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244 views

Discuss how to evaluate $\oint_{C} \frac{e^{z^{2}}}{z^{2}}\,dz$ where C is a simple closed curve enclosing the origin

Discuss how to evaluate $\oint_{C} \frac{e^{z^{2}}}{z^{2}}\,dz$ where C is a simple closed curve enclosing the origin Try using the fact that $e^{z}=\sum_{j=0}^\infty \frac{z_{j}}{j!}$ Could someone ...
0
votes
1answer
409 views

Simple Independence Problem

I have 3 similar problems but I don't quite understand the differences between them and how to approach each. First 2 Problems Lets say you have 2 circuits where all relays function independently: ...
2
votes
1answer
257 views

Constructing finite state automata corresponding to regular expressions. Are my solutions correct?

I have drawn my answers in paint, are they correct? (4c) For the alphabet {0, 1} construct finite state automata corresponding to each of the following regular expressions: (i) 0 My Answer 4ci (ii) ...
1
vote
1answer
1k views

Notation for extracting the index of an element from a set

OK I edited the question. Sorry for the wrong terms. What is the correct notation such that a specific function maps an element of a specific sequence/list/n-tuple to its index? I have researched ...
2
votes
1answer
2k views

Why is the Connect Four gaming board 7x6? (or: algorithm for creating Connect $N$ board)

The Connect Four board is 7x6, as opposed to 8x8, 16x16, or even 4x4. Is there a specific, mathematical reason for this? The reason I'm asking is because I'm developing a program that will be able ...
0
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1answer
77 views

Energy Radiating Off of a Cube Split Into Two Pieces

This is somewhat a Physics question, but ends of being more of an algebra/geometry-related question. Anyway, there is a cube that radiates with a power $P_0$ and it is "cut" into two pieces. These ...
2
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1answer
243 views

Question on Lie's theorem

I am looking a Lie's theorem in Lie algebra liturature but I do not fully understand one part of the proof. The following proof is given in these notes on page 12. Thm. Let $\mathfrak{g}\subset \...
2
votes
0answers
441 views

Matrix Exponential with time parameter

Could someone please expand on Method 9. Lagrange interpolation (page 17) at http://www.cs.cornell.edu/cv/researchpdf/19ways+.pdf because the summation runs from 0 to (n-1) but the eigenvalues ...
0
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2answers
23k views

rate of infection in a population?

I was given this word problem by a friend, and it's stumped me on how to set it up. Hopefully you guys can help. Here it is: "Exactly one person is an isolated population of 10,000 people comes down ...
-1
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4answers
4k views

Find the power series expansion of $f(z)=z^2$ around $z = 2$

Find the power series expansion of $f(z)=z^2$ around $z = 2$. My result was $z^2 = 4 + 4(z − 2) + (z − 2)^2$ But I need different ways to solve Could someone help me through this problem?
5
votes
3answers
188 views

calculate $\int_{0}^{i} e^z\, dz$

calculate $\displaystyle\int_{0}^{i} e^z\, dz$ Could someone help me through this problem?
3
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2answers
227 views

Is it always true that $A^o=(\overline{A})^{o}$?

I'm asking, in a topological space, does a set $A$'s interior equals to $A$'s closure's interior? Or to ask that do all the sets that have the same closure also have the same interior?
5
votes
1answer
331 views

Product of affine schemes

For any ring $A$, define a functor $\text{Spec}(A)$ from rings to sets by $$\text{Spec}(A)(R) = Hom_{\text{Rings}}(A,R)$$ Call a functor $X$ an affine scheme if it is isomorphic to a functor of the ...
2
votes
4answers
10k views

Find tangent line of curve that intersects point.

How do I find the tangent line of the curve $y=x^2$ that intersects the point $(8,2)$?
2
votes
1answer
164 views

DeRham Cohomology

Let $p$ and $q$ be two points of $\mathbb{R}^n$ where let $n\geq 1$. Then $$\dim H^k(\mathbb R^n - p - q) = \begin{cases}0, &\text{ if }k\text{ is not equal to }n-1,\\ 2,&\text{ if }k = n-1.\...
8
votes
2answers
290 views

Prove that $\sum\limits_{n=0}^{\infty}\frac{F_{n}}{2^{n}}= \sum\limits_{n=0}^{\infty}\frac{1}{2^{n}}$

I came up with this identity in high school, and I can't remember how I proved it :P Does anyone know how I would go about doing this? $$\sum_{n=0}^{\infty}\frac{F_{n}}{2^{n}}= \sum_{n=0}^{\infty}\...
2
votes
1answer
68 views

Show that $|\int_{C} \frac{1}{z^{3}+1}\, dz|\leq \frac{\pi}{3}(\frac{R}{R^{3}-1})$

Could someone help me through this problem? Let C be an arc of the circle $|z|=R$, with $R>1$ of angle $\frac{\pi}{3}$. Show that $\left|\displaystyle\int_{C} \frac{1}{z^{3}+1}\, dz\right|\leq \...
3
votes
2answers
298 views

Is the set of conditions for Devaney's definition of chaos minimal?

I am reading Devaney's definition of chaos. Which says: Let $V$ be a set. $f:V \rightarrow V$ is said to be chaotic on V if $f$ has sensitive dependence on initial conditions $f$ is topologically ...
6
votes
2answers
2k views

How to show that if two integral domains are isomorphic, then their corresponding field of quotients are isomorphic? [closed]

If two integral domains $D$ and $D'$ are isomorphic show that their corresponding field of quotients (fractions) $Q(D)$ and $Q(D')$ are isomorphic.
1
vote
0answers
767 views

Prove that if $A$ is null and $f: \mathbb{R} \longrightarrow \mathbb{R}$ has a continuous derivative, then $f(A)$ is null

Prove that if $A$ is null and $f: \mathbb{R} \longrightarrow \mathbb{R}$ has a continuous derivative, then $f(A)$ is null. I think it has something to do with the fact that $f'$ is bounded in any ...
1
vote
0answers
84 views

$\Delta_0$-formulas in the Boolean-valued model $V^B$

I want to show that $\Vert \check x = \check y \Vert = 1$ implies $x = y$, where $\check x$ is the canonical name for $x$ in $V^B$. I'd like try induction over the ranks of $\check x$ and $\check y$ ...
2
votes
1answer
461 views

The Lebesgue differentiation Theorem for Radon measures

Well, I am looking for references of the Lebesgue differentiation theorem generalization for Radon measures. I want also know about results that give us information of uniformity in the limit of the ...
3
votes
2answers
494 views

Box topology on $\prod_{n=1}^\infty\mathbb{R}$

Let $X$ denote $\prod_{n=1}^\infty\mathbb{R}$, the Cartesian product of countably infinitely many copies of $\mathbb R$ (which is just the set of all infinite sequences of real numbers), endowed with ...
0
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1answer
1k views

Meaning of “Radial Separation”?

I'm reading conflicting uses of the term "Radial Separation." On this math forum it is implied to be "the distance between circles" as well as "the distance between the circles of the spiral". A ...
1
vote
0answers
191 views

Regularity of a a transmission problem

How to prove? Let $a_{ij} \in C^{0, \alpha}(B_1 \cap \mathbb{R}^{n}_{+}), b_{ij} \in C^{0,\alpha}(B_1 \cap \mathbb{R}^{n}_{-})$ elliptic matrices and $$ A_{ij}(x) = a_{ij}(x)\chi_{\{ x_n \ge 0 \}} + ...
2
votes
1answer
623 views

basic moment generating function

I came across a piecewise function which seems pretty basic, but I don't know how to find the moment-generating function. If $X$ has the pdf $f_X(x)=x$ for $0\leq x\leq 1$, $2-x$ for $1\leq x \leq 2$ ...
1
vote
1answer
391 views

Find the constant in Weibull distribution.

If $f(x) = kx^{\beta-1}e^{-\alpha x^\beta}$, $x>0$, how do I find the constant $k$ in terms of $\alpha$ and $\beta$? I know that the entire integral needs to evaluate to $1$. Also How do I find the ...
8
votes
5answers
210 views

Let $a_{n}$ be a sequence such that $(a_{n})^{2}=ca_{n-1}$ where ($c>0,a_{1}>0$).Prove that $a_n$ converges to $c$.

Let $a_{n}$ be a sequence such that $(a_{n})^{2}=ca_{n-1}$ where ($c>0,a_{1}>0$).Prove that the sequence converges to $c$. My first problem was to find some terms of the sequence to verify that ...
5
votes
1answer
223 views

A combinatorial number theory proof

How can I prove the following identity: $$\sum_{k=1}^{n}{\sigma_{\ 0} (k^2)} = \sum_{k=1}^{n}{\left\lfloor \frac{n}{k}\right\rfloor \ 2^{\omega(k)}}$$ where $\omega(k)$ is the number of distinct ...
2
votes
0answers
204 views

Projecting onto subspace

Question Find the orthogonal projection of $$x = \begin{bmatrix}7 \\ 0 \\ -4 \\ -4 \end{bmatrix}$$ onto the subspace of $\mathbb R^4$ spanned by $$v_1 = \begin{bmatrix}-4 \\ 2 \\ 2 \\ -4 \...
0
votes
1answer
299 views

P-adically Cauchy sequences

I am trying to do the following question Find a p-adically Cauchy sequence which converges p-adically to $-1/6$ in $\mathbb{Z}_7$. In general in $\mathbb{Q}_p$ what is the stronger condition, to be ...
3
votes
1answer
165 views

Is $\bigwedge(V)$ self-injective?

For a vector space $V$, is the grassman algebra $\bigwedge(V)$ always an injective module over itself? Is there a proof, or even just a brief explanation?
2
votes
0answers
716 views

Riemann Surface of sin(z)

A question on my last homework asks us to discuss the Riemann surface of $\sin(z)$. Since this isn't multivalued, the Riemann surface is trivial right? Either I am missing something, which is ...
4
votes
2answers
898 views

Lie Algebra Homomorphism Question

So this is a bit of a follow-up to my recent question. I don't mean to inundate the feed with my quandaries, but as I move through the theory I keep hitting stumbling blocks (which y'all so kindly ...
2
votes
0answers
102 views

Cohomology of Grassman manifolds

I am considering the restriction homomorphism $H^p(G_n(\mathbb{R}^\infty)) \leftarrow H^p(G_n(\mathbb{R}^{n+k}))$, where the $G_n(-)$ are the relevant Grassman manifolds. Does anyone know of ...
2
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0answers
103 views

A question about hypercomplex numbers: quaternions, octonions etc [duplicate]

Possible Duplicate: Why is 8 so special? First of all let me state that I am not a mathematician but I work in computer science and engineering. I was reading about hypercomplex numbers, and in ...
1
vote
1answer
1k views

Darboux sums inequality

Let $f$ be a continuous function on $[a,b]$, and continuously differentiable on $(a,b)$. Assume that $f'$ is bounded on $(a,b)$ and $\sup_{(a,b)}|f'(x)| = K$. We'll also denote the upper and lower ...
3
votes
2answers
2k views

Non-symmetric simple random walk stopping time

Say there is a random walk $\{S_n\}$ with $S_0=0$ and $0<p=P(S_1=1)<\frac{1}{2}$. We know such a random walk would go to $-\infty$ eventually. Define the stopping time $T=\inf\{n: S_n=-\infty\}$, ...
3
votes
1answer
525 views

Futures pricing and futures price process under the real world measure

This is something that keeps bothering me about the Benchmark approach of Platen, which (very) shortly is as follows: Compare the development of an economic value with a growth optimal portfolio. ...
0
votes
1answer
88 views

Area of a geometric configuration

How to find the area of the triangle in the plane R2 bounded by the lines y=x, y=-3x+8 and 3y+5x=0. How can I solve this? I'm thinking i can take y=x as the origin and just use y=-3x+8 and 3y+5x=0 ...
9
votes
1answer
463 views

Exponentiation of a Dirichlet series

I'm trying to understand a proof in Chandrasekharan's Introduction to Analytic Number Theory. Specifically, the proof of the lemma on p.118 before Dirichlet's theorem on primes in arithmetic ...
4
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0answers
2k views

Why is Cholesky factorization numerically stable

It's often stated (eg: in Numerical Recipes in C) that Cholesky factorization is numerically stable even without column pivoting, unlike LU decomposition, which usually need pivoting schemes. But I'...
5
votes
1answer
286 views

Bergman's Diamond Lemma: do these rules lead to a normal form?

Last week I was recommended Bergman's Diamond Lemma in these comments. I read through the paper, and was working on an exercise in it on page 193: Examine for termination each of the following ...
1
vote
1answer
182 views

A uniqueness proposition involving Erf, the error function

This is a MathOverflow cross-post (currently no answer there) and a generalization of a previous MathOverflow question, "Reducing system of equations involving Erf, Error Function". Consider the ...
9
votes
3answers
6k views

Continuity proof.

I want to prove that $\exp x$ and $\sin x$ are continuous. This means I want to show that $$\lim\limits_{x\to a}e^x=e^a$$ $$\lim\limits_{x\to a}\sin x=\sin a$$ for any fixed $a \in \Bbb R$. Then I ...
19
votes
2answers
4k views

Partial sums of exponential series

What is known about $f(k)=\sum_{n=0}^{k-1} \frac{k^n}{n!}$ for large $k$? Obviously it is is a partial sum of the series for $e^k$ -- but this partial sum doesn't reach close to $e^k$ itself because ...
4
votes
1answer
371 views

Existence of a binary symmetric matrix from row sums

Give a sequence of column/row sums, is it possible to determine (other than by brute force) whether there exists a binary, symmetric matrix with the same column/row sums? I did find this article, ...
0
votes
2answers
154 views

Showing groups of the form $\mathbb{Z}_n$ are isomorphic

I'm trying to understand why how you can determine whether two groups of the form $\mathbb{Z}_n$ are isomorphic to each other. More specifically why is $\mathbb{Z}_2 \oplus \mathbb{Z}_3 \cong \mathbb{...
2
votes
2answers
6k views

Calculating derivative of $\operatorname{Log}(z)$

Show that $e^{\operatorname{Log}(z)} = z$ and use this to evaluate the derivative of the function Log(z). I have done the first part like this: Letting $z = re^{i\theta}$, $$ \begin{align} e^{\...
2
votes
3answers
1k views

Showing that $\cos\left(\frac{\pi}{5}\right)=\frac{1}{2}\phi?$

What is the usual way of proving things like $$\cos\left(\frac{\pi}{5}\right)=\frac{1}{2}\phi?$$ I know that there is an identity which claims the above, but how was it derived? Are other identities ...

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