Estimating χtop\chi_\mathrm{top} Lattice Artifacts from Flowed SU(2) Calorons

Lattice computations of the high-temperature topological susceptibility of QCD receive lattice-spacing corrections and suffer from systematics arising from the type and depth of gradient flow. We study the lattice spacing corrections to $\chi_\mathrm{top}$ semi-analytically by exploring the behavior of discretized Harrington-Shepard calorons under the action of different forms of gradient flow. From our study we conclude that $N_\tau = 6$ is definitely too small of a time extent to study the theory at temperatures of order $4~T_\mathrm{c}$ and we explore how the amount of gradient flow influences the continuum extrapolation.Comment: 10 pages, 8 figures (published version

A Sixth-Order Extension to the MATLAB Package bvp4c of J. Kierzenka and L. Shampine

A new two-point boundary value problem algorithm based upon the MATLAB bvp4c package of Kierzenka and Shampine is described. The algorithm, implemented in a new package bvp6c, uses the residual control framework of bvp4c (suitably modified for a more accurate finite difference approximation) to maintain a user specified accuracy. The new package is demonstrated to be as robust as the existing software, but more efficient for most problems, requiring fewer internal mesh points and evaluations to achieve the required accuracy

Quantization of anomaly coefficients in 6D N=(1,0)\mathcal{N}=(1,0) supergravity

We obtain new constraints on the anomaly coefficients of 6D $\mathcal{N}=(1,0)$ supergravity theories using local and global anomaly cancellation conditions. We show how these constraints can be strengthened if we assume that the theory is well-defined on any spin space-time with an arbitrary gauge bundle. We distinguish the constraints depending on the gauge algebra only from those depending on the global structure of the gauge group. Our main constraint states that the coefficients of the anomaly polynomial for the gauge group $G$ should be an element of $2 H^4(BG;\mathbb{Z}) \otimes \Lambda_S$ where $\Lambda_S$ is the unimodular string charge lattice. We show that the constraints in their strongest form are realized in F-theory compactifications. In the process, we identify the cocharacter lattice, which determines the global structure of the gauge group, within the homology lattice of the compactification manifold.Comment: 42 pages. v3: Some clarifications, typos correcte

Control of Networked Multiagent Systems with Uncertain Graph Topologies

Multiagent systems consist of agents that locally exchange information through a physical network subject to a graph topology. Current control methods for networked multiagent systems assume the knowledge of graph topologies in order to design distributed control laws for achieving desired global system behaviors. However, this assumption may not be valid for situations where graph topologies are subject to uncertainties either due to changes in the physical network or the presence of modeling errors especially for multiagent systems involving a large number of interacting agents. Motivating from this standpoint, this paper studies distributed control of networked multiagent systems with uncertain graph topologies. The proposed framework involves a controller architecture that has an ability to adapt its feed- back gains in response to system variations. Specifically, we analytically show that the proposed controller drives the trajectories of a networked multiagent system subject to a graph topology with time-varying uncertainties to a close neighborhood of the trajectories of a given reference model having a desired graph topology. As a special case, we also show that a networked multi-agent system subject to a graph topology with constant uncertainties asymptotically converges to the trajectories of a given reference model. Although the main result of this paper is presented in the context of average consensus problem, the proposed framework can be used for many other problems related to networked multiagent systems with uncertain graph topologies.Comment: 14 pages, 2 figure

The Power of Strong Fourier Sampling: Quantum Algorithms for Affine Groups and Hidden Shifts

Many quantum algorithms, including Shor's celebrated factoring and discrete log algorithms, proceed by reduction to a hidden subgroup problem, in which an unknown subgroup $H$ of a group $G$ must be determined from a quantum state $\psi$ over $G$ that is uniformly supported on a left coset of $H$. These hidden subgroup problems are typically solved by Fourier sampling: the quantum Fourier transform of $\psi$ is computed and measured. When the underlying group is nonabelian, two important variants of the Fourier sampling paradigm have been identified: the weak standard method, where only representation names are measured, and the strong standard method, where full measurement (i.e., the row and column of the representation, in a suitably chosen basis, as well as its name) occurs. It has remained open whether the strong standard method is indeed stronger, that is, whether there are hidden subgroups that can be reconstructed via the strong method but not by the weak, or any other known, method. In this article, we settle this question in the affirmative. We show that hidden subgroups $H$ of the $q$-hedral groups, i.e., semidirect products ${\mathbb Z}_q \ltimes {\mathbb Z}_p$, where $q \mid (p-1)$, and in particular the affine groups $A_p$, can be information-theoretically reconstructed using the strong standard method. Moreover, if $|H| = p/ {\rm polylog}(p)$, these subgroups can be fully reconstructed with a polynomial amount of quantum and classical computation. We compare our algorithms to two weaker methods that have been discussed in the literature—the “forgetful” abelian method, and measurement in a random basis—and show that both of these are weaker than the strong standard method. Thus, at least for some families of groups, it is crucial to use the full power of representation theory and nonabelian Fourier analysis, namely, to measure the high-dimensional representations in an adapted basis that respects the group's subgroup structure. We apply our algorithm for the hidden subgroup problem to new families of cryptographically motivated hidden shift problems, generalizing the work of van Dam, Hallgren, and Ip on shifts of multiplicative characters. Finally, we close by proving a simple closure property for the class of groups over which the hidden subgroup problem can be solved efficiently