A positive integer n is called a CLT number if every group of order n satisfies the converse of Lagrange’s Theorem. In this note, we find all CLT and supersolvable numbers up to 1000. We also formulate some questions about the distribution of these numbers. RÉSUMÉ: Un nombre entier naturel n est appelé nombre CLT si tout groupe d’ordre n satisfait la réciproque du Théorme de Lagrange. Nous déte...

We consider Wick’s Theorem for finite temperature and finite volume systems. Working at an operator level with a path ordered approach, we show that contrary to claims in the literature, expectation values of normal ordered products can be chosen to be zero and that results obtained are independent of volume. Thus the path integral and operator approaches to finite temperature and finite volume...

So far in the course we have been concerned almost exclusively with functions on the set {−1, 1}n (sometimes written F2 ). What about other product spaces? As alluded to in Problem #1 on Homeworks #4 and #5, we can develop some kind of “orthogonal decomposition” for functions on any product probability space X. This will give us some subset of our Fourier analysis, and it’s useful when one does...

This bachelor thesis provides a short overview of some major results in model theory concerning the spectrum function I(T, κ), where T is a firstorder theory and κ is a cardinal. In short, I(T, κ) tells us how many models of cardinality κ T has up to isomorphism. After summarising some basic definitions and results of elementary predicate logic, we turn our attention to types, both from a model...

A celebrated theorem of Marshall Hall implies that finitely generated free groups are subgroup separable and that all of their finitely generated subgroups are retracts of finite-index subgroups. We use topological techniques inspired by the work of Stallings to prove that all limit groups share these two properties. This answers a question of Sela and provides a purely topological proof that f...

and the last two (actually Lorentz transformations already include both) can be written x → x = ∑ ν Λ μ νx ν = Λνx ν , (using the Einstein summation convention), where the matrix Λ is a real matrix satisfying the pseudoorthogonality condition ΛμνgμρΛ ρ τ = gντ , which is required so that the length of a four-vector is preserved, x′ := xxμ = x . All together, this symmetry group is called the in...

We prove the following theorem: For a partially ordered set Q such that every countable subset has a strict upper bound, there is a forcing notion satisfying ccc such that, in the forcing model, there is a basis of the meager ideal of the real line which is order-isomorphic to Q with respect to set-inclusion. This is a variation of Hechler’s classical result in the theory of forcing.

If an integrable function f on the Heisenberg group is supported on the set B × R where B ⊂ Cn is compact and the group Fourier transform f̂(λ) is a finite rank operator for all λ ∈ R \ {0}, then f ≡ 0.

In this lecture, we show two results dealing with lower bounds in communication complexity. The first lower bound is an Ω(n) lower bound on the distributional complexity of Disjointness due to [3, 8]. Here we will present the simplified proof presented in [8]. In the second part, we will show how to obtain lower bounds on the unbounded error probabilistic communication complexity by Forster’s m...