SOME NEW IDENTITIES FOR THE SECOND COVARIANT DERIVATIVE OF THE CURVATURE TENSOR

In this paper we study the second covariant derivative of Riemannian curvature tensor. Some new identities for the second covariant derivative are given. Namely, identities obtained by cyclic sum with respect to three indices are given. In the first case, two curvature tensor indices and one covariant derivative index participate in the cyclic sum, while in the second case one curvature tensor index and two covariant derivative indices participate in the cyclic sum.

Coordinate Systems and Vectors

This chapter introduces the various types of coordinate systems that exist in three dimensions and develops the basic concept of ‘metric’ to describe their properties. It introduces vectors in these coordinate systems and develops the notions of the ‘index language,’ dependence on the metric, and the covariance of vectors. Early familiarity with the metric tensor, index or component notation, symmetric and anti-symmetric manipulation is intended.

Training Decision Trees as Replacement for Convolution Layers

We present an alternative layer to convolution layers in convolutional neural networks (CNNs). Our approach reduces the complexity of convolutions by replacing it with binary decisions. Those binary decisions are used as indexes to conditional distributions where each weight represents a leaf in a decision tree. This means that only the indices to the weights need to be determined once, thus reducing the complexity of convolutions by the depth of the output tensor. Index computation is performed by simple binary decisions that require fewer cycles compared to conventionally used multiplications. In addition, we show how convolutions can be replaced by binary decisions. These binary decisions form indices in the conditional distributions and we show how they are used to replace 2D weight matrices as well as 3D weight tensors. These new layers can be trained like convolution layers in CNNs based on the backpropagation algorithm, for which we provide a formalization. Our results on multiple publicly available data sets show that our approach performs similar to conventional neuronal networks. Beyond the formalized reduction of complexity and the improved qualitative performance, we show the runtime improvement empirically compared to convolution layers.

DIFFERENTIAL FORMS AND WAVE EQUATIONS FOR GENERAL RELATIVITY

In recent papers, Choquet–Bruhat and York and Abrahams, Anderson, Choquet–Bruhat, and York (we refer to both works jointly as AACY) have cast the 3 + 1 evolution equations of general relativity in gauge-covariant and causal "first-order symmetric hyperbolic form," thereby cleanly separating physical from gauge degrees of freedom in the Cauchy problem for general relativity. A key ingredient in their construction is a certain wave equation which governs the light-speed propagation of the extrinsic curvature tensor. Along a similar line, we construct a related wave equation which, as the key equation in a system, describes vacuum general relativity. Whereas the approach of AACY is based on tensor-index methods, the present formulation is written solely in the language of differential forms. Our approach starts with Sparling's tetrad-dependent differential forms, and our wave equation governs the propagation of Sparling's two-form, which in the "time-gauge" is built linearly from the "extrinsic curvature one-form." The tensor-index version of our wave equation describes the propagation of (what is essentially) the Arnowitt–Deser–Misner gravitational momentum.

Bhabha’s equation for a particle of two mass states in Rarita—Schwinger form

The irreducible relativistic wave equation for a particle having two different mass states and positive charge, given by Bhabha, has been written in a form similar to that given by Rarita & Schwinger for the Dirac-Fierz-Pauli equation for a particle of spin 3/2 . The components of the wave function are written as Dirac four-component wave functions, having in addition a tensor index, and one ordinary Dirac four-component wave function. The only matrices which enter into the formulation are the Dirac matrices. An explicit representation of Bhabha’s matrices in terms of the Dirac matrices is obtained. The solutions for spin 3/2 are just those given by the Dirac-Fierz-Pauli equation, but the solutions for spin ½ differ from the Dirac solutions in having additional non-vanishing components.