Selective Categories and Linear Canonical Relations

A construction of Wehrheim and Woodward circumvents the problem that compositions of smooth canonical relations are not always smooth, building a category suitable for functorial quantization. To apply their construction to more examples, we introduce a notion of highly selective category, in which only certain morphisms and certain pairs of these morphisms are "good". We then apply this notion to the category $\mathbf{SLREL}$ of linear canonical relations and the result ${\rm WW}(\mathbf{SLREL})$ of our version of the WW construction, identifying the morphisms in the latter with pairs $(L,k)$ consisting of a linear canonical relation and a nonnegative integer. We put a topology on this category of indexed linear canonical relations for which composition is continuous, unlike the composition in $\mathbf{SLREL}$ itself. Subsequent papers will consider this category from the viewpoint of derived geometry and will concern quantum counterparts

A priori bounds for co-dimension one isometric embeddings

We prove a priori bounds for the trace of the second fundamental form of a $C^4$ isometric embedding into $R^{n+1}$ of a metric $g$ of non-negative sectional curvature on $S^n$, in terms of the scalar curvature, and the diameter of $g$. These estimates give a bound on the extrinsic geometry in terms of intrinsic quantities. They generalize estimates originally obtained by Weyl for the case $n=2$ and positive curvature, and then by P. Guan and the first author for non-negative curvature and $n=2$. Using $C^{2,\alpha}$ interior estimates of Evans and Krylov for concave fully nonlinear elliptic partial differential equations, these bounds allow us to obtain the following convergence theorem: For any $\epsilon>0$, the set of metrics of non-negative sectional curvature and scalar curvature bounded below by $\epsilon$ which are isometrically embedable in Euclidean space $R^{n+1}$ is closed in the H\"older space $C^{4,\alpha}$, $0<\alpha<1$. These results are obtained in an effort to understand the following higher dimensional version of the Weyl embedding problem which we propose: \emph{Suppose that $g$ is a smooth metric of non-negative sectional curvature and positive scalar curvature on \S^n$which is locally isometrically embeddable in$R^{n+1}$. Does$(S^n,g)$then admit a smooth global isometric embedding into$R^{n+1}$?

Template-based Gravitational-Wave Echoes Search Using Bayesian Model Selection

The ringdown of the gravitational-wave signal from a merger of two black holes has been suggested as a probe of the structure of the remnant compact object, which may be more exotic than a black hole. It has been pointed out that there will be a train of echoes in the late-time ringdown stage for different types of exotic compact objects. In this paper, we present a template-based search methodology using Bayesian statistics to search for echoes of gravitational waves. Evidence for the presence or absence of echoes in gravitational-wave events can be established by performing Bayesian model selection. The Occam factor in Bayesian model selection will automatically penalize the more complicated model that echoes are present in gravitational-wave strain data because of its higher degree of freedom to fit the data. We find that the search methodology was able to identify gravitational-wave echoes with Abedi et al.'s echoes waveform model about 82.3% of the time in simulated Gaussian noise in the Advanced LIGO and Virgo network and about 61.1% of the time in real noise in the first observing run of Advanced LIGO with $\geq 5\sigma$ significance. Analyses using this method are performed on the data of Advanced LIGO's first observing run, and we find no statistical significant evidence for the detection of gravitational-wave echoes. In particular, we find $<1\sigma$ combined evidence of the three events in Advanced LIGO's first observing run. The analysis technique developed in this paper is independent of the waveform model used, and can be used with different parametrized echoes waveform models to provide more realistic evidence of the existence of echoes from exotic compact objects.Comment: 16 pages, 6 figure

Equilibrium Configuration of Black Holes and the Inverse Scattering Method

The inverse scattering method is applied to the investigation of the equilibrium configuration of black holes. A study of the boundary problem corresponding to this configuration shows that any axially symmetric, stationary solution of the Einstein equations with disconnected event horizon must belong to the class of Belinskii-Zakharov solutions. Relationships between the angular momenta and angular velocities of black holes are derived.Comment: LaTeX, 14 pages, no figure

Uniqueness of radial solutions for the fractional Laplacian

We prove general uniqueness results for radial solutions of linear and nonlinear equations involving the fractional Laplacian $(-\Delta)^s$ with $s \in (0,1)$ for any space dimensions $N \geq 1$. By extending a monotonicity formula found by Cabre and Sire \cite{CaSi-10}, we show that the linear equation $(-\Delta)^s u+ Vu = 0$ in $\mathbb{R}^N$ has at most one radial and bounded solution vanishing at infinity, provided that the potential $V$ is a radial and non-decreasing. In particular, this result implies that all radial eigenvalues of the corresponding fractional Schr\"odinger operator $H=(-\Delta)^s + V$ are simple. Furthermore, by combining these findings on linear equations with topological bounds for a related problem on the upper half-space $\mathbb{R}^{N+1}_+$, we show uniqueness and nondegeneracy of ground state solutions for the nonlinear equation $(-\Delta)^s Q + Q - |Q|^{\alpha} Q = 0$ in $\mathbb{R}^N$ for arbitrary space dimensions $N \geq 1$ and all admissible exponents $\alpha >0$. This generalizes the nondegeneracy and uniqueness result for dimension N=1 recently obtained by the first two authors in \cite{FrLe-10} and, in particular, the uniqueness result for solitary waves of the Benjamin--Ono equation found by Amick and Toland \cite{AmTo-91}.Comment: 38 pages; revised version; various typos corrected; proof of Lemma 8.1 corrected; discussion of case \kappa_* =1 in the proof of Theorem 2 corrected with new Lemma A.2; accepted for publication in Comm. Pure. Appl. Mat

Finding diamonds in the rough: Targeted Sub-threshold Search for Strongly-lensed Gravitational-wave Events

Strong gravitational lensing of gravitational waves can produce duplicate signals separated in time with different amplitudes. We consider the case in which strong lensing produces identifiable gravitational-wave events and weaker sub-threshold signals hidden in the noise background. We present a search method for the sub-threshold signals using reduced template banks targeting specific confirmed gravitational-wave events. We apply the method to all events from Advanced LIGO's first and second observing run O1/O2. Using GW150914 as an example, we show that the method effectively reduces the noise background and raises the significance of (near-) sub-threshold triggers. In the case of GW150914, we can improve the sensitive distance by $2.0\% - 14.8\%$. Finally, we present the top $5$ possible lensed candidates for O1/O2 gravitational-wave events that passed our nominal significance threshold of False-Alarm-Rate $\leq 1/30$ days

Experimental Heat-Bath Cooling of Spins

Algorithmic cooling (AC) is a method to purify quantum systems, such as ensembles of nuclear spins, or cold atoms in an optical lattice. When applied to spins, AC produces ensembles of highly polarized spins, which enhance the signal strength in nuclear magnetic resonance (NMR). According to this cooling approach, spin-half nuclei in a constant magnetic field are considered as bits, or more precisely, quantum bits, in a known probability distribution. Algorithmic steps on these bits are then translated into specially designed NMR pulse sequences using common NMR quantum computation tools. The $algorithmic$ cooling of spins is achieved by alternately combining reversible, entropy-preserving manipulations (borrowed from data compression algorithms) with $selective$ $reset$, the transfer of entropy from selected spins to the environment. In theory, applying algorithmic cooling to sufficiently large spin systems may produce polarizations far beyond the limits due to conservation of Shannon entropy. Here, only selective reset steps are performed, hence we prefer to call this process "heat-bath" cooling, rather than algorithmic cooling. We experimentally implement here two consecutive steps of selective reset that transfer entropy from two selected spins to the environment. We performed such cooling experiments with commercially-available labeled molecules, on standard liquid-state NMR spectrometers. Our experiments yielded polarizations that $bypass$ $Shannon's$ $entropy$-$conservation$ $bound$, so that the entire spin-system was cooled. This paper was initially submitted in 2005, first to Science and then to PNAS, and includes additional results from subsequent years (e.g. for resubmission in 2007). The Postscriptum includes more details.Comment: 20 pages, 8 figures, replaces quant-ph/051115

Multi-Dialect Speech Recognition With A Single Sequence-To-Sequence Model

Sequence-to-sequence models provide a simple and elegant solution for building speech recognition systems by folding separate components of a typical system, namely acoustic (AM), pronunciation (PM) and language (LM) models into a single neural network. In this work, we look at one such sequence-to-sequence model, namely listen, attend and spell (LAS), and explore the possibility of training a single model to serve different English dialects, which simplifies the process of training multi-dialect systems without the need for separate AM, PM and LMs for each dialect. We show that simply pooling the data from all dialects into one LAS model falls behind the performance of a model fine-tuned on each dialect. We then look at incorporating dialect-specific information into the model, both by modifying the training targets by inserting the dialect symbol at the end of the original grapheme sequence and also feeding a 1-hot representation of the dialect information into all layers of the model. Experimental results on seven English dialects show that our proposed system is effective in modeling dialect variations within a single LAS model, outperforming a LAS model trained individually on each of the seven dialects by 3.1 ~ 16.5% relative.Comment: submitted to ICASSP 201