# Tag Info

3

Let $f(x) = e^{k_1/x}+e^{k_2/x}+\cdots+e^{k_N/x}-1 =\sum_{i=1}^N e^{k_i/x}-1$, and let $K = \sum_{i=1}^N k_i$. The restrictions that $x > 0$ and $k_i < 0$ are important in what follows. $f'(x) =\sum -\frac{k_i}{x^2}e^{k_i/x} =-\frac1{x^2}\sum k_ie^{k_i/x} =\frac1{x^2}\sum |k_i|e^{k_i/x}$, so $f'(x) > 0$. This means that your function has at ...

0

Write an algorithm to solve the problem... Let $$Q = \left( \sum_{m=1}^{n} \exp(k_m y) - 1 \right)^2.$$ In case we have the solution, we have $$Q = 0$$. For changes in $Q$ we have $$\delta Q = 2 Q \left( \sum_{m=1}^{n} k_m \exp(k_m y) \right) \delta y.$$ For each step use $$\delta y = - 2 Q \left( \sum_{m=1}^{n} k_m \exp(k_m y) ... 0 This is equivalent to finding roots of the polynomial$$y^{-k_1}+\cdots +y^{-k_N}-1=0$$(by making y:= e^{-\frac{1}{x}}) so your problem is as difficult as finding roots of polynomials. It depends on the degree of that polynomial, in this case, it depends on the max of the -k_i. 3 This is related to Jacobi Theta functions since$$\sum_{n=-\infty}^\infty e^{-n^2\pi x}=\vartheta _3\left(0,e^{-\pi x}\right)\sum_{n=-\infty}^\infty e^{-n^2\pi / x}=\vartheta _3\left(0,e^{-\frac{\pi }{x}}\right)$$and the function$$\Psi(x)=\frac 12 \Big(\vartheta _3\left(0,e^{-\pi x}\right)-1\Big)$$satisfies$$\frac{1+2\Psi(x)}{1+2\Psi(\frac ...

2

The sequence is closely related to what is called a Zadoff-Chu sequence, or Frank-Zadoff-Chu (FZC) sequence or just a Chu sequence depending on who is doing the name-calling. One of the properties of FZC sequences is that their Discrete Fourier Transforms are another FZC sequence, conjugated, scaled, and possibly time-scaled as well. These sequences are ...

0

You could probably use the generalized quadratic Gauss sum reciprocity formula, defined for integers $a, b, c$ satisfying $ac \neq 0$ and $ac+b$ even. In this case, we set $\displaystyle S(a,b,c) = \sum_{x=0}^{|c|-1} e\left(\frac{ax^2+bx}{2c}\right)$, and the theorem states \begin{eqnarray*} S(a,b,c) = \left|\frac{c}a\right| ...

Top 50 recent answers are included