On the nucleus of the Grassmann embedding of the symplectic dual polar space DSp(2n,F)DSp(2n,F).

Let \geq 3$and let$ be a field of characteristic 2. Let (2n,F)$denote the dual polar space associated with the building of Type$ over $and let$\mathcal{G}_{n-2}$denote the$(n-2) of type $. Using the bijective correspondence between the points of$\mathcal{G}_{n-2}$and the quads of (2n,F)$, we construct a full projective embedding of $\mathcal{G}_{n-2}$ into the nucleus of the Grassmann embedding of (2n,F)$. This generalizes a result of Cardinali and Lunardon which contains an alternative proof of this fact in the case when =3$ and $ is finite

A property of isometric mappings between dual polar spaces of type DQ(2n,K)

Let f be an isometric embedding of the dual polar space Delta = DQ(2n, K) into Delta' = DQ(2n, K'). Let P denote the point-set of Delta and let e' : Delta' -> Sigma' congruent to PG(2(n) - 1, K') denote the spin-embedding of Delta'. We show that for every locally singular hyperplane H of Delta, there exists a unique locally singular hyperplane H' of Delta' such that f(H) = f(P) boolean AND H'. We use this to show that there exists a subgeometry Sigma congruent to PG(2(n) - 1, K) of Sigma' such that: (i) e' circle f (x) is an element of Sigma for every point x of Delta; (ii) e := e' circle f defines a full embedding of Delta into Sigma, which is isomorphic to the spin-embedding of Delta

An outline of polar spaces: basics and advances

This paper is an extended version of a series of lectures on polar spaces given during the workshop and conference 'Groups and Geometries', held at the Indian Statistical Institute in Bangalore in December 2012. The aim of this paper is to give an overview of the theory of polar spaces focusing on some research topics related to polar spaces. We survey the fundamental results about polar spaces starting from classical polar spaces. Then we introduce and report on the state of the art on the following research topics: polar spaces of infinite rank, embedding polar spaces in groups and projective embeddings of dual polar spaces

Veronesean embeddings of dual polar spaces of orthogonal type

Given a point-line geometry P and a pappian projective space S,a veronesean embedding of P in S is an injective map e from the point-set of P to the set of points of S mapping the lines of P onto non-singular conics of S and such that e(P) spans S. In this paper we study veronesean embeddings of the dual polar space \Delta_n associated to a non-singular quadratic form q of Witt index n >= 2 in V = V(2n + 1; F). Three such embeddings are considered,namely the Grassmann embedding gr_n,the composition vs_n of the spin (projective) embedding of \Delta_n in PG(2n-1; F) with the quadric veronesean map of V(2n; F) and a third embedding w_n defined algebraically in the Weyl module V (2\lambda_n),where \lambda_n is the fundamental dominant weight associated to the n-th simple root of the root system of type Bn. We shall prove that w_n and vs_n are isomorphic. If char(F) is different from 2 then V (2\lambda_n) is irreducible and w_n is isomorphic to gr_n while if char(F) = 2 then gr_n is a proper quotient of w_n. In this paper we shall study some of these submodules. Finally we turn to universality,focusing on the case of n = 2. We prove that if F is a finite field of odd order q > 3 then sv_2 is relatively universal. On the contrary,if char(F) = 2 then vs_2 is not universal. We also prove that if F is a perfect field of characteristic 2 then vs_n is not universal,for any n>=2

Duality Symmetry

Symmetry is one of the most general concepts in physics. Symmetry arguments are used to explain and predict observations at all length scales, from elementary particles to cosmology. The generality of symmetry arguments, combined with their simplicity, makes them a powerful tool for both fundamental and applied investigations. In electrodynamics, one of the symmetries is the invariance of the equations under exchange of electric and magnetic quantities. The continuous version of this symmetry is most commonly known as electromagnetic duality symmetry. This concept has been accepted for more than a century, and, throughout this time, has influenced other areas of physics, like high energy physics and gravitation. This Special Issue is devoted to electromagnetic duality symmetry and other vareities of dualities in physics. It contains four Articles, one Review and one Perspective. The context of the contributions ranges from string theory to applied nanophotonics, which, as anticipated, shows that duality symmetries in general and electromagnetic duality symmetry in particular are useful in a wide variety of physics fields, both theoretical and applied. Moreover, a number of the contributions show how the use of symmetry arguments and the quantification of symmetry breaking can successfully guide our theoretical understanding and provide us with guidelines for system design

Dynamic Hyperspectral and Polarized Endoscopic Imaging

The health of rich, developed nations has seen drastic improvement in the last two centuries. For it to continue improving at a similar rate new or improved diagnostic and treatment technologies are required, especially for those diseases such as cancer which are forecast to constitute the majority of disease burden in the future. Optical techniques such as microscopy have long played their part in the diagnostic process. However there are several new biophotonic modalities that aim to exploit various interactions between light and tissue to provide enhanced diagnostic information. Many of these show promise in a laboratory setting but few have progressed to a clinical setting. We have designed and constructed a flexible, multi-modal, multi-spectral laparoscopic imaging system that could be used to demonstrate several different techniques in a clinical setting. The core of this system is a dynamic hyperspectral illumination system based around a supercontinuum laser and Digital Micromirror Device that can provide specified excitation light in the visible and near infra-red ranges. This is a powerful tool for spectroscopic techniques as it is not limited to interrogating a fixed range of wavelengths and can switch between excitation bands instantaneously. The excitation spectra can be customised to match particular fluorophores or absorption features, introducing new possibilities for spectral imaging. A standard 10 mm diameter rigid endoscope was incorporated into the system to reduce cost and demonstrate compatibility with existing equipment. The polarization properties of two commercial endoscopes were characterised and found to be unsuited to current polarization imaging techniques as birefringent materials used in their construction introduce complex, spatially dependent transformations of the polarization state. Preliminary exemplar data from phantoms and ex vivo tissue was collected and the feasibility and accuracy of different analysis techniques demonstrated including multiple class classification algorithms. Finally, a novel visualisation method was implemented in order to display the complex hyperspectral data sets in a meaningful and intuitive way to the user

Decomposability of Tensors

Tensor decomposition is a relevant topic, both for theoretical and applied mathematics, due to its interdisciplinary nature, which ranges from multilinear algebra and algebraic geometry to numerical analysis, algebraic statistics, quantum physics, signal processing, artificial intelligence, etc. The starting point behind the study of a decomposition relies on the idea that knowledge of elementary components of a tensor is fundamental to implement procedures that are able to understand and efficiently handle the information that a tensor encodes. Recent advances were obtained with a systematic application of geometric methods: secant varieties, symmetries of special decompositions, and an analysis of the geometry of finite sets. Thanks to new applications of theoretic results, criteria for understanding when a given decomposition is minimal or unique have been introduced or significantly improved. New types of decompositions, whose elementary blocks can be chosen in a range of different possible models (e.g., Chow decompositions or mixed decompositions), are now systematically studied and produce deeper insights into this topic. The aim of this Special Issue is to collect papers that illustrate some directions in which recent researches move, as well as to provide a wide overview of several new approaches to the problem of tensor decomposition