Arithmetic is a branch of mathematics that consists of the study of numbers, especially the properties of the traditional operations on them—addition, subtraction, multiplication and division.

2808 Results for the subject "Arithmetic":

AbstractTo verify computation results of double precision arithmetic, a high precision arithmetic environment is needed. However, it is difficult to use high precision arithmetic in ordinary computing environments without any special hardware or libraries. Hence, we designed the quadruple precision arithmetic environment QuPAT on Scilab to satisfy the following requirements: (i) to enable programs to be written simply using quadruple precision arithmetic; (ii) to enable the use of both double and quadruple precision arithmetic at the same time; (iii) to be independent of any hardware and operating systems.To confirm the effectiveness of QuPAT, we applied the GCR method for ill-conditioned matrices and focused on the scalar parameters α and β in GCR, partially using DD arithmetic. We found that the use of DD arithmetic only for β leads to almost the same results as when DD arithmetic is used for all computations. We conclude that QuPAT is an excellent interactive tool for using doubl *Read more...*

AbstractIn this paper we explore aspects of computer arithmetic from the viewpoint of dynamical systems. We demonstrate the effects of finite precision arithmetic in three uniformly hyperbolic chaotic dynamical systems: Bernoulli shifts, cat maps, and pseudorandom number generators. We show that elementary floating-point operations in binary computer arithmetic possess an inherently fractal structure. Each of these dynamical systems allows us to compare the exact results in integer arithmetic with those obtained by using floating-point arithmetic.

Julian Palmore Publication date: 1990/06/01AbstractGiven a density 0<σ⩽1, we show for all sufficiently large primes p that if S⊆Z/pZ has the least number of three-term arithmetic progressions among all sets with at least σp elements, then S contains an arithmetic progression of length at least log1/4+o(1)p.

Ernie Croot Publication date: 2006/01/01AbstractThis paper describes how a combination of interval derivative arithmetic and interval slope arithmetic can be used with the programming language Pascal-SC to obtain an enclosure of the range of a factorable function ƒ:Rn → R1 with ƒ ∈ C2 (D) which is often narrower than that which is obtained by using slope arithmetic alone and which is always narrower than that which is obtained by using interval derivative arithmetic alone.

Shen Zuhe Publication date: 1990/09/01AbstractWe introduce the notions of arithmetic colorings and arithmetic flows over a graph with labelled edges, which generalize the notions of colorings and flows over a graph.We show that the corresponding arithmetic chromatic polynomial and arithmetic flow polynomial are given by suitable specializations of the associated arithmetic Tutte polynomial, generalizing classical results of Tutte (1954) [9].

Michele DʼAdderio Publication date: 2013/01/01Highlights•This study examines neural bases of verbalized arithmetic principles.•Verbal arithmetic principles have stronger parietal activation than does language.•Verbal arithmetic principles rely on the perisylvian language networks.•Verbal arithmetic principles are associated with parietal- temporal connectivity.•Verbal computation is associated with parietal-occipital connectivity.

Jie Liu Publication date: 2017/02/15Abstract:Roughly speaking, floating-point (FP) arithmetic is the way numerical quantities are handled by the computer. Many different programs rely on FP computations such as control software, weather forecasts, and hybrid systems (embedded systems mixing continuous and discrete behaviors). FP arithmetic corresponds to scientific notation with a limited number of digits for the integer significand. On modern processors, it is specified by the IEEE-754 standard which defines formats, attributes and roundings, exceptional values, and exception handling. FP arithmetic lacks several basic properties of real arithmetic; for example, addition is not associative. FP arithmetic is therefore often considered as strange and unintuitive. This chapter presents some basic knowledge about FP arithmetic, including numbers and their encoding, and operations and rounding. Further readings about FP arithmetic include.

Sylvie Boldo Publication date: 2017/01/01Highlights•Bilingual adults were scanned with fMRI while computing mental arithmetic problems.•Arithmetic problem solving induced distinct activation pattern in each of bilingual's languages.•Language plays a critical role in arithmetic.

Amandine Van Rinsveld Publication date: 2017/07/01AbstractSentient Arithmetic is defined as an extension of Elementary Arithmetic with three more derivation rules and the Incompleteness Theorems are derived within it without using any metalanguage. It is shown that Consistency cannot be chosen as an axiom

K.K. Nambiar Publication date: 1996/07/01AbstractA central part of microprocessors is the ALU (Arithmetic Logic Unit). This block in a processor takes a number of inputs from registers and, as its name suggests, carries out either logic functions (such as NOT, AND, OR, and XOR) on the inputs, or arithmetic functions (addition or subtraction as a minimum), although it must be noted that these will be integer (or fixed point potentially) only and not floating point. This chapter of the book will describe how these types of low level logic and arithmetic functions can be implemented in VHDL and Verilog.

Peter Wilson Publication date: 2016/01/01AbstractGiven an absolutely irreducible horizontal hypersurface Z in a projective space over the ring of integers R of a number field, we give an explicit bound for the product of the norms of the prime ideals of R over which the fibre of Z becomes reducible. This bound is given as a function of a projective height of Z and is obtained using arithmetic intersection theory, in particular, an arithmetic Bézout theorem.

Reinie Erné Publication date: 2000/10/01AbstractThe development of mental arithmetic is approached from a mathematical perspective, focusing on several process models of arithmetic performance which have grown out of the chronometric methods of cognitive psychology. These models, based on hypotheses about the nature of underlying mental operations and structures in arithmetic, generate quantitative predictions about reaction time performance. A review of the research suggests a developmental trend in the mastery of arithmetic knowledge—there is an initial reliance on procedural knowledge and methods such as counting which is followed by a gradual shift to retrieval from a network representation of arithmetic facts. A descriptive model of these mental structures and processes is presented, and quantitative predictions about children's arithmetic performance at various stages of mastery are considered.

Mark H Ashcraft Publication date: 1982/09/01AbstractLet fr(k)=k⋅rk/2 (where r≥2 is fixed) and consider r-colorings of [1,nk]={1,2,…,nk}. We show that fr(k) is a threshold function for k-term arithmetic progressions in the following sense: if nk=ω(fr(k)), then limk→∞P([1,nk] contains a monochromatic k-term arithmetic progression)=1; while, if nk=o(fr(k)), then limk→∞P([1,nk] contains ak-term monochromatic arithmetic progression)=0.

Aaron Robertson Publication date: 2016/01/01AbstractA set of numbers a1<a2<⋯<aL is called a weakly arithmetic progression, if there exist L consecutive intervals Ii=[Xi−1,Xi), i=1,…,L, of equal length with ai∈Ii. We give sufficient conditions for the existence of weakly arithmetic progressions of a given length in certain subsets A of N. As a corollary we obtain: If ∑a∈A 1⧸a=∞, then for every L∈N the set A has a weakly arithmetic progression of length L.

Egbert Harzheim Publication date: 1991/05/03AbstractLet B be a set of real numbers of size n. We prove that the length of the longest arithmetic progression contained in the product set B.B={bibj|bi,bj∈B} cannot be greater than O(n1+1/loglogn) an arithmetic progression of length Ω(nlogn), so the obtained upper bound is close to the optimal.

Dmitry Zhelezov Publication date: 2013/09/05AbstractWe incorporate the string theory into the number theoretic formulation based on arithmetic geometry. The string theory is generalized p-adically and interpreted on an arithmetic surface. A p-adic multi-loop scattering amplitude is constructed.

Hitoshi Yamakoshi Publication date: 1988/06/30AbstractThe possibility of estimating bounds for the econometric likelihood function using balanced random interval arithmetic is experimentally investigated. The experiments on the likelihood function with data from housing starts have proved the assumption that distributions of centres and radii of evaluated balanced random intervals are normal. Balanced random interval arithmetic can therefore be used to estimate bounds for this function and global optimization algorithms based on this arithmetic are applicable to optimize it. The interval branch and bound algorithms with bounds calculated using standard and balanced random interval arithmetic were used to optimize the likelihood function. Results of the experiments show that when reliability is essential the algorithm with standard interval arithmetic should be used, but when speed of optimization is more important, the algorithm with balanced random interval arithmetic should be used which in this case finishes faster and provides *Read more...*

AbstractEstimating the discrepancy of the set of all arithmetic progressions in the first N natural numbers was one of the famous open problems in combinatorial discrepancy theory for a long time, successfully solved by K. Roth (lower bound) and Beck (upper bound). They proved that D(N)=minχmaxA|∑x∈Aχ(x)|=Θ(N1/4), where the minimum is taken over all colorings χ:[N]→{−1,1} and the maximum over all arithmetic progressions in [N]={0,…,N−1}.Sumsets of k arithmetic progressions, A1+⋯+Ak, are called k-arithmetic progressions and they are important objects in additive combinatorics. We define Dk(N) as the discrepancy of the set {P∩[N]:P is a k-arithmetic progression}. The second author proved that Dk(N)=Ω(Nk/(2k+2)) and Přívětivý improved it to Ω(N1/2) for all k≥3. Since the probabilistic argument gives Dk(N)=O((NlogN)1/2) for all fixed k, the case k=2 remained the only case with a large gap between the known upper and lower bounds. We bridge this gap (up to a *Read more...*

AbstractWe introduce two new binary operations on combinatorial species; the arithmetic product and the modified arithmetic product. The arithmetic product gives combinatorial meaning to the product of Dirichlet series and to the Lambert series in the context of species. It allows us to introduce the notion of multiplicative species, a lifting to the combinatorial level of the classical notion of multiplicative arithmetic function. Interesting combinatorial constructions are introduced; cloned assemblies of structures, hyper-cloned trees, enriched rectangles, etc. Recent research of Cameron, Gewurz and Merola, about the product action in the context of oligomorphic groups, motivated the introduction of the modified arithmetic product. By using the modified arithmetic product we obtain new enumerative results. We also generalize and simplify some results of Canfield, and Pittel, related to the enumerations of tuples of partitions with the restrictions met.

Manuel Maia Publication date: 2008/12/06AbstractHardware and software implementations of decimal arithmetic have resurfaced in recent years to overcome the limitations of binary arithmetic. Traditionally, decimal arithmetic units have been designed as application-specific hardware modules. But there is an emerging trend towards the design and implementation of decimal arithmetic operations on reconfigurable structures. This paper contributes to this trend by proposing a reconfigurable architecture, namely DARA, for high performance implementation of decimal arithmetic operations. Some basic decimal arithmetic operations were implemented on DARA and synthesized subsequently. The results show that DARA has a delay overhead of 26% and area overhead of 54% on average compared to an ASIC implementation of the same operations. At the same time, if those basic operations had been implemented on a modern commercial FPGA, DARA would have outperformed the commercial device in terms of delay and area by a factor of almost 4 and 9, resp *Read more...*