Counting zero kernel pairs over a finite field

Helmke et al. have recently given a formula for the number of reachable pairs of matrices over a finite field. We give a new and elementary proof of the same formula by solving the equivalent problem of determining the number of so called zero kernel pairs over a finite field. We show that the problem is, equivalent to certain other enumeration problems and outline a connection with some recent results of Guo and Yang on the natural density of rectangular unimodular matrices over F-qx]. We also propose a new conjecture on the density of unimodular matrix polynomials. (C) 2016 Elsevier Inc. All rights reserved.

Homogeneous Hermitian holomorphic vector bundles and the Cowen-Douglas class over bounded symmetric domains

It is known that all the vector bundles of the title can be obtained by holomorphic induction from representations of a certain parabolic group on finite-dimensional inner product spaces. The representations, and the induced bundles, have composition series with irreducible factors. We write down an equivariant constant coefficient differential operator that intertwines the bundle with the direct sum of its irreducible factors. As an application, we show that in the case of the closed unit ball in C-n all homogeneous n-tuples of Cowen-Douglas operators are similar to direct sums of certain basic n-tuples. (c) 2015 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.

A New Finite Element Method for Strain Gradient Theories and Applications to Fracture Analyses

A new compatible finite element method for strain gradient theories is presented. In the new finite element method, pure displacement derivatives are taken as the fundamental variables. The new numerical method is successfully used to analyze the simple strain gradient problems – the fundamental fracture problems. Through comparing the numerical solutions with the existed exact solutions, the effectiveness of the new finite element method is tested and confirmed. Additionally, an application of the Zienkiewicz–Taylor C1 finite element method to the strain gradient problem is discussed. By using the new finite element method, plane-strain mode I and mode II crack tip fields are calculated based on a constitutive law which is a simple generalization of the conventional J2 deformation plasticity theory to include strain gradient effects. Three new constitutive parameters enter to characterize the scale over which strain gradient effects become important. During the analysis the general compressible version of Fleck–Hutchinson strain gradient plasticity is adopted. Crack tip solutions, the traction distributions along the plane ahead of the crack tip are calculated. The solutions display the considerable elevation of traction within the zone near the crack tip.

Fracture criterion based on the higher-order asymptotic fields

A complete development for the higher-order asymptotic solutions of the crack tip fields and finite element calculations for mode I loading of hardening materials in plane strain are performed. The results show that in the higher-order asymptotic solution (to the twentieth order), only three coefficients are independent. These coefficients are determined by matching with the finite element solutions carried out in the present paper (our attention is focused on the first five terms of the higher-order asymptotic solution). We obtain an analytic characterization of crack tip fields, which conform very well to the finite element solutions over wide range. A modified two parameter criterion based on the asymptotic solution of five terms is presented. The upper bound and lower bound fracture toughness curves predicted by modified two parameter criterion are given. These two curves agree with most of the experimental data and fully capture the proper trend.

Higher-order analysis of crack tip fields in elastic power-law hardening materials

A HIGHER-ORDER asymptotic analysis of a stationary crack in an elastic power-law hardening material has been carried out for plane strain, Mode 1. The extent to which elasticity affects the near-tip fields is determined by the strain hardening exponent n. Five terms in the asymptotic series for the stresses have been derived for n = 3. However, only three amplitudes can be independently prescribed. These are K1, K2 and K5 corresponding to amplitudes of the first-, second- and fifth-order terms. Four terms in the asymptotic series have been obtained for n = 5, 7 and 10; in these cases, the independent amplitudes are K1, K2 and K4. It is found that appropriate choices of K2 and K4 can reproduce near-tip fields representative of a broad range of crack tip constraints in moderate and low hardening materials. Indeed, fields characterized by distinctly different stress triaxiality levels (established by finite element analysis) have been matched by the asymptotic series. The zone of dominance of the asymptotic series extends over distances of about 10 crack openings ahead of the crack tip encompassing length scales that are microstructurally significant. Furthermore, the higher-order terms collectively describe a spatially uniform hydrostatic stress field (of adjustable magnitude) ahead of the crack. Our results lend support to a suggestion that J and a measure of near-tip stress triaxiality can describe the full range of near-tip states.

On a Class of Self-Adjoint Compact Operators in Hilbert Spaces and Their Relations with Their Finite-Range Truncations

This paper investigates a class of self-adjoint compact operators in Hilbert spaces related to their truncated versions with finite-dimensional ranges. The comparisons are established in terms of worst-case norm errors of the composite operators generated from iterated computations. Some boundedness properties of the worst-case norms of the errors in their respective fixed points in which they exist are also given. The iterated sequences are expanded in separable Hilbert spaces through the use of numerable orthonormal bases.

Symmetric representation of an integral domain over a subdomain

Let F(θ) be a separable extension of degree n of a field F. Let Δ and D be integral domains with quotient fields F(θ) and F respectively. Assume that Δ ᴝ D. A mapping φ of Δ into the n x n D matrices is called a Δ/D rep if (i) it is a ring isomorphism and (ii) it maps d onto dI_n whenever d ϵ D. If the matrices are also symmetric, φ is a Δ/D symrep.

Every Δ/D rep can be extended uniquely to an F(θ)/F rep. This extension is completely determined by the image of θ. Two Δ/D reps are called equivalent if the images of θ differ by a D unimodular similarity. There is a one-to-one correspondence between classes of Δ/D reps and classes of Δ ideals having an n element basis over D.

The condition that a given Δ/D rep class contain a Δ/D symrep can be phrased in various ways. Using these formulations it is possible to (i) bound the number of symreps in a given class, (ii) count the number of symreps if F is finite, (iii) establish the existence of an F(θ)/F symrep when n is odd, F is an algebraic number field, and F(θ) is totally real if F is formally real (for n = 3 see Sapiro, “Characteristic polynomials of symmetric matrices” Sibirsk. Mat. Ž. 3 (1962) pp. 280-291), and (iv) study the case D = Z, the integers (see Taussky, “On matrix classes corresponding to an ideal and its inverse” Illinois J. Math. 1 (1957) pp. 108-113 and Faddeev, “On the characteristic equations of rational symmetric matrices” Dokl. Akad. Nauk SSSR 58 (1947) pp. 753-754).

The case D = Z and n = 2 is studied in detail. Let Δ’ be an integral domain also having quotient field F(θ) and such that Δ’ ᴝ Δ. Let φ be a Δ/Z symrep. A method is given for finding a Δ’/Z symrep ʘ such that the Δ’ ideal class corresponding to the class of ʘ is an extension to Δ’ of the Δ ideal class corresponding to the class of φ. The problem of finding all Δ/Z symreps equivalent to a given one is studied.

Energy stable and momentum conserving hybrid finite element method for the incompressible Navier-Stokes equations

A hybrid method for the incompressible Navier-Stokes equations is presented. The method inherits the attractive stabilizing mechanism of upwinded discontinuous Galerkin methods when momentum advection becomes significant, equal-order interpolations can be used for the velocity and pressure fields, and mass can be conserved locally. Using continuous Lagrange multiplier spaces to enforce flux continuity across cell facets, the number of global degrees of freedom is the same as for a continuous Galerkin method on the same mesh. Different from our earlier investigations on the approach for the Navier-Stokes equations, the pressure field in this work is discontinuous across cell boundaries. It is shown that this leads to very good local mass conservation and, for an appropriate choice of finite element spaces, momentum conservation. Also, a new form of the momentum transport terms for the method is constructed such that global energy stability is guaranteed, even in the absence of a pointwise solenoidal velocity field. Mass conservation, momentum conservation, and global energy stability are proved for the time-continuous case and for a fully discrete scheme. The presented analysis results are supported by a range of numerical simulations. © 2012 Society for Industrial and Applied Mathematics.

ℓasso MPC: Smart regulation of over-actuated systems

In this paper, a novel MPC strategy is proposed, and referred to as asso MPC. The new paradigm features an 1-regularised least squares loss function, in which the control error variance competes with the sum of input channels magnitude (or slew rate) over the whole horizon length. This cost choice is motivated by the successful development of LASSO theory in signal processing and machine learning. In the latter fields, sum-of-norms regularisation have shown a strong capability to provide robust and sparse solutions for system identification and feature selection. In this paper, a discrete-time dual-mode asso MPC is formulated, and its stability is proven by application of standard MPC arguments. The controller is then tested for the problem of ship course keeping and roll reduction with rudder and fins, in a directional stochastic sea. Simulations show the asso MPC to inherit positive features from its corresponding regressor: extreme reduction of decision variables' magnitude, namely, actuators' magnitude (or variations), with a finite energy error, being particularly promising for over-actuated systems. © 2012 AACC American Automatic Control Council).

Analysis of fields in an air-cored superconducting synchronous motor with an HTS racetrack field winding

High temperature superconducting (HTS) synchronous motors can offer significant weight and size reductions, as well as improved efficiency, over conventional copper-wound machines due to the higher current density of high temperature superconducting (HTS) materials. In order to optimise the design parameters and performance of such a machine, this paper proposes a basic physical model of an air-cored HTS synchronous motor with a copper armature winding and HTS field winding. An analytical method for the field analysis in the synchronous motor is then presented, followed by a numerical finite element analysis (FEA) model to verify the analytical solution. The model is utilised to study the influence of the geometry of the HTS coils on the magnetic field at the armature winding, and geometrical parameter optimisation is carried out using this theoretical model to obtain a more sinusoidal magnetic field at the armature, which has a major influence on the performance of the motor.

Modelling and comparison of trapped fields in (RE)BCO bulk superconductors for activation using pulsed field magnetization

The ability to generate a permanent, stable magnetic field unsupported by an electromotive force is fundamental to a variety of engineering applications. Bulk high temperature superconducting (HTS) materials can trap magnetic fields of magnitude over ten times higher than the maximum field produced by conventional magnets, which is limited practically to rather less than 2 T. In this paper, two large c-axis oriented, single-grain YBCO and GdBCO bulk superconductors are magnetized by the pulsed field magnetization (PFM) technique at temperatures of 40 and 65 K and the characteristics of the resulting trapped field profile are investigated with a view of magnetizing such samples as trapped field magnets (TFMs) in situ inside a trapped flux-type superconducting electric machine. A comparison is made between the temperatures at which the pulsed magnetic field is applied and the results have strong implications for the optimum operating temperature for TFMs in trapped flux-type superconducting electric machines. The effects of inhomogeneities, which occur during the growth process of single-grain bulk superconductors, on the trapped field and maximum temperature rise in the sample are modelled numerically using a 3D finite-element model based on the H-formulation and implemented in Comsol Multiphysics 4.3a. The results agree qualitatively with the observed experimental results, in that inhomogeneities act to distort the trapped field profile and reduce the magnitude of the trapped field due to localized heating within the sample and preferential movement and pinning of flux lines around the growth section regions (GSRs) and growth sector boundaries (GSBs), respectively. The modelling framework will allow further investigation of various inhomogeneities that arise during the processing of (RE)BCO bulk superconductors, including inhomogeneous Jc distributions and the presence of current-limiting grain boundaries and cracks, and it can be used to assist optimization of processing and PFM techniques for practical bulk superconductor applications. © 2014 IOP Publishing Ltd.

When Does a Polynomial Over a Finite Field Permute the Elements of the Field?

We present a class of indecomposable polynomials of non prime-power degree over the finite field of two elements which are permutation polynomials on infinitely many finite extensions of the field. The associated geometric monodromy groups are the simple ...