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1. Combinatorial anti-concentration inequalities, with applications

   https://doi.org/10.1017/S0305004120000183  

   FOX, JACOB; KWAN, MATTHEW; SAUERMANN, LISA (September 2021, Mathematical Proceedings of the Cambridge Philosophical Society)

   Abstract We prove several different anti-concentration inequalities for functions of independent Bernoulli-distributed random variables. First, motivated by a conjecture of Alon, Hefetz, Krivelevich and Tyomkyn, we prove some “Poisson-type” anti-concentration theorems that give bounds of the form 1/ e + o (1) for the point probabilities of certain polynomials. Second, we prove an anti-concentration inequality for polynomials with nonnegative coefficients which extends the classical Erdős–Littlewood–Offord theorem and improves a theorem of Meka, Nguyen and Vu for polynomials of this type. As an application, we prove some new anti-concentration bounds for subgraph counts in random graphs. 

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2. Design of Electrostatic Aberration Correctors for Scanning Transmission Electron Microscopy

   Ribet, S. (January 2023, Condensed matter)

   In a scanning transmission electron microscope (STEM), producing a high-resolution image generally requires an electron beam focused to the smallest point possible. However, the magnetic lenses used to focus the beam are unavoidably imperfect, introducing aberrations that limit resolution. Modern STEMs overcome this by using hardware aberration correctors comprised of many multipole lenses, but these devices are complex, expensive, and can be difficult to tune. We demonstrate a design for an electrostatic phase plate that can act as an aberration corrector. The corrector is comprised of annular segments, each of which is an independent two-terminal device that can apply a constant or ramped phase shift to a portion of the electron beam. We show the improvement in image resolution using an electrostatic corrector. Engineering criteria impose that much of the beam within the probe-forming aperture be blocked by support bars, leading to large probe tails for the corrected probe that sample the specimen beyond the central lobe. We also show how this device can be used to create other STEM beam profiles such as vortex beams and beams with a high degree of phase diversity, which improve information transfer in ptychographic reconstructions. 

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3. Generating Sequences by Learning to Self-Correct

   Welleck, Sean; Lu, Ximing; West, Peter; Brahman, Faeze; Shen, Tianxiao; Khashabi, Daniel; Choi, Yejin (July 2023, The Eleventh International Conference on Learning Representations)

   Sequence generation applications require satisfying semantic constraints, such as ensuring that programs are correct, using certain keywords, or avoiding undesirable content. Language models, whether fine-tuned or prompted with few-shot demonstrations, frequently violate these constraints, and lack a mechanism to iteratively revise their outputs. Moreover, some powerful language models are of extreme scale or inaccessible, making it inefficient, if not infeasible, to update their parameters for task-specific adaptation. We present Self-Correction, an approach that decouples an imperfect base generator (an off-the-shelf language model or supervised sequence-to-sequence model) from a separate corrector that learns to iteratively correct imperfect generations. To train the corrector, we propose an online training procedure that can use either scalar or natural language feedback on intermediate imperfect generations. We show that Self-Correction improves upon the base generator in three diverse generation tasks - mathematical program synthesis, lexically-constrained generation, and toxicity control - even when the corrector is much smaller than the base generator. 

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4. Exit time asymptotics for dynamical systems with fast random switching near an unstable equilibrium

   https://doi.org/10.1142/S0219493720500264  

   Bakhtin, Yuri; Gaál, Alexisz (August 2020, Stochastics and Dynamics)

   null (Ed.)

   We consider the exit problem for a one-dimensional system with random switching near an unstable equilibrium point of the averaged drift. In the infinite switching rate limit, we show that the exit time satisfies a limit theorem with a logarithmic deterministic term and a random correction converging in distribution. Thus, this setting is in the universality class of the unstable equilibrium exit under small white-noise perturbations. 

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5. Non-stationary Itô–Kawada and ergodic theorems for random isometries

   https://doi.org/10.4171/GGD/856  

   Monakov, Grigorii V (January 2025, Groups, Geometry, and Dynamics)

   We consider a non-stationary sequence of independent random isometries of a compact metrizable space. Assuming that there are no proper closed subsets with deterministic image, we establish a weak-* convergence to the unique invariant under isometries measure, ergodic theorem and large deviation type estimate. We also show that all the results can be carried over to the case of a random walk on a compact metrizable group. In particular, we prove a non-stationary analog of classical Itô–Kawada theorem and give a new alternative proof for the stationary case. 

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