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Abstract This article revisits the problem of global well-posedness for the generalized parabolic Anderson model on$$\mathbb {R}^+\times \mathbb {T}^2$$ $R^{+}\times T^2$within the framework of paracontrolled calculus (Gubinelli et al. in Forum Math, 2015). The model is given by the equation:$$\begin{aligned} (\partial _t-\Delta ) u=F(u)\eta \end{aligned}$$ $\begin{array}{c}({\partial }_t-\Delta )u=F\left(u\right)\eta \end{array}$where$$\eta \in C^{-1-\kappa }$$ $\eta \in C^{-1-\kappa }$with$$1/6>\kappa >0$$ $1/6>\kappa >0$, and$$F\in C_b^2(\mathbb {R})$$ $F\in C_b^2\left(R\right)$. Assume that$$\eta \in C^{-1-\kappa }$$ $\eta \in C^{-1-\kappa }$and can be lifted to enhanced noise, we derive new a priori bounds. The key idea follows from the recent work by Chandra et al. (A priori bounds for 2-d generalised Parabolic Anderson Model,,2024), to represent the leading error term as a transport type term, and our techniques encompass the paracontrolled calculus, the maximum principle, and the localization approach (i.e. high-low frequency argument). 

Catalytic DNA has gained significant attention in recent decades as a highly efficient and tunable catalyst, thanks to its flexible structures, exceptional specificity, and ease of optimization. Despite being composed of just four monomers, DNA’s complex conformational intricacies enable a wide range of nuanced functions, including scaffolding, electrocatalysis, enantioselectivity, and mechano-electro spin coupling. DNA catalysts, ranging from traditional DNAzymes to innovative DNAzyme hybrids, highlight the remarkable potential of DNA in catalysis. Recent advancements in spectroscopic techniques have deepened our mechanistic understanding of catalytic DNA, paving the way for rational structural optimization. This review will summarize the latest studies on the performance and optimization of traditional DNAzymes and provide an in-depth analysis of DNAzyme hybrid catalysts and their unique and promising properties. 

Abstract We define a state space and a Markov process associated to the stochastic quantisation equation of Yang–Mills–Higgs (YMH) theories. The state space$$\mathcal{S}$$ $S$is a nonlinear metric space of distributions, elements of which can be used as initial conditions for the (deterministic and stochastic) YMH flow with good continuity properties. Using gauge covariance of the deterministic YMH flow, we extend gauge equivalence ∼ to$$\mathcal{S}$$ $S$and thus define a quotient space of “gauge orbits”$$\mathfrak {O}$$ $O$. We use the theory of regularity structures to prove local in time solutions to the renormalised stochastic YMH flow. Moreover, by leveraging symmetry arguments in the small noise limit, we show that there is a unique choice of renormalisation counterterms such that these solutions are gauge covariant in law. This allows us to define a canonical Markov process on$$\mathfrak {O}$$ $O$(up to a potential finite time blow-up) associated to the stochastic YMH flow. 

Titanium dioxide nanoparticles (TiO2 NPs) have traditionally been utilized as industrial catalysts, finding widespread application in various chemical processes due to their exceptional stability and minimal toxicity. However, quantitatively assessing the reactive sites on TiO2 NPs remains a challenge. In this study, we employed a fluorogenic reaction to probe the apparent reactivity of TiO2 NPs. By manipulating the number of defect sites through control of hydrolysis speed and annealing temperature, we determined that the Ti(Ⅲ) content is positively correlated with the reactivity of TiO2 NPs. Additionally, these Ti(Ⅲ) sites could be introduced by reducing commercial TiO2 NPs using NaBH4. Our findings suggest that fluorogenic oxidation of Amplex Red is an effective method for probing defect site densities on TiO2 NPs. Utilizing single-molecule fluorescence imaging, we demonstrated the ability to map defect site density within TiO2 nanowires. Achieving sub-nanoparticle spatial resolution, we observed significant intraparticle and interparticle variations in the defect site distribution, leading to substantial reactivity heterogeneity. 

Catalytic processes are used in about 1/3 of US manufacturing, from the field of chemical engineering to renewable energy. Assessing the activity of single-molecules, or individual molecules, is necessary to the development of efficient catalysts. Their heterogeneity structure leads to particle-specific catalytic activity. Experimentation with single-molecules can be time consuming and difficult. We purpose a Machine learning (ML) model that allows chemical researchers to run shorter single-molecule experiments to obtain the same level of results. We use common and widely understood ML methods to reduce complexity and enable accessibility to the chemical engineering community. We reduce the experiment time by up to 83%. Our evaluation shows that a small data set is sufficient to train an acceptable model. 300 experiments are needed, including the validation set. We use a well understood multilayer perceptron (MLP) model. We show that more complex models are not necessary and simpler methods are not sufficient.