Interactions, space presentations, blocks and cross products

Physics counts four basic forces, the electromagnetic EMI, weak WI, strong SI interactions and gravity GR. The first three are provided with a unified theory which partly needs revision and has the symmetry U(1)xSU(2)xSU(3). In this article their space presentations are described in order to inlcude a theory for gravity which cannot be added directly to the standrd model. There are many instances of gravitational actions which are different from the other three interactions. Gravity uses geometrical models beside spactime, often projective, including stereographic and spiralic orthogonal subspace projections. Real and complex cross products, symmetries which belong to the complex Moebius transformation subgroups, complex cross ratios, Gleason frame GF measures, dihedrals nth roots of unity with symmetris are some new tools (figure 14) for a new gravity model. The basic vector space is 8-dimensional, but beside the usual vector addition and calculus there are different multiplications added. The author uses complex multiplications in the complex 4-dimensional space C4 for calculus. The SU (3) multiplication of GellMann 3x3-matrices is used for C³ and its three 4-dimensional C² projections. Projective spaces are CP² for nucleons and a GR Higgs plane P² and projective measuring GF‘s which have 3-dimensional, orthogonal base vectors like spin. The doubling of quaternionic spacetime to octonians has a different multiplication and seven GF‘s which partly occur in physics as cross product equations. Beside the real, the complex cross product extends the spacetime dimensions from 4 to 8. Consequences are that there are many 3-dimensional, many 4-dimensional, some 6-dimensional and also projective 5-dimensional spaces in which the actions of gravity can then be described. Spacetime is for this not sufficient. No symmetry can be muliplied to the standard model since the new symmetries belong to different geometries and are not directly related to a set of field quantums like one photon for EMI, three weak bosons (or four) for WI, eight gluons for SI. GR has graviton waves similar to EMI waves and in quasiparticle form rgb-graviton whirls, for mass Higgs bosons, maybe also solitons (density as mass per volume changing). They attribute to a distance metric between two points (kept fixed) an amplitude density (operator} which changes the metrical diameter of the volume, but not the mass.

A CUDA fast multipole method with highly efficient M2L far field evaluation

Solving an N-body problem, electrostatic or gravitational, is a crucial task and the main computational bottleneck in many scientific applications. Its direct solution is an ubiquitous showcase example for the compute power of graphics processing units (GPUs). However, the naïve pairwise summation has [Formula: see text] computational complexity. The fast multipole method (FMM) can reduce runtime and complexity to [Formula: see text] for any specified precision. Here, we present a CUDA-accelerated, C++ FMM implementation for multi particle systems with [Formula: see text] potential that are found, e.g. in biomolecular simulations. The algorithm involves several operators to exchange information in an octree data structure. We focus on the Multipole-to-Local (M2L) operator, as its runtime is limiting for the overall performance. We propose, implement and benchmark three different M2L parallelization approaches. Approach (1) utilizes Unified Memory to minimize programming and porting efforts. It achieves decent speedups for only little implementation work. Approach (2) employs CUDA Dynamic Parallelism to significantly improve performance for high approximation accuracies. The presorted list-based approach (3) fits periodic boundary conditions particularly well. It exploits FMM operator symmetries to minimize both memory access and the number of complex multiplications. The result is a compute-bound implementation, i.e. performance is limited by arithmetic operations rather than by memory accesses. The complete CUDA parallelized FMM is incorporated within the GROMACS molecular dynamics package as an alternative Coulomb solver.

Pipelined vedic multiplier with manifold adder complexity levels

Recently, the increased use of portable devices, has driven the research world to design systems with low power-consumption and high throughput. Vedic multiplier provides least delay even in complex multiplications when compared to other conventional multipliers. In this paper, a 64-bit multiplier is created using the Urdhava Tiryakbhyam sutra in Vedic mathematics. The design of this 64-bit multiplier is implemented in five different ways with the pipelining concept applied at different stages of adder complexities. The different architectures show different delay and power consumption. It is noticed that as complexity of adders in the multipliers reduce, the systems show improved speed and least hardware utilization. The architecture designed using 2 x 2 – bit pipelined Vedic multiplier is, then, compared with existing Vedic multipliers and conventional multipliers and shows least delay.

Double Precision Floating Point Fft Processor using Vedic Mathematics

There should be rapid, efficient and simple process for every scenario now a day. To compute the N point DFT, Fast Fourier Transform (FFT) is a productive algorithm. It has great applications in communication, signal and image processing and instrumentation. In the implementation of FFT one of the challenges is the complex multiplications, so to make this process rapid and simple it’s necessary for a multiplier to be fast and power efficient. To tackle this problem Karatsuba sutra and Nikhilam sutra are an efficient method of multiplication in Vedic Mathematics. This paper will present a design methodology of Double Precision Floating Point Fast Fourier Transform (FFT) Processor.The execution time and complexity can be reduced by the algorithm which is there in Vedic.The main aim is to make FFT Processor process rapid and simple by designing a multiplier which is fast and power efficient by using double precision floating point and Vedic Mathematics concepts.

Low-Complexity Time-Domain Ranging Algorithm with FMCW Sensors

A time-domain ranging algorithm is proposed for a frequency-modulated continuous wave (FMCW) short-range radar sensor with high accuracy and low complexity. The proposed algorithm estimates the distance by calculating the ratio of the beat frequency signal to its derivative and thereby eliminates the restriction of frequency bandwidth on ranging accuracy. Meanwhile, we provide error analysis of the proposed algorithm under different distances, integral lengths, relative velocities, and signal-to-noise ratios (SNRs). Finally, we fabricate FMCW sensor prototype and construct a measurement system. Testing results demonstrate that the proposed time-domain algorithm could achieve range error within 0.8 m. Compared with the conventional fast Fourier transform (FFT) estimation scheme, the proposed method performs ranging without the requirement of complex multiplications, which makes it reasonable to be implemented in real-time and low-cost systems.

Beamforming with Kerdock Codes in LTE System for High-Speed Railway

Beamforming technique based the reciprocity of LTE communication system for high-speed railway is a practical way to improve the system performance in this paper. Kerdock codes, which facilitate efficient codebook and codeword search, are implemented in the LTE system for high-speed railway as the beamforming precoding codebook. The analysis of complexity and simulation results about the communication system of the beamforming with Kerdock codes (KBF) show that the proposed KBF can eliminate complex multiplications and improve the Bit Error Rate (BER) performance at large SNR compared with the existing beamfoming communication system for high-speed railway.

Working Memory, Strategy Execution, and Strategy Selection in Mental Arithmetic

A total of 72 participants estimated products of complex multiplications of two-digit operands (e.g., 63 × 78), using two strategies that differed in complexity. The simple strategy involved rounding both operands down to the closest decades (e.g., 60 × 70), whereas the complex strategy required rounding both operands up to the closest decades (e.g., 70 × 80). Participants accomplished this estimation task in two conditions: a no-load condition and a working-memory load condition in which executive components of working memory were taxed. The choice/no-choice method was used to obtain unbiased strategy execution and strategy selection data. Results showed that loading working-memory resources led participants to poorer strategy execution. Additionally, participants selected the simple strategy more often under working-memory load. We discuss the implications of the results to further our understanding of variations in strategy selection and execution, as well as our understanding of the impact of working-memory load on arithmetic performance and other cognitive domains.

Explicit 3D depth migration with a constrained operator

Numerical anisotropy is one of the main problems in the design of explicit 3D depth-extrapolation operators. This paper introduces a new method based on constraining the number of independent coefficients for the full 3D extrapolation operator. The extrapolation operator is divided into two regions. The coefficients for the inner part of the extrapolation operator are treated the same as the full 3D extrapolation operator. The coefficients for the outer part of the extrapolation operator are constrained to be constant as a function of azimuth for a given radius. This strategy reduces the number of floating-point operations because, for each extrapolation step, the number of complex multiplications are reduced and replaced by complex additions. The numerical workload of this alternative scheme is comparable to the Hale-McClellan scheme. Impulse responses are compared with finite-difference solutions for the two-way acoustic-wave equation. It is demonstrated that the numerical anisotropy for the proposed scheme is negligible and that the constrained-depth-extrapolation operator can be used in media with large lateral velocity contrasts. The design of constrained-depth-extrapolation operators with different maximum propagation angles in inline and crossline directions is explained and exemplified. These types of operators can be used to suppress the propagation of aliased energy in the crossline direction during depth extrapolation while reducing numerical cost.