Triangle Seminars
Tuesday, 27 Nov 2012
Sums, Restricted Sums and Differences
David Penman
(Essex)
Abstract:
Given a (nonempty) set \(A\) of integers, two of the most obvious things to do with it are to form the sumset \(A+A=\{a+b:\,a,b\in A\}\) and the difference set \(A-A=\{a-b:\,a,b\in A\}\). One might also wish to consider the restricted sumset \(A\hat{+}A=\{a+b:\,a,b\in A,\,a\neq b\}\). One can then ask various obvious questions about the relationships between the sizes of various of these sets and what this implies about structure, and I shall discuss some known results on this, including generalisations to more general contexts, e.g. in group theory. An intuition one might have is that the sumset/restricted sumset will be smaller than the difference set as addition is commutative but subtraction isn't: I shall survey various known results showing that this intuition is non-trivially wrong. At the end I shall discuss some recent constructions of sets \(A\) which give new record large values of \(\log(|A+A|)/\log(|A-A|)\). The original part of the talk is based on joint work with my research student Matthew Wells.
Given a (nonempty) set \(A\) of integers, two of the most obvious things to do with it are to form the sumset \(A+A=\{a+b:\,a,b\in A\}\) and the difference set \(A-A=\{a-b:\,a,b\in A\}\). One might also wish to consider the restricted sumset \(A\hat{+}A=\{a+b:\,a,b\in A,\,a\neq b\}\). One can then ask various obvious questions about the relationships between the sizes of various of these sets and what this implies about structure, and I shall discuss some known results on this, including generalisations to more general contexts, e.g. in group theory. An intuition one might have is that the sumset/restricted sumset will be smaller than the difference set as addition is commutative but subtraction isn't: I shall survey various known results showing that this intuition is non-trivially wrong. At the end I shall discuss some recent constructions of sets \(A\) which give new record large values of \(\log(|A+A|)/\log(|A-A|)\). The original part of the talk is based on joint work with my research student Matthew Wells.
Posted by: KCL
Wednesday, 28 Nov 2012
Prospects of Supersymmetry at the LHC: Recent LHC Results
📍 London
Ben Allanach
(Cambridge, DAMTP)
Abstract:
I will summarise the Higgs search results and searches for supersymmetric particles from the LHC, commenting on the
recent evidence found for rare decays of the Bs meson into mu+ mu-. I shall comment on what the Higgs means for the
discovery prospects for supersymmetry.
I will summarise the Higgs search results and searches for supersymmetric particles from the LHC, commenting on the
recent evidence found for rare decays of the Bs meson into mu+ mu-. I shall comment on what the Higgs means for the
discovery prospects for supersymmetry.
Posted by: KCL
Prospects of Supersymmetry at the LHC: Why Should we believe in Supersymmetry?
📍 London
Matt Dolan
(Durham)
Abstract:
I will summarise why SUSY may be a good idea phenomenologically, and what
theoretical and phenomenological problems it can solve for us. I'll argue
that there's more to life than the CMSSM, and discuss what we might learn
from the continuing absence of BSM signals at the LHC.
I will summarise why SUSY may be a good idea phenomenologically, and what
theoretical and phenomenological problems it can solve for us. I'll argue
that there's more to life than the CMSSM, and discuss what we might learn
from the continuing absence of BSM signals at the LHC.
Posted by: KCL
Sunday, 2 Dec 2012
Quantum mechanical models, cellular automata and a discrete action principle
Thomas Elze
(Pisa University)
Abstract:
It will be shown that the dynamics of discrete (integer-valued) Hamiltonian cellular automata can only be consistently defined, if it is linear in the same sense that unitary evolution in quantum mechanics is linear. This suggests us to look for an invertible map between such automata and continuous quantum mechanical models. Based on sampling theory, such a map can indeed be constructed and leads to quantum mechanical models which incorporate a fundamental scale. The admissible observables, the one-to-one correspondence of the respective conservation laws, and the existence of solutions of the modified dispersion relation for stationary states are discussed.
References:
H.-T. Elze, Action principle for cellular automata and the linearity of quantum mechanics, Phys. Rev. A 89, 012111 (2014) [arXiv:1312.1615];
do., Journal of Physics: Conference Series 504 (2014) 012004 [arXiv:1403.2646].
It will be shown that the dynamics of discrete (integer-valued) Hamiltonian cellular automata can only be consistently defined, if it is linear in the same sense that unitary evolution in quantum mechanics is linear. This suggests us to look for an invertible map between such automata and continuous quantum mechanical models. Based on sampling theory, such a map can indeed be constructed and leads to quantum mechanical models which incorporate a fundamental scale. The admissible observables, the one-to-one correspondence of the respective conservation laws, and the existence of solutions of the modified dispersion relation for stationary states are discussed.
References:
H.-T. Elze, Action principle for cellular automata and the linearity of quantum mechanics, Phys. Rev. A 89, 012111 (2014) [arXiv:1312.1615];
do., Journal of Physics: Conference Series 504 (2014) 012004 [arXiv:1403.2646].
Posted by: IC