The affine preservers of non-singular matrices

When K is an arbitrary field, we study the affine automorphisms of M_n(K) that stabilize GL_n(K). Using a theorem of Dieudonn\'e on maximal affine subspaces of singular matrices, this is easily reduced to the known case of linear preservers when n>2 or #K>2. We include a short new proof of the more general Flanders' theorem for affine subspaces of M_{p,q}(K) with bounded rank. We also find that the group of affine transformations of M_2(F_2) that stabilize GL_2(F_2) does not consist solely of linear maps. Using the theory of quadratic forms over F_2, we construct explicit isomorphisms between it, the symplectic group Sp_4(F_2) and the symmetric group S_6.Comment: 13 pages, very minor corrections from the first versio

A note on the minimum distance of quantum LDPC codes

We provide a new lower bound on the minimum distance of a family of quantum LDPC codes based on Cayley graphs proposed by MacKay, Mitchison and Shokrollahi. Our bound is exponential, improving on the quadratic bound of Couvreur, Delfosse and Z\'emor. This result is obtained by examining a family of subsets of the hypercube which locally satisfy some parity conditions

Graphene-Based Nanomaterials for Neuroengineering: Recent Advances and Future Prospective

Graphene unique physicochemical properties made it prominent among other allotropic forms of carbon, in many areas of research and technological applications. Interestingly, in recent years, many studies exploited the use of graphene family nanomaterials (GNMs) for biomedical applications such as drug delivery, diagnostics, bioimaging, and tissue engineering research. GNMs are successfully used for the design of scaffolds for controlled induction of cell differentiation and tissue regeneration. Critically, it is important to identify the more appropriate nano/bio material interface sustaining cells differentiation and tissue regeneration enhancement. Specifically, this review is focussed on graphene-based scaffolds that endow physiochemical and biological properties suitable for a specific tissue, the nervous system, that links tightly morphological and electrical properties. Different strategies are reviewed to exploit GNMs for neuronal engineering and regeneration, material toxicity, and biocompatibility. Specifically, the potentiality for neuronal stem cells differentiation and subsequent neuronal network growth as well as the impact of electrical stimulation through GNM on cells is presented. The use of field effect transistor (FET) based on graphene for neuronal regeneration is described. This review concludes the important aspects to be controlled to make graphene a promising candidate for further advanced application in neuronal tissue engineering and biomedical use

A local-global principle for linear dependence of noncommutative polynomials

A set of polynomials in noncommuting variables is called locally linearly dependent if their evaluations at tuples of matrices are always linearly dependent. By a theorem of Camino, Helton, Skelton and Ye, a finite locally linearly dependent set of polynomials is linearly dependent. In this short note an alternative proof based on the theory of polynomial identities is given. The method of the proof yields generalizations to directional local linear dependence and evaluations in general algebras over fields of arbitrary characteristic. A main feature of the proof is that it makes it possible to deduce bounds on the size of the matrices where the (directional) local linear dependence needs to be tested in order to establish linear dependence.Comment: 8 page

Retrodiction of Generalised Measurement Outcomes

If a generalised measurement is performed on a quantum system and we do not know the outcome, are we able to retrodict it with a second measurement? We obtain a necessary and sufficient condition for perfect retrodiction of the outcome of a known generalised measurement, given the final state, for an arbitrary initial state. From this, we deduce that, when the input and output Hilbert spaces have equal (finite) dimension, it is impossible to perfectly retrodict the outcome of any fine-grained measurement (where each POVM element corresponds to a single Kraus operator) for all initial states unless the measurement is unitarily equivalent to a projective measurement. It also enables us to show that every POVM can be realised in such a way that perfect outcome retrodiction is possible for an arbitrary initial state when the number of outcomes does not exceed the output Hilbert space dimension. We then consider the situation where the initial state is not arbitrary, though it may be entangled, and describe the conditions under which unambiguous outcome retrodiction is possible for a fine-grained generalised measurement. We find that this is possible for some state if the Kraus operators are linearly independent. This condition is also necessary when the Kraus operators are non-singular. From this, we deduce that every trace-preserving quantum operation is associated with a generalised measurement whose outcome is unambiguously retrodictable for some initial state, and also that a set of unitary operators can be unambiguously discriminated iff they are linearly independent. We then examine the issue of unambiguous outcome retrodiction without entanglement. This has important connections with the theory of locally linearly dependent and locally linearly independent operators.Comment: To appear in Physical Review

Random geometric complexes

We study the expected topological properties of Cech and Vietoris-Rips complexes built on i.i.d. random points in R^d. We find higher dimensional analogues of known results for connectivity and component counts for random geometric graphs. However, higher homology H_k is not monotone when k > 0. In particular for every k > 0 we exhibit two thresholds, one where homology passes from vanishing to nonvanishing, and another where it passes back to vanishing. We give asymptotic formulas for the expectation of the Betti numbers in the sparser regimes, and bounds in the denser regimes. The main technical contribution of the article is in the application of discrete Morse theory in geometric probability.Comment: 26 pages, 3 figures, final revisions, to appear in Discrete & Computational Geometr

Large random simplicial complexes, I

In this paper we introduce a new model of random simplicial complexes depending on multiple probability parameters. This model includes the well-known Linial - Meshulam random simplicial complexes and random clique complexes as special cases. Topological and geometric properties of a multi-parameter random simplicial complex depend on the whole combination of the probability parameters and the thresholds for topological properties are convex sets rather than numbers (as in all previously known models). We discuss the containment properties, density domains and dimension of the random simplicial complexes.Comment: 21 pages, 6 figure