Connections of Higher Order and Product Preserving Functors

In this paper we consider a product preserving functor F of order r and a connection of order r on a manifold M. We introduce horizontal lifts of tensor fields and linear connections from M to F(M) with respect to . Our definitions and results generalize the particular cases of the tangent bundle and the tangent bundle of higher order.     

Preservation of Linear Constraints in Approximation of Tensors

For an arbitrary tensor (multi-index array) with linear constraints at each direction, it is proved that the factors of any minimal canonical tensor approximation to this tensor satisfy the same linear constraints for the corresponding directions.     

Numerical multilinear algebra and its applications

Numerical multilinear algebra (or called tensor computation), in which instead of matrices and vectors the higher-order tensors are considered in numerical viewpoint, is a new branch of computational mathematics. Although it is an extension of numerical linear algebra, it has many essential differences from numerical linear algebra and more difficulties than it. In this paper, we present a survey on the state of the art knowledge on this topic, which is incomplete, and indicate some new trends for further research. Our survey also contains a detailed bibliography as its important part. We hope that this new area will be receiving more attention of more scholars.      

Preconditioners for higher order edge finite element discretizations of Maxwell's equations

In this paper,we are concerned with the fast solvers for higher order edge finite element discretizations of Maxwell's equations.We present the preconditioners for the first family and second family of higher order N′ed′elec element equations,respectively.By combining the stable decompositions of two kinds of edge finite element spaces with the abstract theory of auxiliary space preconditioning,we prove that the corresponding condition numbers of our preconditioners are uniformly bounded on quasi-uniform grids.We also present some numerical experiments to demonstrate the theoretical results.     

Four-loop verification of an algorithm for summing Feynman diagrams in the N=1 supersymmetric electrodynamics

By calculating some four-loop diagrams in the N=1 supersymmetric electrodynamics regularized by higher derivatives, we verify a method for summing Feynman diagrams based on using Schwinger-Dyson equations and Ward identities. In particular, for the diagrams considered, we prove the correctness of an additional identity for Green’s functions not reduced to the gauge Ward identity. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 2, pp. 290–302, May, 2006.     

Anomaly Problem in the N=1 Supersymmetric Electrodynamics as a Consequence of the Inconsistency of the Dimensional Reduction Method

We compare the calculation of the two-loop -function in the N=1 supersymmetric electrodynamics regularized via higher derivatives and via dimensional reduction. We show that the renormalized effective action is the same for both regularizations. But in the method of higher derivatives, unlike in the dimensional reduction, the -function defined as the derivative of the renormalized coupling constant with respect to log turns out to be purely one-loop. The anomaly problem therefore does not occur in this regularization, because in the method of higher derivatives, the diagrams with counterterm insertions make a nonzero contribution, which is evaluated exactly in all orders of the perturbation theory. When dimensional reduction is used, this contribution is zero. We argue that this result is a consequence of the mathematical inconsistency of the dimensional reduction method and that just this inconsistency leads to the anomaly problem.     

Three-Loop β-Function of N=1 Supersymmetric Electrodynamics Regularized by Higher Derivatives

We calculate three-loop corrections to the effective action for N=1 supersymmetric electrodynamics regularized by higher derivatives. Using the obtained results, we investigate the anomaly problem in the considered model.     

Investigating the anomaly puzzle in <Emphasis Type="Italic">N</Emphasis>=1 supersymmetric electrodynamics

Using Schwinger-Dyson equations and Ward identities in the N=1 supersymmetric electrodynamics regularized by higher derivatives, we find that we can calculate some contributions to the two-point Greens function of the gauge field and to the -function exactly in all orders of the perturbation theory. We use the results to investigate the anomaly puzzle in the considered theory.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 142, No. 1, pp. 35–56, January, 2005.     

Verifying a new identity for Green’s functions of the n=1 supersymmetric non-Abelian Yang-Mills theory with matter fields

We use the higher covariant derivative regularization to investigate a new identity for Green’s functions. It relates certain coefficients of the matter superfield vertex function for which one of the external matter legs is not chiral. Calculations in the first nontrivial order (for the two-loop vertex function) demonstrate that the new identity also holds in the non-Abelian Yang-Mills theory with matter fields. We demonstrate that the new identity follows because the three-loop integrals determining the Gell-Mann-Low function are integrals of total derivatives. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 2, pp. 270–281, August, 2008.