10
votes
6answers
14k views

“Where” exactly are complex numbers used “in the real world”?

I've always enjoyed solving problems in the complex world during my undergrad. However, I've always wondered where are they used and for what? In my domain (computer science) I've rarely seen it be ...
2
votes
1answer
448 views

Bijection between Cartesian products.

Problem: Show that there is a bijective correspondence of $A \times B$ with $B \times A$. Here is my proof. I want to see if I'm correct about the injective portion of my proof and some ...
3
votes
1answer
81 views

Showing that $(A_{ij})=\left(\frac1{1-x_ix_j}\right)$ is positive semidefinite

Consider the matrix $A$ whose elements are $A_{ij} = \frac{1}{1-x_i x_j} $ where we have $ -1 < x_i < 1$ and $ -1 < x_j < 1$ for $ i,j=1,2,...n$. For example, when $n=3$ the matrix ...
1
vote
0answers
50 views

Alternative explanation for $\iint_D \left|\log \left( \frac{e}{1-z} \right) \right|^2 \ dA = \frac{\pi^3}{6}$?

I thought up a curious definite integral. Let $D = \{ z \in \mathbb{C} : |z|<1\}$. Let $A$ denote area measure on $D$, normalized so that $A(D) = \pi$. I claim that $$\iint_D \left|\log \left( ...
13
votes
3answers
512 views

How can I calculate $\sin\left(10^{10^{100}} - 10\right)^\circ$?

How can I calculate the sine of a googolplex minus 10 degrees?
-1
votes
1answer
221 views

How can I prove it is a martingale when there is a jump process

Let $N_t$ be a Possion process, $M_t=N_t - \lambda t$ we can easily show that $M_t$ is a martingale. Now $\int_0^t\Phi_udM_u=.....=\sum_{i=1}^{N(t)}\Phi_{\tau_i}-\lambda\int_0^t \Phi(u)du $ $\tau_i$ ...
0
votes
2answers
97 views

One stochastic integrability problem

On a lecture notes, there is a following arguement: To make $\int_0^T \pi_t dW_t$ well-defined, (maybe it means to make $\int_0^T \pi_t dW_t<\infty \ \ a.s.$) we only need $\int_0^T \pi_t^2 ...
5
votes
5answers
340 views

Quintic diophantine equation

How can I find non trivial primitive integer solutions, to the Diophantine equation $$a^4+b^4+c^4=d^5$$ Can anyone find me solutions to this equation? Or if possible a parametric equation that ...
0
votes
1answer
143 views

Can all Venn diagrams be constructed?

I have a question that relates to this question about Venn diagrams. Has anyone shown that all Venn diagrams can (theoretically) be constructed?
2
votes
2answers
89 views

Existence of a matrix whose power is $I$.

Let $m,n \ge 2$ be integers. Does there exist a matrix $A \in \mathbb{R}^{n\times n}$ such that $$ A^m=I \ne A^k \ \forall\ k \in\{1,\ldots, m-1\}? $$ For $n=2,3,4$ and $m=3$ the answer is yes (see, ...
3
votes
3answers
514 views

The transpose of a linear injection is surjective.

Let $$T:V\longrightarrow W$$ be a linear map (of vector spaces), and let \begin{eqnarray} T^*:W^* &\longrightarrow& V^* \\ f\ &\longmapsto& f\circ T \end{eqnarray} be its transpose ...
1
vote
2answers
103 views

Reflection around a plane, parallel to a line

I'm supposed to determine the matrix of the reflection of a vector $v \in \mathbb{R}^{3}$ around the plane $z = 0$, parallel to the line $x = y = z$. I think this means that, denoting the plane by $E$ ...
2
votes
2answers
487 views

Distinguishing equality and isomorphism as relations

Is this relational characterization of equality in Wikipedia accepted? The identity relation is the archetype of the more general concept of an equivalence relation on a set: those binary ...
1
vote
2answers
225 views

Von Neumann's ergodic theorem

Where can I find the proof of Von Neumann's ergodic theorem? Please, give me references or write names of books where I can find it.
5
votes
1answer
345 views

Krull dimension of Noetherian local rings is finite

Does anyone know an "elementary" proof of the fact that a Noetherian local ring has finite Krull dimension? The one I know is from Atiyah&Macdonald's book Introduction to Commutative Algebra, ...
3
votes
1answer
242 views

Integration Antiderivative vertical bar [duplicate]

Possible Duplicate: What is the name of the vertical bar? When taking a definite integral, the first step is finding the anti-derivative. Once you have gone through all the steps to ...
1
vote
0answers
155 views

Convergence of the reciprocal of a function whose derivative tends to infinity

If $$\lim_{x\to\infty} f'(x) = \infty$$ prove that $$\int_1^\infty \frac{1}{f(x)}\neq\infty$$ if $f'(x) \geq 1$ and $f(x) \geq1$ for all values of $x$. I'm thinking I can find a way to write it in big ...
3
votes
1answer
233 views

Is the Sobolev Space $H^k(0,1)$ a banach algebra?

In Adams'book:Sobolev Spaces, I know that if $kp>n,\Omega\subset R^n$ is boundary domain and has cone property, then $W^{k,p}(\Omega)$ could see as a banach algebra. My question is that does it ...
1
vote
0answers
53 views

A simple question about a limit

I have the following data: 1) $u(t),u′(t)$ and $u′′(t)$ are bounded 2) $u''(t)+cu'(t)+f(u(t))=0$ where $c>0$ and $f$ is continuous 3) $\int_{0}^{+\infty}{u'(t)^2}<\infty$ I want to prove ...
1
vote
3answers
591 views

Open and closed mapping are not necessarily continuous

In the link http://en.wikipedia.org/wiki/Open_and_closed_maps, it says "To every point on the unit circle we can associate the angle of the positive x-axis with the ray connecting the point with ...
4
votes
2answers
10k views

How can I convert 2's complement to decimal?

Suppose I have the 2's complement, negative number 1111 1111 1011 0101 (0xFFBB5). How can I represent this as a decimal number in base 10?
2
votes
1answer
54 views

Simple question about a limit

I have the following data: 1) $u(t), u'(t)$ and $u''(t)$ are bounded 2) $\lim\limits_{t\rightarrow+\infty}{u(t)}=l$ 3) $\lim\limits_{t\rightarrow+\infty}{u'(t)}=0$ 4) $u''(t)+cu'(t)+f(u)=0$ where ...
0
votes
1answer
44 views

How to prove that $R/(R\cap I)\cong (R+I)/I$

Let $A$ be a unitary ring, $R$ a subring and $I$ an ideal of $A$. It is easy to prove that $R\cap I$ is an ideal of $R$ and that $R+I$ is a sub ring of $A$, but how do I show the following ...
14
votes
3answers
655 views

Are $\mathbb{R}$ and $\mathbb{Q}$ the only nontrivial subfields of $\mathbb{R}$?

I've been asked to prove that any subfield of $\mathbb{R}$ contains $\mathbb{Q}$, and I know how to do it, but it made me wonder if there were subfields of $\mathbb{R}$ that strictly contained ...
0
votes
2answers
650 views

Obtaining wrong quotient when dividing by negative number

I'm having a big moment of ignorance, since my math teacher in college showed us that no one from my class knew to divide two integers right, without knowing anything about the knowledge of my class, ...
0
votes
1answer
70 views

Expectation of a uniform distribution

$X$ is uniform over $(0,1)$. What is $E[X|X<\frac12]$? Here's what I did so far, but I'm not sure it's right: $f_X(x|X<1/2)=2$, which is also uniform, so the expected value is just ...
1
vote
1answer
989 views

Expectation and variance of this stochastic process

I am trying to compute the expectation and variance of the following stochastic process: $$ Z_t = \exp \left( \frac{1}{2} \int_0^t W_s \, dW_s \right) $$ where $W_t$ is a standard Brownian motion. I ...
2
votes
0answers
162 views

A dynamic Stackelberg game - general characterization

my question is about general representation of a dynamic Stackelberg game which is played in continuous time. We have maximization problems of two agents who play this game. Agents are 'Leader' and ...
2
votes
2answers
234 views

How do i show that an equation has a singular solution?

Given the equation $$ y = xy' + \sqrt{1 + y'^2} $$ with a one-parameter family of solutions being $$ y = cx + \sqrt{1 + c^2} $$ how would I go about showing that a relation $x^2 + y^2 = 1$ defines a ...
2
votes
3answers
315 views

Why does the integral of cot x have absolute value?

There's a proof of that integral on this website: http://math2.org/math/integrals/more/cot.htm I just don't understand why we need the absolute value.
0
votes
2answers
159 views

Please solve the following equation for x:

Please solve the following equation for $x$: $5^{2x}+(17/2)^x=9$
1
vote
1answer
490 views

double integral of an absolute function

I'm just a little unsure of how to tackle this one. I understand that typically you would separate the integral into two for where x is positive or negative, I'm just unsure of how to separate it for ...
2
votes
1answer
130 views

Are these two definitions of Riemann-Stieltjes Integral equivalent

I actually posted this question before, but didn't get a correct answer. Two distinct definitions follow; ($f$ is assumed to be a real-valued function and $\alpha$ is assumed to be a monotonically ...
4
votes
1answer
689 views

If $f,g:X \to Y$ are continuous and $Y$ is $T_2$, then $\{x \in X\,|\,f(x)=g(x)\}$ is closed

I'd like to know if the following proof is valid. The only thing I'm not sure about (though I can't see why it's invalid if it is) is if we can always use the Hausdorfness of $Y$ to separate an open ...
2
votes
1answer
177 views

Zeros of holomorphic function have limit point outside domain

The following is exercise 10.20 in Rudin's R&CA. Suppose $f \in H(U)$, $g \in H(U)$, and neither $f$ nor $g$ has a zero in $U$. If $$ \frac{f'}{f}\left(\frac{1}{n}\right) = ...
0
votes
3answers
144 views

Is $\dim\left(U+V\right)=\dim\left(U\cup V\right)$?

If $U$ and $V$ are proper subspaces of a vector space $W$, is $\dim\left(U+V\right)=\dim\left(U\cup V\right)$?
2
votes
0answers
64 views

Measure takes the value of infinity

If a measure $P$ can take value of $+\infty $, does countable additivity ($\sigma$-additivity) imply continuity at $\emptyset$? If not, what is a good example?
2
votes
1answer
80 views

Why is $\lim_{x \to 0} {\rm li}(n^x)-{\rm li}(2^x)=\log\left(\frac{\log(n)}{\log(2)}\right)$?

I'm trying to give at least some partial answers for one of my own questions (this one). There the following arose: $\hskip1.7in$ Why is $\lim_{x \to 0} {\rm li}(n^x)-{\rm ...
2
votes
1answer
78 views

A way to codify (pre-calculatate) if a one Tree Node is a descendant of another

I have a simple, 1-directional tree representing the veins in a human body. It looks somewhat like this (red dots are nodes, blood flow is always downwards, sorry for my drawing): What I need is a ...
5
votes
5answers
2k views

What is the norm of a complex number?

I'm in a number theory class and I'm trying to understand what the norm is... For some complex number $Z = a +bi$, $Z$ times the conjugate of z is equal to $(a^2)+(b^2)$. Most of what I've read about ...
3
votes
2answers
61 views

Find intermediary extension

Let $P$ be the polynomial $$ P=64X^9 + 192X^7 + 240X^6 + 36X^5 + 552X^4 - 847X^3 - 540X^2 + 273X + 143 $$ Then $P$ is irreducible over $\mathbb Q$, and has three real roots. Let $\alpha$ be one of ...
3
votes
2answers
84 views

Are nonsquares actually squares in extensions of even degree?

I was playing around with finite fields, and noticed that if $F$ is a finite field, and $E$ an extension of odd degree, then every nonsquare $a\in F$ is again a nonsquare in $E$. I found this out by ...
12
votes
2answers
396 views

Computation of Laplace-Beltrami operator in a conformally equivalent metric

Could anyone tell me where I'm wrong with the following elementary calculation? Given a smooth Riemannian manifold $(M, g)$, I'd prove that if $\tilde{g}$ is conformally equivalent to $g$ (that is, ...
2
votes
1answer
108 views

Is a finite field extension of a imperfect field imperfect

Let $K$ be a imperfect field. Let $L/K$ be a finite field extension. Is $L$ imperfect? Suppose that $L/K$ is separable. Is $L$ imperfect? Suppose that $L/K$ is Galois. Is $L$ imperfect? I'm ...
2
votes
2answers
295 views

Partial differential equation with chain rule

Problem statement: Consider the PDE: $x\frac{\partial f}{\partial x}+y\frac{\partial f}{\partial y}=\frac{1}{\sqrt{x^2+y^2}}, (x,y)\neq (0,0) $ Determine all solutions to the equation of the form ...
0
votes
1answer
69 views

If we know the mean value of $x$, what is the mean value of the inverse $\frac{1}{x}$?

Random variable $x$ follows normal distribution and we know the mean value of $x$. Is there a well-known way to compute the mean value of $\frac{1}{x}$? Perhaps, some way to estimate the integral of ...
2
votes
1answer
157 views

Solve for Angles in a Matrix

So, I have a linear equation, and I need to solve for some variables in said equation. However, since I don't know much about matrices, I don't know how to solve for the variables. The equation in ...
-2
votes
1answer
134 views

Equivalent conditions in a topological space

Let $X$ is a topological space prove that the following conditions are equivalent for every $A \subseteq X$, $A$ is open in the subspace of $\operatorname{cl}A$ for every $A \subseteq X$, there ...
3
votes
2answers
1k views

Null-recurrence of a random walk

In a random walk on $\mathbb{Z}$ starting at $0$, with probability 1/3 we go +2, with probability 2/3 we go -1. Please prove that all states in this Markov Chain are null-recurrent. Thoughts: it is ...
11
votes
6answers
2k views

Choosing two random numbers in $(0,1)$ what is the probability that sum of them is more than $1$?

Choosing two random numbers in $(0,1)$ what is the probability that sum of them is more than $1$? Also what is probability of sum of them being less than $1$? I think the answer should be ...

15 30 50 per page