Showing papers on "Division (mathematics) published in 1984"

Log Depth Circuits For Division And Related Problems

Abstract: We present optimal depth Boolean circuits (depth O(log n)) for integer division, powering, and multiple products. We also show that these three problems are of equivalent uniform depth and space complexity. In addition, we describe an algorithm for testing divisibility that is optimal for both depth and space.

Choice of Operation in Verbal Arithmetic Problems: The Effects of Number Size, Problem Structure and Context.

Abstract: 12-and 13-year-olds were tested with two types of tasks to test their understanding of applications of the multiplication and division of positive numbers: (i) writing down calculations required to solve verbal problems, and (ii) making up stories to fit given calculations. Selected pupils were interviewed to investigate further the thinking processes involved. The results indicate (a) the pervasive nature of certain numerical misconceptions, (b) the effects of structural differences among the items; particularly whether multiplication can be conceived as repeated addition or not, and whether division has the structure of partition, quotition or rate, (c) specific effects of context attributable to such aspects as relative familiarity, and (d) various interactions between these three sets of factors.

A Problem Solving Approach to Mathematics for Elementary School Teachers

Abstract: 1. An Introduction to Problem Solving 1-1 Mathematics and Problem Solving 1-2 Explorations with Patterns 1-3 Reasoning and Logic: An Introduction 2. Numeration Systems and Sets 2-1 Numeration Systems 2-2 Describing Sets 2-3 Other Set Operations and Their Properties 3. Whole Numbers and Their Operations 3-1 Addition and Subtraction of Whole Numbers 3-2 Algorithms for Whole-Number Addition and Subtraction 3-3 Multiplication and Division of Whole Numbers 3-4 Algorithms for Whole-Number Multiplication and Division 3-5 Mental Mathematics and Estimation for Whole-Number Operations 4. Algebraic Thinking 4-1 Variables 4-2 Equations 4-3 Functions 5. Integers and Number Theory 5-1 Integers and the Operations of Addition and Subtraction 5-2 Multiplication and Division of Integers 5-3 Divisibility 5-4 Prime and Composite Numbers 5-5 Greatest Common Divisor and Least Common Multiple 5-6 *Clock and Modular Arithmetic 6. Rational Numbers as Fractions 6-1 The Set of Rational Numbers 6-2 Addition and Subtraction of Rational Numbers 6-3 Multiplication and Division of Rational Numbers 7. Decimals and Real Numbers 7-1 Introduction to Decimals 7-2 Operations on Decimals 7-3 Nonterminating Decimals 7-4 Real Numbers 7-5 Using Real Numbers in Equations 8. Proportional Reasoning, Percents and Applications 8-1 Ratios, Proportions and Proportional Reasoning 8-2 Percents 8-3 Computing Interest 9. Probability 9-1 How Probabilities are Determined 9-2 Multistage Experiments with Tree Diagrams and Geometric Probabilities 9-3 Using Simulations in Probability 9-4 Odds, Conditional Probability, and Expected Value 9-5 Using Permutations and Combinations in Probability 10. Data Analysis/Statistics: An Introduction 10-1 Displaying Data: Part I 10-2 Displaying Data: Part II 10-3 Measures of Central Tendency and Variation 10-4 Designing Experiments/Collecting Data 10-5 Abuses of Statistics 11. Introductory Geometry 11-1 Basic Notions 11-2 Polygons 11-3 More about Angles 11-4 Geometry in Three Dimensions 11-5 *Networks 12. Constructions, Congruence, and Similarity 12-1 Congruence through Constructions 12-2 Other Congruence Properties 12-3 Other Constructions 12-4 Similar Triangles and Similar Figures 12-5 *Trigonometry Ratios via Similarity 12-6 Lines and Linear Equations in a Cartesian Coordinate System 13. Concepts of Measurement 13-1 Linear Measure 13-2 Areas of Polygons and Circles 13-3 The Pythagorean Theorem, Distance Formula, and Equation of a Circle 13-4 Surface Areas 13-5 Volume, Mass, and Temperature 14. Motion Geometry and Tessellations 14-1 Translations and Rotations 14-2 Reflections and Glide Reflections 14-3 Size Transformations 14-4 Symmetries 14-5 Tessellations of the Plane The Appendices are now full chapters to help students with specific technology tools: Using a Spreadsheet, Graphing Calculators, and Using a Geometry Drawing Utility. These chapters are available in the Technology Manual, which can be shrink-wrapped at no additional cost with the new textbook. See 'Packages' tab for ordering information. Answers to All/Selected Exercises Index Photo Acknowledgments *Optional Sections

The Division of Frictional Heat—A Guide to the Nature of Sliding Contact

Abstract: University of Newcastle upon Tyne, Newcastle upon Tyne, NE1 7RU, England; Now, University of Michigan, Ann Arbor, MI 48109 The work vf Jaeger [l] and Barber [2] indicates that the division of frictional heat between sliding solids should be sensitive to the interfacial boundary conditions. Thus it should provide a guide to the nature of asperity interactions and a more accurate statement of the boundary conditions for any subsequent analysis. Measurement of the division of heat in a symmetric "cylinder on cylinder" system was frustrated by the widespread occurrence of thermoelastic instability. An alternative specimen geometry has been developed which has permitted the division of heat between sliding solids of various materials to be investigated experimentally. The characteristics of the microscopic thermal resistance at the surface of a sliding solid have been investigated theoretically for several types of asperity interaction. An approximate method has been employed to estimate the thermal resistance of an oxidized surface. The observed division of heat is interpreted with reference to the characteristic behavior associated with the various types of asperity interaction. It is shown that in the mild wear regime oxide films have an appreciable effect on the microscopic thermal resistance and hence, on the thermal behavior of the sliding solids, particularly the division of heat between them.

Managing device of copying machine

Abstract: PURPOSE:To sum up automatically the number of use of each division with respect to a lot of copying machines by connecting the copying machine which becomes usable by inserting a magnetic card and is capable of executing a copy number count of each division, to a central managing device. CONSTITUTION:Plural copying machines 1-4 are provided with input terminals 5-8, and when a magnetic card is inserted, a division code is read, and when it is registered, cophing can be executed. The number of copies of each division, and the number of generation of a jam, etc. are stored by a microcomputer and a memory in the terminal. Its data is sent to a central managing device 9 through a circuit 10, and a data of each copying machine is summed up by each used division. From the central managing device 9, a guidance information such as time, a countermeasure at the time of a trouble, etc. is sent to each copying machine 1-4, and displayed on the terminals 5-8.

Biased model reduction by factor division

Abstract: Model reduction by the factor division method is extended to produce biased models. A simple Routh-type algorithm avoids the necessity to calculate the retained system time moments and Markov parameters beforehand and solving the Pade equations for the reduced numerator. An example demonstrates its use.

Electric storage batteries

Abstract: A battery includes a plurality of frames of plastics material, each frame comprising a perimeter member and one or more division elements together defining two or more active material support spaces. The frames are arranged in a stack and the perimeter member and division elements of each frame are connected to those of adjacent frames. Each support space receives an electrode, and every alternate division element has on one side two connector recesses extending over a minor proportion of its length, connector recesses in adjacent frames being in division elements that are offset from one another. The spaces on each side of each division element receive electrodes which are connected together by connectors which are received in the connector recesses. Preferably that portion of each division element which overlies a connector recess in the adjacent frame has an elongate recess in it, the connector recess and elongate recess containing a sealing material.

Manufacture of screen for projection

Abstract: PURPOSE:To uniformly form a reflecting surface and a light absorbing surface over the entire screen by forming a screen base of a light absorbing material and making division step surfaces of division reflecting parts of its curved surface into light absorbing surfaces CONSTITUTION:Aluminum films 4 are vapor-deposited only on division reflecting surfaces 3 of the linear Fresnel type reflection part 2 of the screen base 1 formed by adding a light absorbing agent to polyester resin Then, the curved reflecting surface 10 is so slanted as to put an observation line 7 and a video light beam 9 reflected on the screen 8 close to each other, and the reflecting surface 10 is divided horizontally; and step grooves 11 are formed in the outer peripheral surface of a molding roll 6 in conformity with the sectional shape of the linear Fresnel reflection part 2, and oriented mat work is carried out for division surfaces 12 of the grooves 11 Consequently, division step parts of the reflection part 2 become light absorbing surfaces 5

Computer modelling of cell division during development using a topological approach.

Abstract: Development in multicellular animals consists of a constant progression of cell division, differentiation and morphogenesis. Our understanding of the relationship between division and the acquisition of shape and form is not well understood, and this paper describes a computer representation of cell division processes with possible applications to the modelling of developmental events. This representation is not itself a model in the true sense, but is a scaffolding onto which a set of model assumptions and parameters can be built. We discuss one such set of assumptions, used to model cell sorting, describe the extension of the framework to represent sheets of cells in three dimensions, and make some observations on the incorporation of mechanical forces into the representation.

Arithmetic in Complex Bases

Abstract: In this note, we introduce a novel way of doing complex arithmetic that does not involve separating the complex numbers into their real and imaginary parts. This method uses the representation of complex numbers in positional notation using a complex base - n + i, for a positive integer n, with natural numbers as digits. Addition, subtraction and multiplication can be performed directly in this positional notation and is similar to real decimal arithmetic; the main difference is in the carry digits. However, division is more complicated and the construction of a good algorithm for long division is a challenging unsolved problem. We say that an integer z (real or complex) is represented in the base b with digits from the set 6D

Method for transmitting binary signals of at least two channels

Abstract: To transmit binary signals of at least two channels with the aid of block codes, the binary signals are multiplexed at the transmission end by means of a time division multiplexer, scrambled by means of a scrambler, re-coded by means of a block coder and provided, by means of a code violation generator, with a frame identification signal. At the receiving end, the transmitted, re-coded signals are re-coded into binary representation by means of a decoder, maintained in frame synchronisation with the transmitting end by means of a code violation detector, descrambled by means of a descrambler and demultiplexed by means of a time division demultiplexer. The block length of the coder or decoder, the frame length of the time division multiplex signal generated by the time division multiplexer or time division demultiplexer and the word length of the scrambler or descrambler have a fixed integral relationship with one another, so that only one synchronisation device is to be provided in each case at the transmitting and receiving ends.

Reduction of 2-D rational functions

Abstract: This paper presents a method of reducing a two-dimensional (2-D) rational function to an irreducible one. It is achieved by searching the first linearly dependent row, in order from top to bottom, of a generalized resultant. Some properties of the resultant are discussed. The result can be used to compute the greatest common divisor (g.c.d.) of two 2-D polynomials by carrying out one additional division. This procedure does not require the computation of primitive polynomials, which is required by all existing methods, and thus provides a potentially attractive method of computing g.c.d. The method can be readily extended to three-or higher-dimensional cases.

Digital signal distributor

Abstract: A digital signal distributor for time division multiplex signals having the same frame format and for discrete channel signals has frame adaptors for synchronizing and storing the time division multiplex signals, and memories for the discrete channel signals, the frame adaptors and memories being provided following the inputs for the time division multiplex channels. The circuit further includes a distributor unit having an operator interface connected to the frame adaptors and the memories by first bus lines and first discrete lines. A clock pulse generator controls cyclical feeding of the stored code words into the first bus and first discrete lines. Second bus and second discrete lines are connected to an output of the distributor unit and transmit the time division multiplex output signals and the discrete signals to outputs of the circuit.

Exhaust chamber device of car mounted with two cycle engine

Abstract: PURPOSE:To enable to adjust to a suitable length comforming to an engine speed, by a method wherein two parts of an exhaust pipe obtained by dividing the pipe into two are connected movably with each other, the full length of the pipe is shortened at the time of high speed running of an engine and similarly the pipe is extended at the time of low speed running. CONSTITUTION:An exhaust chamber 1 is provided with a diffuser pipe 2, a reflecting taper part 3, an expansion chamber 4, an expanding taper part 5 and an exhaust pipe 6. In this instance, a first pipe 7 to be connected with an engine side is inserted into a second pipe 8 after division of the above pipe 6 into two and both the pipes are connected movably with each other so as to enable the full length to expand and contract. At the time of low speed running of an engine, the full length of the pipe 6 is extended (l+l') by moving the second pipe 8 to the left as shown in a sketch through a pin 11 by a cam lever 14 by pulling an operating wire 13 constituting an operating device 9, and on the one hand, at the time of high speed running, the full length of the pipe 6 is contracted (l) by returning the second pipe 8 through a spring 10 by making the operating wire 13 slacken.

Indication control for workpiece picture in nc apparatus

Abstract: PURPOSE: To reduce the useless no-indication area by installing a work-dimension calculating and judging part and obtaining the division rate of a picture from the max. dimension of a workpiece to be indicated onto a display and division- indicating two pictures for the workpiece onto the display on the basis of the division rate. CONSTITUTION: Each max. dimension MX, MX 1 , and MX 2 of a workpiece is calculated for each drawing to be displayed in the variable direction of a display picture from the working data shown on the working-program memory 5 into which the working program PRO is stored, display 10, indication control part 6, and PRO. A work-dimension calculation and judging part 7 for obtaining the division rate DX of the picture in case of indication on the display 10 from the above-described result is installed. A control part 6 division-indicates two pictures of the front view, side view, etc. related to the workpiece on the display 10 on the basis of the division rate DX. Therefore, two pictures are displayed in the form proportional to the size of the work, and useless no-indication area can be reduced. COPYRIGHT: (C)1986,JPO&Japio

Time division optical exchange

Abstract: PURPOSE:To miniaturize the exchange by an optical switch connected to a semiconductor laser operating bistably between an input optical waveguide and an output optical waveguide and a current drive circuit so as to set optionally the length of a frame exchanged by the exchange of a time division optical signal. CONSTITUTION:A switch 320 for write and a switch 330 for readout are provided between an input optical waveguide 310 inputting a time division optical signal 100 and an output optical wavguide 380 outputting a time division optical signal 190. Semiconductor lasers 340, 350, 360, 370 operating bistably are connected between an output side of the respective switch 320 and the input side of the switch 330 via an optical waveguide, and current drive circuits 345, 355, 365, 375 injecting respectively a bias current are connected to the lasers 340, 350, 360 and 370. Further, each of the switches 320, 330 is driven respectively by optical switch driving circuits 321, 331, the length of the frame exchanged by the exchange is set optionally and the exchange is miniaturized.

Dividing method of light-emitting diode

Abstract: PURPOSE:To obtain a dividing method for a light-emitting diode having high yield on a division by previously executing a scribing method for acquiring a photoemissive surface and dividing and processing other surfaces after said process. CONSTITUTION:Since an electrode 3 on a main surface is deviated to the one end surface 2 side from a center S, a photoemissive surface can be obtained by dividing a section extremely close by the electrode on the division of a water. The wafter 10 to which a P-N junction 11 and electrodes 14, 13, 13... are formed completely is stuck to a thin-film sheet 15, and scribing grooves 16, 16... for obtaining the photoemissive surface are shaped. The wafer 10 to which the scribing grooves 16, 16... are formed completely is pressed by a roller 19 and divided. Divided surfaces take a rectangular form and are shaped in a mirror surface, and broken pieces and cracks are hardly generated. Residual division is conducted in order to form light-emitting diodes to an approximately dice shape, but the division is conducted by forming kerfs 20, 20... in sufficient depth or cutting the wafer through a dicing method.

Time division type light transmitter

Abstract: PURPOSE:To collect and transmit/receive many pieces of ON/OFF or work information, by providing a multi-spot processing function, a series-parallel or parallel-series converting functions, and a light emitting-photodetecting function to the transmitting and receiving parts. CONSTITUTION:A transmitting part contains (n) units of input terminals 5, (m) units of parallel gate circuits 6, a selector circuit 9 which scans successively the circuit 5, and a parallel-serial conversion synchronizing circuit 13 respectively. The circuit 9 is driven by an n-notation counter 10 which counts the clocks supplied from a quartz oscillator part 11. Then the circuit 9 transmits the ON/ OFF or word information given from (n) units of input terminals to an optical cable 17 via the circuit 13, At the receiving side, the output is successively supplied to (n) units of (d) output terminals after carrying out the processing opposite to that of the transmitting part.

Method of producing curved surface

Abstract: A method of producing a curved surface includes a step of inputting section data on two sections (11, 12) of a three-dimensional curved-surface object, data on first and second section curves (11a, 12a) formed by the two sections (11, 12), respectively, and data on a three-dimensional curve (31a) which includes points (P1, P1') on the first and second section curves, respectively, and specifies the outer shape of the three-dimensional curved-surface object; a step of generating an intermediate section (41) for each of division points (Si) (i = 1, 2, 3 ... ) which divide the three-dimensional curve (31a) into a large number of line elements, each intermediate section (41) including the corresponding division point; a step of calculating an intermediate section curve (41a) of each intermediate section using the data on the two section curves (11a, 12a) and positional information about the division points (Si); and a step of generating the curved surface of the three-dimensional curved-surface object using this set of a plurality of intermediate section curves.