How to stop fractions from simplifying, when inside a Manipulate[] [duplicate]

I want to stop this fraction from simplifying inside a Manipulate[]

Manipulate[            x/y            , {x, 7, 10, 1}            , {y, 7, 10, 1} ] 

i.e. when this program is evaluated the fraction should look like:

$ $ \frac{7}{7}$ $


I have tried many different wrapping configurations of ToString, Hold, HoldForm:

but to no avail:

enter image description here

Surprisingly mathematica will even simplify a fraction containing strings to $ 1$

Circuit depth of computing the continued fractions of a rational number

If you want to convert a rational number into its continued fraction, what is the circuit depth of this process, in terms of the total number of bits of input?

I was reading through some notes which mentioned that the work being done while computing the continued fraction is basically the same as the work being done while computing a GCD. Are their circuit depths similar?

notes

Egyptian fractions in Python

Here’s my code for the Egyptian fraction written in Python. I need your feedback if there is any room for improvement.

def eg_fr(n, d):     ls = []     ls_str = []     fraction = n / d     total = 0     for i in range(1, 1000):         ls.append((1/i, str(1) + '/' + str(i)))     for i in ls:         while i[0] + total <= fraction:             ls_str.append(i[1])             total += i[0]             if total >= fraction:                 break     print('%s/%s = ' % (n, d), end='')     for i in range(len(ls_str) - 1):         print('', ls_str[i], '+ ', end='')     print(ls_str[-1], end='')     return ls_str   if __name__ == '__main__':     eg_fr(7, 133) 

Homology of linear groups over integral domains and their field of fractions

Let $ A$ be a noetherian integral domain of finite Krull dimension with the field of fractions of $ F$ . Consider the natural injections $ i_n:GL_n(A)\hookrightarrow GL_{n+1}(A)$ , $ j_n:GL_n(F)\hookrightarrow GL_{n+1}(F)$ , $ \varphi_n:GL_n(A)\hookrightarrow GL_n(F)$ and similarly $ \varphi_{n+1}$ . Let $ x\in H_k(BGL_{n+1}(A))$ and assume that $ \varphi_{*(n+1)}(x)$ lies in the image of $ j_{*n}$ i.e. for some $ y\in H_k(BGL_n(F))$ , $ j_{*n}(y)=\varphi_{*(n+1)}(x)$ . Does that imply that $ y$ is in the image of $ \varphi_{*n}$ ?

How to select best k fractions out of n fractions (k

For example, given 4 fractions $ \frac{4}{2}$ , $ \frac{2}{3}$ , $ \frac{1}{2}$ , $ \frac{10}{20}$ , I have to select 3 fractions out of these 4 so that the value of $ \frac{\text{numerator sum}}{\text{denominator sum}}$ is maximized. As in this case, selecting $ \frac{4}{2}$ , $ \frac{2}{3}$ , $ \frac{1}{2}$ will result in $ \frac{4+2+1}{2+3+2}$ = 1 which is maximized.

How can I select $ k$ fractions which will always result in the maximum $ \frac{\text{numerator sum}}{\text{denominator sum}}$ value?

Edit:

  • Attempt 1: Sorting fractions in descending order. This passes a few test cases but that it’s. This can also be proven by example that it is not the optimal solution. Let’s say we have a list of 3 fractions sorted in descending order, $ \frac{1}{2}$ , $ \frac{20}{59}$ , $ \frac{1}{3}$ and $ k = 2$ . The optimal solutions is $ \frac{1+1}{2+3}$ , not $ \frac{1+20}{2+59}$ .

Egyptian fractions of the form `2/n` with smallest sum of denominators

I realize this has been asked for the more general $ p/q$ case; however, I am restricting myself to the case of $ p=2$ and $ q$ is odd (see the Rhind Mathematical Papyrus 2/n table).

It’s easy to show that any fraction of the form $ 2/(2N+1)$ can be written as the fraction

$ $ \frac{2}{2N+1} = \frac{1}{N+1} + \frac{1}{(N+1)(2N+1)}$ $

which appears to be optimal for prime numbers. However, I was trying to express the problem as an optimization problem (KKT) and am not getting a solution. I’m not sure if I’m formulating it correctly, or something else. The problem can be formulated as

$ $ f(x) = x_1 + x_2$ $ $ $ g(x) = \frac{2}{2N+1} – \frac{1}{x_1} – \frac{1}{x_2} = 0$ $ $ $ h(x) = [ 2-x_1 , x_1+1 – x_2 ]^T \le 0$ $

This means the Lagrangian is

$ $ \mathcal{L} = x_1 + x_2 – \lambda \left[ (2N+1)(x_1+x_2)-2x_1x_2 \right] – \mu_1(2-x_1)-\mu_2(x_1+1 – x_2)$ $

So the KKT conditions are

$ $ \frac{\partial \mathcal{L}}{\partial x_1} = \lambda (2 n – 2 x_2 + 1 )+1+\mu_1-\mu_2=0$ $

$ $ \frac{\partial \mathcal{L}}{\partial x_2} = \lambda (2 n-2 x_1+1) + 1 +\mu_2 = 0$ $

$ $ \mu_1 ( 2 – x_1 ) =0$ $

$ $ \mu_1 \ge 0$ $

$ $ \mu_2 ( x_1 + 1 – x_2 ) =0$ $

$ $ \mu_2 \ge 0$ $

$ $ \frac{2}{2N+1} =\frac{1}{x_1} + \frac{1}{x_2} = 0$ $

Solving for $ x_1$ and $ x_2$

\begin{array}{c|c|c|c|c} \hline \mu_1>0 & \mu_2>0 & x_1=2 & x_2=3 & \mathrm{violates\ constraint} \ \hline \mu_1>0 & \mu_2=0 & x_1=2 & x_2=6 & \mathrm{correct} \ \hline \mu_1=0 & \mu_2>0 & x_1=n + \frac{1}{2} \sqrt{2+4n+4n^2} & x_2=x_1+1 & \mathrm{not\ a\ solution\ over\ the\ natural\ numbers} \ \hline \mu_1=0 & \mu_2=0 & x_1=2n+1 & x_2=x_1 & \mathrm{violates\ constraint} \ \hline \end{array}

So is this problem not suited for KKT? Have I done something incorrect?

Any guidance would be appreciated

Local behaviour of fractions with bounded denominator / Was it already studied?

My question is about a point process that I feel it would be natural to study, but that I have never heard of… This point process would represent, morally, the local behaviour of the set of fractions with bounded (but large) denominator, when seen from a random point. I would like to know if any of you has already heard about such a process…?

Let me give a formal definition of the point process that I would like to know more about. For $ n$ a (large) integer, denote by $ \mathbf{Q}_n$ the set of fractions with denominator $ \leq n$ : $ $ \mathbf{Q}_n := \bigl\{x \in \mathbf{R} \,\big|\, (\exists q \in \{1, \ldots, n\}) \ q x \in \mathbf{Z}\bigr\}\text.$ $ So, morally this is the set of “simple enough” fractions having a bounded denominator.

You can look at $ \mathbf{Q}_n$ from a (uniform) random point of the real line (note that $ \mathbf{Q}_n$ is invariant by $ 1$ -translations, so the notion of “uniform random point on the line” makes sense in this context): namely, for $ X_0$ a real random variable uniform on $ [0, 1)$ , consider the point process defined by $ $ \mathfrak{N}_n := n^2 \times (\mathbf{Q}_n – X_0)$ $ (here I see $ \mathfrak{N}_n$ as a random variable whose values are discrete subsets of $ \mathbf{R}$ ), whose law is completely well-defined. The factor $ n^2$ in the definition of the process $ \mathfrak{N}_n$ means that we are “zooming” on our set of simple fractions in such a way that the density of points remains of order $ O(1)$ regardless of the value of $ n$ (as it is a classical result that the density of points in $ \mathbf{Q}_n$ is equivalent to $ 3 n^2 / \pi^2$ when $ n \to \infty$ ).

Now, I have strong reasons to expect that when $ n \to \infty$ , the law of the process $ \mathfrak{N}_n$ converges (for some appropriate topology on discrets subsets of $ \mathbf{R}$ ) to the law of a limiting point process $ \mathfrak{N}_\infty$ . This process would somehow represent the local behaviour of the set of not-too-complicated fractions, when seen from a random point of the line. Note that one funny property of this point process in that two distinct points of it must always be apart from a distance $ \geq 1$ ! (because, for $ p / q$ and $ p’ / q’$ two dinstinct fractions of $ \mathbf{Q}_n$ , $ \left|p / q – p’ / q’\right| \geq 1 / q q’ \geq n^{-2}$ ).

My question is: has anyone studied this process? If yes, how is this process called, and could you give me references on it?…