Reciprocal Square Research Articles

二项式系数级数连带奇数倒数平方和

二项式系数级数连带奇数倒数平方和

Influence of grain boundaries with dispersed nanoscale Al2O3 particles on the strength of Al for a wide range of homologous temperatures

The deformation and strengthening behavior of an ultra-fine grained (UFG) Al fabricated via powder metallurgy was investigated over a wide range of homologous temperatures (TH). Our results reveal that the presence of a high density of γ-Al2O3 nanoparticles, located primarily at high angle grain boundaries (HAGBs), promoted remarkable stabilization of the Al grain structure up to TH = 0.94. The 0.2% strain offset yield stress (YS0.2) of the materials annealed at 600 °C for 24 h with grain sizes (d3D) of 0.57, 1.67, 2.33 and 2.99 μm was systematically studied from room temperature (RT) to 600 °C. Our study reveals that with decreasing d3D the strain hardening ability of the materials gradually decreased, and this behavior became pronounced at elevated T, which was attributed to the onset of plastic instability. The YS0.2 of the materials at all T followed the relation YS0.2 = a + k d3D−0.5. As determined on the basis of data interpolation, a positive transition from an established Hall-Petch (HP) relation at RT occurred at d3D = ∼8 μm. As the parameter a was negative at RT and 300 °C the documented strength–structure relationship cannot be explained on the basis of a Hall-Petch relation. The γ-Al2O3 particles located at HAGBs did not contribute notably to RT strengthening, whereas GBs played a significant role in the overall strength. The materials showed a high YS0.2 up to a TH of 0.94. The YS0.2 was observed to be linearly correlated with the reciprocal square root of the d3D and the coefficient k followed k = 224.4–0.376 T relation. The HP model overestimated the YS0.2 values at elevated T namely for the materials with a smaller d3D. It is proposed that softening occurred as a result of the γ-Al2O3 particles densely distributed within a 3D network, which were included in Hansen's strengthening model; particle strengthening played a role at elevated T.

On-chip implementation of a low-latency bit-accurate reciprocal square root unit

Many applications such as gaming, digital signal processing and communications systems, require computation of the reciprocal square root operation (RSR). Although several architectures have been reported for computing the RSR operation, these are mainly focused on accelerating high-precision floating-point units. In mobile-device implementations, fixed-point (FxP) units are preferred due to their low computational cost and power consumption. This article presents an on-chip implementation of a bit-accurate, FxP-RSR unit using a 130 nm CMOS process. The proposed architecture is based on a piecewise-polynomial approximation in a reduced range of the RSR function and the Newton-Raphson method. Experimental results show that the manufactured chip exhibits lower latency and less power consumption than existing standard-cell-based implementations. These characteristics make the proposed chip a useful silicon intellectual property suitable for embedded applications where low power, low latency, and low hardware cost is required.

Crystal Engineering for Mechanical Strength at Nano-Scale Dimensions

The mechanical strengths of nano-scale individual crystal or nanopolycrystalline metals, and other dimensionally-related materials are increased by an order of magnitude or more as compared to those values measured at conventional crystal or polycrystal grain dimensions. An explanation for the result is attributed to the constraint provided at the surface of the crystals or, more importantly, at interfacial boundaries within or between crystals. The effect is most often described in terms either of two size dependencies: an inverse dependence on crystal size because of single dislocation behavior or, within a polycrystalline material, in terms of a reciprocal square root of grain size dependence, designated as a Hall-Petch relationship for the researchers first pointing to the effect for steel and who provided an enduring dislocation pile-up interpretation for the relationship. The current report provides an updated description of such strength properties for iron and steel materials, and describes applications of the relationship to a wider range of materials, including non-ferrous metals, nano-twinned, polyphase, and composite materials. At limiting small nm grain sizes, there is a generally minor strength reversal that is accompanied by an additional order-of-magnitude elevation of an increased strength dependence on deformation rate, thus giving an important emphasis to the strain rate sensitivity property of materials at nano-scale dimensions.

Optimized Linear, Quadratic and Cubic Interpolators for Elementary Function Hardware Implementations

This paper presents a method for designing linear, quadratic and cubic interpolators that compute elementary functions using truncated multipliers, squarers and cubers. Initial coefficient values are obtained using a Chebyshev series approximation. A direct search algorithm is then used to optimize the quantized coefficient values to meet a user-specified error constraint. The algorithm minimizes coefficient lengths to reduce lookup table requirements, maximizes the number of truncated columns to reduce the area, delay and power of the arithmetic units, and minimizes the maximum absolute error of the interpolator output. The method can be used to design interpolators to approximate any function to a user-specified accuracy, up to and beyond 53-bits of precision (e.g., IEEE double precision significand). Linear, quadratic and cubic interpolator designs that approximate reciprocal, square root, reciprocal square root and sine are presented and analyzed. Area, delay and power estimates are given for 16, 24 and 32-bit interpolators that compute the reciprocal function, targeting a 65 nm CMOS technology from IBM. Results indicate the proposed method uses smaller arithmetic units and has reduced lookup table sizes compared to previously proposed methods. The method can be used to optimize coefficients in other systems while accounting for coefficient quantization as well as truncation and rounding effects of multiple arithmetic units.

Hydrogen Peroxide Reduction in Aqueous Electrolytes: Influence of a Heterogeneous Decomposition Step

It has been traditionally assumed that the reduction of hydrogen peroxide proceeds by a simple, first-order, 2-electron reduction, to yield water as the product. Under this model, a plot of the reciprocal of the current vs the reciprocal square root of the rotation rate, known as a Koutecky-Levich plot, should be linear, from which kinetic information can be obtained based on the magnitude of the intercepts as a function of the overpotential. However, close inspection of many such plots reported in the literature do exhibit some curvature especially at high rotation rates and low overpotential, a behavior that has been largely ignored but which may affect the interpretation of such plots1. One reaction seldom considered in this context is the well-documented heterogeneous disproportionation of hydrogen peroxide, a process known to occur on many surfaces that yields dioxygen2, a species that could subsequently reduce and thus contribute to the overall current observed (see Scheme 1). According to this Scheme, the current derived from the mass balances associated with peroxide and dioxygen is expressed in Equation 1. Shown in Panel A, Figure 1, are a series of Koutecky-Levich plots based upon this formalism for three reasonable values of the heterogeneous electron transfer rates for the processes involved for disproportionation rates spanning the entire domain. As clearly indicated, and for certain values of these parameters, the plots display the same type of curvature as those reported in the literature. Shown in Panel B of the Figure are curves for a fictitious set of parameters which were deliberately chosen to obtain a curvature equal in magnitude but opposite in direction. Several methods are currently being considered in order to measure independently the rates of peroxide disproportionation for the same type of rotating disk electrodes used for collecting data of the type discussed herein3. Acknowledgments This work was supported by a grant from NSF, CHE-1412060

A study of densification and phase transformations of nanocomposite Cu-Fe prepared by mechanical alloying and consolidation process

The purpose of this study is an attempt to obtain nanocrystalline bulk Cu-40 at.% Fe alloy by mechanical alloying followed by consolidation process under 7 GPa and annealing. The transformations occurring in the material were studied by the use of X-ray diffraction. The structural investigations revealed that consolidation enabled supersaturated Cu(Fe) to precipitate out Cu4Fe and CuFe4 intermetallics. A limited grain growth was noticed after annealing. The results of microhardness have shown that the hardening increases similarly up to 40 h milling. The variation of microhardness as a function of the reciprocal square root of the grain size demonstrates that the rate of hardening is similar to that predicted by the Hall-Petch behavior.

Hall-Petch Relation in a Spheroidized Carbon Steel with Emphasis on the Effect of Intergranular Particles

The Hall-Petch relation in a spheroidized steel with bimodal cementite particle size distribution has been investigated in this study, with an emphasis on considering the effect of the large particles at ferrite grain boundaries and triple junctions. A medium carbon steel was processed by variable thermomechanical procedures to achieve spheroidized structures with different combinations of microstructrual parameters, but all exhibiting a bimodal particle size distribution, in which large intergranular particles and small intragranular particles coexisted in the ferrite matrix. A quantitative relationship between the Hall-Petch parameter ky and the volume fraction of the intergranular cementite particles is presented, by considering a composite model. The contribution of the large intergranular particles to grain boundary strengthening wa substantiated by the increment of the ky parameter, since the average orientation factor of the composite, is increased. After correction of the ky parameters based on the constants from literatures, the predicted stresses show good agreement with the experimental stresses. A linear fit between the experimental stresses and the reciprocal square root of grain sizes is performed, the slope constant ky derived agrees to within 11 % of the corrected ky parameters based on the constants from literatures.

Capturing lithium-ion battery dynamics with support vector machine-based battery model

During long and high current pulses, diffusion resistance becomes important in lithium-ion batteries. In such diffusion-intense situations, a static support vector machine-based battery model relying on instantaneous current, state-of-charge (SOC), and temperature is not sufficient to capture the time-dependent voltage characteristics. In order to account for the diffusion-related voltage dynamics, we suggest therefore the inclusion of current history in the data-driven battery model by moving averages of the recent current. The voltage estimation performance of six different dynamic battery models with additional current history input is studied during relevant test scenarios. All current history models improve the time-dependent voltage drop estimation compared to the static model, manifesting the beneficial effect of the additional current history input during diffusion-intense situations. The best diffusion resistance estimation results are obtained for the two-step voltage estimation models that incorporate a reciprocal square root of time weighing function for the current of the previous 100 s or an exponential time function with a 20 s time constant (1–8% relative error). Those current history models even improve the overall voltage estimation performance during the studied test scenarios (under 0.25% root-mean-square percentage error).

Polarization of core orbitals and computation of nuclear quadrupole coupling constants using Gaussian basis sets

Most standard Gaussian basis sets for first row atoms, even large sets designed to converge on a ‘complete basis set’ limit, systematically overestimate the electric field gradient at nuclear sites for first row atoms, resulting in errors of up to 15% in the computation of nuclear quadrupole coupling constants. This error results from a failure to include tight d functions, which permit the core 1s orbitals to distort under the influence of the field of the nuclear quadrupole. Augmentation of standard basis sets with a single set of single-exponent d functions, matched to the reciprocal square of the nominal 1s radius, reduces these errors by up to 90%.

Material grain size and crack size influences on cleavage fracturing.

A review is given of the analogous dependence on reciprocal square root of grain size or crack size of fracture strength measurements reported for steel and other potentially brittle materials. The two dependencies have much in common. For onset of cleavage in steel, attention is focused on relationship of the essentially athermal fracture stress compared with a quite different viscoplastic yield stress behaviour. Both grain-size-dependent stresses are accounted for in terms of dislocation pile-up mechanics. Lowering of the cleavage stress occurs in steel because of carbide cracking. For crack size dependence, there is complication of localized crack tip plasticity in fracture mechanics measurements. Crack-size-dependent conventional and indentation fracture mechanics measurements are described also for results obtained on the diverse materials: polymethylmethacrylate, silicon crystals, alumina polycrystals and WC-Co (cermet) composites.

Fast bit-accurate reciprocal square root

The reciprocal square root (RSR) is an operation extensively used in signal processing algorithms, where it is necessary the design of RSR architectures in fixed-point (FxP) representation for using in mobile devices. Currently, RSR implementations are mainly focused on floating point format, which requires long execution time and large area resources. In this paper, an algorithm for designing FxP RSR architectures is proposed, which achieves bit-accurate results in two clock cycles of execution time. The proposed algorithm is based on the Newton–Raphson method, where the seed is provided through piecewise polynomial approximation. A comparison between this RSR proposal and the straightforward approach shows that the proposed algorithm achieves an approximately 10-fold gain in execution time, which allows a speed-up of signal processing algorithms.

التحليل الوراثي لصفة الغلة و مكوناتها و بعض الصفات المورفوفيزيولوجية في هجن فردية و عكسية من الذرة الصفراء = Genetic Analysis for Yield and Its Components and Some Morpho-Physiological Traits in Single and Reciprocal Crosses of Maize Zea Mays L.

Complete diallel mating design experiment with reciprocal crosses was used to determine combining abilitiy and maternal effects of six maize inbred lines and their hybrid combinations with R.C.B.D. in three replications for grain yield, silking date, plant and ear height, shelling percentage, physiological maturity, ear length, ear diameter, during 2009-2010 season, at the Researches Center of Hama (G.C.S.A.R.). Crosses, general (GCA) and specific (SCA) combining ability mean squares were significant for all traits, while reciprocal crosses mean squares were non-significant for all traits except silking date and physiological maturity. Also, the maternal mean squares were non-significant for all traits except physiological maturity. The ratios σ2GCA/σ2SCA were detected for all traits and showed that non-additive gene action was more important than additive gene action in controlling of most traits except ear height and ear length. GCA effects showed that the lines P5 (IL.359-06) and P2 (IL.358-06) were good general combiners for grain yield. Also, SCA effects showed that five hybrids were the best F1 crosses combinations such as (IL.259-06 × IL.358-06) for grain yield. The results of means compared showed that the cross (IL.358-06 × IL.359-06) (13.478 Ton/h) and it's reciprocal cross (IL.359-06 × IL.358-06) (13.223 Ton/h) were the best hybrids compared with the two check varieties (Basel-1 and Ghota-82).

Hall–Petch k dependencies in nanopolycrystals

Measurements at nanopolycrystal grain sizes of the Hall–Petch plastic flow stress dependence on the reciprocal square root of polycrystal grain diameter occasionally fall short of expectations. The present report reviews experimental and theoretical aspects of such behaviours for a number of metal systems. Comparison with the results on copper had led to prediction on a dislocation pile-up mechanics basis of an even stronger grain size dependence that is now shown to carry over to explanation of similar results reported recently for cold-drawn eutectoid steel wires. In the current review, emphasis is given on additional important results achieved on the production of nanoscale zinc and tungsten carbide materials by Professor Jay Narayan and colleagues at North Carolina State University.

Zeta(2) Once Again

This note provides a strikingly efficient evaluation of zeta(2).

Back-transformation of treatment differences—an approximate method

Transformation of outcomes is frequently used in the analysis of studies in clinical nutrition. However, back-transformation of estimated treatment means and differences is complicated by the nonlinear nature of the transformations. It is not straightforward to obtain an estimated treatment difference that can be interpreted without any reference to the additional predictors included in the statistical model; and moreover, standard errors are not easily available. The aim of this work was to provide a generally applicable, yet operational procedure for obtaining back-transformed estimated differences, and corresponding standard errors and 95% confidence intervals. Based on data from two randomized controlled studies and an exemplary data set that had all previously been published, we evaluated our approximate procedure by comparing results for different approaches for showing back-transformed estimated treatment differences. Estimated differences obtained on logarithm, square root and reciprocal square root-transformed scales were back-transformed into estimated differences on the original scales, and these estimates were in good agreement with the results reported by the original studies. The proposed approximate procedure provides a flexible approach for obtaining quite accurate back-transformed estimated differences in terms of medians and for deriving the corresponding standard errors.

Assessing NextGen Operational Improvements: A Case of Airspace Complexity and Aircraft Arrival Management System

The Next Generation Air Transportation System (NextGen) supports multiple operational and technological improvements that enable efficiencies for users and service providers. One of the potential components of the NextGen - the Aircraft Arrival Management System (AAMS) is a tool that jointly optimizes timing of arriving traffic to streamline flight arrivals at congested airports. This study utilizes experimental settings to examine the AAMS effect on the terminal area airspace complexity. During the experiment, the air traffic trajectory data before and after the AAMS implementation were collected at two terminal areas: Charlotte Douglas International Airport (CLT) and Minneapolis-St. Paul International Airport (MSP). Then, the trajectory data were used to estimate the Reciprocal Square Metrics (RSM) airspace complexity measures that were compared between the baseline and active AAMS periods. The results of the study suggest that potential implementation of AAMS concepts could reduce terminal area complexity at congested airports.

Power Allocation for Robust Distributed Best-Linear-Unbiased Estimation Against Sensing Noise Variance Uncertainty

Motivated by the fact that system parameter mismatch occurs in real-world sensing environments, this paper proposes power allocation schemes for robust distributed bestlinear-unbiased estimation (BLUE) that take account of the uncertainty in the local sensing noise levels. Assuming that (i) the sensing noise variance follows a statistical distribution widely used in the literature and (ii) the link channel gains between sensor nodes and the fusion center (FC) are i.i.d. Rayleigh fading, we propose to use the average reciprocal mean square error (ARMSE), averaged with respect to the distributions of sensing noise variance and fading channels, as the distortion measure. A fundamental inequality characterizing the relation between ARMSE and the average mean square error (AMSE) is established to justify the proposed design metric. While the exact formula for ARMSE is difficult to find, we derive an associated closed-form lower bound which involves the incomplete gamma function. To further ease analysis, we further derive a key inequality that specifies the range of the ARMSE lower bound. Particularly, it is shown that the boundary points of this inequality are characterized by a common function, which involves the Gaussian-tail Q(·) and is thus more analytically appealing. By conducting optimization on the basis of such a function, we obtain closed-form robust solutions for two power allocation problems: (i) optimizing distortion metric under a total power constraint, and (ii) minimizing total power under a target distortion requirement. In case that instantaneous channel state information (CSI) is available to the FC, the proposed approach can be easily modified to derive analytic robust power allocation factors best matched to the CSI realizations. Computer simulations evidence the effectiveness of the proposed schemes.

Hall-Petch analysis of dislocation pileups in thin material layers and in nanopolycrystals

Abstract

Effect of Surface and Interface Recombination on Carrier Lifetime in 6H-SiC Layers

Carrier lifetimes in 6H-SiC epilayers were investigated by using numerical simulations and micro-wave photoconductivity decay measurements. The measured carrier lifetimes were significantly increasing with an increased thickness up to 200 μm while it stays almost constant in layers thicker than 200 μm. From a comparison of the simulation and experimental results, we found that if the bulk lifetime in 6H-SiC is around a few microseconds, both the surface recombination and interface recombination influence the carrier lifetime in layers with thickness less than 200 μm while only the surface recombination determines the carrier lifetime in layers with thickness more than 200 μm. In samples with varying thicknesses, a bulk lifetime = 2.93 μs and carrier diffusion coefficient D= 2.87 cm2/s were derived from the linear fitting of reciprocal lifetime vs reciprocal square thickness.