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1. "Manifold curvature from covariance analysis"

   Alvarez-Vizoso, J.; Kirby, M.; Peterson, C. Department of (June 2018, IEEE Statistical Signal Processing Workshop)

   Principal component analysis of cylindrical neighborhoods is proposed to study the local geometry of embedded Riemannian manifolds. At every generic point and scale, a highdimensional cylinder orthogonal to the tangent space at the point cuts out a path-connected patch whose point-set distribution in ambient space encodes the intrinsic and extrinsic curvature. The covariance matrix of the points from that neighborhood has eigenvectors whose scale limit tends to the Frenet-Serret frame for curves, and to what we call the Ricci-Weingarten principal directions for submanifolds. More importantly, the limit of differences and products of eigenvalues can be used to recover curvature information at the point. The formula for hypersurfaces in terms of principal curvatures is particularly simple and plays a crucial role in the study of higher-codimension cases. 

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2. Universal bounds on CFT Distance Conjecture

   https://doi.org/10.1007/JHEP12(2024)154  

   Ooguri, Hirosi; Wang, Yifan (December 2024, Journal of High Energy Physics)

   A<sc>bstract</sc> For any unitary conformal field theory in two dimensions with the central chargec, we prove that, if there is a nontrivial primary operator whose conformal dimension ∆ vanishes in some limit on the conformal manifold, the Zamolodchikov distancetto the limit is infinite, the approach to this limit is exponential ∆ = exp(−αt+O(1)), and the decay rate obeys the universal boundsc^−1/2≤α≤ 1. In the limit, we also find that an infinite tower of primary operators emerges without a gap above the vacuum and that the conformal field theory becomes locally a tensor product of a sigma-model in the large radius limit and a compact theory. As a corollary, we establish a part of the Distance Conjecture about gravitational theories in three-dimensional anti-de Sitter space. In particular, our bounds onαindicate that the emergence of exponentially light states is inevitable as the moduli field corresponding totrolls beyond the Planck scale along the steepest path and that this phenomenon can begin already at the curvature scale of the bulk geometry. We also comment on implications of our bounds for gravity in asymptotically flat spacetime by taking the flat space limit and compare with the Sharpened Distance Conjecture. 

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3. Effects of Pressure and Characteristic Scales on the Structural and Statistical Features of Methane/Air Turbulent Premixed Flames

   https://doi.org/10.1007/s10494-024-00550-6  

   Bowers, Jamie; Durant, Eli; Ranjan, Reetesh (May 2024, Flow, Turbulence and Combustion)

   Abstract In this study, the highly nonlinear and multi-scale flame-turbulence interactions prevalent in turbulent premixed flames are examined by using direct numerical simulation (DNS) datasets to understand the effects of increase in pressure and changes in the characteristic scale ratios at high pressure. Such flames are characterized by length-scale ratio (ratio of integral length scale and laminar thermal flame thickness) and velocity-scale ratio (ratio of turbulence intensity and laminar flame speed). A canonical test configuration corresponding to an initially laminar methane/air lean premixed flame interacting with decaying isotropic turbulence is considered. We consider five cases with the initial Karlovitz number of 18, 37, 126, and 260 to examine the effects of an increase in pressure from 1 to 10 atm with fixed turbulence characteristics and at a fixed Karlovitz number, and the changes to characteristic scale ratios at the pressure of 10 atm. The increase in pressure for fixed turbulence characteristics leads to enhanced flame broadening and wrinkling due to an increase in the range of energetic scales of motion. This further manifests into affecting the spatial and state-space variation of thermo-chemical quantities, single point statistics, and the relationship of heat-release rate to the flame curvature and tangential strain rate. Although these results can be inferred in terms of an increase in Karlovitz number, the effect of an increase in pressure at a fixed Karlovitz number shows differences in the spatial and state-space variations of thermo-chemical quantities and the relationship of the heat release rate with the curvature and tangential strain rate. This is due to a higher turbulent kinetic energy associated with the wide range of scales of motion at atmospheric pressure. In particular, the magnitude of the correlation of the heat release rate with the curvature and the tangential strain rate tend to decrease and increase, respectively, with an increase in pressure. Furthermore, the statistics of the flame-turbulence interactions at high pressure also show sensitivity to the changes in the characteristic length- and velocity-scale ratios. The results from this study highlight the need to accurately account for the effects of pressure and characteristic scales for improved modeling of such flames. 

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4. Semiclassical geometry in double-scaled SYK

   https://doi.org/10.1007/JHEP11(2023)093  

   Goel, Akash; Narovlansky, Vladimir; Verlinde, Herman (November 2023, Journal of High Energy Physics)

   A<sc>bstract</sc> We argue that at finite energies, double-scaled SYK has a semiclassical approximation controlled by a couplingλin which all observables are governed by a non-trivial saddle point. The Liouville description of double-scaled SYK suggests that the correlation functions define a geometry in a two-dimensional bulk, with the 2-point function describing the metric. For small coupling, the fluctuations are highly suppressed, and the bulk describes a rigid (A)dS spacetime. As the coupling increases, the fluctuations become stronger. We study the correction to the curvature of the bulk geometry induced by these fluctuations. We find that as we go deeper into the bulk the curvature increases and that the theory eventually becomes strongly coupled. In general, the curvature is related to energy fluctuations in light operators. We also compute the entanglement entropy of partially entangled thermal states in the semiclassical limit. 

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5. Row-clustering of a Point Process-valued Matrix

   Yin, Lihao; Xu, Ganggang; Sang, Huiyan; Guan, Yongtao (December 2021, Advances in neural information processing systems)

   Structured point process data harvested from various platforms poses new challenges to the machine learning community. To cluster repeatedly observed marked point processes, we propose a novel mixture model of multi-level marked point processes for identifying potential heterogeneity in the observed data. Specifically, we study a matrix whose entries are marked log-Gaussian Cox processes and cluster rows of such a matrix. An efficient semi-parametric Expectation-Solution (ES) algorithm combined with functional principal component analysis (FPCA) of point processes is proposed for model estimation. The effectiveness of the proposed framework is demonstrated through simulation studies and real data analyses. 

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