Showing papers on "Division (mathematics) published in 1998"

Behaviour of recursive division surfaces near extraordinary points

Abstract: The behaviour of the limits surface defined by a recursive division construction can be analysed in terms of the eigenvalues of a set of matrices. This analysis predicts effects actually observed, and leads to suggestions for the further improvement of the method.

A Mechanically Checked Proof of IEEE Compliance of the Floating Point Multiplication, Division and Square Root Algorithms of the AMD-K7™ Processor

Abstract: We describe a mechanically verified proof of correctness of the floating point multiplication, division, and square root instructions of the AMD-K7 microprocessor. The instructions are implemented in hardware and represented here by register-transfer level specifications, the primitives of which are logical operations on bit vectors. On the other hand, the statements of correctness, derived from IEEE Standard 754, are arithmetic in nature and considerably more abstract. Therefore, we begin by developing a theory of bit vectors and their role in floating point representations and rounding. We then present the hardware model and a rigorous proof of its correctness. All of our definitions, lemmas and theorems have been formally encoded in the ACL2 logic, and every step in the proof has been mechanically checked with the ACL2 prover.

A mechanically checked proof of the AMD5/sub K/86/sup TM/ floating-point division program

Abstract: We report on the successful application of a mechanical theorem prover to the problem of verifying the division microcode program used on the AMD5/sub K/86 microprocessor. The division algorithm is an iterative shift and subtract type. It was implemented using floating point microcode instructions. As a consequence, the floating quotient digits have data dependent precision. This breaks the constraints of conventional SRT division theory. Hence, an important question was whether the algorithm still provided perfectly rounded results at 24, 53, or 64 bits. The mechanically checked proof of this assertion is the central topic of the paper. The proof was constructed in three steps. First, the divide microcode was translated into a formal intermediate language. Then, a manually created proof was transliterated into a series of formal assertions in the ACL2 dialect. After many expansions and modifications to the original proof, the theorem prover certified the assertion that the quotient will always be correctly rounded to the target precision.

Ground based beam forming utilizing synchronized code division multiplexing

Abstract: A method permitting beam forming at a ground station for providing a coherent and stable uplink signal to a satellite system employing multiple spot beams by combining orthogonal synchronous code division multiplex codes and pseudo-random spreading codes with an information signal to implement a satellite feeder uplink

The Division of Inventive Labor and the Extent of the Market

Abstract: ing away from incentives issues, we ask which system minimizes costs. This leads immediately to the (true) Smith-Stigler theorem with the right definition of “the extent of the market” and also to the counter-intuitive modern theorem with the other definition. Set-up of the problem, and assumptions about costs. Many of the elements of our very simple model are the same under either system. We posit N distinct uses of the knowledge or input -applications sectors. Each use has a desired rate of production, Q . Since ours is a n theory of cost, we take N and Q to be given and exogenous. n Each sector needs to undertake two activities, which might be understood as production processes or as inventions. One of the two activities is always localized and serves to define the identity of an AS. The other activity might be localized or might be served by a general specialty. These have cost functions given by c (Q ) and K(Q), respectively. We make no assumptions n n about the c ( ), as they do not affect our results. The costs K( ) are the costs of providing the n potentially general input. We assume that there are increasing returns in this activity -that is, the elasticity of K( ) is smaller than 1. Moreover, we assume that the degree of scale economies is falling with the rate of output, i.e. K(Q)/Q K’(Q) declines with Q. These assumptions on the shape of K( ) are crucial. Costs Under Localization and General Specialty. If there is no general specialty, the c L n ' cn(Qn ) % K(Qn)

Fiber optic dense wavelength division multiplexer with a phase differential method of wavelength separation utilizing a polarization beam splitter and a nonlinear interferometer

Abstract: A dense wavelength division multiplexer for separating an optical signal into optical channels is provided. The dense wavelength division multiplexer of the present invention includes a manner for inputting an optical signal where the optical signal comprises a plurality of optical channels; a manner for separating one or more of the plurality of optical channels by introducing a phase difference between at least two of the plurality of optical channels, where the manner of separation includes a polarization beam splitter; and a manner for outputting the separated plurality of optical channels along a plurality of optical paths. The dense wavelength division multiplexer of the present invention provides an ease in alignment and a higher tolerance to drifts due to the increase in the width of the pass band. Its separators may also be placed in a multi-stage parallel cascade configuration to provide for a lower insertion loss. It may also be easily modified to perform the add/drop function as it separates channels. The material required to manufacture and implement the dense wavelength division multiplexer is readily available and do not require special or expensive materials or processes. It is thus cost effective.

Moving image coding method, moving image coder and moving image decoder

Abstract: PROBLEM TO BE SOLVED: To facilitate setting of an area in response to a characteristic of an image by enhancing a degree of freedom of an area shape in the case of dividing an image into a plurality of areas and coding them. SOLUTION: The moving image coder has an area division section 1, a coding section 7, and a memory 9 that is used to predict motion compensation. The area division section 1 has a division processing section and an integral processing section. The division processing section divides an input image based on a reference relating to the propriety of division. The integral processing section integrates areas close to each other based on the reference relating to the propriety of integration. Then each area is coded. The integral processing diversifies the shape of the area.

On finding an envy-free Pareto-optimal division

Abstract: This paper describes an algorithm to find an (α-)envy-free Pareto-optimal division in the case of a finite number of homogeneous infinitely divisible goods and linear utility functions. It is used to find an allocation in the classical cake division problem that is almost Pareto-optimal and α-envy-free.

Threshold circuits of small majority-depth

Abstract: We investigate the complexity of computations with constant-depth threshold circuits. Such circuits are composed of gates that determine if the sum of their inputs is greater than a certain threshold. When restricted to polynomial size, these circuits compute exactly the functions in the class TC$\sp0$. These circuits are usually studied by measuring their efficiency in terms of their total depth. Using this point of view, the best division and iterated multiplication circuits have depth three and four, respectively. In this thesis, we propose a different approach. Since threshold gates are much more powerful than AND-OR gates, we allow the explicit use of AND-OR gates and consider the main measure of complexity to be the majority-depth of the circuit, i.e. the maximum number of threshold gates on any path in the circuit. Using this approach, we obtain division and iterated multiplication circuits of total depth four and five, but of majority-depth two and three. The technique used is called Chinese remaindering. We present this technique as a general tool for computing functions with integer values and use it to obtain depth-four threshold circuits of majority-depth two for other arithmetic problems such as the logarithm and power series approximation. We also consider the iterated multiplication problem for integers modulo q and for finite fields. The notion of majority-depth naturally leads to a hierarchy of subclasses of TC$\sp0$. We investigate this hierarchy and show that it is closely related to the usual depth hierarchy.

Fast recursive division

Abstract: We present a new recursive method for division with remainder of integers. Its running time is 2K(n)+O(n log n) for division of a 2n-digit number by an n-digit number where K(n) is the Karatsuba multiplication time. It pays in practice for numbers with 860 bits or more. Then we show how we can lower this bound to 3=2K(n)+O(n log n) if we are not interested in the remainder. As an application of division with remainder we show how to speedup modular multiplication. We also give practical results of an implementation that allow us to say that we have the fastest integer division on a SPARC architecture compared to all other integer packages we know of.

Efficient way to produce a delayed version of a maximum length sequence using a division circuit

Abstract: Method and apparatus for efficiently producing a delayed version of a maximum length sequence output from a linear feedback shift register. Polynomial (vector) exponentiation is performed instead of matrix exponentiation to calculate the mask coefficients which yield the delayed sequence from the linear feedback shift register. Polynomial (vector) operations are much simpler and faster than the corresponding matrix operations and require substantially less circuitry and computational effort. Modulo exponentiation of polynomials is done by repeated squaring and shifting, and a division circuit built on a linear feedback shift register is provided to perform an efficient modulo squaring of polynomials.

Cascading optical multiplexing device

Abstract: An optical multiplexing device is provided comprising multiple wavelength division multiplexers cascaded together. A first one of the wavelength division multiplexers has a common port and multiple optical ports which are optically coupled to the common port. The common port may be optically coupled to a trunk line of a system employing wavelength division multiplexing, for example, a fiber-optic telecommunication system employing 4, 8, 16 or other number of multiplexed channels. The optical ports include multiple channel ports, each of which is transparent to a corresponding wavelength sub-range and reflective of other wavelengths. The second wavelength division multiplexer has a common port optically coupled to one of the optical ports of the first wavelength division multiplexer. The second wavelength division multiplexer also has multiple optical ports which are optically coupled to its common port and include multiple wavelength-selective channel ports. A waveguide, such as a fiber-optic line, can optically connect the common port of the second wavelength division multiplexer to an optical port of the first wavelength division multiplexer. The cascaded WDMs each may be optically coupled to the output of a passive coupler and a housing may be provided defining an enclosed space in which the optical multiplexing device is mounted. Optionally, additional WDMs may be cascaded with the first two WDMs in a parallel or branched formation, an in-line formation or some combination. Preferably, the channels are interleaved, such that they are removed from the multiplexed signal in certain non-sequential order. The optical multiplexing device also may employ compound interleaving wherein adjacent channels are multiplexed by different ones of the cascaded WDMs. The optical multiplexing devices can operate to add signals, remove signals or a combination of both.

Algorithm for computer control of a digital plotter

Abstract: An algorithm is given for computer control of a digital plotter. The algorithm may be programmed without multiplication or division instructions and is efficient with respect to speed of execution and memory utilization.

Supersimple fields and division rings

Abstract: It is proved that any supersimple field has trivial Brauer group, and more generally that any supersimple division ring is commutative. As prerequisites we prove several results about generic types in groups and fields whose theory is simple.

A developmental multimodal model for multiplication and division

Abstract: This paper presents an analysis of young students’ development of multiplication and division concepts based on a multimodal SOLO model The analysis is drawn from two sources of data: a two-year longitudinal study of 70 Grade 2 to 3 students’ solutions to 24 multiplicative word problems, and examples from a problem-centred teaching project with Grade 3 students An increasingly complex range of counting, additive, and multiplicative strategies based on an equal-grouping structure demonstrated conceptual growth through ikonic and concrete symbolic modes The solutions employed by students to solve any particular problem reflected the mathematical structure they imposed on it A SOLO developmental model for multiplication and division is described in terms of developing structure and associated counting and calculation strategies

Program conversion apparatus for constant reconstructing VLIW processor

Abstract: A program conversion apparatus includes: the constant division unit 12 for specifying instructions in the serial assembler code 42 that use large constants which cannot be arranged within the operation fields of object VLIWs and for dividing the specified instructions into divided constant use instructions for storing pieces of the large constants into the specialized constant buffer 107 of a VLIW processor and divided constant use instructions for performing operations using the stored constants; the dependence graph generation unit 20 for generating a dependence graph based on the execution order of each instruction in the serial assembler code 42 after the division process by the constant division unit 12; and the instruction relocation unit 21 for relocating the instructions according to the dependence graph to generate parallel assembler code.

Computation of /spl radic/(x/d) in a very high radix combined division/square-root unit with scaling and selection by rounding

Abstract: A very-high radix digit-recurrence algorithm for the operation /spl radic/(x/d) is developed, with residual scaling and digit selection by rounding. This is an extension of the division and square-root algorithms presented previously, and for which a combined unit was shown to provide a fast execution of these operations. The architecture of a combined unit to execute division, square-root, and /spl radic/(x/d) is described, with inverse square-root as a special case. A comparison with the corresponding combined division and square-root unit shows a similar cycle time and an increase of one cycle for the extended operation with respect to square-root. To obtain an exactly rounded result for the extended operation a datapath of about 2n bits is needed. An alternative is proposed which requires approximately the same width as for square-root, but produces a result with an error of less than one ulp. The area increase with respect to the division and square root unit should be no greater than 15 percent. Consequently, whenever a very high radix unit for division and square-root seems suitable, it might be profitable to implement the extended unit instead.

Base station transceiver for frequency hopping code division multiple access system

Abstract: A small and lightweight base station transceiver utilizes a plurality of low-powered time division duplexing switches instead of a single high-powered time division duplexing switch. The base station transceiver includes a forward signal path, a reverse signal path, and a plurality of time division duplexing switches being switchable to the forward signal path or the reverse signal path according to a time division duplexing timing control signal generated from a time division duplexing timing controller. The plurality of time division duplexing switches corresponds to N time division duplexing switches where the symbol N represents a number. Output signals of the time division duplexing switches are filtered in a plurality of N bandpass filters, coupled in a junction coupler, and then radiated in the air through first and second antennas. The junction coupler couples the outputs of odd-numbered bandpass filters to transfer the coupled signals to the first antenna, and couples the outputs of even-numbered bandpass filters to transfer the coupled signal to the second antenna, so as to reduce mutual interference among the channels.

Method and apparatus for minimizing delay variance via time divided transmissions

Abstract: A local area network for connection between a terrestrial cell/packet based network and a time division based point-to-multipoint network comprising a switch (10), an interface (710) including a time stamp processing unit (701) and a frame counter unit (702), means for forming a time marked cell/packet (720), means for assembling into a time division frame (730), and a modem (740) for transmitting time division frame.

Multilayered substrate and method for its production

Abstract: A multilayered substrate divisible into a plurality of parts at predetermined positions of division is configured such that intermittently formed perforation-like holes or slit-like holes are present at the positions of division of a core layer. Further, V-cut V-grooves are present at the positions of division of surface layers. The multilayered substrate, divisible at predetermined positions of division, is also configured such that first intermittently formed holes (perforation-like holes, slit-like holes, or widely spaced holes) are present at the positions of division of the core layer. Further, second intermittently formed holes (perforation-like holes, or slit-like holes) are present at the positions of division of the surface layers so as to pass through the multilayered substrate. At the positions of division of the core layer, each of the second holes is located between the adjacent first holes, whereby the first holes and the second holes are continuous with each other. Methods for production of these multilayered substrates are also disclosed.

Image processing apparatus and method and storing medium

Abstract: A system which can efficiently compress any one of a multivalue image and a binary image and can preferably keep a high picture quality is provided. To accomplish this object, the system has means for generating conversion coefficients of M values by converting input image data by a first system, means for generating conversion coefficients of N values (N>M) by converting the input image data by a second system and means for outputting first information showing a position where a significant conversion coefficient exists in a block constructed by the conversion coefficients of M values or the conversion coefficients of N values and second information showing the significant conversion coefficient. Block division information which is generated by a block dividing process as an element technique of the image encoding system is efficiently entropy encoded. To accomplish it, the system has means for outputting division information showing whether each of a plurality of blocks including an encoding subject block has further been divided into a plurality of blocks or not and means for entropy encoding the division information corresponding to the encoding subject block on the basis of a presumption probability according to a division situation of a neighboring block of the encoding subject block.

The labour of division: the manager as ‘self’ and ‘other’

Abstract: Do join me in the space which our editors have allocated to me in their editorial labour of division and engage with this story I have to tell. But in accepting, for a moment, your identity as reader and mine as writer, note that I said 'engage with' this story. This means moving beyond these identities so that we not only subvert the division of reader and writer but transcend it in order to work together on a story about managers. We will necessarily work separately too, however. Whether reading or writing, each of us brings to a story our own experiences of life and self in organizational settings and what we know of managers. And we will also bring forward what we might have read in other divisions of this book or from chapters in other books. Allowing categorical divisions to appear and fade in this way is not just part of my introductory recognition of the divisions with which we are both working. It is, first, to apply to ourselves the sort of deconstructive work that we are about to do to others. We can call it work, yet it is work which, true to its own spirit, recognizes its playing with categories. And this is what the manager does too-or so my story will have it. Will yours? Let's see. Why is the manager an important figure for study? One answer to this lies in the very ambiguous situation of this organizational employee who is treated to the identity of 'manager' by virtue of the fact that he or she can be distinguished from 'the managed', and yet who is in many ways more 'managed' than those whom they allegedly manage. Being a manager at the same time as not being a manager, we might add, is similar to that other ambiguity where the managers find themselves playing the part of a hard-headed officer of the corporation at the same time as acting out the part of

Semi-logarithmic number systems

Abstract: We present a new class of number systems, called Semi-Logarithmic Number Systems, that constitute a family of various compromises between floating-point and logarithmic number systems. This allows trade between the speed of the arithmetic operations and the size of the required tables. We give arithmetic algorithms (addition/subtraction, multiplication, division) for the Semi-Logarithmic Number Systems, and we compare these number systems to the classical floating-point or logarithmic number systems.

Picture information providing system, image information display terminal, and server device

Abstract: PROBLEM TO BE SOLVED: To provide a map information providing system capable of smoothly performing scroll operation. SOLUTION: When coordinates are designated on a map providing screen, a terminal unit 3 of this map information providing system generates division information in a display range on the basis of a position of the coordinates and requests picture data to be inputted into the division to the server device 2. The server device 2 transmits the picture data for a division information train requested from the terminal unit 3 to the terminal unit 3. Then, when the alteration of a display area is instructed on a menu display for providing a map in the terminal unit 3 (scroll operation), the terminal unit 3 generates a division information train at the time of executing the display alteration and requests picture data transfer to the server device 2 only about a division except the division having the picture data already. COPYRIGHT: (C)2000,JPO

Compression of data transmission by use of prime exponents

Abstract: A method of compressing data utilizes the prime number series to generate unique compression parameters that may be used to recover an original data stream. The original data is converted from a binary form to a decimal form. Various compression parameters are selected to initialize the system. The compression parameters include the number of prime numbers which will be used in the compression process and an exponential value corresponding to each prime number. A header is constructed which includes the compression parameters. The data is compressed by a compression algorithm which performs successive division operations by the series of prime numbers selected. The compression algorithm generates a plurality of exponential values corresponding to each of the prime numbers. The header is then assembled with the exponential values and transmitted to a receiving station. In decompressing the data, the prime numbers are raised to the exponential value generated by the compression algorithm, and their product is taken. A fault parameter may be generated to compensate for numbers not readily divisible by the prime number selected in the initialization parameters.

Apparatus for beam splitting, combining wavelength division multiplexing and demultiplexing

Abstract: The present invention uses a light-pipe for beam splitting, combining, hybrid space-wavelength multiplexing, hybrid space-wavelength demultiplexing, dense wavelength division multiplexing, and dense wavelength division demultiplexing. A light-pipe has a first end where light enters the light-pipe and a second end where the light exits the light-pipe. The light-pipe is a polygonal rod or a hollow reflective cavity. To perform beam splitting, light enters the first end of the light-pipe and reflects off of the side surfaces inside the light-pipe producing additional light beams. By placing an imaging lens at the output end of the light-pipe, all of the virtual images (n-1) of the additional light beams and the original light entering the light-pipe become the object of the imaging lens. As a result, the light-pipe acts as a 1-to-n splitter. If the input signal, that is, the light entering the light-pipe, has more than one wavelength and a grating is added to the second end of the light-pipe, the device becomes a hybrid space-wavelength division multiplexer/demultiplexer. If the spacing between adjacent wavelengths is small, the light-pipe becomes a Dense Wavelength Division Demultiplexer. If the input and output are reversed, the beam splitter becomes a combiner. If the inputs have different wavelengths, the light-pipe becomes a Wavelength Division Multiplexer.