10 research outputs found

    Presemifields, bundles and polynomials over GF (pn)

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    The content of this thesis is first and foremost about presemifields and the equivalence classes they may be categorized by. This equivalence has been termed “bundle equivalence'' by Horadam. Bundle equivalence is inherited from multiplicative orthogonal cocycles, and the final Chapter is devoted entirely to coboundaries and cocycles. In this thesis we provide a complete computational classification of the bundles of presemifields in all presemifield isotopism classes of order p n , provide a formula for the number of bundles in the presemifields isotopism class of GF (p 2 ) and give a representative of each bundle, for any prime p . We provide computational classification of the bundles of presemifields in the isotopism class of GF (p 3 )  for the cases  p =3,5,7,11 and give representatives, give formulae for two of the three possible size bundles in the presemifield isotopism class of  GF (p 3 )   which we call the minimum and the mid-size bundles. We provide a Conjecture which states the total number of mid-size bundles in the isotopism class of  GF (p 3 ) and give a computational classification of the bundles of presemifields in the isotopism class of  GF (2 5 ) and  GF (3 4 ) . We provide a measurement of the differential uniformity of functions derived from the diagonal map of presemifield multiplications with order p n < 16 and derive bivariate polynomial formulae for cocycles and coboundaries in We produce a basis for the ( p n - 1 - n ) - dimensional -space of coboundaries. When p = 2 we give a recursive definition of the basis coboundaries. We use the Kronecker product to explain the self-similarity of the binomial coefficients modulo a prime and use the Kronecker product to define recursively the basis coboundaries for p odd, and we demonstrate this holds for the case p = 2. We show that each cocycle has a unique decomposition as a direct sum of a coboundary and a multiplicative cocycle of restricted form when  p = 2.  The results of this thesis have been published in the Proceedings of the International Workshop on Coding and Cryptography, Designs, Codes and Cryptography and the Proceedings of IEEE International Symposium on Information Theory and will appear in the Journal of the Australian Mathematical Society

    Weighted complex projective 2-designs from bases: optimal state determination by orthogonal measurements

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    We introduce the problem of constructing weighted complex projective 2-designs from the union of a family of orthonormal bases. If the weight remains constant across elements of the same basis, then such designs can be interpreted as generalizations of complete sets of mutually unbiased bases, being equivalent whenever the design is composed of d+1 bases in dimension d. We show that, for the purpose of quantum state determination, these designs specify an optimal collection of orthogonal measurements. Using highly nonlinear functions on abelian groups, we construct explicit examples from d+2 orthonormal bases whenever d+1 is a prime power, covering dimensions d=6, 10, and 12, for example, where no complete sets of mutually unbiased bases have thus far been found.Comment: 28 pages, to appear in J. Math. Phy

    Model Theory and Groups

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    The aim of the workshop was to discuss the connections between model theory and group theory. Main topics have been the interaction between geometric group theory and model theory, the study of the asymptotic behaviour of geometric properties on groups, and the model theoretic investigations of groups of finite Morley rank around the Cherlin-Zilber Conjecture

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Natural Communication

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    In Natural Communication, the author criticizes the current paradigm of specific goal orientation in the complexity sciences. His model of "natural communication" encapsulates modern theoretical concepts from mathematics and physics, in particular category theory and quantum theory. The author is convinced that only by looking to the past is it possible to establish continuity and coherence in the complexity science

    Parity duality of super rr-matrices via O\mathcal O-operators and pre-Lie superalgebras

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    This paper studies super rr-matrices and operator forms of the super classical Yang-Baxter equation. First by a unified treatment, the classical correspondence between rr-matrices and O\mathcal{O}-operators is generalized to a correspondence between homogeneous super rr-matrices and homogeneous O\mathcal{O}-operators. Next, by a parity reverse of Lie superalgebra representations, a duality is established between the even and the odd O\mathcal{O}-operators, giving rise to a parity duality among the induced super rr-matrices. Thus any homogeneous \OO-operator or any homogeneous super rr-matrix with certain supersymmetry produces a parity pair of super rr-matrices, and generates an infinite tree hierarchy of homogeneous super rr-matrices. Finally, a pre-Lie superalgebra naturally defines a parity pair of O\mathcal{O}-operators, and thus a parity pair of super rr-matrices.Comment: 26 pages; to appear in Math Research Letter

    Topological Photonics

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    Topological photonics is a rapidly emerging field of research in which geometrical and topological ideas are exploited to design and control the behavior of light. Drawing inspiration from the discovery of the quantum Hall effects and topological insulators in condensed matter, recent advances have shown how to engineer analogous effects also for photons, leading to remarkable phenomena such as the robust unidirectional propagation of light, which hold great promise for applications. Thanks to the flexibility and diversity of photonics systems, this field is also opening up new opportunities to realize exotic topological models and to probe and exploit topological effects in new ways. This article reviews experimental and theoretical developments in topological photonics across a wide range of experimental platforms, including photonic crystals, waveguides, metamaterials, cavities, optomechanics, silicon photonics, and circuit QED. A discussion of how changing the dimensionality and symmetries of photonics systems has allowed for the realization of different topological phases is offered, and progress in understanding the interplay of topology with non-Hermitian effects, such as dissipation, is reviewed. As an exciting perspective, topological photonics can be combined with optical nonlinearities, leading toward new collective phenomena and novel strongly correlated states of light, such as an analog of the fractional quantum Hall effect.Comment: 87 pages, 30 figures, published versio

    Extended Field Theories as higher Kaluza-Klein theories.

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    PhD ThesesExtended Field Theories (ExFTs) include Double Field Theory (DFT) and Exceptional Field Theory, which are respectively the T- and U-duality covariant formulations of the supergravity limit of String Theory and M-theory. Extended Field Theories do not live on spacetime, but on an extended spacetime, locally modelled on the space underlying the fundamental representation of the duality group. Despite its importance in M-theory, however, the global understanding of Extended Field Theories is still an open problem. In this thesis we propose a global geometric formulation of Extended Field Theory. Recall that ordinary Kaluza-Klein theory unifies a metric with a gauge field on a principal bundle. We propose a generalisation of the Kaluza-Klein principle which unifies a metric and a higher gauge field on a principal infinity-bundle. This is achieved by introducing an atlas for the principal infinity-bundle, whose local charts can be naturally identified with the ones of Extended Field Theory. Thus, DFT is interpreted as a higher Kaluza-Klein theory set on the total space of a bundle gerbe underlying Kalb-Ramond field. As first application, we define the higher Kaluza-Klein monopole by naturally generalising the ordinary Gross-Perry monopole. Then we show that this monopole is exactly the NS5-brane of String Theory. Secondly, we show that our higher geometric formulation gives automatically rise to global abelian T-duality and global Poisson-Lie T-duality. In particular, we globally recover the abelian T-fold and we define the notion of Poisson-Lie T-fold. Crucially, we will investigate the global geometric formulation of tensor hierarchies and gauged supergravity. In particular, we will provide a global formulation of generalised Scherk-Schwarz reductions and we will discuss the global non-geometric properties of tensor hierarchies. Finally, we explore the T-duality covariant geometric quantisation of DFT by transgressing its underlying bundle gerbe to a U(1)-bundle on the loop space of its base manifold
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