Bounds for graph regularity and removal lemmas

We show, for any positive integer k, that there exists a graph in which any equitable partition of its vertices into k parts has at least ck^2/\log^* k pairs of parts which are not \epsilon-regular, where c,\epsilon>0 are absolute constants. This bound is tight up to the constant c and addresses a question of Gowers on the number of irregular pairs in Szemer\'edi's regularity lemma. In order to gain some control over irregular pairs, another regularity lemma, known as the strong regularity lemma, was developed by Alon, Fischer, Krivelevich, and Szegedy. For this lemma, we prove a lower bound of wowzer-type, which is one level higher in the Ackermann hierarchy than the tower function, on the number of parts in the strong regularity lemma, essentially matching the upper bound. On the other hand, for the induced graph removal lemma, the standard application of the strong regularity lemma, we find a different proof which yields a tower-type bound. We also discuss bounds on several related regularity lemmas, including the weak regularity lemma of Frieze and Kannan and the recently established regular approximation theorem. In particular, we show that a weak partition with approximation parameter \epsilon may require as many as 2^{\Omega(\epsilon^{-2})} parts. This is tight up to the implied constant and solves a problem studied by Lov\'asz and Szegedy.Comment: 62 page

Infinite matrices may violate the associative law

The momentum operator for a particle in a box is represented by an infinite order Hermitian matrix $P$. Its square $P^2$ is well defined (and diagonal), but its cube $P^3$ is ill defined, because $P P^2\neq P^2 P$. Truncating these matrices to a finite order restores the associative law, but leads to other curious results.Comment: final version in J. Phys. A28 (1995) 1765-177

Testing formula satisfaction

We study the query complexity of testing for properties defined by read once formulae, as instances of massively parametrized properties, and prove several testability and non-testability results. First we prove the testability of any property accepted by a Boolean read-once formula involving any bounded arity gates, with a number of queries exponential in \epsilon and independent of all other parameters. When the gates are limited to being monotone, we prove that there is an estimation algorithm, that outputs an approximation of the distance of the input from satisfying the property. For formulae only involving And/Or gates, we provide a more efficient test whose query complexity is only quasi-polynomial in \epsilon. On the other hand we show that such testability results do not hold in general for formulae over non-Boolean alphabets; specifically we construct a property defined by a read-once arity 2 (non-Boolean) formula over alphabets of size 4, such that any 1/4-test for it requires a number of queries depending on the formula size

How does an interacting many-body system tunnel through a potential barrier to open space?

The tunneling process in a many-body system is a phenomenon which lies at the very heart of quantum mechanics. It appears in nature in the form of alpha-decay, fusion and fission in nuclear physics, photoassociation and photodissociation in biology and chemistry. A detailed theoretical description of the decay process in these systems is a very cumbersome problem, either because of very complicated or even unknown interparticle interactions or due to a large number of constitutent particles. In this work, we theoretically study the phenomenon of quantum many-body tunneling in a more transparent and controllable physical system, in an ultracold atomic gas. We analyze a full, numerically exact many-body solution of the Schr\"odinger equation of a one-dimensional system with repulsive interactions tunneling to open space. We show how the emitted particles dissociate or fragment from the trapped and coherent source of bosons: the overall many-particle decay process is a quantum interference of single-particle tunneling processes emerging from sources with different particle numbers taking place simultaneously. The close relation to atom lasers and ionization processes allows us to unveil the great relevance of many-body correlations between the emitted and trapped fractions of the wavefunction in the respective processes.Comment: 18 pages, 4 figures (7 pages, 2 figures supplementary information

Atom interferometry with trapped Bose-Einstein condensates: Impact of atom-atom interactions

Interferometry with ultracold atoms promises the possibility of ultraprecise and ultrasensitive measurements in many fields of physics, and is the basis of our most precise atomic clocks. Key to a high sensitivity is the possibility to achieve long measurement times and precise readout. Ultra cold atoms can be precisely manipulated at the quantum level, held for very long times in traps, and would therefore be an ideal setting for interferometry. In this paper we discuss how the non-linearities from atom-atom interactions on one hand allow to efficiently produce squeezed states for enhanced readout, but on the other hand result in phase diffusion which limits the phase accumulation time. We find that low dimensional geometries are favorable, with two-dimensional (2D) settings giving the smallest contribution of phase diffusion caused by atom-atom interactions. Even for time sequences generated by optimal control the achievable minimal detectable interaction energy $\Delta E^{\rm min}$ is on the order of 0.001 times the chemical potential of the BEC in the trap. From there we have to conclude that for more precise measurements with atom interferometers more sophisticated strategies, or turning off the interaction induced dephasing during the phase accumulation stage, will be necessary.Comment: 28 pages, 13 figures, extended and correcte

Randomization in Criminal Justice: A Criminal Law Conversation

In this Criminal Law Conversation (Robinson, Ferzan & Garvey, eds., Oxford 2009), the authors debate whether there is a role for randomization in the penal sphere - in the criminal law, in policing, and in punishment theory. In his Tanner lectures back in 1987, Jon Elster had argued that there was no role for chance in the criminal law: “I do not think there are any arguments for incorporating lotteries in present-day criminal law,” Elster declared. Bernard Harcourt takes a very different position and embraces chance in the penal sphere, arguing that randomization is often the only way to avoid the pitfalls of ideology and unconscious bias. Alon Harel challenges Harcourt’s position, arguing that he is overly skeptical and that instead of embracing chance by default, he should abandon his skepticism for the sake of defending randomization. Ken Levy argues that Harcourt confuses power with right and that it is not possible to embrace randomization without first addressing the proper justification for punishment. Michael O’Hear acknowledges the significant role of luck in contemporary punishment practices, but he argues for channeling chance in more appropriate and useful directions. Alice Ristroph, while also acknowledging the significant role of chance in the criminal law, argues that instead of embracing chance at moments of indeterminacy, it would be better simply not to punish. In a reply, Harcourt responds to these criticisms and argues that we should think of randomization in the punishment field as a way to get beyond punishment as a form of social engineering – as a practice intended to change humans, to correct delinquents, to treat the deviant, or to deter the super-predator. The increased use of chance to resolve issues at moments of indeterminacy, Harcourt argues, could usher in a world in which punishment is chastened by critical reason – an idea, he suggests, worth taking seriously

Harmonic generation by atoms in circularly polarized two-color laser fields with coplanar polarizations and commensurate frequencies

The generation of harmonics by atoms or ions in a two-color, coplanar field configuration with commensurate frequencies is investigated through both, an analytical calculation based on the Lewenstein model and the numerical ab initio solution of the time-dependent Schroedinger equation of a two-dimensional model ion. Through the analytical model, selection rules for the harmonic orders in this field configuration, a generalized cut-off for the harmonic spectra, and an integral expression for the harmonic dipole strength is provided. The numerical results are employed to test the predictions of the analytical model. The scaling of the cut-off as a function of both, one of the laser intensities and frequency ratio $\eta$, as well as entire spectra for different $\eta$ and laser intensities are presented and analyzed. The theoretical cut-off is found to be an upper limit for the numerical results. Other discrepancies between analytical model and numerical results are clarified by taking into account the probabilities of the absorption processes involved.Comment: 8 figure