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Lifting theorems are theorems that relate the query complexity of a function f : {0, 1}^n â {0, 1} to the communication complexity of the composed function f â ¦ g^n, for some â gadgetâ g : {0, 1}^b à {0, 1}^b â {0, 1}. Such theorems allow transferring lower bounds from query complexity to the communication complexity, and have seen numerous applications in the recent years. In addition, such theorems can be viewed as a strong generalization of a direct-sum theorem for the gadget g. We prove a new lifting theorem that works for all gadgets g that have logarithmic length and exponentially-small discrepancy, for both deterministic and randomized communication complexity. Thus, we increase the range of gadgets for which such lifting theorems hold considerably. Our result has two main motivations: First, allowing a larger variety of gadgets may support more applications. In particular, our work is the first to prove a randomized lifting theorem for logarithmic-size gadgets, thus improving some applications the theorem. Second, our result can be seen a strong generalization of a direct-sum theorem for functions with low discrepancy. Joint work with Arkadev Chattopadhyay, Yuval Filmus, Or Meir, Toniann Pitassi
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The transfer tensor method [1] is a compact and intuitive tool for the analysis and simulation of general open quantum systems. By extracting the information contained in short samples of the initial dynamics, it has the ability to extend the simulation power of existing exact approaches, like the chain-mapping DMRG-based simulation method TEDOPA [2] or stochastic methods [3]. Crucially, it can treat problems with initial system-environment correlations, such as emission and absorption spectra of multichromophoric molecules [3]. In combination with the hierarchy of equations of motion, transfer tensors that contain information about energetic and particle currents of the environment may be derived, facilitating quantum transport studies in the strong-coupling and non-Markovian regimes. $$ $$ [1] J. Cerrillo, J. Cao, Phys Rev. Lett. 112, 110401 (2014). $$ $$ [2] R. Rosenbach, J. Cerrillo, S.F. Huelga, J. Cao, M.B. Plenio, New J. Phys. 18, 023035 (2016). $$ $$ [3] M. Buser, J. Cerrillo, G. Schaller, and J. Cao, Phys. Rev. A 96, 062122 (2017).
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A recently proposed mechanism suggests that transient compartmentalization could have preceded cell division in prebiotic scenarios. Here, we study various classes of transient compartmentalization dynamics. We show that two regimes are possible: In a diffusion-limited regime (e.g. simple autocatalysis), a large noise is generated at the population level due to asynchronous growth. In contrast, in a replication-limited regime with many steps (e.g. polymerization), a low noise is generated at the population level. Since strong noise will yield many unviable population compositions, polymerization can present a strong fitness advantage. For deterministic growth dynamics, we introduce mutations that turn functional replicators into parasites. This can either lead to coexistence or parasite dominance, and we derive the phase boundary separating these two phases as a function of relative growth, inoculation size and mutation rate. We show that transient compartmentalization allows coexistence beyond the classical error threshold.
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We describe new ways of constructing pseudorandom generators for Boolean functions that satisfy certain bounds on their Fourier spectrum. We discuss the possibility of using this approach to construct pseudorandom generators for complexity classes that have eluded researches for decades. Based on joint works with Pooya Hatami, Kaave Hosseini, Shachar Lovett and Avishay Tal.
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I will present a multiscale generalisation of the Bakry--Emery criterion for a measure to satisfy a Log-Sobolev inequality. It relies on the control of an associated PDE well known in renormalisation theory: the Polchinski equation. It implies the usual Bakry--Emery criterion, but we show that it remains effective for measures which are far from log-concave. Indeed, we prove that the massive continuum Sine-Gordon model on $R^2$ with $\beta < 6\pi$ satisfies asymptotically optimal Log-Sobolev inequalities for Glauber and Kawasaki dynamics. These dynamics can be seen as singular SPDEs recently constructed via regularity structures, but our results are independent of this theory. This is joint work with T. Bodineau.
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The advent of a new modeling paradigm known as â differentiable programmingâ makes possible bespoke machine-learned models of biological phenomena that are partly learned from data and partly informed by human-derived biophysical knowledge. In this talk I will describe three instantiations of this new approach for (i) de novo protein structure prediction, (ii) elucidation of the combinatorial grammar underlying metazoan signaling networks, and (iii) design of new protein function. In all cases qualitative improvements in model accuracy or speed, or both, are achieved using differentiable programming, enabling new scientific insights into biological macromolecules and the networks they comprise.
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About a third of the proteome consists of intrinsically disordered proteins (IDPs) that fold, whether fully or partially, upon binding to their partners [1]. IDPs use their inherent flexibility to play key regulatory roles in many biological processes [2]. Such flexibility makes their structural analysis extremely challenging, being nuclear magnetic resonance (NMR) the most suitable high-resolution technique. However, conventional NMR structure determination methods, which seek to determine a single high-resolution structure [3], are inadequate for IDPs. There are several tools available for the structural analysis of IDPs using NMR data and primarily Chemical Shifts (CS) [4-6]. However, a persistent problem is how to effectively sample the extensive, but not random, conformational space of IDPs. We have implemented a novel relational database, termed Glutton, that links all existing CS data with corresponding protein 3D structures with the goal of enabling the conformational analysis of IDPs directly from their experimental CS. Gluttonâ s uniqueness is in its focus on dihedral angle distributions consistent with a given set of CS rather than with unique structures. Such dihedral distributions define how native-like is the ensemble and lead to the effective calculation of large ensembles of structures that efficiently sample the available conformational space. With Glutton, we examined Nuclear Coactivator Binding Domain (NCBD), an IDP with NMR structure obtained using osmolyte stabilizers that is partly disordered in native conditions [7]. As means of comparison, we produced a 60ï ­s long MD simulation of NCBD in explicit solvent starting from the NMR structure and using the CHARMM36m force field with modified TIP3P water which was suggested as a good combination to explore the conformational space of IDPs [8]. The structural ensembles obtained from Glutton are based only on geometric considerations and CS restraints, but they can be further refined using additional computational (force field) and/or experimental (distance restraints) information. <p> <p> Acknowledgments: This work was supported by grants: the startup fund at the University of New Mexico, the W.M. Keck Foundation, the National Science Foundation [NSF-MCB-161759 and NSF-CREST-1547848] and the European Research Council [ERC-2012-AdG-323059]. <p> <p> [1] H.J. Dyson, P.E. Wright, Chem. Rev., 104, 3607â 3622, 2004 <p> [2] M.M. Babu, Biochem. Soc. Trans., 44, 1185â 1200, 2016 <p> [3] A.M., Gronenborn, G.M. Clore, Anal. Chem., 62, 2â 15, 1990 <p> [4] V. Ozenne, F. Bauer, L. Salmon, J. Huang, M.R. Jensen, S. Segard, P. Bernadó, C. Charavay, M. Blackledge, Bioinformatics, 28, 1463â 1470, 2012 <p> [5] M. Krzeminski, J.A. Marsh, C. Neale, W. Choy, J.D. Forman-Kay, Bioinformatics, 29, 398â 399, 2013 <p> [6] D.H. Brookes, T. Head-Gordon, J. Am. Chem. Soc., 138, 4530â 4538, 2016 <p> [7] A. Naganathan, M. Orozco, J. Am. Chem. Soc., 133, 12154â 12161, 2011 <p> [8] J. Huang, S. Rauscher, G. Nawrocki, T. Ran, M. Feig, B. L de Groot, H. Grubmüller, A.D. MacKerell Jr, Nat Methods, 14, 71â 73, 2017
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