Arithmetic Systems Research Articles

A Survey of Paradigms for Building and Designing Parallel Computing Machines

In this paper we describe paradigms for building and designing parallel computing machines. Firstly we elaborate the uniqueness of MIMD model for the execution of diverse applications. Then we compare the General Purpose Architecture of Parallel Computers with Special Purpose Architecture of Parallel Computers in terms of cost, throughput and efficiency. Then we describe how Parallel Computer Architecture employs parallelism and concurrency through pipelining. Since Pipelining improves the performance of a machine by dividing an instruction into a number of stages, therefore we describe how the performance of a vector processor is enhanced by employing multi pipelining among its processing elements. Also we have elaborated the RISC architecture and Pipelining in RISC machines After comparing RISC computers with CISC computers we observe that although the high speed of RISC computers is very desirable but the significance of speed of a computer is dependent on implementation strategies. Only CPU clock speed is not the only parameter to move the system software from CISC to RISC computers but the other parameters should also be considered like instruction size or format, addressing modes, complexity of instructions and machine cycles required by instructions. Considering all parameters will give performance gain . We discuss Multiprocessor and Data Flow Machines in a concise manner. Then we discuss three SIMD (Single Instruction stream Multiple Data stream) machines which are DEC/MasPar MP-1, Systolic Processors and Wavefront array Processors. The DEC/MasPar MP-1 is a massively parallel SIMD array processor. A wide variety of number representations and arithmetic systems for computers can be implemented easily on the DEC/MasPar MP-1 system. The principal advantages of using such 64×64 SIMD array of 4-bit processors for the implementation of a computer arithmetic laboratory arise out of its flexibility. After comparison of Systolic Processors with Wave front Processors we found that both of the Systolic Processors and Wave front Processors are fast and implemented in VLSI. The major drawback of Systolic Processors is the problem of availability of inputs when clock ticks because of propagation delays in connection buses. The Wave front Processors combine the Systolic Processor architecture with Data Flow machine architecture. Although the Wave front processors use asynchronous data flow computing structure, the timing in the interconnection buses, at input and at output is not problematic..

Guest Editors’ Introduction: Special Sectionon Computer Arithmetic

The articles in this special issue focus on current trends and developments in the field of computer arithmetic. This is a field that encompasses the definition and standardization of arithmetic system for computers. The field also deals with issues of hardware and software implementations and their subsequent testing and verification. Many practitioners of the field also focus on the art and science of using computer arithmetic to carry out scientific and engineering computations. Computer arithmetic is therefore an interdisciplinary field that draws upon mathematics, computer science and electrical engineering. Advances in this field span from being highly theoretical (for instance, new exotic number systems) to being highly practical (for instance, new floating-point units for microprocessors).

Proportional Design Systems in Seventeenth-Century Holland

This article focuses on architectural drawings, some only recently discovered, that contain indications of authentic proportional systems from the ‘inner circle’ of Dutch seventeenth-century classical architecture. These drawings demonstrate how these architects used arithmetical grid systems as well as geometrical constructions in the hope of achieving their classical ideal of beauty and harmony. In light of these proportional systems we may propose a method for analysing other drawings from this period that do not have any proportional systems inscribed.

Intertextuality and sense production in the learning of algebraic methods

In studies carried out in the 1980s the algebraic symbols and expressions are revealed through prealgebraic readers as non-independent texts, as texts that relate to other texts that in some cases belong to the reader’s native language or to the arithmetic sign system. Such outcomes suggest that the act of reading algebraic texts submerges the reader into a network of intertextual relations derived from that reader’s prior mathematics and linguistic experiences. In this article we propose an analytical perspective of algebraic activity that is based on intertextuality and that makes it possible to advance our knowledge concerning the production of sense in said activity. In particular, we resort to the notion of intertextuality as a means of theoretically describing and explaining sense production processes in the learning of two algebraic methods: the Cartesian Method (for the algebraic solution of word problems) and the Substitution Method (for the solution of systems of two linear equations with two unknowns). We support our arguments via a series of episodes of empirical study interviews with secondary school students, in order to demonstrate the pertinence and relevance of carrying out their analysis from the perspective proposed.

Review: The Geometry of Creation: Architectural Drawing and the Dynamics of Gothic Design, by Robert Bork

Robert Bork The Geometry of Creation: Architectural Drawing and the Dynamics of Gothic Design Farnham, UK, and Burlington, Vt.: Ashgate, 2011, 484 pp., 240 b/w illus. $119.95, ISBN 9780754660620 In 1383 the Troyes cathedral chapter paid Henry de Bruisselles 20 sous “for a portrait for the … choir screen made on parchment … to show to the bourgeois and to the laborers of the city side by side with another portrait made by Michelin, the mason. The said bourgeois and laborers held that the portrait of Henry de Bruisselles was the better.”1 At about the same time, Peter Parler or his son, Wenzel, designed the upper stories of Prague cathedral’s south transept, revising a drawing begun three decades earlier. The same drawing, taken to Vienna by Wenzel, informed his work on the south tower of the Stephansdom. Drawings on parchment, complemented by wood or paper templates, small sketches, and full-scale engravings on walls and pavements, embody one intricate facet of the robust technology of architectural representation. Vehicles of communication, construction guides, and instruments of public persuasion, they informed all aspects of the late medieval building process. Robert Bork’s The Geometry of Creation: Architectural Drawing and the Dynamics of Gothic Design presents case studies of the use of drawings in architectural projects from the thirteenth to the sixteenth century. Although most of the six hundred surviving Gothic drawings are connected to building projects in the Holy Roman Empire, Bork extends his inquiry to examples in France and Italy to argue that the same design procedures were shared by masons working throughout Europe. Reconstructions of the geometry used to design plans, elevations, towers, tabernacles, and tombs are presented in precise diagrams using Vectorworks CAD software that allows information, gathered through meticulous observation of the original parchments, to be processed “with unprecedented flexibility and rigor” (20). Bork, whose account features masons at their drafting tables with compass and straightedge, resists excursions into the swamps of symbolic interpretation, and although he illuminates …

Stability of Number-Processing Based on Information Granularity Associative with Parietal Cortex

Stability, the ability to automatically extract and produce the efficient and accurate results of a defined problem without making epistemic assumptions, is discussed here as a possible memory system for understanding complex cognitive functions of the arithmetical learning. Stability is of top priority because it may typify organization of granule (knowledge-based information unit) structure. Memory efficiencies are that they depend on both linguistic factors and exposure to arithmetic training during granule formation or consolidation, supporting the idea of analog coding of numerical representations. Neuroimaging studies suggest that the parietal lobe as a potential substrate for a domain-specific representation of numeric quantities and associative memory mechanisms in stability, and results from these studies indicate that there may be the organization of number-related processes of stability in the parietal lobe. Stability seems to depend on the automatic information-processing system's response to experiential knowledge combining granularity (degree of detail or precision), maturational constraints, spatial factors (mental number line) and linguistic factors, making it an ideal candidate for understanding how these interactions play out in the cognitive arithmetic system.

Stability of Number-Processing Based on Information Granularity Associative with Parietal Cortex

Stability, the ability to automatically extract and produce the efficient and accurate results of a defined problem without making epistemic assumptions, is discussed here as a possible memory system for understanding complex cognitive functions of the arithmetical learning. Stability is of top priority because it may typify organization of granule (knowledge-based information unit) structure. Memory efficiencies are that they depend on both linguistic factors and exposure to arithmetic training during granule formation or consolidation, supporting the idea of analog coding of numerical representations. Neuroimaging studies suggest that the parietal lobe as a potential substrate for a domain-specific representation of numeric quantities and associative memory mechanisms in stability, and results from these studies indicate that there may be the organization of number-related processes of stability in the parietal lobe. Stability seems to depend on the automatic information-processing system's response to experiential knowledge combining granularity (degree of detail or precision), maturational constraints, spatial factors (mental number line) and linguistic factors, making it an ideal candidate for understanding how these interactions play out in the cognitive arithmetic system.

Remarks on numerical algorithms for computing the inverse Laplace transform

Algorithms for computing the inverse Laplace transform that consist essentially in choosing a series expansion for the original function are particularly effective in many cases and are widely used. The main purpose of this paper is to review these algorithms in the context of regularization. We relate this viewpoint to the design of reliable algorithms destined to be run on finite precision arithmetic systems.

Nonstandard second-order arithmetic and Riemannʼs mapping theorem

In this paper, we introduce systems of nonstandard second-order arithmetic which are conservative extensions of systems of second-order arithmetic. Within these systems, we do reverse mathematics for nonstandard analysis, and we can import techniques of nonstandard analysis into analysis in weak systems of second-order arithmetic. Then, we apply nonstandard techniques to a version of Riemannʼs mapping theorem, and show several different versions of Riemannʼs mapping theorem.

VME Bus-Based Multiprocessor Parallel Data Acquisition System

Synchronization data acquistition system with multiple processors is the trend of the modern manufacturing equipment development. Complex high-precision equipment even has hundreds of sensors and they are acquiring data at the same time, the sensors are distributed in dozens of data acquisition cards. The key technology is the acquisition system enable to synchro-gather data from data acquisition cards which base on data bus between them and then sent the gather data to arithmetic system. In order to synchronic data integration, it is required a high-speed and low-latency data channels and a suitable data protocol between data acquisition cards and main control card. This paper introduce a new type of synchronous data acquisition system, through the self-defined data bus and a specific memory allocation mechanism, the data acquisition system can sent the integrate data to arithmetic system through main control card after it get the data from acquisition cards, the transfer delay of data exchange is nanoseconds.

Implicit Learning of Arithmetic Regularities Is Facilitated by Proximal Contrast

Natural number arithmetic is a simple, powerful and important symbolic system. Despite intense focus on learning in cognitive development and educational research many adults have weak knowledge of the system. In current study participants learn arithmetic principles via an implicit learning paradigm. Participants learn not by solving arithmetic equations, but through viewing and evaluating example equations, similar to the implicit learning of artificial grammars. We expand this to the symbolic arithmetic system. Specifically we find that exposure to principle-inconsistent examples facilitates the acquisition of arithmetic principle knowledge if the equations are presented to the learning in a temporally proximate fashion. The results expand on research of the implicit learning of regularities and suggest that contrasting cases, show to facilitate explicit arithmetic learning, is also relevant to implicit learning of arithmetic.

Interactive Realizers

We propose a realizability interpretation of a system for quantier free arithmetic which is equivalent to the fragment of classical arithmetic without nested quantifiers, called here EM 1 -arithmetic. We interpret classical proofs as interactive learning strategies, namely as processes going through several stages of knowledge and learning by interacting with the “nature,” represented by the standard interpretation of closed atomic formulas, and with each other. We obtain in this way a program extraction method by proof interpretation, which is faithful with respect to proofs, in the sense that it is compositional and that it does not need any translation.

Paradox and Potential Infinity

We describe a variety of sets internal to models of intuitionistic set theory that (1) manifest some of the crucial behaviors of potentially infinite sets as described in the foundational literature going back to Aristotle, and (2) provide models for systems of predicative arithmetic. We close with a brief discussion of Church’s Thesis for predicative arithmetic.

Twin-float arithmetic

We present a heuristically certified form of floating-point arithmetic and its implementation in CoCoALib. This arithmetic is intended to act as a fast alternative to exact rational arithmetic, and is developed from the idea of paired floats expounded by Traverso and Zanoni (2002). As prerequisites we need a source of (pseudo-)random numbers, and an underlying floating-point arithmetic system where the user can set the precision. Twin-float arithmetic can be used only where the input data are exact, or can be obtained at high enough precision. Our arithmetic includes a total cancellation heuristic for sums and differences, and so can be used in classical algebraic algorithms such as Buchberger’s algorithm. We also present a (new) algorithm for recovering an exact rational value from a twin-float, so in some cases an exact answer can be obtained from an approximate computation.The ideas presented here are implemented as a ring in CoCoALib, called RingTwinFloat, allowing them to be used easily in a wide variety of algebraic computations (including Gröbner bases).

On the Sophie Germain prime conjecture

By extending the operations +,× on natural numbers to the operations on finite sets of natural numbers, we founded a new formal system of a second order arithmetic 〈P(N), N,+,×,0,1, ∈〉. We designed...

Resettable Multiple-Mode Molecular Arithmetic Systems Based on Spectral Properties of 2-Quinolin-2-ylmethylene-malonic Acids

The development of molecular arithmetic systems is very crucial for the realization of a molecule-scale calculator (a Moleculator). In this work, two compounds with pH-sensitive functional groups, ...

History and its racial legacies: quotas in South African rugby and cricket

South African identity has always been shaped by racial quotas; that is, divisions, assignments, allowances and allocations based on socially created ideas of race and difference. Both law and custom assigned a hierarchy which separated the rulers from the ruled, and allocated and rationed goods, services and enjoyments in all spheres of life, including sport. ‘Superior’ whites were layered above the Cape coloured people, followed by the Indian community and, lastly, the Africans, the black majority. This article looks briefly at the historical context of racial divisions and, with the downfall of apartheid, the rhetoric of an avowedly de-racialized ‘new South Africa’. Given the chronic history of negative discrimination, it is understandable that affirmative action has become a major policy framework in the building of a post-apartheid society. But sport is a sobering example of how a domain can be ‘re-racialized’ in this quest. How does the African National Congress justify the (re)introduction into sport of a proportional or numerical quota system based on racial categories? Is there a need for demographic representativeness in white-dominated sports like cricket and rugby, but seemingly not in black-dominated soccer? Is an arithmetic quota system not merely a logical extension of the reviled racial genres and divides of previous centuries?

Indexed Natural Numbers in Mind: A Formal Model of the Basic Mature Number Competence

The paper undertakes three interdisciplinary tasks. The first one consists in constructing a formal model of the basic arithmetic competence, that is, the competence sufficient for solving simple arithmetic story-tasks which do not require any mathematical mastery knowledge about laws, definitions and theorems. The second task is to present a generalized arithmetic theory, called the arithmetic of indexed numbers (INA). All models of the development of counting abilities presuppose the common assumption that our simple, folk arithmetic encoded linguistically in the mind is based on the linear number representation. This classical conception is rejected and a competitive hypothesis is formulated according to which the basic mature representational system of cognitive arithmetic is a structure composed of many numerical axes which possess a common constituent, namely, the numeral zero. Arithmetic of indexed numbers is just a formal tool for modelling the basic mature arithmetic competence. The third task is to develop a standpoint called temporal pluralism, which is motivated by neo-Kantian philosophy of arithmetic.

Proof theory and mathematical meaning of paraconsistent C-systems

A proof-theoretic analysis and new arithmetical semantics are proposed for some paraconsistent C-systems, which are a relevant sub-class of Logics of Formal Inconsistency ( LFIs) introduced by W.A. Carnielli et al. (2002, 2005) [8,9]. The sequent versions BC, CI, CIL of the systems bC, Ci, Cil presented in Carnielli et al. (2002, 2005) [8,9] are introduced and examined. BC, CI, CIL admit the cut-elimination property and, in general, a weakened sub-formula property. Moreover, a formal notion of constructive paraconsistent system is given, and the constructivity of CI is proven. Further possible developments of proof theory and provability logic of CI-based arithmetical systems are sketched, and a possible weakened Hilbertʼs program is discussed. As to the semantical aspects, arithmetical semantics interprets C-system formulas into Provability Logic sentences of classical Arithmetic PA (Artemov and Beklemishev (2004) [2], Japaridze and de Jongh (1998) [19], Gentilini (1999) [15], Smorynski (1991) [22]): thus, it links the notion of truth to the notion of provability inside a classical environment. It makes true infinitely many contradictions B ∧ ¬ B and falsifies many arbitrarily complex instances of non-contradiction principle ¬ ( A ∧ ¬ A ) . Moreover, arithmetical models falsify both classical logic LK and intuitionistic logic LJ, so that a kind of metalogical completeness property of LFI-paraconsistent logic w.r.t. arithmetical semantics is proven. As a work in progress, the possibility to interpret CI-based paraconsistent Arithmetic PACI into Provability Logic of classical Arithmetic PA is discussed, showing the role that PACI arithmetical models could have in establishing new meta-mathematical properties, e.g. in breaking classical equivalences between consistency statements and reflection principles.

Understanding economical outcomes with the mental number line

With this study we provide evidence that the cognitive processes involved in addition/subtraction, mapped along the mental number line, seem to mediate our understanding of trading verbs. When left-to-right culture participants read "loss" verbs, cognitive activation moves "leftward" as in arithmetical subtraction, while reading "gain" verbs activates a mental rightward space as in addition.We test this hypothesis by asking to a group of participants to use their left and right hand in judging (as correct of not) the syntactic form of several verbs meaning financial outcomes. Results show that processing “gain verbs” was associated with shorter latencies when responding with the right hand similarly when performing an addition task, while processing “loss verbs” was associated with shorter latencies when responding with the left, similarly when performing a subtraction task. This finding suggests that understanding language denoting economics outcomes covertly engages the arithmetical system in a spatially left-right dimension.