We propose a unified approach to addition of some physical quantities (among which resistors and capacitors are the most well-known) that are usually encountered in introductory physics such that the formulæ required to solve problems are always simply additive. This approach has the advantage of being consistent with the intuition of students. To demonstrate the effectiveness of our approach, ...

Suppose each of k ≤ no(1) players holds an n-bit number xi in its hand. The players wish to determine if ∑ i≤k xi = s. We give a public-coin protocol with error 1% and communication O(k lg k). The communication bound is independent of n, and for k ≥ 3 improves on the O(k lg n) bound by Nisan (Bolyai Soc. Math. Studies; 1993). Our protocol also applies to addition modulo m. In this case we give ...

It is shown that there exists an absolute constant H such that for every h > H, every prime p, and every set A ⊆ Zp such that 10 ≤ |A| ≤ p(lnh)/(9h) and |hA| ≤ h3/2|A|/(8(lnh)1/2), the set A is contained in an arithmetic progression modulo p of cardinality max1≤j≤h−1 |hA|−Pj(|A|) h−j + 1, where Pj(n) = (j+1)j 2 n − j + 1. This result can be viewed as a generalization of Freiman’s “2.4-theorem”.

Small Finding a shorter addition/subtraction-chain for an integer is an important problem for many cryptographic systems based on number theory. Especially, execution time of multiplication on an elliptic curve cryptosystem is directly proportional to the length of the addition/subtraction-chain. In this paper, we propose an algorithm to find an addition/subtractionchain. The proposed algorithm...

Ordinal addition and multiplication can be extended in a natural way to all sets. I survey the structure of the sets under these operations. In particular, the natural partial ordering associated with addition of sets is shown to be a tree. This allows us to prove that any set has a unique representation as a sum of additively irreducible sets, and that the non-empty elements of any model of se...

Polycyclic aromatic hydrocarbons (PAHs) are omnipresent in the interstellar medium (ISM) and also in carbonaceous meteorites (CM) such as Murchison. However, the basic reaction routes leading to the formation of even the simplest PAH-naphthalene (C10 H8 )-via the hydrogen-abstraction/acetylene-addition (HACA) mechanism still remain ambiguous. Here, by revealing the uncharted fundamental chemist...

We study the effect of addition on the Hamming weight of a positive integer. Consider the first 2n positive integers, and fix an α among them. We show that if the binary representation of α consists of Θ(n) blocks of zeros and ones, then addition by α causes a constant fraction of low Hamming weight integers to become high Hamming weight integers. This result has applications in complexity theo...

The integer addition is often applied in ciphers as a cryptographic means. In this paper we will present some results about the linear approximating for the integer addition.