相关论文: Adding digit vectors
To represent real $m$-dimensional vectors, a positional vector system given by a non-singular matrix $M \in \mathbb{Z}^{m \times m}$ and a digit set $\mathcal{D} \subset \mathbb{Z}^m$ is used. If $m = 1$, the system coincides with the well…
In a series of ten two-dimensional graphical vector addition questions with varying visual representations, most students stuck to a single solution method, be it correct or incorrect. Changes to the visual representation include placing…
Suppose that a random n-bit number V is multiplied by an odd constant M, greater than or equal to 3, by adding shifted versions of the number V corresponding to the 1s in the binary representation of the constant M. Suppose further that the…
When numbers are added in base $b$ in the usual way, carries occur. If two random, independent 1-digit numbers are added, then the probability of a carry is $\frac{b-1}{2b}$. Other choices of digits lead to less carries. In particular, if…
One can recover vectors from $\mathbb{R}^m$ with arbitrary precision, using only $\lceil \log_2(m+1)\rceil +1$ continuous measurements that are chosen adaptively. This surprising result is explained and discussed, and we present…
If we want to represent integers in base $m$, we need a set $A$ of digits, which needs to be a complete set of residues modulo $m$. When adding two integers with last digits $a_1, a_2 \in A$, we find the unique $a \in A$ such that $a_1 +…
Parallel addition, i.e., addition with limited carry propagation, has been so far studied for complex bases and integer alphabets. We focus on alphabets consisting of integer combinations of powers of the base. We give necessary conditions…
The number of positive and negative carries in the addition of two independent random signed digit expansions of given length is analyzed asymptotically for the $(q, d)$-system and the symmetric signed digit expansion. The results include…
Today's PCs can directly manipulate numbers not longer than 64 bits because the size of the CPU registers and the data-path are limited. Consequently, arithmetic operations such as addition, can only be performed on numbers of that length.…
We study the combinatorics of addition using balanced digits, deriving an analog of Holte's "amazing matrix" for carries in usual addition. The eigenvalues of this matrix for base b balanced addition of n numbers are found to be…
The implementation of a quaternary 1-digit adder composed of a 2-bit binary adder, quaternary to binary decoders and binary to quaternary encoders is compared with several recent implementations of quaternary adders. This simple…
Bit addition arises virtually everywhere in digital circuits: arithmetic operations, increment/decrement operators, computing addresses and table indices, and so on. Since bit addition is such a basic task in Boolean circuit synthesis, a…
Relativistic addition of velocities in one dimension, though a mainstay of introductory physics, contributes much less physical insight than it could. For such calculations, we propose the use of velocity factors (two-way doppler factors).…
Adding a column of numbers produces "carries" along the way. We show that random digits produce a pattern of carries with a neat probabilistic description: the carries form a one-dependent determinantal point process. This makes it easy to…
In Carry Propagate Adders, carry propagation is the critical delay. For the 1-digit adders that they use, the most efficient scheme is to generate two intermediate carries: C$_{out0}$ ($C_{in}$=0) and $C_{out1}$($C_{in}$=1). Then multiplex…
A system of $m$ nonzero vectors in $\mathbb{Z}^n$ is called an $m$-icube if they are pairwise orthogonal and have the same length. The paper describes $m$-icubes in $\mathbb{Z}^4$ for $2\le m\le 4$ using Hurwitz integral quaternions, counts…
In this short note, we show a simple characterization of integers that reach records for a sequence described by adding binary strings to runs of 1's and 0's in a binary representation. In particular, we show that this set does not depend…
The notion of Carry Value Transformation (CVT) is a model of Discrete Deterministic Dynamical System. In this paper, we have studied some interesting properties of CVT and proved that (1) the addition of any two non-negative integers is…
The well known binary and decimal representations of the integers, and other similar number systems, admit many generalisations. Here, we investigate whether still every integer could have a finite expansion on a given integer base b, when…
Let $A$ be an expanding $2 \times 2$ matrix with rational entries and $\mathbb{Z}^2[A]$ be the smallest $A$-invariant $\mathbb{Z}$-module containing $\mathbb{Z}^2$. Let $\mathcal{D}$ be a finite subset of $\mathbb{Z}^2[A]$ which is a…