Find the smallest group of numbers with sum bigger then $X$

Given a list of numbers $ S$ where $ 0 < s_i < 100$ , find the smallest group of numbers with sum bigger than $ X$ .
Each number can be used multiple times.

Ex: for $ S = [3,4.1], X = 10$ the solution is $ [3, 3, 4.1]$

Is it a known problem? What will be the best way of solving it?

For now, my best solution is to randomly pick numbers and repeat the process multiple times.

Finding Smallest Frontier for Graphs of bounded “width”

Let $ G$ be a graph and $ X=x_1,x_2,…,x_n$ be an permutation/ordering of the vertex set of $ G$ . We then let $ S_i = \{x_j:j\le i\}$ , and $ F_i$ be the number vertices $ v\in S_i$ that are adjacent to some vertex $ u(v) \not\in S_i$ . We finally define $ F$ to be a list of values $ F_i$ sorted from largest to smallest. e.g. if $ F_1=2,F_2=1,F_3=6, F_4=2$ we’d have $ F = 6,2,2,1$ (we caution that in reality $ F_{i+1}-F_i\le 1$ so the sequence features in the example could not occur)

In general, finding $ X$ such that $ F$ is lexicographically minimal is a task which I’d assume is NP-Hard.

However, letting $ \mathcal{G}_{k,t}$ denote the family of graphs $ G$ such that the vertex set of $ G$ is partitioned in to $ t$ parts $ V_1,\dots,V_t$ such that $ V_i \le k$ for all $ i$ , and $ |a-b|\ge 2$ implies there is no edge $ (u,v)$ in $ G$ where $ u\in V_a$ and $ v\in V_b$ .

For fixed $ k$ , and given $ G\in \ mathcal{G}_{k,t}$ , is there an algorithm that finds $ X$ such that $ F$ is lexicographically minimal, whose worst case run time is polynomial in $ t$ ?

Smallest subarray problem

Say you have an array of integers like [1, 2, 3, 4, 5, 6], the problem is to find the smallest way to break up the array into sub-arrays where each sub-array satisfies the following requirement:

  • sub-array.first_integer and sub-array.last_integer must have a common divisor that is not 1.

So for [1, 2, 3, 4, 5, 6] the answer would be 2, because you can break it up like [1] and [2, 3, 4, 5, 6], where 2 and 6 have a common divisor of 2 (which is > 1 so it meets the requirement).

You can assume the array can be huge but the numbers are not too big. Is there a way to do this in n or n*log(n) time? I think with dp and caching n^2 is possible but not sure how to do it faster.

Finding the smallest number that scales a set of irrational numbers to integers

Say we have a set $ S$ of $ n$ irrational numbers $ \left\{a_1, …, a_n\right\}$ .

Are there any known algorithms that can determine a scaling factor $ s \in \mathcal{R}$ such that $ s * a_i \in \mathcal{N} \;\forall a_i \in S$ , assuming that such factor exists? If multiple exist, how about the smallest one?

Moreover, I wonder, under what input conditions could one assume that an algorithm for this problem can’t (or can) return a valid scaling factor?

If no known algorithms to this problem exist, are there any known classes of “scaling algorithms” that may solve a similar problems?

Write the smallest positive number that can be represented by the floating point system

Using a normalised floating point representation box with an 8-bit mantissa and a 4-bit exponent, both stored using two’s complement.

(a) Write the smallest positive number that can be represented by the floating point system in the boxes below. The result is: Mantissa 0.1000000 and exponent 1000

Do not see how this can could someone please explain.

Finding the smallest integer such that a given condition holds with “binary search”

Setup. Suppose we are given a function $ $ f:\mathbb N\to\{\text{False},\text{True}\}$ $ such that $ $ f(n)=\text{True}\implies f(n+1)=\text{True}$ $ and such that $ f(n)=\text{True}$ for some $ n$ large enough.

In natural language. The function $ f$ imposes a condition on the natural numbers which is fulfilled once $ n$ is large enough.

My question. How can I, for any given $ f$ , find the smallest $ n$ such that $ f(n)=\text{True}$ ?

A first idea would be to start with $ n=1$ and to increment $ n$ by one until $ f(n)$ is True. However, this is fairly slow. So I designed a sort of “binary search” for this task. How can I translate this to Mathematica?

Here is the Code in Python:

def bin_search(cond):     n = 1     while not cond(n):         n *= 2     lower_bound = n//2     upper_bound = n     middle = (lower_bound + upper_bound)//2     while upper_bound - lower_bound > 1:         if cond(middle):             upper_bound = middle         else:             lower_bound = middle         middle = (lower_bound + upper_bound)//2     return upper_bound 

For example, one such condition would be $ $ f(n)=[H_n\geq 10],$ $

where $ $ H_n=\sum_{i=1}^n \frac 1i$ $ is the $ n$ th harmonic number.

What are the smallest and biggest negative floating point numbers in IEEE 754 32 bit?

I am stuck with a question that asks for smallest and biggest negative floating point numbers in IEEE 754 32-bit (their representation and decimal numerical value from which one can approximate the precision of the number)? So -0, NaN and Infinity do not belong to negative rational numbers.

I have stumbled upon -3.403 x 10^38 and 2^-126. I came close to the first one actually. I tried to do some calculations but got kind of lost in the process as floating point representation is counter-intutive for me, especially when calculating negative numbers. Can someone help me to clarify my thought process for the calculations so that I can find the numbers?