Four lectures on secant varieties

This paper is based on the first author's lectures at the 2012 University of Regina Workshop "Connections Between Algebra and Geometry". Its aim is to provide an introduction to the theory of higher secant varieties and their applications. Several references and solved exercises are also included.Comment: Lectures notes to appear in PROMS (Springer Proceedings in Mathematics & Statistics), Springer/Birkhause

On Comon's and Strassen's conjectures

Comon's conjecture on the equality of the rank and the symmetric rank of a symmetric tensor, and Strassen's conjecture on the additivity of the rank of tensors are two of the most challenging and guiding problems in the area of tensor decomposition. We survey the main known results on these conjectures, and, under suitable bounds on the rank, we prove them, building on classical techniques used in the case of symmetric tensors, for mixed tensors. Finally, we improve the bound for Comon's conjecture given by flattenings by producing new equations for secant varieties of Veronese and Segre varieties.Comment: 12 page

Effective criteria for specific identifiability of tensors and forms

In applications where the tensor rank decomposition arises, one often relies on its identifiability properties for interpreting the individual rank-$1$ terms appearing in the decomposition. Several criteria for identifiability have been proposed in the literature, however few results exist on how frequently they are satisfied. We propose to call a criterion effective if it is satisfied on a dense, open subset of the smallest semi-algebraic set enclosing the set of rank-$r$ tensors. We analyze the effectiveness of Kruskal's criterion when it is combined with reshaping. It is proved that this criterion is effective for both real and complex tensors in its entire range of applicability, which is usually much smaller than the smallest typical rank. Our proof explains when reshaping-based algorithms for computing tensor rank decompositions may be expected to recover the decomposition. Specializing the analysis to symmetric tensors or forms reveals that the reshaped Kruskal criterion may even be effective up to the smallest typical rank for some third, fourth and sixth order symmetric tensors of small dimension as well as for binary forms of degree at least three. We extended this result to $4 \times 4 \times 4 \times 4$ symmetric tensors by analyzing the Hilbert function, resulting in a criterion for symmetric identifiability that is effective up to symmetric rank $8$, which is optimal.Comment: 31 pages, 2 Macaulay2 code

A theoretical study of the semiconductor laser structures with lateral discontinuity in the optical cavity

A theoretical study of the semiconductor laser structures with lateral discontinuity in the optical resonant cavity is presented in this thesis. Specifically, the lateral discontinuity is referred to the lateral expansion structure newly invented for increasing the power output of semiconductor lasers while keeping single mode operation. In the first part of the thesis (Chapter 2 and Chapter 3), an explicit expression for calculating the lateral discontinuity problems is formulated by the incorporation of mode-matching method. Our approach is based on the mode expansion theory developed for lossless micro-wave and optical fiber waveguides, but the effect of gain in active laser media is discussed. Direct numerical evaluation to various discontinuities problems in semiconductor laser stripes are feasible, as long as the discontinuity lies entirely in a surface perpendicular to the direction of propagation. The second part of the thesis is devoted to applications of the theoretical approach. A computer program written in Fortran-5.0 is used to calculate the efficiency of lateral expansion laser structure. Graphs of the electric field patterns are produced by a program written in programming language C

Elias Allen and the role of instruments in shaping the mathematical culture of seventeenth-century England

Elias AlIen (c.1588-1653) was known as the best mathematical instrument maker of his day. He lived and worked in London, creating a thriving business - he was the first English instrument maker to support himself solely through the production of instruments - and teaching his skills to many apprentices who became the core of the trade during the latter part of the century. My thesis provides a full biography of Allen, set within the framework of the community of people who were in some way connected to mathematics. The second part of the dissertation is devoted to the most important of Allen's instruments: the Gunter quadrant aqd Gilnter sector, and various of William Oughtred's designs - the circles of proportion; the horizontal instrument and double horizontal dial, and the universal equinoctial ring dial. These are described, their uses explained, and used as the base for discussions of the ways in which instruments influenced and were influenced by the development of mathematics. This section concludes with a catalogue of all the Allen instruments in British museums. As well as a comprehensive literature survey of the mathematical texts printed in England during Allen's lifetime, I have given considerable time to a 'reading' of the instruments themselves - through study of the originals, through production of my own versions, through reconstructions of the methods,of use, and, in the case of the sector, through computer analysis of the accuracy in use. The conclusion of my thesis is that the mathematical culture of seventeenth century England was far broader than that which is normally portrayed in histories of mathematics, involving a wide range of people with very different backgrounds and very different approaches to and understandings of mathematics. Above all, it is shown to be rooted strongly in a geometrical interpretation of mathematics and one which is inherently practical. In such a culture the role of instruments is fundamental and thus instrument makers like Elias Alien have a place at the heart of the mathematical community

Application de la tomographie sonique au diagnostic du béton

Recognition of the many problems in concrete structures requires the reconstruction of their internal image. For this reason, some years ago, a new nondestructive method, sonic tomography, has been developed for scanning concrete structures. This method is most often based on transient stress wave propagation for verifying a concrete body, as well as measuring the wave velocity. The relative variation in wave propagation velocity in the material provides information about changes in the structure, and therefore the state of degradation. Capacity and limitation of this method for reconstructing the internal image of the concrete structures are addressed in this thesis. This study discusses some experiments in which sonic tomography were performed on various concrete models in laboratory and the structures in service in-situ. In this case, two important aims are the main focus: (1) Degraded area is usually marked by fracture and delaminations or poorly performed sections within the structure. Unfortunately, most of the degradation in concrete structures remain undetectable to the naked eye. Sonic tomography is a strong technique to show the damaged areas in body concrete. In addition, this technique is able to evaluate internal reparation (e.g. control of grout-injection zones). (2) This is limited by the measurement and reconstruction of image conditions and their parameters (frequency, pixels size, rays type, measurement step,...). For illustrating the real image it is necessary to know precisely these conditions and the quality of their influence on the tomographic image