# Tagged Questions

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### Evaluation of a class of continued fractions

Is there a closed-form way of writing the continued fraction: $$1 + \frac{2}{3+ \frac{4}{5 + \frac{6}{7 + ...}}}$$ EDIT: Since the above has been determined as $\frac{1}{\sqrt{e}-1}$, is there a ...
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### Convergence of sum of antiderivative and derivative

This question is inspired by this question: Solutions for $\frac{dy}{dx}=y$?. It makes me wonder if there are any function where the sum of all antiderivative and derivative converges. The ...
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### $f_{n+1}(x)=f_n(x+1)-f_n(x)$ functional equation and “classification of functions”

Doing a quiz I found a question of this kind "given $a_0, a_1, a_2, ...,a_n$ find $a_{n+1}$" In order to find the $f$ such that $f(a_n)=a_{n+1}$ I tryed for a function like $f(x)=k+x$ ...
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### The “trick” in the Herglotz trick

In How does the Herglotz trick work?, is explained as in "Proofs from THE BOOK" by Aigner and Ziegler, but the "trick" itself I found to be not so clear. The trick says: It follows from (4) ...
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### How to solve this infinite set of equations?

If I can find a solution to the following set of equations then, with a bit of luck, I should be able to derive all sorts of nifty new results in non-equilibrium statistical mechanics. However, I'm ...
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### Summarize my formula?

I would like to summarize my formula. $p$ and $y$ are constant value, $10000$ and $0.65$. When $n = 3$, my formula recalculate the result of $n = 2$. I don't want to recalculate. Is there way to ...
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### Complexity of $T(n)=\sqrt{n}T(\sqrt{n})+n$

I tried to find the complexity of this recursion equation: $T(n)=\sqrt{n}T(\sqrt{n})+n$, by doing couple of iterations and getting a general idea, but I completely got lost. I'd really love your ...
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### Transform the sample to make it more similar to a given

$X=\{x_{i}\}$ and $Y=\{y_{i}\}$ are numeric samples: $y_i \ge 0, x_i \ge 0, i \in [0..N]$. I need to find the mapping $F(X)=\{F(x_i)\}$ with fairly simple formula such that: Euclidean distance ...
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### Looking for a function $f$ such that $f(i)=2(f(i-1)+f(\lceil i/2\rceil))$

I'm looking for a solution $f$ to the difference equation $$f(i)=2(f(i-1)+f(\lceil i/2\rceil))$$ with $f(2)=4$. Very grateful for any ideas. PS. I've tried plotting the the initial values into ...
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Let $f : \mathbb R \to \mathbb R$ be a given function with $\lvert f(x) \rvert \le 1$ and $f(0) = 1$. Is there a nice simplified expression for \begin{align}F(x) &= f(x) f(x/2) f(x/4) f(x/8) ... 2answers 131 views ### How to solve the following system? I need to find the function c(k), knowing that\sum_{k=0}^{\infty} \frac{c(k)}{k!}=1\sum_{k=0}^{\infty} \frac{c(2k)}{(2k)!}=0\sum_{k=0}^{\infty} \frac{c(2k+1)}{(2k+1)!}=1 ...
I have the following series of equations (n+2 equations n+2 variables): $k_0q_0+\lambda q_0 + c_0 = 0$ $k_1q_1+\lambda q_1 + c_1 = 0$ $k_nq_n+\lambda q_n + c_n = 0$ $q_1+q_2+....+q_n = 1$ ...
I have a set of quadratic equations of the form.. $2S_0q_0(q_0S_0 + q_1S_1 + ...... + q_iS_n)+k_0 = 0$ $2S_1q_1(q_0S_0 + q_1S_1 + ...... + q_iS_n)+k_1 = 0$ . . . \$ 2S_nq_n(q_0S_0 + q_1S_1 ...