Clock Synchronization based on Second-Order Quantum Coherence of Entangled Photons

We present an algorithm for synchronizing two clocks based on second-order quantum interference between entangled photons generated by parametric down-conversion. The procedure is distinct from the standard Einstein two-way clock synchronization method in that photon correlations are used to define simultaneous events in the frame of reference of a Hong-Ou-Mandel (HOM) interferometer. Once the HOM interferometer is balanced, by use of an adjustable optical delay in one arm, arrival times of simultaneously generated photons are recorded by each clock. Classical information on the arrival times is sent from one clock to the other, and a correlation of arrival times is done to determine the clock offset.Comment: 5 pages, 2 figures. ReVTeX

Unconditional regularity and trace results for the isentropic Euler equations with γ=3\gamma = 3

In this paper, we study the regularity properties of bounded entropy solutions to the isentropic Euler equations with $\gamma = 3$. First, we use a blow-up technique to obtain a new trace theorem for all such solutions. Second, we use a modified De Giorgi type iteration on the kinetic formulation to show a new partial regularity result on the Riemann invariants. We are able to conclude that in fact for any bounded entropy solution $u$, the density $\rho$ is almost everywhere upper semicontinuous away from vacuum. To our knowledge, this is the first example of a nonlinear hyperbolic system, which fails to be Temple class, but has the property that generic $L^\infty$ initial data give rise to bounded entropy solutions with a form of near classical regularity. This provides one example that $2\times 2$ hyperbolic systems can possess some of the more striking regularizing effects known to hold generically in the genuinely nonlinear, multidimensional scalar setting. While we are not able to use our regularity results to show unconditional uniqueness, the results substantially lower the likelihood that current methods of convex integration can be used in this setting.Comment: introduction reworked, included new application of method (Theorem 1.2) replacing construction of generalized characteristics in previous version, 25 page

Nonlinear asymptotic stability in L∞L^\infty for Lipschitz solutions to scalar conservation laws

In this note, we show nonlinear stability in $L^\infty$ for Lipschitz solutions to genuinely nonlinear, multi-dimensional scalar conservation laws. As an application, we are able to compute explicit algebraic decay rates of the $L^\infty$ norm of perturbations of global-in-time Lipschitz solutions, including perturbations of planar rarefaction waves. Our analysis uses the De Giorgi method applied to the kinetic formulation and is an extension of the method introduced recently by Silvestre in [Comm. Pure Appl. Math., 72(6):1321-1348, 2019].Comment: 10 pages, Corollary 2 removed, typos fixed, and additional references adde

Local-in-time strong solutions of the homogeneous Landau-Coulomb with LpL^p initial datum

In this article, we show local-in-time existence of strong solutions to the homogeneous Landau equation with Coulomb potential for general initial datum $f_{in} \in L^p$ for $p$ arbitrarily close to $3/2$. The constraint $p > 3/2$ has appeared in several related works and appears to be the minimal integrability assumption achievable with current techniques. We adapt recent ODE methods and conditional regularity results appearing in [arXiv:2303.02281] to produce new short time a priori smoothing estimates for large $L^p$ data, provided $p > 3/2$. These estimates enable us to construct local-in-time strong solutions for the corresponding initial data, but also enable us to show directly a large number of unweighted Prodi-Serrin type results, providing a form of weak-strong uniqueness for our solutions.Comment: 19 pages, 1 figur

Think Different: Applying the Old Macintosh Mantra to the Computability of the SUSY Auxiliary Field Problem

Starting with valise supermultiplets obtained from 0-branes plus field redefinitions, valise adinkra networks, and the "Garden Algebra," we discuss an architecture for algorithms that (starting from on-shell theories and, through a well-defined computation procedure), search for off-shell completions. We show in one dimension how to directly attack the notorious "off-shell auxiliary field" problem of supersymmetry with algorithms in the adinkra network-world formulation.Comment: 28 pages, 1 figur

Nonlinear regularization estimates and global well-posedness for the Landau-Coulomb equation near equilibrium

We consider the Landau equation with Coulomb potential in the spatially homogeneous case. We show short time propagation of smallness in $L^p$ norms for $p>3/2$ and instantaneous regularization in Sobolev spaces. This yields new short time quantitative a priori estimates that are unconditional near equilibrium. We combine these estimates with existing literature on global well-posedness for regular data to extend the well-posedness theory to small $L^p$ data with $p$ arbitrarily close to $3/2$. The threshold $p = 3/2$ agrees with previous work on conditional regularity for the Landau equation in the far from equilibrium regime.Comment: 25 pages; minor revisions to introduction and changes to the title/abstrac

Global smooth solutions to the Landau-Coulomb equation in L3/2L^{3/2}

We consider the homogeneous Landau equation in $\mathbb{R}^3$ with Coulomb potential and initial data in polynomially weighted $L^{3/2}$. We show that there exists a smooth solution that is bounded for all positive times. The proof is based on short-time regularization estimates for the Fisher information, which, combined with the recent result of Guillen and Silvestre, yields the existence of a global-in-time smooth solution. Additionally, if the initial data belongs to $L^p$ with $p>3/2$, there is a unique solution. At the crux of the result is a new $\varepsilon$-regularity criterion in the spirit of the Caffarelli-Kohn-Nirenberg theorem: a solution which is small in weighted $L^{3/2}$ is regular. Although the $L^{3/2}$ norm is a critical quantity for the Landau-Coulomb equation, using this norm to measure the regularity of solutions presents significant complications. For instance, the $L^{3/2}$ norm alone is not enough to control the $L^\infty$ norm of the competing reaction and diffusion coefficients. These analytical challenges caused prior methods relying on the parabolic structure of the Landau-Coulomb to break down. Our new framework is general enough to handle slowly decaying and singular initial data, and provides the first proof of global well-posedness for the Landau-Coulomb equation with rough initial data.Comment: 37 pages. Added a corollary on the convergence to equilibrium in $L^\infty$ (Corollary 1.5

Sharp a-contraction estimates for small extremal shocks

In this paper, we study the $a$-contraction property of small extremal shocks for 1-d systems of hyperbolic conservation laws endowed with a single convex entropy, when subjected to large perturbations. We show that the weight coefficient $a$ can be chosen with amplitude proportional to the size of the shock. The main result of this paper is a key building block in the companion paper, [{arXiv:2010.04761}, 2020], in which uniqueness and BV-weak stability results for $2\times 2$ systems of hyperbolic conservation laws are proved.Comment: 41 pages, 2 figures, minor correction to lemma 4.2 and corresponding changes to section

Site-Selective Cross-Coupling of Remote Chlorides Enabled by Electrostatically-Directed Palladium Catalysis

Control of site-selectivity in chemical reactions that occur remote from existing functionality remains a major challenge in synthetic chemistry. We describe a strategy that enables three of the most commonly used cross-coupling processes to occur with high site-selectivity on dichloroarenes which bear acidic functional groups. We have achieved this by repurposing an established sulfonylated phosphine ligand to exploit its inherent bifunctionality. Mechanistic studies suggest that the sulfonate group engages in attractive electrostatic interactions with the associated cation of deprotonated substrate, guiding cross-coupling to the chloride at the arene meta-position. This counterintuitive combination of anionic ligand and anionic substrate demonstrates an alternative design principle when considering applying non-covalent interactions to direct catalysis.We are grateful to AstraZeneca for PhD studentships through the AZ‐Cambridge PhD program (W.A.G. and R.P.), the Royal Society for a University Research Fellowship (R.J.P.), the EPSRC (EP/N005422/1) and the ERC (StG 757381)