arXiv subjects

Michael Kelly

Publications and source records attributed to Michael Kelly.

At least 19 recordsLinked to original sources

arXiv2024

Periodic orbit theory of Bethe-integrable quantum systems: an $N$-particle Berry-Tabor trace formula

One of the fundamental results of semiclassical theory is the existence of trace formulae showing how spectra of quantum mechanical systems emerge from massive interference among amplitudes related with time-periodic structures of the corresponding classical limit. If it displays the properties of Hamiltonian integrability, this connection is given by the celebrated Berry-Tabor trace formula, and the periodic structures it is built on are KAM tori supporting closed trajectories in phase space. Here we show how to extend this connection into the domain of quantum many-body systems displaying integrability in the sense of the Bethe ansatz, where a classical limit cannot be rigorously defined due to the presence of singular potentials. Formally following the original derivation of Berry and Tabor [1, 2], but applied to the Bethe equations without underlying classical structure, we obtain a many-particle trace formula for the density of states of N interacting bosons on a ring, the Lieb-Liniger model. Our semiclassical expressions are in excellent agreement with quantum mechanical results for $N$ = 2, 3 and 4 particles. For N = 2 we relate our results to the quantization of billiards with mixed boundary conditions. Our work paves the way towards the treatment of the important class of integrable many-body systems by means of semiclassical trace formulae pioneered by Michael Berry in the single-particle context.

quant-ph↗

arXiv2023

Advancements in Superconducting Microwave Cavities and Qubits for Quantum Information Systems

Superconducting microwave cavities with ultra-high Q-factors are revolutionizing the field of quantum computing, offering long coherence times exceeding 1 ms, which is critical for realizing scalable multi-qubit quantum systems with low error rates. In this work, we provide an in-depth analysis of recent advances in ultra-high Q-factor cavities, integration of Josephson junction-based qubits, and bosonic-encoded qubits in 3D cavities. We examine the sources of quantum state dephasing caused by damping and noise mechanisms in cavities and qubits, highlighting the critical challenges that need to be addressed to achieve even higher coherence times. We critically survey the latest progress made in implementing single 3D qubits using superconducting materials, normal metals, and multi-qubit and multi-state quantum systems. Our work sheds light on the promising future of this research area, including novel materials for cavities and qubits, modes with nontrivial topological properties, error correction techniques for bosonic qubits, and new light-matter interaction effects.

quant-ph↗

arXiv2018

HG-DAgger: Interactive Imitation Learning with Human Experts

Imitation learning has proven to be useful for many real-world problems, but approaches such as behavioral cloning suffer from data mismatch and compounding error issues. One attempt to address these limitations is the DAgger algorithm, which uses the state distribution induced by the novice to sample corrective actions from the expert. Such sampling schemes, however, require the expert to provide action labels without being fully in control of the system. This can decrease safety and, when using humans as experts, is likely to degrade the quality of the collected labels due to perceived actuator lag. In this work, we propose HG-DAgger, a variant of DAgger that is more suitable for interactive imitation learning from human experts in real-world systems. In addition to training a novice policy, HG-DAgger also learns a safety threshold for a model-uncertainty-based risk metric that can be used to predict the performance of the fully trained novice in different regions of the state space. We evaluate our method on both a simulated and real-world autonomous driving task, and demonstrate improved performance over both DAgger and behavioral cloning.

cs.RO↗

arXiv2018

Pattern Equivariant Mass Transport in Aperiodic Tilings and Cohomology

Suppose that we have a repetitive and aperiodic tiling $\bf T$ of $\mathbb{R}^n$, and two mass distributions $f_1$ and $f_2$ on $\mathbb{R}^n$, each pattern equivariant with respect to $\bf T$. Under what circumstances is it possible to do a bounded transport from $f_1$ to $f_2$? When is it possible to do this transport in a strongly or weakly pattern-equivariant way? We reduce these questions to properties of the \v Cech cohomology of the hull of $\bf T$, properties that in most common examples are already well-understood.

math.DS↗

arXiv2018

Morphology of AGN Emission Line Regions in SDSS-IV MaNGA Survey

Extended narrow-line regions (NLRs) around active galactic nuclei (AGN) are shaped by the distribution of gas in the host galaxy and by the geometry of the circumnuclear obscuration, and thus they can be used to test the AGN unification model. In this work, we quantify the morphologies of the narrow-line regions in 308 nearby AGNs ($z=0-0.14$, \lbol $\sim 10^{42.4-44.1}$ \erg{}) from the MaNGA survey. Based on the narrow-line region maps, we find that a large fraction (81\%) of these AGN have bi-conical NLR morphology. The distribution of their measured opening angles suggests that the intrinsic opening angles of the ionization cones has a mean value of 85--98$^\circ$ with a finite spread of 39-44$^\circ$ (1-$\sigma$). Our inferred opening angle distribution implies a number ratio of type I to type II AGN of 1:1.6--2.3, consistent with other measurements of the type I / type II ratio at low AGN luminosities. Combining these measurements with the WISE photometry data, we find that redder mid-IR color (lower effective temperature of dust) corresponds to stronger and narrower photo-ionized bicones. This relation is in agreement with the unification model that suggests that the bi-conical narrow-line regions are shaped by a toroidal dusty structure within a few pc from the AGN. Furthermore, we find a significant alignment between the minor axis of host galaxy disks and AGN ionization cones. Together, these findings suggest that obscuration on both circumnuclear ($\sim $pc) and galactic ($\sim$ kpc) scales are important in shaping and orienting the AGN narrow-line regions.

astro-ph.GA↗

arXiv2018

Unitarity violation in non-integer dimensional Gross-Neveu-Yukawa model

We construct an explicit example of unitarity violation in fermionic quantum field theories in non-integer dimensions. We study the two-point correlation function of four-fermion operators. We compute the one-loop anomalous dimensions of these operators in the Gross-Neveu-Yukawa Model. We find that at one loop order, the four-fermion operators split into three classes with one class having negative norms. This implies that the theory violates unitarity following the definition in arXiv:1512.00013 [hep-th].

hep-th↗

arXiv2017

The Beurling-Selberg Box Minorant Problem via Linear Programming Bounds

In this paper we investigate a high dimensional version of Selberg's minorant problem for the indicator function of an interval. In particular, we study the corresponding problem of minorizing the indicator function of the box $Q_{N}=[-1,1]^N$ by a function whose Fourier transform is supported in the same box $Q_N$. We show that when the dimension is sufficiently large there are no minorants with positive mass and we give an explicit lower bound for such dimension. On the other hand, we explicitly construct minorants for dimensions $1,2,3,4$ and $5$ and, as an application, we use them to produce an improved diophantine inequality for exponential sums.

math.CA↗

arXiv2016

Zooming into local active galactic nuclei: The power of combining SDSS-IV MaNGA with higher resolution integral field unit observations

Ionised gas outflows driven by active galactic nuclei (AGN) are ubiquitous in high luminosity AGN with outflow speeds apparently correlated with the total bolometric luminosity of the AGN. This empirical relation and theoretical work suggest that in the range L_bol ~ 10^43-45 erg/s there must exist a threshold luminosity above which the AGN becomes powerful enough to launch winds that will be able to escape the galaxy potential. In this paper, we present pilot observations of two AGN in this transitional range that were taken with the Gemini North Multi-Object Spectrograph Integral Field Unit (IFU). Both sources have also previously been observed within the Sloan Digital Sky Survey-IV (SDSS) Mapping Nearby Galaxies at Apache Point Observatory (MaNGA) survey. While the MaNGA IFU maps probe the gas fields on galaxy-wide scales and show that some regions are dominated by AGN ionization, the new Gemini IFU data zoom into the centre with four times better spatial resolution. In the object with the lower L_bol we find evidence of a young or stalled biconical AGN-driven outflow where none was obvious at the MaNGA resolution. In the object with the higher L_bol we trace the large-scale biconical outflow into the nuclear region and connect the outflow from small to large scales. These observations suggest that AGN luminosity and galaxy potential are crucial in shaping wind launching and propagation in low-luminosity AGN. The transition from small and young outflows to galaxy-wide feedback can only be understood by combining large-scale IFU data that trace the galaxy velocity field with higher resolution, small scale IFU maps.

astro-ph.GA↗

arXiv2016

Separated Nets in Nilpotent Groups

In this paper we generalize several results on separated nets in Euclidean space to separated nets in connected simply connected nilpotent Lie groups. We show that every such group $G$ contains separated nets that are not biLipschitz equivalent. We define a class of separated nets in these groups arising from a generalization of the cut-and-project quasi-crystal construction and show that generically any such separated net is bounded displacement equivalent to a separated net of constant covolume. In addition, we use a generalization of the Laczkovich criterion to provide `exotic' perturbations of such separated nets.

math.MG↗

arXiv2016

Effective equidistribution of translates of maximal horospherical measures in the space of lattices

Recently Mohammadi and Salehi-Golsefidy gave necessary and sufficient conditions under which certain translates of homogeneous measures converge, and they determined the limiting measures in the cases of convergence. The class of measures they considered includes the maximal horospherical measures. In this paper we prove the corresponding effective equidistribution results in the space of unimodular lattices. We also prove the corresponding results for probability measures with absolutely continuous densities in rank two and three. Then we address the problem of determining the error terms in two counting problems also considered by Mohammadi and Salehi-Golsefidy. In the first problem, we determine an error term for counting the number of lifts of a closed horosphere from an irreducible, finite-volume quotient of the space of positive definite $n\times n$ matrices of determinant one that intersect a ball with large radius. In the second problem, we determine a logarithmic error term for the Manin conjecture of a flag variety over $\mathbb{Q}$.

math.DS↗

arXiv2016

Constructing bounded remainder sets and cut-and-project sets which are bounded distance to lattices, II

Recent results of several authors have led to constructions of parallelotopes which are bounded remainder sets for totally irrational toral rotations. In this brief note we explain, in retrospect, how some of these results can easily be obtained from a geometric argument which was previously employed by Duneau and Oguey in the study of deformation properties of mathematical models for quasicrystals.

math.DS↗

arXiv2015

Lower Bound on the Rate of Adaptation in an Asexual Population

We consider a model of asexually reproducing individuals with random mutations and selection. The rate of mutations is proportional to the population size, $N$. The mutations may be either beneficial or deleterious. In a paper by Yu, Etheridge and Cuthbertson (2009) it was conjectured that the average rate at which the mean fitness increases in this model is $O(\log N/(\log\log N)^2)$. In this paper we show that for any time $t > 0$ there exist values $\epsilon_N \rightarrow 0$ and a fixed $c > 0$ such that the maximum fitness of the population is greater than $cs\log N/(\log\log N)^2$ for all times $s \in [\epsilon_N,t]$ with probability tending to 1 as $N$ tends to infinity.

math.PR↗

arXiv2014

One-sided Band-limited Approximations of Some Radial Functions

We construct majorants and minorants of a Gaussian function in Euclidean space that have Fourier transforms supported in a box. The majorants that we construct are shown to be extremal and our minorants are shown to be asymptotically extremal as the sides of the box become uniformly large. We then adapt the Distribution and Gaussian Subordination methods of Carneiro-Littmann-Vaaler to the multidimensional setting to obtain majorants and minorants for a class of radial functions. Periodic analogues of the main results are proven and applications to Hilbert-type inequalities are given.

math.CA↗

arXiv2014

Pattern Equivariant Cohomology and Theorems of Kesten and Oren

In 1966 Harry Kesten settled the Erd\H os-Sz\"usz conjecture on the local discrepancy of irrational rotations. His proof made heavy use of continued fractions and Diophantine analysis. In this paper we give a purely topological proof Kesten's theorem (and Oren's generalization of it) using the pattern equivariant cohomology of aperiodic tiling spaces.

math.DS↗

arXiv2014

A Variation on Selberg's Approximation Problem

Let $\alpha\in\mathbb{C}$ in the upper half-plane and let $I$ be an interval. We construct an analogue of Selberg's majorant of the characteristic function of $I$ that vanishes at the point $\alpha$. The construction is based on the solution to an extremal problem with positivity and interpolation constraints.

math.CA↗

arXiv2013

The Blaschke-Santalo Inequality

The Blaschke-Santal\'o Inequality is the assertion that the volume product of a centrally symmetric convex body in Euclidean space is maximized by (and only by) ellipsoids. In this paper we give a Fourier analytic proof of this fact.

math.MG↗

arXiv2012

Equivalence relations on separated nets arising from linear toral flows

In 1998, Burago-Kleiner and McMullen independently proved the existence of separated nets in $\mathbb{R}^d$ which are not bi-Lipschitz equivalent (BL) to a lattice. A finer equivalence relation than BL is bounded displacement (BD). Separated nets arise naturally as return times to a section for minimal $\mathbb{R}^d$-actions. We analyze the separated nets which arise via these constructions, focusing particularly on nets arising from linear $\mathbb{R}^d$-actions on tori. We show that generically these nets are BL to a lattice, and for some choices of dimensions and sections, they are generically BD to a lattice. We also show the existence of such nets which are not BD to a lattice.

math.DS↗

arXiv2012

Uniform Dilations in Higher Dimensions

A theorem of Glasner says that if $X$ is an infinite subset of the torus $\mathbb{T}$, then for any $\epsilon>0$, there exists an integer $n$ such that the dilation $nX=\{nx: x \in \mathbb{T} \}$ is $\epsilon$-dense (i.e, it intersects any interval of length $2\epsilon$ in $\mathbb{T}$). Alon and Peres provided a general framework for this problem, and showed quantitatively that one can restrict the dilation to be of the form $f(n)X$ where $f \in \mathbb{Z}[x]$ is not constant. Building upon the work of Alon and Peres, we study this phenomenon in higher dimensions. Let ${\bf A}(x)$ be an $L \times N$ matrix whose entries are in $\mathbb{Z}[x]$, and $X$ be an infinite subset of $\mathbb{T}^N$. Contrarily to the case $N=L=1$, it's not always true that there is an integer $n$ such that $\bA(n)X$ is $\epsilon$-dense in a translate of a subtorus of $\mathbb{T}^{L}$. We give a necessary and sufficient condition for matrices ${\bf A}$ for which this is true. We also prove an effective version of the result.

math.NT↗