BACKGROUND Although it is believed that children with cerebral palsy are at high risk for learning difficulties and arithmetic difficulties in particular, few studies have investigated this issue. METHODS Arithmetic ability was longitudinally assessed in children with cerebral palsy in special (n = 41) and mainstream education (n = 16) and controls in mainstream education (n = 16). Second gra...

This research addresses system reliability analysis using weakest t-norm based approximate intuitionistic fuzzy arithmetic operations, where failure probabilities of all components are represented by different types of intuitionistic fuzzy numbers. Due to the incomplete, imprecise, vague and conflicting information about the component of system, the present study evaluates the reliability of sy...

Interval arithmetic is arithmetic for continuous sets. Floating-point intervals are intervals of real numbers with floating-point bounds. Operations for intervals can be efficiently implemented. Hence, the time is ripe for standardization. In this paper we present an interval model that is mathematically sound and closed for the 4 basic operations. The model allows for exception free interval a...

Z = {integers} Q = {rational numbers} R = {real numbers} C = {complex numbers} Z+ = { positive integers} q = a power of a prime p Fq = A Finite field with q elements A = Fq[t] A+ = {monics in A} K = Fq(t) K∞ = Fq((1/t)) = completion of K at ∞ C∞ = completion of algebraic closure of K∞ [n] = tq n − t dn = ∏n−1 i=0 (t qn − tqi) `n = ∏n i=1(t− tq i ) deg = function assigning to a ∈ A its degree in...

We study some arithmetic properties of the Ramanujan function τ(n), such as the largest prime divisor P(τ(n)) and the number of distinct prime divisors ω(τ(n)) of τ(n) for various sequences of n. In particular, we show that P(τ(n)) ≥ (logn)33/31+o(1) for infinitely many n, and P(τ(p)τ(p2)τ(p3)) > (1+o(1)) log log p log log log p loglog log log p for every prime p with τ(p) 6= 0.

FUNCTION VERIFICATION OF COMBINATIONAL ARITHMETIC CIRCUIT MAY 2015 DUO LIU B.S., JIANGNAN UNIVERSITY, WUXI, JIANGSU, CHINA M.S.E.C.E., UNIVERSITY OF MASSACHUSETTS AMHERST Directed by: Professor Maciej Ciesielski Hardware design verification is the most challenging part in overall hardware design process. It is because design size and complexity are growing very fast while the requirement for pe...

A bipartition of n is an ordered pair of partitions (λ, μ) such that the sum of all of the parts equals n. In this article, we concentrate on the function c5(n), which counts the number of bipartitions (λ, μ) of n subject to the restriction that each part of μ is divisible by 5. We explicitly establish four Ramanujan type congruences and several infinite families of congruences for c5(n) modulo 3.

• An arithmetic function takes positive integers as inputs and produces real or complex numbers as outputs. • If f is an arithmetic function, the divisor sum Df(n) is the sum of the values of f at the positive divisors of n. • τ (n) is the number of positive divisors of n; σ(n) is the sum of the positive divisors of n. • The Möbius function μ(n) is 1 if n = 1 and 0 if n has a repeated prime fac...