Abstract—We take a new look at the issue of network capacity. It is shown that network coding is an essential ingredient in achieving the capacity of a network. Building on recent work by Li et al., who examined the network capacity of multicast networks, we extend the network coding framework to arbitrary networks and robust networking. For networks which are restricted to using linear network codes, we find necessary and sufficient conditions for the feasibility of any given set of connections over a given network. We also consider the problem of network recovery for nonergodic link failures. For the multicast setup we prove that there exist coding strategies that provide maximally robust networks and that do not require adaptation of the network interior to the failure pattern in question. The results are derived for both delay-free networks and networks with delays. Index Terms—Algebraic coding, network information theory, network robustness. I.

These are the lecture notes for ten lectures to be given at the CBMS

Symmetric tensors of order larger than two arise more and more often in signal and image processing and automatic control, because of the recent complementary use of High-Order Statistics (HOS). However, very few special purpose tools are at disposal for manipulating such objects in engineering problems. In this paper, the decomposition of a symmetric tensor into a sum of simpler ones is focused on, and links with the theory of homogeneous polynomials in several variables (i.e. quantics) are pointed out. This decomposition may be seen as a formal extension of the Eigen Value Decomposition (EVD), known for symmetric matrices. By reviewing the state of the art, quite surprising statements are emphasized, that explain why the problem is much more complicated in the tensor case than in the matrix case. Very few theoretical results can be applied in practice, even for cubics or quartics, because proofs are not constructive. Nevertheless in the binary case, we have more freedom to devise numerical algorithms. Keywords. Tensors, Polynomials, Diagonalization, EVD, High-Order Statistics, Cumulants. 1

We first review the basic properties of the well known classes of Toeplitz, Hankel, Vandermonde, and other related structured matrices and reexamine their correlation to operations with univariate polynomials. Then we define some natural extensions of such classes of matrices based on their correlation to multivariate polynomials. We describe the correlation in terms of the associated operators of multiplication in the polynomial ring and its dual space, which allows us to generalize these structures to the multivariate case. Multivariate Toeplitz, Hankel, and Vandermonde matrices, Bezoutians, algebraic residues and relations between them are studied. Finally, we show some applications of this study to root-finding problems for a system of multivariate polynomial equations, where the dual space, algebraic residues, Bezoutians and other structured matrices play an important role. The developed techniques enable us to obtain a better insight into the major problems of multivariate polynomial computations and to improve substantially the known techniques of the study of these problems. In particular, we simplify and/or generalize the known reduction of the multivariate polynomial systems to matrix eigenproblem, the derivation of the Bézout and Bernshtein bounds on the number of the roots, and the construction of multiplication tables. From the algorithmic and computational complexity point, we yield acceleration by one order of magnitude of the known methods for some fundamental problems of solving multivariate polynomial systems of equations.

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In this paper the dioeerent algebraic varieties that can be generated from multiple view geometry with uncalibrated cameras have been investigated. The natural descriptor, Vn , to work with is the image of IP 3 in IP 2 \Theta IP 2 \Theta \Delta \Delta \Delta \Theta IP 2 under a corresponding product of projections, (A1 \Theta A2 \Theta : : : \Theta Am). Another descriptor, the variety Vb , is the one generated by all bilinear forms between pairs of views, which consists of all points in IP 2 \Theta IP 2 \Theta \Delta \Delta \Delta \Theta IP 2 where all bilinear forms vanish. Yet another descriptor, the variety V t , is the variety generated by all trilinear forms between triplets of views. It has been shown that when m = 3, Vb is a reducible variety with one component corresponding to V t and another corresponding to the trifocal plane. Furthermore, when m = 3, V t is generated by the three bilinearities and one trilinearity, when m = 4, V t is generated by the six bil...

There are very close connections between the arithmetic of integer lattices, algebraic properties of the associated ideals, and the geometry and the combinatorics of corresponding polyhedra. In this paper we investigate the generating sets ("Gröbner bases") of integer lattices that correspond to the Gröbner bases of the associated binomial ideals. Extending results by Sturmfels & Thomas, we obtain a geometric characterization of the universal Gröbner basis in terms of the vertices and edges of the associated corner polyhedra. In the special case where the lattice has finite index, the corner polyhedra were studied by Gomory, and there is a close connection to the "group problem in integer programming." We present exponential lower and upper bounds for the maximal size of a reduced Grobner basis. The initial complex of (the ideal of) a lattice is shown to be dual to the boundary of a certain simple polyhedron.

We consider the problem of minimizing a polynomial over a semialgebraic set defined by polynomial equations and inequalities, which is NP-hard in general. Hierarchies of semidefinite relaxations have been proposed in the literature, involving positive semidefinite moment matrices and the dual theory of sums of squares of polynomials. We present these hierarchies of approximations and their main properties: asymptotic/finite convergence, optimality certificate, and extraction of global optimum solutions. We review the mathematical tools underlying these properties, in particular, some sums of squares representation results for positive polynomials, some results about moment matrices (in particular, of Curto and Fialkow), and the algebraic eigenvalue method for solving zero-dimensional systems of polynomial equations. We try whenever possible to provide detailed proofs and background.

We introduce the multigraded Hilbert scheme, which parametrizes all homogeneous ideals with fixed Hilbert function in a polynomial ring that is graded by any abelian group. Our construction is widely applicable, it provides explicit equations, and it allows us to prove a range of new results, including Bayer’s conjecture on equations defining Grothendieck’s classical Hilbert scheme and the construction of a Chow morphism for toric Hilbert schemes. 1.

Abstract—We consider distributed parameter systems where the underlying dynamics are spatially invariant, and where the controls and measurements are spatially distributed. These systems arise in many applications such as the control of vehicular platoons, flow control, microelectromechanical systems (MEMS), smart structures, and systems described by partial differential equations with constant coefficients and distributed controls and measurements. For fully actuated distributed control problems involving quadratic criteria such as linear quadratic regulator (LQR), P and, optimal controllers can be obtained by solving a parameterized family of standard finite-dimensional problems. We show that optimal controllers have an inherent degree of decentralization, and this provides a practical distributed controller architecture. We also prove a general result that applies to partially distributed control and a variety of performance criteria, stating that optimal controllers inherit the spatial invariance structure of the plant. Connections of this work to that on systems over rings, and systems with dynamical symmetries are discussed. Index Terms—Distributed control, infinite-dimensional systems, optimal control, robust control, spatially invariant systems.