A coordinate independent derivation of the Eulerian and Lagrangian strain tensors of finite deformation theory is given based on the parallel propagator, the world function, and the displacement vector field as a three-point tensor. The derivation explicitly shows that the Eulerian and Lagrangian strain tensors are two-point tensors, each a function of both the spatial and material coordinates....

This paper studies the possibility of achieving indistinguishabilitybased security in privately outsourcing linear equation systems over real numbers. The particular task is to solve a full-rank n× n system Ax = b. Since the most complex part of this task is inverting A, the problem can be reduced to outsourcing of a square matrix inverse computation. Although outsourcing matrix inverse is triv...

We study the solution of linear systems resulting from the discreitization of unsteady diffusion equations with stochastic coefficients. In particular, we focus on those linear systems that are obtained using the so-called stochastic Galerkin finite element method (SGFEM). These linear systems are usually very large with Kronecker product structure and, thus, solving them can be both timeand co...

The notion of finite zeros of continuous-time positive linear systems is introduced. It is shown that such zeros are real numbers. It is also shown that a square positive strictly proper or proper system of uniform rank with observability matrix of full column rank has no finite zeros. The problem of zeroing the system output for positive systems is defined. It is shown that a square positive s...

As a continuation of the work on linear maps between operator algebras which preserve certain subsets of operators with finite rank, or finite corank, here we consider the problem inbetween, that is, we treat the question of preserving operators with infinite rank and infinite corank. Since, as it turns out, in this generality our preservers cannot be written in a nice form what we have got use...

We analyze a certain class of integral equations associated with Marchenko equations and Gel’fand-Levitan equations. Such integral equations arise through a Fourier transformation on various ordinary differential equations involving a spectral parameter. When the integral operator is perturbed by a finite-rank perturbation, we explicitly evaluate the change in the solution in terms of the unper...

We construct a family of weight functions on finite abelian groups that yield invertible MacWilliams identities for additive codes. The weights are obtained composing a suitable support map with the rank function of a graded lattice that satisfies certain regularity properties. We express the Krawtchouk coefficients of the corresponding MacWilliams transformation in terms of the combinatorial i...

We define the Polish space R of non-degenerate rank-1 systems. Each non-degenerate rank-1 system can be viewed as a measurepreserving transformation of an atomless, σ-finite measure space and as a homeomorphism of a Cantor space. We completely characterize when two non-degenerate rank-1 systems are topologically isomorphic. We also analyze the complexity of the topological isomorphism relation ...

The low-rank matrix recovery (LMR) is a rank minimization problem subject to linear equality constraints, and it arises in many fields such as signal and image processing, statistics, computer vision, system identification and control. This class of optimization problems is NP-hard and a popular approach replaces the rank function with the nuclear norm of the matrix variable. In this paper, we ...

A linear algebra proof is given of the fact that the nullspace of a finite-rank linear projector, on polynomials in two complex variables, is an ideal if and only if the projector is the bounded pointwise limit of Lagrange projectors, i.e., projectors whose nullspace is a radical ideal, i.e., the set of all polynomials that vanish on a certain given finite set. A characterization of such projec...