $4$ glazed?” For this question, I'm supposed to come up with a ...
1
vote
1answer
11 views

Why is $0$ an eigen value of $L_G$?

I am learning Spectral Graph Theory. If the Laplacian Matrix of a graph $G=(V,E)$ is defined by $(a_{ij})=-1 ;(i,j)\in E, d_i ; i=j$ and $0$ otherwise then how does it follow that $0$ is an ...
0
votes
1answer
29 views

The complex integral of the reciprocal of polynomial is constant on sufficiently large circles

Let $P$ be a polynomial of degree $n ≥ 2$. Suppose all the roots of $P$ lie in the disk $D_{r}(0)$. Let $R > r$ and $$I(R)=\int_{|Z|=R}{1\over P(z)}dz$$ Prove that the integral is constant. I ...
0
votes
0answers
11 views

Matlab inverse tangent

I have a system that contain complex conjugate pole pair in his transfer function. If I examine the function with Matlab's bode it works great and I get results I ...

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