Basic arithmetic is the cornerstone of mathematics and computer sciences. In arithmetic, ‘division by zero’ is an undefined operation and any attempt at extending logic for algebraic division to incorporate division by zero has resulted in paradoxes and fallacies. However, there is no proven theorem or mathematical logic that suggests that, defining logic for division by zero would result in br...

Robustness problems of computational geometry algorithms is a topic that has been subject to intensive research efforts from both computer science and mathematics communities. Robustness problems are caused by the lack of precision in computations involving floatingpoint instead of real numbers. This paper reviews methods dealing with robustness and inaccuracy problems. It discusses approaches ...

Programming languages such as Common Lisp and virtu ally every computer algebra system CAS support exact arbitrary precision integer arithmetic as well as exact ra tional number computation Several CAS include interval arithmetic directly but not in the extended form indicated here We explain why changes to the usual rational number system to include in nity and not a number may be use ful espe...

Computer arithmetic is predominantly performed using binary arithmetic because the hardware implementations of the operations are simpler than those for decimal computation. However, many decimal fractions cannot be represented exactly as binary fractions with a finite number of bits. The value 0.1, for example, can only be represented as an infinitely recurring binary number. If a binary appro...

let $f$ be a finite extension of $bbb q$‎, ‎${bbb q}_p$ or a global‎ ‎field of positive characteristic‎, ‎and let $e/f$ be a galois extension‎. ‎we study the realization fields of‎ ‎finite subgroups $g$ of $gl_n(e)$ stable under the natural‎ ‎operation of the galois group of $e/f$‎. ‎though for sufficiently large $n$ and a fixed‎ algebraic number field $f$ every its finite extension $e$ is‎ ‎re...

the concept of geometric-arithmetic indices (ga) was put forward in chemical graph theoryvery recently. in spite of this, several works have already appeared dealing with these indices.in this paper we present lower and upper bounds on the second geometric-arithmetic index(ga2) and characterize the extremal graphs. moreover, we establish nordhaus-gaddum-typeresults for ga2.

continuing the work k. c. das, i. gutman, b. furtula, on second geometric-arithmetic indexof graphs, iran. j. math chem., 1(2) (2010) 17-28, in this paper we present lower and upperbounds on the third geometric-arithmetic index ga3 and characterize the extremal graphs.moreover, we give nordhaus-gaddum-type result for ga3.

The purpose of this survey is to present the main concepts and results in non-standard number representation, and to give some examples of practical applications. This domain lies at the interface between discrete mathematics (dynamical systems, number theory, combinatorics) and computer science (computer arithmetic, cryptography, coding theory, algorithms). It also plays an important role in t...

This paper describes our new satisfyability (SAT) modulo theory (SMT) solver STABLE for the quantifier-free logic over fixedsized bit vectors. Our main application domain is formal verification of system-on-chip (SoC) modules designed for complex computational tasks, for example, in signal processing applications. Ensuring proper functional behavior for such modules, including arithmetic correc...

[email protected] 1 , [email protected] 2 , Abstract—the most timing critical part of logic design usually contains one or more arithmetic operations, in which addition is commonly involved. Addition is a fundamental arithmetic operation and it is the base for arithmetic operations such as multiplication and the basic adder cell can be modified to function as subtractor by adding ano...