# All Questions

1,274,174 questions
Filter by
Sorted by
Tagged with
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 ...
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: ...
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) ...
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 ...
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 ...
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 ...
243 views

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 ...
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.
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 ...
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 $\check y$ ...
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 ...
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 ...
1k views

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 ...
191 views

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 $$A_{ij}(x) = a_{ij}(x)\chi_{\{ x_n \ge 0 \}} + ... 1answer 623 views ### basic moment generating function I came across a piecewise function which seems pretty basic, but I don't know how to find the moment-generating function. If X has the pdf f_X(x)=x for 0\leq x\leq 1, 2-x for 1\leq x \leq 2 ... 1answer 391 views ### Find the constant in Weibull distribution. If f(x) = kx^{\beta-1}e^{-\alpha x^\beta}, x>0, how do I find the constant k in terms of \alpha and \beta? I know that the entire integral needs to evaluate to 1. Also How do I find the ... 5answers 210 views ### Let a_{n} be a sequence such that (a_{n})^{2}=ca_{n-1} where (c>0,a_{1}>0).Prove that a_n converges to c. Let a_{n} be a sequence such that (a_{n})^{2}=ca_{n-1} where (c>0,a_{1}>0).Prove that the sequence converges to c. My first problem was to find some terms of the sequence to verify that ... 1answer 223 views ### A combinatorial number theory proof How can I prove the following identity:$$\sum_{k=1}^{n}{\sigma_{\ 0} (k^2)} = \sum_{k=1}^{n}{\left\lfloor \frac{n}{k}\right\rfloor \ 2^{\omega(k)}}$$where \omega(k) is the number of distinct ... 0answers 204 views ### Projecting onto subspace Question Find the orthogonal projection of$$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 \...
299 views

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 ...
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?
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 ...
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 ...
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 ...
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 ...
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 ...
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 $T=\inf\{n: S_n=-\infty\}$, ...
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. ...
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 ...
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 ...
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'...
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 ...
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 ...
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 ...
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 ...