Levi-Civita Symbol Research Articles

Low-energy effective field theory below the electroweak scale: one-loop renormalization in the ’t Hooft-Veltman scheme

The low-energy effective field theory below the electroweak scale (LEFT) describes the effects at low energies of both the weak interaction and physics beyond the Standard Model. We study the one-loop renormalization of the LEFT in the ’t Hooft-Veltman scheme, which offers an algebraically consistent definition of the Levi-Civita symbol and γ5 in dimensional regularization. However, in connection with minimal subtraction this scheme leads to a spurious breaking of chiral symmetry in intermediate steps of the calculation. Based on the ’t Hooft-Veltman prescription, we define a renormalization scheme that restores chiral symmetry by including appropriate finite counterterms. To this end, we extend the physical LEFT operator basis by a complete set of off-shell and one-loop-evanescent operators and we perform the renormalization at one loop. We determine the finite counterterms to the physical parameters that compensate both the insertions of evanescent operators, as well as the chiral-symmetry-breaking terms from the renormalizable part of the Lagrangian in D dimensions. Our results can be applied in next-to-leading-log calculations in the ’t Hooft-Veltman scheme: using our renormalization scheme instead of pure minimal subtraction separates the physical sector from the unphysical evanescent sector and leads to results that are manifestly free of spurious chiral-symmetry-breaking terms.

Calculus: High-Dimensional Numerical and Symbolic Calculus in R

The R package calculus implements C++ optimized functions for numerical and symbolic calculus, such as the Einstein summing convention, fast computation of the Levi-Civita symbol and generalized Kronecker delta, Taylor series expansion, multivariate Hermite polynomials, high-order derivatives, ordinary differential equations, differential operators and numerical integration in arbitrary orthogonal coordinate systems. The library applies numerical methods when working with R functions or symbolic programming when working with characters or expressions. The package handles multivariate numerical calculus in arbitrary dimensions and coordinates and implements the symbolic counterpart of the numerical methods whenever possible, without depending on external computer algebra systems. Except for Rcpp, the package has no strict dependencies in order to provide a stable self-contained toolbox that invites re-use.

Efficient Parallel Decoding Architecture for Cluster Erasure Correcting 2-D LDPC Codes for 2-D Data Storage

Native 2-D low-density parity-check (LDPC) codes provide 2-D burst erasure correction capability and have promising applications in two-dimensional magnetic recording (TDMR) technology. Though carefully constructed rastered 1-D LDPC codes can provide 2-D burst erasure correction, they are not as efficient as 2-D native codes constructed for handling the 2-D span of burst erasures. Our contributions are twofold. First, we propose a new 2-D LDPC code with girth greater than 4 by generating a parity-check tensor through stacking permutation tensors of size $p\times p\times p$ along with the $i,j,k$ -axes. The permutations are achieved through circular shifts on an identity tensor along different coordinate axes in such a way that it provides a burst erasure correction capability of at least $p\times p$ . Second, we propose a fast, efficient, and scalable hardware architecture for a parallel 2-D LDPC decoder based on the proposed code construction for data storage applications. Through efficient indexing of the received messages in an RAM, we propose novel routing mechanisms for messages between the check nodes and variable nodes through a set of two barrel shifters, producing shifts along two axes. Through simulations, we show that the performance of the proposed 2-D LDPC codes matches a 1-D QC-LDPC code, with a sharp waterfall drop of three to four orders of magnitude over ~0.3 dB, for random errors over code sizes of ~32 kbits or equivalently ~ $180\times 180 2$ -D arrays. Furthermore, we prove that the proposed native 2-D LDPC codes outperform their 1-D counterparts in terms of 2-D cluster erasure correction ability. For $p=16$ and code arrays of size $48\times 48$ , we implemented the proposed design architecture on a Kintex-7 KC-705 field-programmable gate array (FPGA) kit, achieving a significantly high worst case throughput of 12.52 Gb/s at a clock frequency of 163 MHz.

On the extreme points of slice-stochastic tensors

In this paper, we study the properties of extreme points of slice-stochastic tensors, and present an analogue of the Birkhoff–von Neumann theorem for slice-stochastic tensors. Some equivalent characterizations for a slice-stochastic tensor to be an extreme point are presented. The number of permutation tensors is also discussed.

Fourth-order tensor Riccati equations with the Einstein product

ABSTRACT In this paper, the fourth-order tensor Riccati equation with the Einstein product is investigated. Using the Einstein product, we introduce the -tensor, the spectrum and the block tensor multiplication. We also present an expression of the fourth-order permutation tensors and generalize the Riccati equation from the matrix to the tensor case. The existence and uniqueness of the solution for the fourth-order -tensor Riccati equation are studied. Furthermore, we provide a perturbation analysis of the mixed and componentwise condition numbers for the -tensor Riccati equation. An algorithm for solving the equation is proposed. Numerical examples are presented to show the effectiveness of our algorithm.

ПСЕВДОТЕНЗОРНАЯ ФОРМУЛИРОВКА МЕХАНИКИ ГЕМИТРОПНЫХ МИКРОПОЛЯРНЫХ СРЕД

The possibility of applications of relative tensors concepts to the mechanics of micropolar continuum and, in particular, for the hemitropic micropolar continua is considered. The fundamental tensors and orienting relative scalars in three-dimensional space are introduced. Permutation symbols and absolute Levi-Civita tensors are investigated in further details. Algebraic and differential properties of the relative tensors are discussed. The weights of the fundamental kinematic tensors are determined. The wryness tensor and the asymmetric strain tensor are constructed in terms of the vectors of micro-rotation and displacements. Notions of force and couple traction vectors, associated force and associated couple stress vector, force and couple stresses tensors are discussed in the frameworks of relative tensors algebra. The weights of the basic micropolar elasticity tensors are determined and discussed. The constitutive form of the micropolar elastic potential is introduced as an absolute scalar in order to obtain micropolar constitutive equations. In the linear case, the elastic potential is a quadratic form whose coefficients are pseudoscalars. The weights of the constitutive pseudoscalars are calculated. The dimensionless constitutive micropolar constants and constitutive constants with physical dimensions are discriminated. Statics and dynamics of micropolar elastic continua are developed in terms of relative tensors. Dynamic equations involving displacements and microrotations in the case of semi-isotropic (hemitropic) symmetry are derived and represented by the pseudotensor technique. The paper can be considered as a script of fundamental formulas and concepts related to the algebra and differentiation of relative tensors of arbitrary rank.

Birkhoff-von Neumann theorem and decomposition for doubly stochastic tensors

The well-known Birkhoff-von Neumann theorem states that a doubly stochastic matrix is a convex combination of permutation matrices. In this paper, we present a new concept for doubly stochastic tensors and study a generalization of this theorem for doubly stochastic tensors. Particularly, we prove that each permutation tensor is an extreme point of the set of doubly stochastic tensors, and the Birkhoff-von Neumann theorem holds for doubly stochastic tensors. Furthermore, an algorithm is proposed to find a convex combination of permutation tensors for any doubly stochastic tensor.

Mathematical Approach and Implementation of Frequency Mapping Techniques in Power-Line Communications Channel

Abstract Power-line channel is considered to be a very hostile channel compared to other channels in view of the different types of noise that could exist. Therefore, the choice of the error correcting code and the modulation scheme can play a big role in combating the noise in such a channel. M -FSK modulation has shown its robustness for such a type of channel. Two frequency mappings techniques are presented in this paper. In the first technique, M orthogonal frequencies are arranged in sequences based on the value and the position of permutation symbols, while in the second technique, the frequencies are rearranged based on the sign changes of the Walsh-Hadamard transform (WHT). The obtained M-FSK modulation is combined to codes based on Viterbi decoding algorithms since Viterbi decoder is considered to be the maximum-likelihood decoding algorithm for convolutional codes and codes with state machine representation. A mathematical approach and implementation of frequency mappings is introduced to investigate the performance of the new designed communication system in the presence of permanent frequency disturbances, also known as narrow-band interference (NBI), such as those encountered in power line communications (PLC) channel.

Premetric teleparallel theory of gravity and its local and linear constitutive law

We continue to investigate the premetric teleparallel theory of gravity (TG) with the coframe (tetrad) as gravitational potential. We start from the field equations and a local and linear constitutive law. We create a Tonti diagram of TG in order to disclose the structure of TG. Subsequently we irreducibly decompose the 6th order constitutive tensor under the linear group. Moreover, we construct the most general constitutive tensors from the metric and the totally antisymmetric Levi-Civita symbol, and we demonstrate that they encompass nontrivial axion and skewon type pieces. Using these tools, we derive for TG in the geometric-optics approximation propagating massless spin 0, 1, and 2 waves, including the special case of Einstein’s general relativity.

Spontaneous parametric downconversion of light by a dielectric nanoparticle

In the present paper, we theoretically consider the generation of entangled photon pairs in the process of spontaneous parametric downconversion of light by a nonlinear dielectric nanoparticle. This nanoparticle is the only source of nonlinearity in the considered model. We assume that the particle is small compared to the wavelength of light so that the process is non-resonant. An intuitive theoretical description is developed for the case when the nonlinear susceptibility tensor of the material of a nanoparticle has a form of the Levi-Civita symbol. In this case, a two-photon amplitude which characterizes a spatial distribution of entangled photon pairs can be interpreted as a result of interference between two induced dipoles, both being orthogonal to the direction of polarization of the pump photon. A generation of entangled surface plasmon-polaritons by a dielectric nanoparticle located near the metal-dielectric interface is also considered.

Polytopes of stochastic tensors

Considering $n\times n\times n$ stochastic tensors $(a_{ijk})$ (i.e., nonnegative hypermatrices in which every sum over one index $i$, $j$, or $k$, is 1), we study the polytope ($\Omega_{n}$) of all these tensors, the convex set ($L_n$) of all tensors in $\Omega_{n}$ with some positive diagonals, and the polytope ($\Delta_n$) generated by the permutation tensors. We show that $L_n$ is almost the same as $\Omega_{n}$ except for some boundary points. We also present an upper bound for the number of vertices of $\Omega_{n}$.

Method of message authentication code formation based on permutations

The task of controlling the integrity of transmitted messageswhich is provided by the use of message authenticationis relevant for constructing of protected telecommunicationssystems. Given the growth of computing meansproductivity, and improving the methods of hacking ofinformation protection systems including protectionagainst false data imposing requirements for messageauthentication methods and means increase. In this paperthe structural diagram of device of random permutationssequence formation is developed and shown. On thebasis of this device construction principles the methodand device of message authentication code formation isproposed. The essence of the method lies in the fact thatas a message authentication code is used a part of large dimension permutation symbols in some chosen order.This permutation is formed from the sequence of charactersof the message converted into a sequence of related numbers in factorial number system. To hide the law offormation of message authentication code a replaceabletransformation key is used. The resistance of permutationand message authentication code generated from it whentrying to break a key by "brute force" is determined.

Birkhoff--von Neumann Theorem for Multistochastic Tensors

In this paper, we study the Birkhoff--von Neumann theorem for a class of multistochastic tensors. In particular, we give a necessary and sufficient condition such that a multistochastic tensor is a convex combination of finitely many permutation tensors. It is well-known that extreme points in the set of doubly stochastic matrices are just permutation matrices. However, we find that extreme points in the set of multistochastic tensors are not just permutation tensors. We provide the other types of tensors contained in the set of extreme points.

VEST: Abstract vector calculus simplification in Mathematica

We present a new package, VEST (Vector Einstein Summation Tools), that performs abstract vector calculus computations in Mathematica. Through the use of index notation, VEST is able to reduce three-dimensional scalar and vector expressions of a very general type to a well defined standard form. In addition, utilizing properties of the Levi-Civita symbol, the program can derive types of multi-term vector identities that are not recognized by reduction, subsequently applying these to simplify large expressions. In a companion paper Burby et al. (2013) [12], we employ VEST in the automation of the calculation of high-order Lagrangians for the single particle guiding center system in plasma physics, a computation which illustrates its ability to handle very large expressions. VEST has been designed to be simple and intuitive to use, both for basic checking of work and more involved computations. Program summaryProgram title: VEST (Vector Einstein Summation Tools)Catalogue identifier: AEQN_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEQN_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 10469No. of bytes in distributed program, including test data, etc.: 72539Distribution format: tar.gzProgramming language: Mathematica.Computer: Any computer running Mathematica.Operating system: Linux, Unix, Windows, Mac OS X.RAM: Usually under 10 MbytesClassification: 5, 12, 19.Nature of problem:Large scale vector calculus computationsSolution method:Reduce expressions to standard form in index notation, automatic derivation of multi-term vector identities.Restrictions:Current version cannot derive vector identities without cross products or curlAdditional comments:Intuitive user input and output in a combination of vector and index notationRunning time:Reduction to standard form is usually less than one second. Simplification of very large expressions can take much longer.

Order Relation on the Permutation Symbols in the Ehresmann Subvariety Class Associated to the Distinguished Monomials of Flag Manifolds

In this paper, we use the theory of lexicographical and graded lexicographical orders to compare two distinguished monomials through their codes of invariants and study the effect of this comparison on their respective defining permutation symbols in the Ehresmann subvariety classes.

On the distinguished monomials associated to the permutation symbols in the Ehresmann cycle class [(a_0,..., a_n),F] of the flag manifold

On the distinguished monomials associated to the permutation symbols in the Ehresmann cycle class [(a_0,..., a_n),F] of the flag manifold

A Comparative Study on the Distance-Optimality for Distance-Preserving Mappings Constructions with Cyclic-Shift Prefix Technique

This paper presents a comparative study on the distance optimality for Distance-Preserving Mappings codes generated from different constructions. Few constructions are considered to investigate the reasons behind making codes optimum when they reach the upper bound on the sum of the Hamming distances for certain lengths of the permutation sequences. The technique of cyclic-shift prefix or suffix of permutation symbols is used to investigate the conditions making distance-preserving mappings codes optimum.

The divergence theorem of a triangular integral

This divergence theorem of a triangular integral demands the antisymmetric symbol to derive the inner product of the nabla and a vector. Replacing a cuboid (a rectangular solid) by a tetrahedron (a triangular pyramid) as the finite volume element, a single limit is only demanded for triple sums in our theory of a triple integral. The divergence theorem of a triangular integral is derived by substituting the total differentials into our new method of a triangular integral. We thus infer that our new method of a triangular integral must be the inverse operation of the total differential.

Generation of spectral nulls with permutation sequences

Abstract This paper describes the construction of classes of spectral null codes, which are obtained by shaping permutation sequences. The properties and the construction techniques of binary spectral null codes are used. The power spectral densities (PSD) of the proposed codes show nulls at the zero frequency and at submultiples of the frequencies of the transmitted symbols. The channel symbols or voltage levels mapped to each transmitted permutation symbol are chosen to satisfy the spectral null equation. Interesting results show the elimination of the DC component at the lower frequencies. The well-shaped power spectral densities of these codes may overcome some communications problem like zero frequency components. The proposed technique is generalized and the corresponding spectral null permutation sequences that generate spectral null codes are presented.

The First Acquaintance with the Tensor

The main concepts of the theory of tensors are presented. The emphasis is on the basic notions of tensor algebra and practical skills in culculations involving the Kronecker delta and Levi-Civita symbol. Sixty routine exercises are included. This article is intended for junior students of physical, mathematical and geophysical departments of Universities