# All Questions

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### 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 ...
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### Check computation of conditional covariance

Note: HERE YOU CAN SEE THIS PAGE. Explanation of an integral formula for the expectation of $(X_1-X_2)(Y_1-Y_2)$
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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: ...
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### 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.
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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: $$\lim_{n\to\infty} ... 1answer 19 views ### The tangent hyperplane to the graph of harmonic function This is an interesting question I found online about Laplace equation. We define a function u :R^N\to R's graph to be the set \{x,u(x):\,x\in R^N\}\subset R^{N+1}. Then I want to prove that ... 0answers 47 views ### Equivalence Relation between sets X \times Y If X and Y are sets a relation between X and Y, is a subset R\subset X \times Y . For a relation R\subset X \times Y we have \{(a,b):a\in X, b\in Y\}. if (a,b)\in R or (a,b)\notin R. ... 0answers 34 views ### Need formula to calculate probabilities for a forex trading system [on hold] I'm developing a forex trading system and I'm trying to understand how to calculate the probabilities involved. Unfortunately, my math skills are lacking. Here are the system basics: The stop loss ... 2answers 56 views ### Clarification regarding Drinker's paradox This is the informal proof of Drinker's paradox The proof begins by recognising it is true that either everyone in the pub is drinking (in this particular round of drinks), or at least one ... 1answer 22 views ### Stopped process of Brownian motion I am baffled about the following problem: Let (B_t) be a standard Brownian motion. Let$$ \tau:= \inf\{ t \geq 0 :B_t = x \} \wedge \inf\{ t \geq 0 :B_t = -y \}$$be a stopping time, where x,y ... 1answer 38 views ### Preferred way to write elements of the direct sum of vector spaces Suppose V and W are vector spaces over the same field and V\oplus W is their direct sum. Reading through the literature I found essentially two ways of writing elements of V\oplus W. 1.) We ... 3answers 243 views ### Calculating a limit of integral Computing the limit:$$\lim_{n\rightarrow\infty}\left(\frac{1}{3\pi}\int_\pi^{2\pi}\frac{x}{\arctan(nx)} \ dx\right)^n$$I made the substiution t=nx then, we have: ... 0answers 17 views ### Calculus of Variations-First and Second Order Deviations I'm new to Calculus of Variations and the Method of Least Action (L=T-V) What I'm unsure about is how first and second order deviations are used in finding the least action. I know it's used to find ... 0answers 20 views ### Solve Karush–Kuhn–Tucker conditions solving a constrained optimizing problem with equality constraints can be done with the lagrangian multiplier. (http://en.wikipedia.org/wiki/Lagrange_multiplier) This approach leads to a system of ... 2answers 43 views ### Maple: How do I type “solve” with an arrow under? I am trying to learn using Maple 18 (Mac). I have defined a function with a list of X and Y values. f := x->LinReg(X, Y, x) Now I would like to output the unknown "x" value that correlates with ... 3answers 41 views ### Limit of a function to the power of another function Is there a theorem in real analysis for \underset{n\rightarrow \infty}\lim f(n)^{g(n)}, where f(n) and g(n) are arbitrary functions of n? Under what conditions on f(n) and g(n) does the ... 1answer 28 views ### How to draw simple lattice diagrams (MathJax syntax) I recently asked a question on TeX SE about how to draw lattice diagrams with MathJax (as in, the TeX commands for creating one, once I already drew it on paper and know what it should look like). ... 1answer 37 views ### Problem with system \dot x = 2x + y +e^t; \dot y = -2x + 2t. I am solving system of differential equations,$$ \begin{cases} \dot x = 2x + y +e^t; \\ \dot y = -2x + 2t. \end{cases} $$It's matrix, eigenvalues and eigenvectors are as follows: \qquad A = ... 0answers 27 views ### Evaluate the following sum using the hyperbola method This is an exercise from Iwaniec and Kowalski's book Analytic Number Theory: Prove that$$\displaystyle \sum_{n \leq x} \tau(n^2 + 1) = \frac{3}{\pi}x \log x + O(x).$$The constant 3/\pi is quite ... 0answers 19 views ### Eliminating equality constains The following text derived from book convex optimization by Boyd, page 143. For a convex problem the equality constraints must be linear, i.e., of the form Ax = b. In this case they can be ... 1answer 43 views ### elementary number theory (greater integer function ) problem? [on hold] 5(b): For what values of n does n! terminates in 37 zeros ? This is from here. 0answers 19 views ### What is the dual of L^{\infty}(K) with K a compact subset of R^n? I know it's probably hard to describe the dual of L^{\infty}(X) for a general X. But can we describe it when X is just a compact subset of a Euclidean space? 3answers 26 views ### At the carnival there are 5 times as many elephants as camels , if there are a total of 54 elephants and camels , how many camels are there? [on hold] At the carnival there are 5 times as many elephants as camels. If there are a total of 54 elephants and camels, how many camels are there ? 2answers 24 views ### Equilibrium distribution of Ehrenfest's urn (I'll post my own answer to this, but others may be of interest, so post your own if you have one.) The physicist Paul Ehrenfest posted the problem of two urns containing some marbles. At each step ... 2answers 40 views ### Consider the function with its first and second derivative help please [duplicate]$$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 ...
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 $n=1,...,10$. ...
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 ...