Quantum unique ergodicity – QUE – is concerned with the distributional properties of high-frequency eigenfunctions of the Laplacian on a domain Ω, that is of solutions to the equation ∆φ j + λ j φ j = 0 with the Dirichlet boundary conditions φ j | ∂Ω = 0 and normalization Ω φ 2 j dx dy = 1, where ∆ is the appropriate Laplacian. There is a connection between the high-frequency states and the cla...

In this paper we apply some new and some old methods in order to construct classical and intuitionistic models for theories of bounded arithmetic. We use these models to obtain proof theoretic consequences. In particular, we construct an ωchain of models of BASIC such that the union of its worlds satisfies S2 but none of its worlds satisfies the sentence ∀x∃y(x = 0 ∨ x = y + 1). Interpreting th...

Monier and Rabin proved that an odd composite can pass the Strong Probable Prime Test for at most 1 4 of the possible bases. In this paper, a probable prime test is developed using quadratic polynomials and the Frobenius automor-phism. The test, along with a xed number of trial divisions, ensures that a composite n will pass for less than 1 7710 of the polynomials x 2 ? bx ? c with b 2 +4c n = ...

It is well-known that all finite connected graphs have a unique prime factor decomposition (PFD) with respect to the strong graph product which can be computed in polynomial time. Essential for the PFD computation is the construction of the so-called Cartesian skeleton of the graphs under investigation. In this contribution, we show that every connected thin hypergraph H has a unique prime fact...

As an extension of known correlations, galaxies over four decades in stellar mass are shown to obey a strong correlation between u − K colors and gas-to-stellar mass (G/S) ratios, using neutral H I gas masses and stellar mass-to-light ratios derived from optical colors. The correlation holds for G/S ratios ranging from nearly 10:1 to 1:100 for a sample obtained by merging the SDSS DR2, 2MASS, a...

ETH Workshop on Software Correctness and Reliability, Zürich, October 2–3, 2015 © P. Cousot Software correctness proofs • Any formal proof of a non-trivial program requires a reasoning by mathematical induction (e.g., following Turing, on the number of program execution steps): • Invent an inductive argument (e.g. invariant, variant function), the hardest part • Prove the base case and inductiv...

Experiments on pure electron plasmas have found that the decay of two-dimensional ~2D! turbulence can lead to spontaneous formation of ‘‘vortex crystals,’’ which are symmetric arrays of strong vortices within a background of weaker vorticity. This paper presents a theory of these novel equilibrium states of 2D turbulence. The paper consists of two parts. In the first part, we show that the vort...

We develop an algebraic underpinning of backtracking monad transformers in the general setting of monoidal categories. As our main technical device, we introduce Eilenberg–Moore monoids, which combine monoids with algebras for strong monads. We show that Eilenberg–Moore monoids coincide with algebras for the list monad transformer (‘done right’) known from Haskell libraries. From this, we obtai...

The Markov Decision Problem (MDP) plays a central role in AI as an abstraction of sequential decision making. We contribute to the theoretical analysis of MDP planning, which is the problem of computing an optimal policy for a given MDP. Specifically, we furnish improved strong worstcase upper bounds on the running time of MDP planning. Strong bounds are those that depend only on the number of ...