Non-existence of (3,2)-Equicolourings in C k -Designs

Abstract A block colouring of a C k -design Σ = (X, B) of order v (odd) is a mapping φ : B → C, where blocks and d(x) is called, using graph theoretic terminology, the degree of the vertex x. A partition of degree D into s parts defines a colouring of type s in which the blocks containing a vertex x are coloured exactly with s colours. For a vertex x and for i = 1, 2, . . . , s, indicate by B x,i the set of all the blocks incident in x and coloured with the colour C i . A colouring of type s is equitable if, for every vertex x, it is |B x,i −B x,j | ≤ 1, for all i, j = 1, . . . , s. If |C| = r, such a colouring will said an (r, s)-equiblock-colouring. In this paper we prove the non-existence of (r, s)-equiblock-colourings, having s = 2 and r = 3, for some classes of C 4 -designs. Mathematics Subject Classification: 05B0

The full Quantum Spectral Curve for AdS4/CFT3AdS_4/CFT_3

The spectrum of planar N=6 superconformal Chern-Simons theory, dual to type IIA superstring theory on $AdS_4 \times CP^3$, is accessible at finite coupling using integrability. Starting from the results of [arXiv:1403.1859], we study in depth the basic integrability structure underlying the spectral problem, the Quantum Spectral Curve. The new results presented in this paper open the way to the quantitative study of the spectrum for arbitrary operators at finite coupling. Besides, we show that the Quantum Spectral Curve is embedded into a novel kind of Q-system, which reflects the OSp(4|6) symmetry of the theory and leads to exact Bethe Ansatz equations. The discovery of this algebraic structure, more intricate than the one appearing in the $AdS_5/CFT_4$ case, could be a first step towards the extension of the method to $AdS_3/CFT_2$.Comment: 43 + 27 pages, 7 figures. v4: text improved, more details and App D included. This is the same as the published version JHEP09(2017)140, with small typos corrected in App

On the Construction of Scattering Amplitudes for Spinning Massless Particles

In this paper the general form of scattering amplitudes for massless particles with equal spins s ($s s \to s s$) or unequal spins ($s_a s_b \to s_a s_b$) are derived. The imposed conditions are that the amplitudes should have the lowest possible dimension, have propagators of dimension $m^{-2}$, and obey gauge invariance. It is shown that the number of momenta required for amplitudes involving particles with s > 2 is higher than the number implied by 3-vertices for higher spin particles derived in the literature. Therefore, the dimension of the coupling constants following from the latter 3-vertices has a smaller power of an inverse mass than our results imply. Consequently, the 3-vertices in the literature cannot be the first interaction terms of a gauge-invariant theory. When no spins s > 2 are present in the process the known QCD, QED or (super) gravity amplitudes are obtained from the above general amplitudes.Comment: 19 pages, Late

Nonlinear surface plasmons

We derive an asymptotic equation for quasi-static, nonlinear surface plasmons propagating on a planar interface between isotropic media. The plasmons are nondispersive with a constant linearized frequency that is independent of their wavenumber. The spatial profile of a weakly nonlinear plasmon satisfies a nonlocal, cubically nonlinear evolution equation that couples its left-moving and right-moving Fourier components. We prove short-time existence of smooth solutions of the asymptotic equation and describe its Hamiltonian structure. We also prove global existence of weak solutions of a unidirectional reduction of the asymptotic equation. Numerical solutions show that nonlinear effects can lead to the strong spatial focusing of plasmons. Solutions of the unidirectional equation appear to remain smooth when they focus, but it is unclear whether or not focusing can lead to singularity formation in solutions of the bidirectional equation

Graphs with few spanning substructures

In this thesis, we investigate a number of problems related to spanning substructures of graphs. The first few chapters consider extremal problems related to the number of forest-like structures of a graph. We prove that one can find a threshold graph which contains the minimum number of spanning pseudoforests, as well as rooted spanning forests, amongst all graphs on n vertices and e edges. This has left the open question of exactly which threshold graphs have the minimum number of these spanning substructures. We make progress towards this question in particular cases of spanning pseudoforests. The final chapter takes on a different flavor---we determine the complexity of a problem related to Hamilton cycles in hypergraphs. Dirac\u27s theorem states that graphs with minimum degree at least half the size of the vertex set are guaranteed to have a Hamilton cycle. In 1993, Karpinksi, Dahlhaus, and Hajnal proved that for any c\u3c1/2, the problem of determining whether a graph with minimum degree at least cn has a Hamilton cycle is NP-complete. The analogous problem in hypergraphs, for both a Dirac-type condition and complexity, are just as interesting. We prove that for classes of hypergraphs with certain minimum vertex degree conditions, the problem of determining whether or not they contain an l-Hamilton cycle is NP-complete. Advisor: Professor Jamie Radcliff