2
votes
2answers
44 views

Does there exist a subfield $S$ of $\mathbb C$ such that $\mathbb R \subset S \subset \mathbb C$?

Does there exist a subfield $S$ of $\mathbb C$ such that $\mathbb R \subset S \subset \mathbb C$ ? ; I kind of have a feeling that there does not exist any such $S$ but cannot prove . Thanks in ...
-1
votes
0answers
35 views

Check computation of conditional covariance

Note: HERE YOU CAN SEE THIS PAGE. Explanation of an integral formula for the expectation of $(X_1-X_2)(Y_1-Y_2)$
1
vote
1answer
50 views

Proving this equivalence relation

If $X,Y$ are reflexive, symmetric, and transitive, then $X \times Y$ is an equivalence relation where ${(a,b):a\in X, b\in Y}$. I am trying to self learn these topics. I do know what an ...
5
votes
5answers
206 views

For every positive integer $n, n^2 + 4n + 3$ is not a prime

Prove: For every positive integer $n, n^2 + 4n + 3$ is not a prime. I tried to disprove the statement, which I could not using several number examples with constructive proof. However I am not sure ...
0
votes
1answer
32 views

Dot “power” of a matrix

By analogy with the matrix product is there a name for the matrix "power" operation defined by $$y_i = \prod_j x_j^{a_{ij}}?$$ For example: $$\left( \begin{array}{lll} x_1 & x_2 & ...
2
votes
3answers
44 views

Limit of $a_n$, where $a_1=-\frac32$ and $3a_{n+1}=2+a_n^3$.

Let $a_1=-\frac32$, and $3a_{n+1}=2+a_n^3$. I need to show that $\displaystyle \lim_{n\to \infty} a_n = 1$. I can show that the sequence is monotonically increasing and bounded as follows: By the ...
-4
votes
0answers
26 views

Negative binomial distribution [on hold]

Ann plays a casino game and receives a token whenever a six or a one comes up when a die is rolled. The games ends when she get y tokens ; she receives x dollars where x is the number of rolls made. ...
0
votes
0answers
9 views

Universal hash function when size of hash is p^m

Can we define universal hash function from $U \rightarrow T$ when $T=\{0,1,2,..,m-1\}$ and $m=p^a$? (where $p$ is a prime and a is an integer) I know that we can define universal hash funciton when ...
2
votes
1answer
19 views

Hitting time process of Brownian motion [on hold]

I am stuck with this problem: Let $(B_t)$ be a standard Brownian motion in $\mathbb{R}$. For $t \geq 0$, let $$ H_t = \inf \{ s \geq 0 : B_s = t \}, \quad S_t = \inf \{ s \geq 0 : B_s > t \}. $$ ...
2
votes
1answer
8 views

Convergence in probability given that covariance matrix goes to $0$

Suppose I have a sequence of random vectors $\{X_n\}$ each of dimension $2\times 1$. Suppose also that I know $$ ...
0
votes
0answers
52 views

Prove $X\times Y$ is an equivalence relation

(Relation between two sets) If $X$ and $Y$ are sets, a relation between $X$ and $Y$ is a subset $R \subset X \times Y.$ For a relation $R \subset X\times Y$ and $a \in X$ and $b \in Y$ if $(x,y) \in ...
1
vote
3answers
76 views

What's wrong with this derivative for $\left(1−\frac{1}{x}\right)^x$?

I asked a question about how to differentiate $(1−1/x)^x$ before, for $x>1$. The derivative I was told is ...
-2
votes
2answers
50 views

How do I take the 100th derivative of a polynomial [on hold]

How could I find $$f^{100}(x)$$ for $$f(x)=2x^{100}-7x^{80}+15x^{60}-27x^{40}-18x^{20}+300$$
0
votes
1answer
16 views

What is meant by $AB$ in boolean algebra?

I am endeavoring to teach myself Boolean Algebra. Oh what fun! From the questions I've read on this site, one of the most common notations I've seen is $AB$ (examples: here, here, and here). Problem ...
0
votes
2answers
34 views

Is there any standard terminology for this property?

Let $f$ be a map whose domain is $X$. If $f$ satisfies the property that for all $x\in X$, $$f(f(x))=f(x)\text{,}$$ is there any standard name for such a function? Not sure if "projection" is the ...
0
votes
1answer
18 views

Expectation of 1/Y. Y~Gam

I need to find the expectation E$\big[\frac{1}{Y}\big]$ where Y is distributed by Gamma with parameters $\alpha$ and $y$. Do I need to find the pdf of $\frac{1}{Y}$ by using some transformation ...
1
vote
1answer
10 views

Help understanding the output from Apache Commons Math's Discrete Fourier Transform

I'm using a discrete Fourier transform to translate a finite set of samples to the frequency domain. I'm trying to start with a very simple set, but am still getting confused. I'm starting with this ...
3
votes
2answers
38 views

Help identifying the singularities of $\csc(\cos z) = \frac{1}{\sin(\cos z)}$

I am really stuck with this one: $\frac{1}{\sin(\cos z)}$ has a singularity when $\cos z = k\pi $ since $\sin(k\pi) = 0$ but how do I solve for the value of $z$, how can i evaluate $\cos z = k\pi $?
0
votes
0answers
11 views

maximum principle for $p$-Laplace equation

Consider $\Omega \subset R^n$ a bounded domain. Let $\varphi \in W^{1,p}(\Omega) \cap L^{\infty}(\Omega)$. Let $u \in W^{1,p}(\Omega)$ with $\Delta_p u = 0$ in $\Omega$ with $u - \varphi\in ...
1
vote
1answer
16 views

Derivation of energy integral - harmonic functions

I am following the solution of the following problem on the topic of the energy integral of a surface. For a real-valued continuously differentiable function $u(x,y)$ on a closed domain $D$, the ...
0
votes
0answers
23 views

Stadium billiard reflection angles

Given a boundary and a massless particle with constant velocity with a certain direction, a billiard consists of an experiment where the particle collides with the walls conserving its velocity ...
2
votes
2answers
70 views

How to create a computationally cheap function passing through given points?

I am trying to develop a function which goes through the follow points. The function will be calculated on a microprocessor which has 20 mHz. List of given points: ...
1
vote
5answers
70 views

If $a,b\in\mathbb R$ with $a<b$, then there is some rational $r$ with $a<r<b$. [duplicate]

How do you prove this question? I was thinking proving contrapositive. But I was stuck..Thanks guys.
1
vote
1answer
22 views

singularity and degeneracy of an ODE

I have trouble distinguishing the difference of singularity and degeneracy in the context of ODE theory. Could anyone give me a couple of examples in illustrating the difference of singular point and ...
1
vote
2answers
19 views

How to replace a complex term in an equation using a function?

I have recently been working on a few models that look at mosquito predation. Now one of the peers wants me to add the complete equation of my model in the manuscript. I previously had the equation ...
1
vote
1answer
17 views

Is the image of a $*$-homomorphism $\pi:\mathcal{A}\to\mathcal{B}$ closed if $\pi(1)\neq 1$?

Setting Given C*-algebras $\mathcal{A}$ and $\mathcal{B}$ with unit $1\in\mathcal{A}$. Consider a morphism: $\pi:\mathcal{A}\to\mathcal{B}$ without $\pi[1]=1\in\mathcal{B}$. Especially, it is a ...
2
votes
3answers
43 views

Arithmetic functions of particular type

Any there any natural functions real valued single variable that: changes (increases) values only at primes but otherwise stay constant (like a non periodic increasing staircase)? whose increase in ...
1
vote
2answers
43 views

Limit radius of convergence $ S = \sum^\infty_{n=0} \frac{(n + 1)!}{8^n} $

Here's the problem as given: Let: $$ S = \sum^\infty_{n=0} \frac{(n + 1)!}{8^n} $$ If I use the Ratio Test to determine whether S converges, I need to determine: $$ \lim_{n\to\infty} ...
0
votes
1answer
19 views

The tangent hyperplane to the graph of harmonic function

This is an interesting question I found online about Laplace equation. We define a function $u$ :$R^N\to R$'s graph to be the set $\{x,u(x):\,x\in R^N\}\subset R^{N+1}$. Then I want to prove that ...
-3
votes
0answers
47 views

Equivalence Relation between sets $X \times Y$

If $X$ and $Y$ are sets a relation between $X$ and $Y,$ is a subset $R\subset X \times Y$ . For a relation $R\subset X \times Y$ we have $\{(a,b):a\in X, b\in Y\}.$ if $(a,b)\in R$ or $(a,b)\notin R.$ ...
-2
votes
0answers
34 views

Need formula to calculate probabilities for a forex trading system [on hold]

I'm developing a forex trading system and I'm trying to understand how to calculate the probabilities involved. Unfortunately, my math skills are lacking. Here are the system basics: The stop loss ...
1
vote
2answers
56 views

Clarification regarding Drinker's paradox

This is the informal proof of Drinker's paradox The proof begins by recognising it is true that either everyone in the pub is drinking (in this particular round of drinks), or at least one ...
4
votes
1answer
22 views

Stopped process of Brownian motion

I am baffled about the following problem: Let $(B_t)$ be a standard Brownian motion. Let $$ \tau:= \inf\{ t \geq 0 :B_t = x \} \wedge \inf\{ t \geq 0 :B_t = -y \}$$ be a stopping time, where $x,y ...
1
vote
1answer
38 views

Preferred way to write elements of the direct sum of vector spaces

Suppose $V$ and $W$ are vector spaces over the same field and $V\oplus W$ is their direct sum. Reading through the literature I found essentially two ways of writing elements of $V\oplus W$. 1.) We ...
8
votes
3answers
243 views

Calculating a limit of integral

Computing the limit: $$\lim_{n\rightarrow\infty}\left(\frac{1}{3\pi}\int_\pi^{2\pi}\frac{x}{\arctan(nx)} \ dx\right)^n$$ I made the substiution $t=nx$ then, we have: ...
0
votes
0answers
17 views

Calculus of Variations-First and Second Order Deviations

I'm new to Calculus of Variations and the Method of Least Action (L=T-V) What I'm unsure about is how first and second order deviations are used in finding the least action. I know it's used to find ...
0
votes
0answers
20 views

Solve Karush–Kuhn–Tucker conditions

solving a constrained optimizing problem with equality constraints can be done with the lagrangian multiplier. (http://en.wikipedia.org/wiki/Lagrange_multiplier) This approach leads to a system of ...
1
vote
2answers
43 views

Maple: How do I type “solve” with an arrow under?

I am trying to learn using Maple 18 (Mac). I have defined a function with a list of X and Y values. f := x->LinReg(X, Y, x) Now I would like to output the unknown "x" value that correlates with ...
-1
votes
3answers
41 views

Limit of a function to the power of another function

Is there a theorem in real analysis for $\underset{n\rightarrow \infty}\lim f(n)^{g(n)}$, where $f(n)$ and $g(n)$ are arbitrary functions of $n$? Under what conditions on $f(n)$ and $g(n)$ does the ...
1
vote
1answer
28 views

How to draw simple lattice diagrams (MathJax syntax)

I recently asked a question on TeX SE about how to draw lattice diagrams with MathJax (as in, the TeX commands for creating one, once I already drew it on paper and know what it should look like). ...
1
vote
1answer
37 views

Problem with system $\dot x = 2x + y +e^t; \dot y = -2x + 2t$.

I am solving system of differential equations, $$ \begin{cases} \dot x = 2x + y +e^t; \\ \dot y = -2x + 2t. \end{cases} $$ It's matrix, eigenvalues and eigenvectors are as follows: $\qquad A = ...
0
votes
0answers
27 views

Evaluate the following sum using the hyperbola method

This is an exercise from Iwaniec and Kowalski's book Analytic Number Theory: Prove that $$\displaystyle \sum_{n \leq x} \tau(n^2 + 1) = \frac{3}{\pi}x \log x + O(x).$$ The constant $3/\pi$ is quite ...
0
votes
0answers
19 views

Eliminating equality constains

The following text derived from book convex optimization by Boyd, page 143. For a convex problem the equality constraints must be linear, i.e., of the form $Ax = b$. In this case they can be ...
-5
votes
1answer
43 views

elementary number theory (greater integer function ) problem? [on hold]

5(b): For what values of n does n! terminates in 37 zeros ? This is from here.
0
votes
0answers
19 views

What is the dual of $L^{\infty}(K)$ with K a compact subset of $R^n$?

I know it's probably hard to describe the dual of $L^{\infty}(X)$ for a general $X$. But can we describe it when $X$ is just a compact subset of a Euclidean space?
-4
votes
3answers
26 views

At the carnival there are 5 times as many elephants as camels , if there are a total of 54 elephants and camels , how many camels are there? [on hold]

At the carnival there are 5 times as many elephants as camels. If there are a total of 54 elephants and camels, how many camels are there ?
0
votes
2answers
24 views

Equilibrium distribution of Ehrenfest's urn

(I'll post my own answer to this, but others may be of interest, so post your own if you have one.) The physicist Paul Ehrenfest posted the problem of two urns containing some marbles. At each step ...
-4
votes
2answers
40 views

Consider the function with its first and second derivative help please [duplicate]

$$f(x)=\frac{4x^2}{x^2+3} $$ $$f'(x)=\frac{24x}{(x^2+3)^2} $$ $$f''(x)=\frac{72(1-x^2)}{(x^2+3)^3}$$ a)What are the critical numbers(if any)? b)On what intervals is the function increasing and on ...
4
votes
1answer
71 views

Analytic solutions to a simple math trick

As proven here $3816547290$ is the only positive integer in which every digit is used; each digit is used only once; the first $n$ digits are divisible by $n$, for $n=1,...,10$. ...
-4
votes
0answers
29 views

Function continuous at end-points [on hold]

If we have a function $f$ that is absolutely continuous on $(-1,1)$ (and also the derivative of $f$ is absolutely continuous on $(-1,1)$) and we have that for the derivative the following limits ...

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