Pointwise equidistribution for almost smooth functions with an error rate and Weighted L\'evy-Khintchin theorem

The purpose of this article is twofold: to prove a pointwise equidistribution theorem with an error rate for almost smooth functions, which strengthens the main result of Kleinbock, Shi and Weiss (2017); and to obtain a L\'evy-Khintchin theorem for weighted best approximations, which extends the main theorem of Cheung and Chevallier (2019). To do so, we employ techniques from homogeneous dynamics and the methods developed in the work of Cheung-Chevallier (2019) and Shapira-Weiss (2022).Comment: 32 page

Calculation of static transmission errors associated with thermo-elastic coupling contacts of spur gears

The static transmission error is one of the key incentives of gear dynamics and addressed by many scholars. However, the traditional calculation method of static transmission errors of spur gears does not take into account the influence of thermo-elastic coupling caused by the gear temperature field, and it limits the accuracy of the dynamic characteristic analysis. Thus, in this study, the calculation method of static transmission errors associated with thermo-elastic coupling is proposed. Furthermore, the differences between static transmission errors associated with thermo-elastic coupling contacts and traditional method of gear is discussed. The study is helpful to improve the accuracy of dynamic analysis of gear transmission system

Collective behavior of squirmers in thin films

Bacteria in biofilms form complex structures and can collectively migrate within mobile aggregates, which is referred to as swarming. This behavior is influenced by a combination of various factors, including morphological characteristics and propulsive forces of swimmers, their volume fraction within a confined environment, and hydrodynamic and steric interactions between them. In our study, we employ the squirmer model for microswimmers and the dissipative particle dynamics method for fluid modeling to investigate the collective motion of swimmers in thin films. The film thickness permits a free orientation of non-spherical squirmers, but constraints them to form a two-layered structure at maximum. Structural and dynamic properties of squirmer suspensions confined within the slit are analyzed for different volume fractions of swimmers, motility types (e.g., pusher, neutral squirmer, puller), and the presence of a rotlet dipolar flow field, which mimics the counter-rotating flow generated by flagellated bacteria. Different states are characterized, including a gas-like phase, swarming, and motility-induced phase separation, as a function of increasing volume fraction. Our study highlights the importance of an anisotropic swimmer shape, hydrodynamic interactions between squirmers, and their interaction with the walls for the emergence of different collective behaviors. Interestingly, the formation of collective structures may not be symmetric with respect to the two walls. Furthermore, the presence of a rotlet dipole significantly mitigates differences in the collective behavior between various swimmer types. These results contribute to a better understanding of the formation of bacterial biofilms and the emergence of collective states in confined active matter.Comment: 17 pages, 12 figure

Fast Conditional Mixing of MCMC Algorithms for Non-log-concave Distributions

MCMC algorithms offer empirically efficient tools for sampling from a target distribution $\pi(x) \propto \exp(-V(x))$. However, on the theory side, MCMC algorithms suffer from slow mixing rate when $\pi(x)$ is non-log-concave. Our work examines this gap and shows that when Poincar\'e-style inequality holds on a subset $\mathcal{X}$ of the state space, the conditional distribution of MCMC iterates over $\mathcal{X}$ mixes fast to the true conditional distribution. This fast mixing guarantee can hold in cases when global mixing is provably slow. We formalize the statement and quantify the conditional mixing rate. We further show that conditional mixing can have interesting implications for sampling from mixtures of Gaussians, parameter estimation for Gaussian mixture models and Gibbs-sampling with well-connected local minima.Comment: Camera ready versio

Discrete forecast reconciliation

While forecast reconciliation has seen great success for real valued data, the method has not yet been comprehensively extended to the discrete case. This paper defines and develops a formal discrete forecast reconciliation framework based on optimising scoring rules using quadratic programming. The proposed framework produces coherent joint probabilistic forecasts for count hierarchical timeTwo discrete reconciliation algorithms are proposed and compared to generalisations of the top-down and bottom-up approaches to count data. Two simulation experiments and two empirical examples are conducted to validate that the proposed reconciliation algorithms improve forecast accuracy. The empirical applications are to forecast criminal offences in Washington D.C. and the exceedance of thresholds in age-specific mortality rates in Australia. Compared to the top-down and bottom-up approaches, the proposed framework shows superior performance in both simulations and empirical studies

A supramolecular radical cation: folding-enhanced electrostatic effect for promoting radical-mediated oxidation.

We report a supramolecular strategy to promote radical-mediated Fenton oxidation by the rational design of a folded host-guest complex based on cucurbit[8]uril (CB[8]). In the supramolecular complex between CB[8] and a derivative of 1,4-diketopyrrolo[3,4-c]pyrrole (DPP), the carbonyl groups of CB[8] and the DPP moiety are brought together through the formation of a folded conformation. In this way, the electrostatic effect of the carbonyl groups of CB[8] is fully applied to highly improve the reactivity of the DPP radical cation, which is the key intermediate of Fenton oxidation. As a result, the Fenton oxidation is extraordinarily accelerated by over 100 times. It is anticipated that this strategy could be applied to other radical reactions and enrich the field of supramolecular radical chemistry in radical polymerization, photocatalysis, and organic radical battery and holds potential in supramolecular catalysis and biocatalysis

Convergence of AdaGrad for Non-convex Objectives: Simple Proofs and Relaxed Assumptions

We provide a simple convergence proof for AdaGrad optimizing non-convex objectives under only affine noise variance and bounded smoothness assumptions. The proof is essentially based on a novel auxiliary function $\xi$ that helps eliminate the complexity of handling the correlation between the numerator and denominator of AdaGrad's update. Leveraging simple proofs, we are able to obtain tighter results than existing results \citep{faw2022power} and extend the analysis to several new and important cases. Specifically, for the over-parameterized regime, we show that AdaGrad needs only $\mathcal{O}(\frac{1}{\varepsilon^2})$ iterations to ensure the gradient norm smaller than $\varepsilon$, which matches the rate of SGD and significantly tighter than existing rates $\mathcal{O}(\frac{1}{\varepsilon^4})$ for AdaGrad. We then discard the bounded smoothness assumption and consider a realistic assumption on smoothness called $(L_0,L_1)$-smooth condition, which allows local smoothness to grow with the gradient norm. Again based on the auxiliary function $\xi$, we prove that AdaGrad succeeds in converging under $(L_0,L_1)$-smooth condition as long as the learning rate is lower than a threshold. Interestingly, we further show that the requirement on learning rate under the $(L_0,L_1)$-smooth condition is necessary via proof by contradiction, in contrast with the case of uniform smoothness conditions where convergence is guaranteed regardless of learning rate choices. Together, our analyses broaden the understanding of AdaGrad and demonstrate the power of the new auxiliary function in the investigations of AdaGrad.Comment: COLT 202