10 research outputs found
Presemifields, bundles and polynomials over GF (pn)
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
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
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
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Natural Communication
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 -matrices via -operators and pre-Lie superalgebras
This paper studies super -matrices and operator forms of the super
classical Yang-Baxter equation. First by a unified treatment, the classical
correspondence between -matrices and -operators is generalized
to a correspondence between homogeneous super -matrices and homogeneous
-operators. Next, by a parity reverse of Lie superalgebra
representations, a duality is established between the even and the odd
-operators, giving rise to a parity duality among the induced
super -matrices. Thus any homogeneous \OO-operator or any homogeneous
super -matrix with certain supersymmetry produces a parity pair of super
-matrices, and generates an infinite tree hierarchy of homogeneous super
-matrices. Finally, a pre-Lie superalgebra naturally defines a parity pair
of -operators, and thus a parity pair of super -matrices.Comment: 26 pages; to appear in Math Research Letter
Topological Photonics
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.
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