Showing papers on "Division (mathematics) published in 1997"

Young children's intuitive models of multiplication and division

Abstract: In this study, an intuitive model was defined as an internal mental structure corresponding to a class of calculation strategies. A sample of female students was observed 4 times during Grades 2 and 3 as they solved the same set of 24 word problems. From the correct responses, 12 distinct calculation strategies were identified and grouped into categories from which the children’s intuitive models of multiplication and division were inferred. It was found that the students used 3 main intuitive models: direct counting, repeated addition, and multiplicative operation. A fourth model, repeated subtraction, only occurred in division problems. All the intuitive models were used with all semantic structures, their frequency varying as a complex interaction of age, size of numbers, language, and semantic structure. The results are interpreted as showing that children acquire an expanding repertoire of intuitive models and that the model they employ to solve any particular problem reflects the mathematical structure they impose on it. Several recent studies have shown that students can solve a variety of multiplicative problems long before formal instruction on the operations of multiplication and division. For example, Kouba (1989) found that 30% of Grade 1 and 70% of Grade 2 students could solve simple equivalent group problems and Mulligan (1992) found a steady increase in success rate on similar problems from over 50% at the beginning of Grade 2 to nearly 95% at the end of Grade 3. More recently Carpenter, Ansell, Franke, Fennema, and Weisbeck (1993) found that even kindergarten students could

Self-replicating spots in reaction-diffusion systems

Abstract: In this article we discuss the self-replicating spot, a particlelike phenomenon that occurs in reaction-diffusion ~RD! systems @1#. The spots consist of localized regions in which the concentrations of the reactants differ from the surrounding concentration fields. They grow, reaching a critical size at which time they divide in two. The two resulting spots again grow and divide. This process, which is visually similar to cell division, continues indefinitely. The long-time behavior depends on the precise values of the external control parameters, but typically consists of a chaotic ‘‘soup’’ in which many spots compete for resources as illustrated in Fig. 1. Those spots that find adequate resources continue to grow and divide. Those that are unable to find adequate resources decay into the background. The spots observed in @1# were found during an attempt to model labyrinthine patterns in the ferrocyanide-iodate-sulfate ~FIS! reaction @2#. Since then, replicating spot patterns have been observed both numerically and experimentally in the FIS reaction @3,4#. There are obvious differences in the Gaspar-Showalter model of the FIS reaction @5# and of the models discussed by us and others. This fact suggests that replication is a generic feature characterizing a broad class of reaction-diffusion systems. In @6#, we presented some arguments in support of this proposition. These arguments included both a heuristic description of the process of replication and demonstrations of analytic features common to several related model RD systems. It turns out that replication is more general than our analysis accounts for. Nevertheless, we think it worthwhile to spell out the details of the theory presented in @6#. We remark here that various aspects of the replication phenomenon have been discussed by other authors @7,8#. Kerner and Osipov have a large body of work on largeamplitude dissipative structures including an analysis of the static division of one-dimensional pulses as the system size is changed. Gurevich and Mints have a body of work on replication of thermal hot spots in composite superconductors. In the article by Petrov, Scott, and Showalter replication

On the relation between skilled performance of simple division and multiplication

Abstract: Are corresponding multiplication and division facts (e.g., 7 x 8, 56 divided by 7) based on common or on independent memory processes? University students received division problems alternated with multiplication problems under instructions for speeded responses. Response times were highly correlated for corresponding division and multiplication problems, and error characteristics indicated parallel retrieval structures. Specifically, division errors were constrained by the distance between the dividend and the product implied by the error, rather than by distance from the correct quotient. This suggests that division memory is organized in terms of multiplicative relationships. Multiplication errors (e.g., 7 x 9 = 56) were primed by previous division trials (56 divided by 7 = 8), but division errors were not primed by previous multiplications. The error priming results suggest that multiplication is often used at least to check division.

High-precision division and square root

Abstract: We present division and square root algorithm for calculations with more bits than are handled by the floating-point hardware. These algorithms avoid the need to multiply two high-precision numbers, speeding up the last iteration by as much as a factor of 10. We also show how to produce the floating-point number closest to the exact result with relatively few additional operations.

A Moving-knife solution to the four-person envy-free cake-division problem

Abstract: We present a moving-knife procedure, requiring only 11 cuts, that produces an envy-free allocation of a cake among four players and discuss possible extensions to five players.

Mechanism of energy transportation and turbulent flow in a 3D channel

Abstract: This paper presents an analytical approach for the partitioning of the flow cross-sectional area in steady and uniform three-dimensional channels. The mechanism and direction of energy transport from the main flow are analyzed first. Based on the availability of surplus energy from the main flow, a concept of energy transportation through a minimum relative distance toward a unit area on the wetted perimeter is proposed. This leads to a novel method to analytically divide the flow area into various parts corresponding to the channel shape and roughness composition of its wetted perimeter. The demarcated boundary or “division line” can be evaluated using a presented equation. The existence of the division lines, which are lines of zero Reynolds shear stress within the flow region, has been proven through comparison with turbulence characteristics measurements. The method is illustrated in a study of the shear stress distribution in smooth rectangular open channels. Analytical solutions, valid for all aspec...

Spread spectrum communication system

Abstract: Low rate data D1-D4 and middle rate data D5 and D6 are time division multiplexed independently according to their rates by a time division multiplexer TDM(A) (110) so that their rates are converted into the rate of high rate data D7 and D8 which are not passed through the time division multiplexer TDM(A) (110). Subsequently, the entire data are converted into biorthogonal signals by biorthogonal signal generators BORTs (121, 122a-125a), and code division multiplexed by a code division multiplexer CDM(A) (130a). The code division multiplexed signal undergoes modulation by a spreading modulator (20) and a carrier modulator (30), and transmitted.

Optical network, optical division and insertion node and recovery system from network failure

Abstract: Disclosed is an optical network which has a plurality of optical division and insertion nodes and one or a plurality of hub (central) nodes which are circularly connected by using an optical transfer system where a clockwise or counterclockwise data transfer is conducted by using a plurality of optical fibers or an optical wavelength-multiplexing technique; wherein the hub (central) node and the plurality of division and insertion nodes individually use signal lights with wavelengths different from each other and are provided with a transmission system composed of two or more clockwise or counterclockwise independent paths, and the optical division and insertion node selects one of signals obtained from the two or more paths.

Parcelization of geographic data for storage and use in a navigation application

Abstract: A system and method for arranging and storing a plurality of records of geographic data, wherein each record corresponds to a physical feature having a physical location in a geographic region. The method and system comprise arranging the records of geographic data into a plurality of parcels. Each parcel includes records of geographic data that represent features having physical locations encompassed within a corresponding associated rectangular area located in the geographic region. The size and location of each such rectangular area associated with a parcel is determined by a series of divisions of a bounding rectangle that encompasses all of the features represented by the plurality of records into further rectangular areas. Each division, subsequent to an initial division, is made on a rectangular area resulting from the preceding division. Each such division of a rectangular area is made at a location along the rectangular area based upon an assessment of one or more trial divisions of the rectangular area at one or more locations. A division is selected based upon a comparison of the quantities of data encompassed by the rectangular area and each of the further rectangular areas formed by the one or more trial divisions. The assessment is based upon a comparison of these quantities of data for each such trial division to a plurality of ranges of acceptable data quantities. These acceptable sizes are derived from a desired fill percentage of parcels with data.

Process for making a mechanical fastener

Abstract: A method for forming a plurality of fastener components comprises providing a composite web which includes a web of hook material. The hook material includes a hook base layer which has first and second longitudinally extending side sections and has a plurality of hook elements which are integrally formed with the base layer and extend away from a base plane of the hook base layer. The hook elements are configured to operably engage a selected, cooperating loop material to provide an operative fastening, and the web of hook material has an extending section of carrier web material attached to extend laterally outboard from each of the side sections of the web of hook material. Each section of carrier web material has an extending web of panel material attached to extend laterally outboard from each section of carrier web material. The composite web is divided along first and second, non-intersecting, serpentine division lines which repeatedly traverse across a complete lateral width of the hook web and extend generally longitudinally along a medial region of the composite web to provide for a serpentine strip which can be removed away and separated from the composite web. The first serpentine division line has a first plurality of laterally inboard hill regions and relatively outboard valley regions which alternate in occurrence along said longitudinal dimension of said composite web. The second serpentine division line has a second plurality of laterally inboard hill regions and relatively outboard valley regions which alternate in occurrence along the longitudinal dimension of the composite web. The hill regions of the first and second sepentine division lines do not extend into the webs of panel material.

Efficient initial approximation for multiplicative division and square root by a multiplication with operand modification

Abstract: An efficient initial approximation method for multiplicative division and square root is proposed. It is a modification of the piecewise linear approximation. The multiplication and the addition required for the linear approximation are replaced by only one multiplication with a slight modification of the operand. The same accuracy is achieved. The modification of the operand requires only a bit-wise inversion and a one-bit shift, and can be implemented by a very simple circuit. One clock cycle may be saved, because the addition is removed. The required table size is also reduced, because only one coefficient instead of two has to be stored.

A low power zero-overhead self-timed division and square root unit combining a single-rail static circuit with a dual-rail dynamic circuit

Abstract: An asynchronous pipeline scheme that combines a low power static circuit with a high-speed dual-rail dynamic circuit is proposed. The scheme utilizes a dual-rail circuit only in the critical path of an SRT division and square root calculation unit. The proposed implementation of the calculation unit reduced power consumption by more than 1/2 of the full-dynamic implementation while maintaining the calculation speed. Because of the elimination of spurious transitions, the proposed implementation showed even less power consumption over synchronous static circuit implementations. By using 0.3 /spl mu/m triple metal CMOS technology, the calculation time of floating point 56-b full mantissa division and square root is expected to be 45 ns in the worst case.

Laser machining method

Abstract: In the present invention, a laser beam with a prespecified oscillation frequency is divided into the arbitrary number of pulses to be irradiated to all holes in an area, and this step is repeated until the irradiation reaches a desired number of pulses, whereby an interactive time can be decided according to the number of divided pulses and a cooling period of time can be obtained, and also the number of pulses per one division is set to be a plurality of pulses, which allows an etching rate to be increased, so that the number of pulses required for machining can be reduced as compared to a case where a laser is irradiated to all the holes pulse by pulse.

Understanding Mathematics in the Lower Primary Years: A Guide for Teachers of Children 3 - 8

Abstract: Making Connections Understanding Number Understanding Addition and Subtraction Understanding Multiplication and Division Number Patterns and Calculations Measurement Shape and Space Handling Data Thinking Mathematically

Method and apparatus for performing arithmetic division with a machine

Abstract: An arithmetic division for implementation using hardware or software is performed by firstly storing the reciprocal of each nth value in a range 0 to X where n is greater than 1 and storing the differences between the reciprocals of the intervening n-1 values. A mantissa and an operand for an arithmetic division are received. one of the stored reciprocals is retrieved in response to the most significant part of the received mantissa and a set of differences is retrieved in response to a least significant part of the received mantissa. The differences and the reciprocal are summed and the result is multiplied by the received operand, other thereby giving the result of the arithmetic division.

Management method and device for information

Abstract: PROBLEM TO BE SOLVED: To improve the safety of secret protection or the like by inhibiting the restoration of information without the permission of plural managers SOLUTION: Source information 1 with the need of the secret protection is divided into the plural pieces of divided information 4A, 4B and 4C corresponding to a prescribed rule by a division device 2 They are respectively stored and kept in different media 5A, 5B and 5C and the media are respectively managed by the different managers In the case of restoring the information, the permission of the managers is obtained, the divided information is read from the respective media and it is restored corresponding to an original, division rule

Air conditioner for a vehicle

Abstract: A temperature adjusting unit portion is constructed by two division cases, and the two division cases have vertically divided type case structure, each of which has a division surface in a vertical direction. An evaporator and a heater are held and accommodated in the division cases. Between the division cases of the division case below the evaporator, there is provided a sealing member formed in a stick shape and in a circular shape in a cross section thereof.

Multiple wavelength division multiplexer with reduced loss

Abstract: A system and method for providing wavelength division multiplexer is disclosed. The wavelength division multiplexer is for separating an optical signal into a plurality of components. The wavelength division multiplexer includes a splitter and a plurality of isolators coupled to the splitter. The splitter is for separating the optical signal into a plurality of groups. Each group contains a portion of the plurality of components. The plurality of isolators correspond to the plurality of groups. Each of the plurality of isolators is for isolating each component of the corresponding group. In one embodiment, the splitter further includes a splitter input. In this embodiment, the wavelength division multiplexer further includes a filter coupled to the splitter input. The filter is for isolating one of the plurality of components.

Hitless reconfiguration of a wavelength division multiplexed optical communication network

Abstract: A method of reconfiguring a multi-wavelength communication network (Fig. 6). The transition includes determining the union of the old and new topologies. During at least the start of the transition, the network supports the union topology. New connections are connected according to the new topology, and the old topology, as its connections expire, is gradually removed. In allocating wavelengths to a new topology, two algorithms are developed which assign wavelengths which do not color clash with the wavelength assignments of the old topology.

The centralizer of invariant functions and division properties of the moment map

Abstract: Let Phi : M --> g^* be a proper moment map associated to an action of a compact connected Lie group, G, on a connected symplectic manifold, (M,\\omega). A collective function is a pullback via \\Phi of a smooth function on g^*. In this paper we present four new results about the relationship between the collective functions and the G-invariant functions in the Poisson algebra of smooth functions on M.

Multiprecision division on an 8-bit processor

Abstract: Small processors can be especially useful in massively parallel architectures. This paper considers multiprecision division algorithms on an 8-bit processor (the Kestrel processor, currently in fabrication) that includes a small amount of memory and an 8-bit multiplier. We evaluate several variations of the Newton-Raphson reciprocal approximation methods for use with division. Our final single-precision algorithm requires 41 cycles to divide two 24-bit numbers to produce a 26-bit result. The double-precision version requires 98 cycles to divide two 53-bit numbers to produce a 55-bit result. This low cycle count is the result of several techniques, including low-precision arithmetic, early introduction of dividends, and simple (yet good) initial reciprocal estimates.

Channel layered optical cross-connect restoration system

Abstract: An optical telecommunications system node terminates a plurality of optical cable routes. Each of the optical cable routes includes a traffic route and a restoration route. The node includes a plurality of traffic wavelength division multiplexers and a plurality of restoration wavelength division multiplexers. A traffic wavelength division multiplexer is connected to each traffic route and a restoration wavelength division multiplexer is connected to each restoration route. The node includes a plurality of optical cross connect layers. Each optical cross connect layer includes a connection to each of the traffic wavelength division multiplexers and to each of the restoration wavelength division multiplexers.

A Problem Solving Approach to Mathematics

Abstract: Chapter 1 An Introduction to Problem Solving 1-1 Mathematics and Problem Solving 1-2 Explorations with Patterns 1-3 Algebraic Thinking 1-4 Logic: An Introduction Chapter 2 Sets, Whole Numbers, and Functions 2-1 Describing Sets 2-2 Other Set Operations and Their Properties 2-3 Addition and Subtraction of Whole Numbers 2-4 Multiplication and Division of Whole Numbers 2-5 Functions Chapter 3 Numeration Systems and Whole-Number Computation 3-1 Numeration Systems 3-2 Algorithms for Whole-Number Addition and Subtraction 3-3 Algorithms for Whole-Number Multiplication and Division 3-4 Mental Mathematics and Estimation for Whole-Number Operations Chapter 4 Integers and Number Theory 4-1 Integers and the Operations of Addition and Subtraction 4-2 Multiplication and Division of Integers 4-3 Divisibility 4-4 Prime and Composite Numbers 4-5 Greatest Common Divisor and Least Common Multiple 4-6 Clock and Modular Arithmetic Chapter 5 Rational Numbers as Fractions 5-1 The Set of Rational Numbers 5-2 Addition and Subtraction of Rational Numbers 5-3 Multiplication and Division of Rational Numbers 5-4 Proportional Reasoning Chapter 6 Decimals, Percents, and Real Numbers 6-1 Introduction to Decimals 6-2 Operations on Decimals 6-3 Nonterminating Decimals 6-4 Real Numbers 6-5 Percents 6-6 Computing Interest Chapter 7 Probability 7-1 How Probabilities Are Determined 7-2 Multistage Experiments with Tree Diagrams and Geometric Probabilities 7-3 Using Simulations in Probability 7-4 Odds, Conditional Probability, and Expected Value 7-5 Using Permutations and Combinations in Probability Chapter 8 Data Analysis/ Statistics: An Introduction 8-1 Statistical Graphs of Categorical and Numerical Data 8-2 Measures of Central Tendency and Variation 8-3 Abuses of Statistics Chapter 9 Introductory Geometry 9-1 Basic Notions 9-2 Polygons 9-3 More About Angles 9-4 Geometry in Three Dimensions 9-5 Networks Chapter 10 Constructions, Congruence, and Similarity 10-1 Congruence Through Constructions 10-2 Other Congruence Properties 10-3 Other Constructions 10-4 Similar Triangles and Similar Figures 10-5 Trigonometry Ratios via Similarity 10-6 Lines in a Cartesian Coordinate System Chapter 11 Concepts of Measurement 11-1 Linear Measure 11-2 Areas of Polygons and Circles 11-3 The Pythagorean Theorem and the Distance Formula 11-4 Surface Areas 11-5 Volume, Mass & Temperature Chapter 12 Motion Geometry and Tessellations 12-1 Translations and Rotations 12-2 Reflections and Glide Reflections 12-3 Size Transformations 12-4 Symmetries 12-5 Tesselations of the Plane Appendix I: Using a Spreadsheet Appendix II: Graphing Calculators Appendix III: Using a Geometry Drawing Utility

Data converter apparatus and method particularly useful for a database-to-object inspection system

Abstract: Data converter apparatus for converting in real time data stored in an input compact format into an output expanded real time format. The data including a plurality of groups and repetition of the groups. The groups including, each, many basic geometric figures (BGF). The data converter includes a processor adapted to selecting data relating to a predetermined division, dividing and processing the data into consecutive bins according to scan order, which bins have structural correspondence to the input format. The data converter is further capable of sub-dividing the division into sub-divisions which have structural correspondence to the output format. The data processor is further capable of processing and allocating the BGFs in the bins to sub-divisions to thereby produce a division data stream in the output expanded real time format.

Mechanizing Verification of Arithmetic Circuits: SRT Division

Abstract: The use of a rewrite-based theorem prover for verifying properties of arithmetic circuits is discussed. A prover such as Rewrite Rule Laboratory (RRL) can be used effectively for establishing numbertheoretic properties of adders, multipliers and dividers. Since verification of adders and multipliers has been discussed elsewhere in earlier papers, the focus in this paper is on a divider circuit. An SRT division circuit similar to the one used in the Intel Pentium processor is mechanically verified using RRL. The number-theoretic correctness of the division circuit is established from its equational specification. The proof is generated automatically, and follows easily using the inference procedures for contextual rewriting and a decision procedure for the quantifier-free theory of numbers (Presburger arithmetic) already implemented in RRL. Additional enhancements to rewrite-based provers such as RRL that would further facilitate verifying properties of circuits with structure similar to that of the SRT division circuit are discussed.