Questions related with (base 2) representation of numbers and their unique properties arising out of number representation.

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7
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208 views

Number of ways to express a binary number in a certain way

So I'm working on a problem where I get to a point where I have to count the number of solutions to an equation or at least find a decent upper bound to be used in an estimate I need later. The ...
4
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33 views

Is there a way to approximate a polynomial as another, binary-coefficient polynomial?

Let's say I have a polynomial: $$p(x) = \sum_{n=0}^N a_n x^n$$ where $x \in \mathbb C$. Does there exist theory and/or methods on approximating $p$ as: $$p(x) \approx \hat p(x) = \sum_{m=0}^M b_n ...
4
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89 views

What is the explicit formula (solution) to this recursively defined binary matrix?

My question concerns the following binary matrix (call it matrix $A$). Or rather the entire family of such matrices, for some number of columns $n$ and rows $2^n$. The ellipses indicate that the ...
4
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872 views

Binary representation of powers of 3

We write a power of 3 in bits in binary representation as follows. For example $3=(11)$, $3^2=(1001)$ which means that we let the $k$-th bit from the right be $1$ if the binary representation of this ...
3
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65 views

Is there any kind of known pattern to $\sqrt 2$ in base 2?

Is there any kind of known pattern to $\sqrt 2$ in base 2? Is there any classification categories for decimal digits of numbers that for example would put $\sqrt 2, \sqrt 3 \cdots \sqrt n$ into ...
3
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317 views

Number of 'unique' one bit binary functions with N-bit inputs

Consider the set of binary functions that takes an N-bit input -> 1 bit output. There are 2^(2^N) elements in this set. Now potentially reduce this set by restricting to only considering functions ...
2
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0answers
114 views

Bounding the global intersection of a family of sets

Suppose that we have a decision tree of height $r + 1$ that describes how to increment an $n$-bit integer in the range $[0, 2^n -1]$. That is, the internal nodes are labelled with a bit position that ...
2
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0answers
104 views

An expression for the number of n-bit binary strings with at most k ones (without summations)

Say we need to find an expression for the number of binary strings of length $n$, which have at most $k$ ones. My solution was to split the problem into $k+1$ cases, where the number of ones, ...
2
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0answers
101 views

Upper limit for adding N numbers in grid of N processors in parallel

We must show how N binary numbers, each consisting of N bits each, can be added in a linear grid of N+logN processors. Suppose that the binary digits can be inserted directly in each processor of the ...
2
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0answers
272 views

Unambiguous expression for binary strings containing some substring

Is there some systematic way for finding an unambiguous expression for a binary string which contains a certain substring? For finding expressions not containing a substring, it is sometimes easy to ...
2
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0answers
259 views

An Upper Bound for an $[n,k,d]$ Linear Binary Code.

I've been reading about the various upper bounds for different types of codes. Recently, I came across a statement that is similar to the Singleton Upper Bound that I am having trouble proving. The ...
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0answers
55 views

Name of this formula or more explantation of the proof?

I have found this formula which is a combinatorial identity for counting binary words. I'd like more information on it, or the name of the proof. I am also not totally clear on the step between the ...
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48 views

Probability of having n more 0's than 1's in a subsequence

We have a number K>2 and a sequence of m binary values $$b_1, ... b_m $$ Given a subsequence $$b_i, ... b_j, i>j $$ we define: W(i,j) is the number of 1's in the sequence L(i,j) is the number ...
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18 views

largest number machine number that this comp has, numerical analysis, mantissa

I am having trouble understanding the basics on how to calculate the m, s and c of the formula. I have looked in my textbook but not much of an explanation, also on the IEEE websites but nothing much. ...
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14 views

How to perform binary transformations?

One of the steps in Binary Index Tree algorithm is to find a node's parent which is done by un-setting the rightmost SET bit. For example: if a node has index 1010 than it's parent is 1000. To apply ...
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0answers
35 views

Finding a Hamiltonian cycle for $Q_4$

A hyper cube $Q_n$ is a graph that have the length-n binary sequences as its vertices. Two vertices are adjacent if they differ in one entry. I found a Hamilton cycle for $Q_3$ as follows $$000 \to ...
1
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0answers
15 views

binary divison with remainder

I wanted to calculate this (binary): 10101.101/1.1. the result is: 1110.01101010101010101010101010... I succeed in calculating the integer part, but I didn't understand how I find the numbers that ...
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0answers
17 views

Canonical forms for matrices with binary elements.

Based on this answer to a combinatorics question I grew curious of results regarding similarities or canonical forms of matrices fulfilling these criteria: Elements of matrix are binary valued ...
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0answers
45 views

Rounding up a binary number

I just converted the fraction $\frac{3}{5}$ to the following floating point binary number: $(1.001100110011001100110011\cdots)_{2}\times 2^{-1}$. Now, I am trying to find the two nearest machine ...
1
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0answers
12 views

solution of equations with inter-dependent variables.

Suppose I have few equations of the form a12*x+a13*y+a14*z+a15*xy+a16*xz+a17*yz=d1 In short I have n free variables and other (n choose 2) dependent variables. ...
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20 views

Binary operations and proofs

$\text{binary}(a)$ = binary representation of a base 10 number $a$ Are the following statements correct? If yes, where can I find the proofs? (1) $\text{binary}(a\times b)=\text{binary}(a)\times ...
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110 views

Generator polynomial creates a 127 bit sequence

I have been reading a paper that states that a generator polynomial $$ G(D)= 1+ D^4+D^7$$ creates a 127 bit sequence which is as follows 00001110 11110010 11001001 00000010 00100110 00101110 ...
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43 views

Learning Booth's Algorithm, I Can't Find the Issue on Final Result

I am practicing using Booth's Algorithm to multiply a positive number and a negative number (specifically the problem is $-12 \times 4$). I have included my attempt, but I can't find the issue. If ...
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0answers
44 views

Binary division using polynomial

I want to do a division of two binaries and take the rest (mod). But I want to do this using polynomials, let's take the example: binary dividend: 010001100101000000000000 binary divisor: 100000111 ...
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71 views

How to determine size and height balance of binary search tree?

I've been reading/ learning binary search trees and I've been stuck on the following question for a while now. I have the following tree, how do I determine the height and size balance of it? How do ...
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0answers
99 views

Generating function from a set of binary strings

So this question is in my textbook and there's no solution, so I'm seeing if I can get a confirmation? Q: Let $S$ be the set of all binary strings of length 4, where for each string $a\in S$, the ...
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0answers
48 views

Has anyone established an upper bound for the least integer $k$ such that infinitely many primes have at most $k$ ones in their binary representation?

Has anyone established an upper bound for the least integer $k$ such that infinitely many primes have at most $k$ ones in their binary representation? $2$ is the only prime with $1$ one, the Fermat ...
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81 views

Leibniz Binary Representation of Squares

Leibniz claims to have found patterns in the square numbers and their binary representations. I cannot see any patterns at all. Here are the first ten squares and their binary representations, can ...
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0answers
43 views

Is there a way to get the n-th of bits of 2^k

I have a large number N=O(2^k). For simplicity, let's say that N=n^k. However, I only need the n-th bits of N, say for example the 10-th to 16-th bit of N... without calculating the full expansion... ...
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166 views

Why doesn't this base 10 number x mod 2^y work for converting base 10 to binary

Okay I tried to convert 1 million to binary by dividing by a power of 2 and taking the remainder and dividing that by a power of 2 and so on and I got this: 1111010000100100000 Google says 1 million ...
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0answers
104 views

conversion of number from base 10 to base 16

I have trouble finding an example of a conversion of a number from base 10 to base 16, where there is a loss of significant digits when using the Euclidean algorithm due to division by large numbers, ...
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0answers
5k views

Calculating minimum hamming distance of a code

We use hamming code of (7,4,3); Given 4 bits of information, we'll add 3 bits of parity, and one more parity bit for the 7-bits code. Given $x_3,x_5,x_6,x_7$ $x_1 = (x_3+x_5+x_7) \mod 2$ $x_2 = ...
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0answers
43 views

How to show that the spectral radius of a binary tree approaches exp(1) as the N tends to infinity?

How can I prove mathematically that the spectral radius of a binary tree approaches e as the number of nodes tends to infinity? I mean it is true that the increase in nodes number is exponential but ...
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0answers
105 views

Probability of getting a Column full rank binary matrix

Suppose I have a $m \times n$ ($m>>n$) zero matrix (all of the elements are $0$). I want to flip $k$ ($k \ge n$ and can be controlled by the user) elements of the matrix randomly. After this ...
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0answers
366 views

Recurrent relation for number of ways to get a balanced n-binary tree

In answering a question related to binary trees, I came up with the following recurrent relation: Base cases: $$ f \left (1 \right ) = 1 $$ $$ f \left (2 \right ) = 2 $$ Recurrent relations: $$ f(n) ...
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0answers
276 views

Finding the maximum XOR metric

I'm trying to find a way to find n keys (x bits) where the XOR distance metric between them would be greatest. By XOR distance metric I just mean the value when two keys are XORed together. So for ...
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0answers
38 views

Predict binary occupancy vector from history of vectors

I have a set of binary vectors where each vector represents one day of occupancy in a house and consists of 48 elements (each element for 30 minutes of the day). Each element can be 1 meaning that ...
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0answers
138 views

Fixed decimal point numbers binary representation

I saw this formula that I cannot understand how it applies (the proof). Q: say we have a $N$ bit register (imagine a binary number with $N$ digits) for representing fixed decimal point numbers. In ...
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57 views

System of equations with binary variables

Is there any way to simplify solving system of equations, assuming that all variables are binary? All equations, however, are seen as equations with real (or, to make things simple, integer) ...
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0answers
118 views

interpret reduced cost of all binary variables

I have prepared and run a Linear Programming model in SAS. I have some questions about the output that I can’t find answers to, and am hoping that someone can help. My model contains the decision ...
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0answers
167 views

1s surpassing 0s in binary strings of odd length

Let $A(k)$ be the number of distinct binary strings of length $2k+1,$ for which the number of $1$s surpasses the number of $0$s for the first time at digit number $2k +1$, i.e., in the final digit in ...
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219 views

Recurrence relation for the digits of the integer square root in binary

I was investigating a question on the Electrical Engineering Stack Exchange site, available here: ...
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10 views

Can $1.0*2^{-1}$ and $1.1*2^0$ both equal -0.5 in twos complement?

I am happy that if you write -0.5 in binary as -0.1 and then convert to twos-complement you get 1.1. You can read that as $-1+1/2=-1/2$. However if you use standard form and write $0.5_{10}$ as ...
0
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0answers
14 views

Floating point binary to half precision floating point

I am trying to convert to $16$ bit half precision floating point however I ran into a possible error and am unsure if a negative exponent is ok. I am trying to convert $0010011100010000$ I separate ...
0
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0answers
16 views

Binary Tree - Determin Number of Nodes

Given a full balanced binary tree where at depth 0, the number of nodes is 1, at depth 1, the number of nodes is 3, at depth 2, the number of nodes is 7. At depth X, what is the number of nodes?
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43 views

Two's complement involving binary numbers [Single-precision floating point number]

Given the following bit pattern: 0010 0100 1001 0010 0100 1001 0010 0100 What decimal number does it represent, assuming it is a single-precision floating point number?
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28 views

Designing a Turing machine for Binary Division

I'm trying to design a TM to binary divide 2 numbers. The best approach I've found is the method: division as repeated subtraction, but I don't know if there's a fastest way or an easier way. ...
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0answers
8 views

How would I round the number $-(1.01101|01)_2$ down?

I'm trying to round $-(1.01101|01)_2$ in binary down. The $|01$ are the guard bits. The answer is $-(1.01110)_2$ My attempt is as follows: $-(1.01101|01)_2$ $-(0.00001|00)$ = $(1.01100)$ Not ...
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0answers
17 views

Is it possible to break a system into two BSCs to find the total capacity of a system?

I have been trying to solve this problem in manor of ways but I cannot seem to find a satisfactory solution. I have tried the basic way of calculating capacity through self information but I was ...
0
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0answers
30 views

How to find the number of subsets with a given length and XOR?

I have A (0<A<500000) elements (up to 10^6) in the set. I need to find in how many ways can I remove a subset, the size of ...