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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192 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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71 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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776 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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160 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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65 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
84 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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247 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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242 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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76 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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25 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
34 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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34 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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77 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
43 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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68 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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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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89 views

Why is this number the smallest positive normalised binary value?

In the AQA A2 Computing textbook (Bond and Langfield, 2009), they say that this number is the smallest positive normalised value, given a 10 bit mantissa and a 6 bit exponent: ...
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126 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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96 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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4k 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
39 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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89 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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280 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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259 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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37 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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118 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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51 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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104 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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161 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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199 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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37 views

minimum number of leaves in a perfect binary tree

I'm trying to prove that the number of leaves in a perfect binary tree is at least H+1 where H is the height of the tree. This is what I've done up til now: No of leaves at height $H = 2^H$ Base ...
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18 views

Closest finite series representation of original number < 1

I have another question related to this Decompose negative power of ten in finite series. Suppose we have number $0.1$ decomposed in finite sum of two in negative power: $10^{-1} = 2^{-4} + 2^{-5} + ...
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13 views

Using the algebraic expression ((x-2)+3) / ((2-(3+y))*(w-8)) show the results of performing a preorder, an inorder, and a postorder search.

Using the algebraic expression ((x-2)+3) / ((2-(3+y))*(w-8)) show the results of performing a preorder, an inorder, and a postorder search. Preorder is root, left, right. Inorder is left, root, ...
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0answers
23 views

Signed Number's Binary Addition

I'm going to discuss about Signed Number's Binary addition, I searched about it and even read books. Now I make little changes in it's logic and start my own logic to solve it. Let me show 4 bit ...
0
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0answers
31 views

Bitwise Operations and the Naming Convention of their Operators

So I just recently came across a bitwise operation on StackOverflow which shifts the bits in a binary number to the right while zero-filling from the left. The left side zero-filling overwrites the ...
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21 views

Binary block code

I don't fully understand how i would construct binary block codes? As far as I know we have a code, say, $(5,4,3)-$code, then $5$ would represent the length, $4$ would represent the number of codes ...
0
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0answers
13 views

Is there a standard to follow when writing a weighted binary code (ex. 6-3-1-1 or 2-4-2-1)?

For example, in 6-3-1-1 code, 7 can be represented as 1001 or 1010. Similarly in 2-4-2-1 code, 6 can be represented as 1100 or 0110. My professor said that there is a standard to follow and there can ...
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46 views

Is there an alternative encoding scheme to binary where similarity of pattern correlates with size of number?

If I compare binary for 7 111 and binary for 8 1000 there is no correlation between these two patterns that suggests that ...
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53 views

binary search worst case for a set database of entries some good some bad

A database has 10,000 sorted entries, 20% known to be good. When looking up a record in the database, the good entries account for 60%. Two design options are considered to store the data in the ...
0
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0answers
72 views

A binary plot of the Catalan numbers and the pseudo-Fibonacci series that can be found inside. Why do they appear?

I was trying to find in Internet a binary plot of the Catalan numbers, and I did not find anyone... so I did it by myself and here it is (about 2000 elements): There are not clear patterns inside ...
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20 views

Find minimum algorithm complexity

So, I have this task: Let us have square matrix $A \ size\ n\ \times\ n$ for which is true: $$A(k,l)<=A(m,p)\ if \ k <=m, l<=p$$ I need to find algorithm, which finds value X in such matrix, ...
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111 views

Looking for a best binary division tutorial

All I can found is such confusing "tutorials" like the below examples.. Even if i search it on Youtube, most videos are having dislikes and with short or unclear explanation. Is it a really hard ...
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0answers
22 views

Determining whether two trees are isomorphic

Is there a (probably recursive) algorithm that can be used to determine whether two not necessarily binary ordered (sub)trees are isomorphic or not?
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0answers
18 views

Distribution of binary sequences?

I have two binary sequence and wanted to test if there is a similar patterns between the two. This is a completely new topic for me. Are there ways to define distribution of a binary sequence ...
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21 views

why use Unpacked BCD

I wonder why someone should ever use unpacked BCD, as packed BCD is a lot more compact, and you can also form each number. Is there a good reason to use it or not?
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25 views

What are the possible relations?

If there are two sets A and B. Set A has two elements so does Set B. How many different possible relations can we have from A to B. For example. Set A has (David, Max) and Set B has (x,y). Would the ...
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37 views

Performing arithmetical operations (with binary numbers) using propositional logic

Clarifying some terms. By arithmetical operations I mean the four basic operations of addition, subtraction, multiplication and division. By binary numbers I mean numbers in the binary system. By ...
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40 views

What are the possible dimensions of a binary cyclic code of length 37?

What about the possible dimensions of a 16-ary cyclic code of length 37? [Edit]: The cyclic codes of length $37$ are in a bijective correspondence with factors of $x^{37}-1$. Furthermore, the ...
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42 views

Binary to Decimal Floating Point Number

I seem to be having difficulties trying to figure this out: I have a Binary 0101011101010000 and I'm trying to compute the decimal floating point number, in the IEEE-754 format Can somebody help? ...
0
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0answers
25 views

Binary Overflow Detection

I am trying to solve several problems, which are binary and encoded using the 2's complement system. One problem has stuck out to me: 0111 + 0001 Both are positive, with 0111 being 1+2+4 or 7, and ...