Simple Mapping Function Research Articles

Single-dimension software pipelining for multidimensional loops

Traditionally, software pipelining is applied either to the innermost loop of a given loop nest or from the innermost loop to outer loops. This paper proposes a three-step approach, called single-dimension software pipelining (SSP) , to software pipeline a loop nest at an arbitrary loop level that has a rectangular iteration space and contains no sibling inner loops in it. The first step identifies the most profitable loop level for software pipelining in terms of initiation rate, data reuse potential, or any other optimization criteria. The second step simplifies the multidimensional data-dependence graph (DDG) of the selected loop level into a one-dimensional DDG and constructs a one-dimensional (1D) schedule. Based on the one-dimensional schedule, the third step derives a simple mapping function that specifies the schedule time for the operation instances in the multidimensional loop. The classical modulo scheduling is subsumed by SSP as a special case. SSP is also closely related to hyperplane scheduling, and, in fact, extends it to be resource constrained. We prove that SSP schedules are correct and at least as efficient as those schedules generated by traditional modulo scheduling methods. We extend SSP to schedule imperfect loop nests, which are most common at the instruction level. Multiple initiation intervals are naturally allowed to improve execution efficiency. Feasibility and correctness of our approach are verified by a prototype implementation in the ORC compiler for the IA-64 architecture, tested with loop nests from Livermore and SPEC2000 floating-point benchmarks. Preliminary experimental results reveal that, compared to modulo scheduling, software pipelining at an appropriate loop level results in significant performance improvement. Software pipelining is beneficial even with prior loop transformations.

Use of total electron content data to analyze ionosphere electron density gradients

In the presence of electron density gradients the thin shell approximation for the ionosphere, used together with a simple mapping function to convert slant total electron content (TEC) to vertical TEC, could lead to TEC conversion errors. These “mapping function errors” can therefore be used to detect the electron density gradients in the ionosphere. In the present work GPS derived slant TEC data have been used to investigate the effects of the electron density gradients in the middle and low latitude ionosphere under geomagnetic quiet and disturbed conditions. In particular the data corresponding to the geographic area of the American Sector for the days 5–7 April 2000 have been used to perform a complete analysis of mapping function errors based on the “coinciding pierce point technique”. The results clearly illustrate the electron density gradient effects according to the locations considered and to the actual levels of disturbance of the ionosphere. In addition, the possibility to assess an ionospheric shell height able to minimize the mapping function errors has been verified.

Analytical solution of the modified Reynolds equation for squeeze film damping in perforated MEMS structures

The squeeze-film damping in perforated structures is modelled using a modified Reynolds equation that includes compressibility and rarefaction effect. This equation is linearized and transformed to the standard two-dimensional diffusion equation using a simple mapping function. The analytical solution is then obtained using Green’s function. The solution thus obtained adds an additional term Γ to the damping and spring force expressions derived by Blech for compressible squeeze flow through non-perforated plates. This additional term contains several parameters related to perforations and rarefaction. Setting Γ=0, one recovers Blech’s formulae. We compute the squeeze film forces using these new formulae and compare the computed forces with the solution of 3D Navier–Stokes equation solved using ANSYS for different perforation ratios (ratio of hole to cell dimensions). The results match very well. The approximate limit of maximum frequencies under which the formulae give reasonable results is also discussed. Although the main result is derived for a rigid plate under transverse motion, we discuss the effect of flexibility of the structure by deriving results for a flexible plate under a specified set of boundary conditions and comparing the results with that of a suitably modified rigid plate result. For small amplitude motion, the results show that a suitably modified rigid plate model can capture the effect of flexibility through a simple scaling factor.

High performance, high throughput turbo/SOVA decoder design

Two efficient approaches are proposed to improve the performance of soft-output Viterbi (1998) algorithm (SOVA)-based turbo decoders. In the first approach, an easily obtainable variable and a simple mapping function are used to compute a target scaling factor to normalize the extrinsic information output from turbo decoders. An extra coding gain of 0.5 dB can be obtained with additive white Gaussian noise channels. This approach does not introduce extra latency and the hardware overhead is negligible. In the second approach, an adaptive upper bound based on the channel reliability is set for computing the metric difference between competing paths. By combining the two approaches, we show that the new SOVA-based turbo decoders can approach maximum a posteriori probability (MAP)-based turbo decoders within 0.1 dB when the target bit-error rate (BER) is moderately low (e.g., BER<10/sup -4/ for 1/2 rate codes). Following this, practical implementation issues are discussed and finite precision simulation results are provided. An area-efficient parallel decoding architecture is presented in this paper as an effective approach to design high-throughput turbo/SOVA decoders. With the efficient parallel architecture, multiple times throughput of a conventional serial decoder can be obtained by increasing the overall hardware by a small percentage. To resolve the problem of multiple memory accesses per cycle for the efficient parallel architecture, a novel two-level hierarchical interleaver architecture is proposed. Simulation results show that the proposed interleaver architecture performs as well as random interleavers, while requiring much less storage of random patterns.

A simple “geometric” mapping function for the hydrostatic delay at radio frequencies and assessment of its performance

The hydrostatic mapping function, m(ε), describes the dependence of the hydrostatic path delay on the elevation angle ε. We have developed a simple mapping function, where the only free parameter is an “effective height” of the atmosphere, Hatm, corresponding to about the first two scale heights above the surface. The value of m(ε) is given by the ratio of the straight‐line ray path length within height Hatm to height Hatm itself. We used simulated delays at GPS (Global Positioning System) frequencies derived from high resolution ECMWF (European Centre for Medium‐Range Weather Forecasts) atmospheric analysis fields to assess the performance of the “geometric” mapping function relative to well established ones (the Davis, Niell, and Herring mapping functions, respectively). At elevations &gt;6° the new mapping function generally exhibits, based on a single parameter, a performance comparable to or better than the other mapping functions.

Tropospheric water vapor imaging by combination of ground‐based and spaceborne GNSS sounding data

The Global Navigation Satellite System (GNSS) comprises the U.S. system GPS (Global Positioning System), its Russian pendant GLONASS, and presumably, in the future, the European system Galileo. The potential of GNSS‐based phase delay measurements for accurately estimating vertically and slant‐path‐integrated water vapor has been demonstrated recently for radio links between GPS satellites and ground‐based GPS receivers. GNSS‐based radio occultation, on the other hand, has been demonstrated via the GPS/Meteorology experiment to deliver accurate near‐vertical profiles of atmospheric variables such as temperature and humidity with high vertical resolution. Height‐resolving imaging of atmospheric water vapor becomes feasible when occultation profiles from spaceborne receivers in Low Earth Orbits (LEO) are combined with ground‐based GNSS data from a colocated receiver network. We developed a two‐dimensional, height‐resolving tomographic imaging technique following the Bayesian approach for optimal combination of information from different sources. Using simulated GNSS‐based water vapor measurements from LEO and ground, we show representative results derived from simple synthetic refractivity fields as well as from a realistic refractivity field based on a European Centre for Medium‐Range Weather Forecasts (ECMWF) analysis. For cases located poleward of ∼40° we found a new simple mapping function to perform best within our forward model scheme, where the only free parameter is the climatological scale height in the troposphere, the exact value of which is not critical. The mapping function exploits the ratio between the straight‐line ray path length within the first two scale heights above surface and the “effective height” defined by these first two scale heights. We found our technique capable of reconstructing atmospheric features like water vapor maxima near the top of the trade wind inversion. Adjustment of the integral over the water vapor profile measurements to the horizontally averaged ground‐based vertical integrated water vapor data efficiently mitigates potential biases in the former data. Accuracies are best in areas with high absolute humidities but also over drier areas such as Finland, useful two‐dimensional information can still be obtained. Thus it is attractive to apply the developed technique in a next step to real data.

A general solution for stress resultants and moments around holes in unsymmetric laminates

The complex variable method of Lekhnitskii and Savin can be applied for solving stress concentration problems in symmetric laminates. Although significant amount of literature is available on heterogeneous aelotropic plates, the stress concentration problems have not been addressed in it until Becker gave his complex potential method for plate problems with bending extension coupling. The present work is an extension of Becker's solution for elliptical hole problem for unsymmetric laminates. In this paper, it is aimed to bring out a general solution to determine the stress resultants and moments around holes of any shape with a simple mapping function under arbitrary biaxial loading condition.

Fitting genetic mapping functions based on sperm typing: results for three chromosomal segments in cattle

Genetic mapping functions translate the observed recombination rate between two loci into the corresponding map distance in Morgan units. Different mapping functions give different weights to multiple crossing over and therefore lead to different results. This points out that not every function is best suited to fit a data set. The data used in this study originated from 2214 sperm from 37 Norwegian bulls, which were genotyped for 11 markers. The optimal functions for the chromosomes 6, 23 and the sex chromosome of cattle were derived using the maximum likelihood method, the likelihood ratio test and empirical discriminant analysis. It became apparent that for each chromosome a different function fitted the data best. These were the function of Rao et al. (Human Heredity 1977, 27, 99-104) with p = 0.63 for chromosome 6, the function of Goldgar & Fain (American Journal of Human Genetics 1988, 43, 38-45) with C0 = 0.42, C1 = 0.47, C2 = 0.07 and C3 = 0.04 for chromosome 23 and the function of Felsenstein (Genetics 1979, 91, 769-75) with K = 0.23 for the sex chromosome. The well known functions of Haldane (Journal of Genetics 1919, 8, 299-309) and Kosambi (Annals of Eugenics 1944, 12, 172-5) were shown to be suboptimal in most cases. A function is said to be multilocus feasible if the evaluation of the probability of all possible recombination events does not lead to negative values. The optimal function for chromosome 23 turned out to be multilocus feasible, whereas the functions for chromosome 6 and the sex chromosome were not. The choice of the correct mapping function is shown to have a considerable impact in mapping studies, when double recombinations have to be taken into account. Since there is no unique best mapping function, it is argued that it might be useful to use a simple parametric mapping function (like the one of Felsenstein 1979) and to estimate the respective parameter specifically for a given data set.

An efficient hybrid direct-vague fuzzy moves system using fuzzy-rules-based precise rules

In this paper, a game transformation is proposed to enable a player to get a globally reasonable goal which is not only as advantageous to his own side as possible but also as disadvantageous to his opponent as possible. The Theory of Fuzzy Moves (TFM), based on the Theory of Moves (TOM) and game theory, is suitable for making a player reach a relatively good fuzzy outcome. Through analysis, we have found that only some of the fuzzy moves' rules are vague (i.e. fuzzy), and the others are direct (i.e. precise). We have found that (1) for the direct rules, the hybrid system is fault-tolerant and stable; (2) for the vague rules, the hybrid system is sensitive for the relevant players' fuzzy payoffs and globally strategic goals since there exist sensitive points or curves. Through the mathematical simplification of the vague rules' inference, we can finally construct the hybrid direct-vague fuzzy moves system by directly using the simple mathematical mapping functions without using the conventional fuzzification, fuzzy inference, and defuzzification. Importantly, the hybrid direct-vague fuzzy moves system with the simple mapping functions can run much more quickly than the fuzzy moves system with the complex fuzzy mapping functions. Finally, some typical examples of globally optimal fuzzy moves have shown that the hybrid direct-vague fuzzy moves system is stable and fault-tolerant enough to deal with the fuzzy moves in dynamic and uncertain situations.

Computation of the potential flow through cascades using conformal mapping and the singularity method.

A new procedure for a numerical analysis of incompressible potential flow through cascades of airfoils with arbitrary cross-sections has been presented. The procedure consists of three steps. First, airfoil rows in a physical plane are mapped into near-circle rows using a simple mapping function; then the flow field through the near-circles on the mapped plane is calculated with the singularity method. Finally, the flow on the mapped plane is transformed to that through the cascades of airfoils. The method for determination of a parameter of the mapping function to obtain near-circles with sufficiently smooth contours on the mapped plane, which is indispensable to achieving the highest accuracy in the calculation with the singularity method, has been proposed. The validity of this procedure is demonstrated by comparing the present results with exact solutions, which indicates that the method proposed is very useful for the numerical analysis of the cascades flow with simple thin airfoils as well as many practical airfoils with a camber and thickness.

A simple function for mapping induced currents

Summary. A linear inverse method that can be applied to the spatial structure of vertical fields has recently been presented. The application of the method to a regional geomagnetic dataset provided bounds on the lateral and vertical distribution of current concentrations, over a known anomaly. The results obtained by this rigorous analysis are compared with the results that can be obtained using a simple mapping function. The function is the anomalous vertical to anomalous horizontal field-ratio. The high degree of correspondence between the two sets of results confirms that the ratio is a useful, quantitative function for mapping geoelectric boundaries.

Development of projections between areas of the nervous system.

A theory of projections in the nervous system (such as the retino-tectal projection) is proposed. Components of axonal growth cones and target tissue interact and cooperate, within the area of contact, to generate a guiding parameter, in the simplest case a "guiding substance" of distribution p. The components which are involved in this production are assumed to have graded distributions with respect to position in the projecting and target area, respectively. The distribution p thus produced guides the growth cone in the direction of maximal slope until the minimal value of p is reached. In this way, each growth cone can be guided to a position on the target tissue which depends on the origin of the fiber in such a manner that a projection results. Adhesive forces could but need not be involved in the guiding mechanism. The slope of p may interfere with an intracellular pattern forming mechanism within the growth cone, determining the polarity of activation (as modelled previously on the basis of autocatalysis and lateral inhibition) and thus the direction of growth. For the generation of a distribution of p leading to a reliable projection, simple graded distributions in the projecting and target area suffice, involving one or two components in each dimension. Their effect on the generation of p may be activatory as well as inhibitory. Exponential gradients give rise to particularly simple mapping functions. The following is an example of this general type of model: Growth cones as well as target tissue contribute to the production of a guiding substance. For each dimension, there is, in the target tissue, an exponentially graded component exerting (directly or indirectly) two functions: it actively produces guiding substance p and it interacts, in an inhibitory fashion, with the production of p by a component of the growth cone (which is, in turn, graded with respect to position of origin in the projecting area). While the theory is proposed as a fair approximation of the primary events in neural projections, superimposed regulatory effects can also be incorporated. These include fiber-fiber interactions, mechanisms smoothing out unequal density distributions of axon terminals and effects of time of arrival of fibers on the projection, which have been proposed previously as primary mechanisms generating projections. A further extension of the model is to assume that crude and more refined positional specificity is determined in a combinatorial fashion, allowing the possibility of interchanges and transformations of parameters.

A mapping function for human chromosomes.

The available simple mapping functions are surveyed, and a new mapping function that provides for positive interference within chromosome arms and no interference across the centromere is proposed, together with the corresponding formula for centromeric linkage. This new function is derived by assuming that all chromosome arms except the short arms of acrocentric chromosomes hav an obligatory chiasma, and that the remaining chiasmata are distributed at random; assumptions which may correspond reasonably well to reality. A method for the comparison of the goodness of fit of mapping functions with family data is given. Low levels of interference seem to be indicated, although as yet insufficient human data is available to allow interference to the specified. Interference has a considerable effect on the estimation of map distances between loci from 3-point lod scores as is shown by the linkage group Rh, UMPK, PGM1, Amy, 1qh, Fy, on chromosome 1.

Transformation in E. coli K12: relation of linkage to distance between markers.

E. coli K12 transformants, selected as leu+ or pyrA+ transformants, were analysed for inheritance of some closely linked unselected markers. Based on the observation that the number of recombinants which require, besides an integration event, one or more crossing-over events was negligible, a simple mapping function was deduced. The function L= e-kd, which directly relates observed linkage of an unselected marker and the relative distance of that unselected marker to the selected marker, gave a consistent interpretation of the experimental results.