Kida's formula and congruences

We prove a formula (analogous to that of Kida in classical Iwasawa theory and generalizing that of Hachimori-Matsuno for elliptic curves) giving the analytic and algebraic p-adic Iwasawa invariants of a modular eigenform over an abelian p-extension of Q to its p-adic Iwasawa invariants over Q. On the algebraic side our methods, which make use of congruences between modular forms, yield a Kida-type formula for a very general class of ordinary Galois representations. We are further able to deduce a Kida-type formula for elliptic curves at supersingular primes

Mazur-Tate elements of non-ordinary modular forms

We establish formulae for the Iwasawa invariants of Mazur--Tate elements of cuspidal eigenforms, generalizing known results in weight 2. Our first theorem deals with forms of "medium" weight, and our second deals with forms of small slope . We give examples illustrating the strange behavior which can occur in the high weight, high slope case

Lattice Supersymmetry in the Open XXZ Model: An Algebraic Bethe Ansatz Analysis

We reconsider the open XXZ chain of Yang and Fendley. This model possesses a lattice supersymmetry which changes the length of the chain by one site. We perform an algebraic Bethe ansatz analysis of the model and derive the commutation relations of the lattice SUSY operators with the four elements of the open-chain monodromy matrix. Hence we give the action of the SUSY operator on off-shell and on-shell Bethe states. We show that this action generally takes one on-shell Bethe eigenstate to another. The exception is that a zero-energy vacuum state will be a SUSY singlet. The SUSY pairings of Bethe roots we obtain are analogous to those found previously for closed chains by Fendley and Hagendorf by analysing the Bethe equations.Comment: 13 pages; dimension counting of SUSY operator image and kernel spaces added; references update

PT Symmetry on the Lattice: The Quantum Group Invariant XXZ Spin-Chain

We investigate the PT-symmetry of the quantum group invariant XXZ chain. We show that the PT-operator commutes with the quantum group action and also discuss the transformation properties of the Bethe wavefunction. We exploit the fact that the Hamiltonian is an element of the Temperley-Lieb algebra in order to give an explicit and exact construction of an operator that ensures quasi-Hermiticity of the model. This construction relys on earlier ideas related to quantum group reduction. We then employ this result in connection with the quantum analogue of Schur-Weyl duality to introduce a dual pair of C-operators, both of which have closed algebraic expressions. These are novel, exact results connecting the research areas of integrable lattice systems and non-Hermitian Hamiltonians.Comment: 32 pages with figures, v2: some minor changes and added references, version published in JP

A Free Field Representation of the Screening Currents of $U_q(\widehat{sl(3)})

We construct five independent screening currents associated with the $U_q(\widehat{sl(3)})$ quantum current algebra. The screening currents are expressed as exponentials of the eight basic deformed bosonic fields that are required in the quantum analogue of the Wakimoto realization of the current algebra. Four of the screening currents are `simple', in that each one is given as a single exponential field. The fifth is expressed as an infinite sum of exponential fields. For reasons we discuss, we expect that the structure of the screening currents for a general quantum affine algebra will be similar to the $U_q(\widehat{sl(3)})$ case.Comment: 21 pages (LaTeX), CRM-126

The Dynamical Correlation Function of the XXZ Model

We perform a spectral decomposition of the dynamical correlation function of the spin $1/2$ XXZ model into an infinite sum of products of form factors. Beneath the four-particle threshold in momentum space the only non-zero contributions to this sum are the two-particle term and the trivial vacuum term. We calculate the two-particle term by making use of the integral expressions for form factors provided recently by the Kyoto school. We evaluate the necessary integrals by expanding to twelfth order in $q$. We show plots of $S(w,k)$, for $k=0$ and $\pi$ at various values of the anisotropy parameter, and for fixed anisotropy at various $k$ around $0$ and $\pi$.Comment: 20 pages (LaTeX), CRM-219

Variation of Iwasawa invariants in Hida families

Let r : G_Q -> GL_2(Fpbar) be a p-ordinary and p-distinguished irreducible residual modular Galois representation. We show that the vanishing of the algebraic or analytic Iwasawa mu-invariant of a single modular form lifting r implies the vanishing of the corresponding mu-invariant for all such forms. Assuming that the mu-invariant vanishes, we also give explicit formulas for the difference in the algebraic or analytic lambda-invariants of modular forms lifting r. In particular, our formula shows that the lambda-invariant is constant on branches of the Hida family of r. We further show that our formulas are identical for the algebraic and analytic invariants, so that the truth of the main conjecture of Iwasawa theory for one form in the Hida family of r implies it for the entire Hida family

Vertex Operators and Matrix Elements of Uq(su(2)k)U_q(su(2)_k) via Bosonization

We construct bosonized vertex operators (VOs) and conjugate vertex operators (CVOs) of $U_q(su(2)_k)$ for arbitrary level $k$ and representation $j\leq k/2$. Both are obtained directly as two solutions of the defining condition of vertex operators - namely that they intertwine $U_q(su(2)_k)$ modules. We construct the screening charge and present a formula for the n-point function. Specializing to $j=1/2$ we construct all VOs and CVOs explicitly. The existence of the CVO allows us to place the calculation of the two-point function on the same footing as $k=1$; that is, it is obtained without screening currents and involves only a single integral from the CVO. This integral is evaluated and the resulting function is shown to obey the q-KZ equation and to reduce simply to both the expected $k=1$ and $q=1$ limits.Comment: 20 pages, LaTex. Minor change

Deep probabilistic methods for improved radar sensor modelling and pose estimation

Radar’s ability to sense under adverse conditions and at far-range makes it a valuable alternative to vision and lidar for mobile robotic applications. However, its complex, scene-dependent sensing process and significant noise artefacts makes working with radar challenging. Moving past classical rule-based approaches, which have dominated the literature to date, this thesis investigates deep and data-driven solutions across a range of tasks in robotics. Firstly, a deep approach is developed for mapping raw sensor measurements to a grid-map of occupancy probabilities, outperforming classical filtering approaches by a significant margin. A distribution over the occupancy state is captured, additionally allowing uncertainty in predictions to be identified and managed. The approach is trained entirely using partial labels generated automatically from lidar, without requiring manual labelling. Next, a deep model is proposed for generating stochastic radar measurements from simulated elevation maps. The model is trained by learning the forward and backward processes side-by-side, using a combination of adversarial and cyclical consistency constraints in combination with a partial alignment loss, using labels generated in lidar. By faithfully replicating the radar sensing process, new models can be trained for down-stream tasks, using labels that are readily available in simulation. In this case, segmentation models trained on simulated radar measurements, when deployed in the real world, are shown to approach the performance of a model trained entirely on real-world measurements. Finally, the potential of deep approaches applied to the radar odometry task are explored. A learnt feature space is combined with a classical correlative scan matching procedure and optimised for pose prediction, allowing the proposed method to outperform the previous state-of-the-art by a significant margin. Through a probabilistic consideration the uncertainty in the pose is also successfully characterised. Building upon this success, properties of the Fourier Transform are then utilised to separate the search for translation and angle. It is shown that this decoupled search results in a significant boost to run-time performance, allowing the approach to run in real-time on CPUs and embedded devices, whilst remaining competitive with other radar odometry methods proposed in the literature