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We construct and study new generalisations to rooted trees and forests of some properties of shuffles of words. First, we build a coproduct on rooted trees which, together with their shuffle, endow them with bialgebra structure. We then caracterize the coproduct dual to the shuffle product of rooted forests and build a product on rooted trees to obtain the bialgebra dual to the shuffle bialgebra. We then characterize and enumerate primitive trees for the dual coproduct. Finally, using modified shuffles of rooted forests, we prove a property in the category of Rota-Baxter algebras., Comment: 28 pages

Nucleic acids such as mRNA have emerged as a promising therapeutic modality with the capability of addressing a wide range of diseases. Lipid nanoparticles (LNPs) as a delivery platform for nucleic acids were used in the COVID-19 vaccines and have received much attention. While modern manufacturing processes which involve rapidly mixing an organic stream containing the lipids with an aqueous stream containing the nucleic acids are conceptually straightforward, detailed understanding of LNP formation and structure is still limited and scale-up can be challenging. Mathematical and computational methods are a promising avenue for deepening scientific understanding of the LNP formation process and facilitating improved process development and control. This article describes strategies for the mechanistic modeling of LNP formation, starting with strategies to estimate and predict important physicochemical properties of the various species such as diffusivities and solubilities. Subsequently, a framework is outlined for constructing mechanistic models of reactor- and particle-scale processes. Insights gained from the various models are mapped back to product quality attributes and process insights. Lastly, the use of the models to guide development of advanced process control and optimization strategies is discussed., Comment: 67 pages, 10 figures

Computational elements in thermodynamics have become increasingly important in contemporary chemical-engineering research and practice. However, traditional thermodynamics instruction provides little exposure to computational thermodynamics, leaving students ill-equipped to engage with the state-of-the-art deployed in industry and academia. The recent rise of easy-to-use open-source thermodynamic codes presents an opportunity for educators to help bridge this gap. In this work, we present a short course that was developed and rolled-out using the Clapeyron.jl package, the material of which is all openly available on GitHub. The course can serve as a foundation for others to similarly integrate computational material in thermodynamics education. The course is structured into three sections. Section one serves as a refresher and covers core material in numerical methods and thermodynamics. Section two introduces a range of thermodynamic models such as activity-coefficient models and cubic equations of state, outlining their implementation. In section three the focus is moved to deployment, guiding students on how to implement computational-thermodynamics methods covering volume solvers, saturation solvers, chemical-stability analysis and flash problems. In a pilot study conducted with both undergraduate and graduate students, participants found the material engaging and highly relevant to their chemical-engineering education.

Thermodynamic models and, in particular, SAFT-type equations are vital in characterizing complex systems. This paper presents a framework for sampling parameter distributions in PC-SAFT and SAFT-VR Mie equations of state to understand parameter confidence intervals and correlations. We identify conserved quantities contributing to significant correlations. Comparing the equations of state, we find that additional parameters introduced in the SAFT-VR Mie equation increase relative uncertainties (1\%-2\% to 3\%-4\%) and introduce more correlations. When incorporating association through additional parameters, relative uncertainties increase, but correlations slightly decrease. We investigate how uncertainties propagate to derived properties and observe small uncertainties for that data with which the parameters were regressed, especially for saturated-liquid volumes. However, extrapolating to saturated-vapour volumes yields larger uncertainties due to the larger isothermal compressibility. Near the critical point, uncertainties in saturated volumes diverge due to increased sensitivity of the isothermal compressibility to parameter uncertainties. This effect significantly impacts bulk properties, particularly isobaric heat capacity, where uncertainties near the critical point become extremely large, even when these uncertainties are small. We emphasize that even small uncertainties near the critical point lead to divergences in predicted properties.

We introduce the notion of a risk-limiting financial auditing (RLFA): given $N$ transactions, the goal is to estimate the total misstated monetary fraction~($m^*$) to a given accuracy $\epsilon$, with confidence $1-\delta$. We do this by constructing new confidence sequences (CSs) for the weighted average of $N$ unknown values, based on samples drawn without replacement according to a (randomized) weighted sampling scheme. Using the idea of importance weighting to construct test martingales, we first develop a framework to construct CSs for arbitrary sampling strategies. Next, we develop methods to improve the quality of CSs by incorporating side information about the unknown values associated with each item. We show that when the side information is sufficiently predictive, it can directly drive the sampling. Addressing the case where the accuracy is unknown a priori, we introduce a method that incorporates side information via control variates. Crucially, our construction is adaptive: if the side information is highly predictive of the unknown misstated amounts, then the benefits of incorporating it are significant; but if the side information is uncorrelated, our methods learn to ignore it. Our methods recover state-of-the-art bounds for the special case when the weights are equal, which has already found applications in election auditing. The harder weighted case solves our more challenging problem of AI-assisted financial auditing., Comment: 23 pages, 8 figures, to appear in the Proceedings of Uncertainty in Artificial Intelligence (UAI) 2023

Real-world graphs can be difficult to interpret and visualize beyond a certain size. To address this issue, graph summarization aims to simplify and shrink a graph, while maintaining its high-level structure and characteristics. Most summarization methods are designed for homogeneous, undirected, simple graphs; however, many real-world graphs are ornate; with characteristics including node labels, directed edges, edge multiplicities, and self-loops. In this paper we propose LM-Gsum, a versatile yet rigorous graph summarization model that (to the best of our knowledge, for the first time) can handle graphs with all the aforementioned characteristics (and any combination thereof). Moreover, our proposed model captures basic sub-structures that are prevalent in real-world graphs, such as cliques, stars, etc. LM-Gsum compactly quantifies the information content of a complex graph using a novel encoding scheme, where it seeks to minimize the total number of bits required to encode (i) the summary graph, as well as (ii) the corrections required for reconstructing the input graph losslessly. To accelerate the summary construction, it creates super-nodes efficiently by merging nodes in groups. Experiments demonstrate that LM-Gsum facilitates the visualization of real-world complex graphs, revealing interpretable structures and high- level relationships. Furthermore, LM-Gsum achieves better trade-off between compression rate and running time, relative to existing methods (only) on comparable settings., Comment: 17 pages, 8 figures, 4 tables

The present exploratory paper deals with tensor products in the locality framework {developed in previous work}, a natural setting for an algebraic formulation of the locality principle in quantum field theory. Locality tensor products of locality vector spaces raise challenging questions, such as whether the locality tensor product of two locality vector spaces is a locality vector space. A related question is whether the quotient of locality vector spaces is a locality vector space, which we first reinterpret in a group theoretic language and then in terms of short exact sequences. We prove a universal property for the locality tensor algebra and for the locality enveloping algebra, the analogs in the locality framework of the tensor algebra and of the enveloping algebra. These universal properties hold under compatibility assumptions between the locality and the multilinearity underlying the construction of tensor products which we formulate in the form of conjectural statements. Assuming they hold true, we generalise the Milnor-Moore theorem to the locality setup and discuss some of its consequences., Comment: 59 pages, 14 figures

We show that any convergent (shuffle) arborified zeta value admits a series representation. This justifies the introduction of a new generalisation to rooted forests of multiple zeta values, and we study its algebraic properties. As a consequence of the series representation, we derive elementary proofs of some results of Bradley and Zhou for Mordell-Tornheim zeta values and give explicit formulas. The series representation for shuffle arborified zeta values also implies that they are conical zeta values. We characterise which conical zeta values are arborified zeta values and evaluate them as sums of multiple zeta values with rational coefficients., Comment: 29 pages. One section added, where tree zeta values are studied. One new author. Change of title

Thermodynamic models are often vital when characterising complex systems, particularly natural gas, electrolyte, polymer, pharmaceutical and biological systems. However, their implementations have historically been abstruse and cumbersome, and as such, the only options available were black-box commercial tools. In this article, we present Clapeyron.jl: a pioneering attempt at an open-source fluid-thermodynamics toolkit to build and make use of thermodynamic models. This toolkit is built in Julia, a modern language for scientific computing known for its ease of use, extensibility, and first-class support for differentiable programming. We currently support more models than any package available, including standard cubic (SRK, PR, PSRK, etc.), activity-coefficient (NRTL, UNIFAC, etc.), COSMO-based, and the venerable SAFT equations. The property-estimation methods supported are extensive, including bulk, VLE, LLE, VLLE and critical properties. With Clapeyron.jl, researchers and enthusiasts alike will be able to focus on the application and worry less about the implementation.

We introduce the concept of TRAP (Traces and Permutations), which can roughly be viewed as a wheeled PROP (Products and Permutations) without unit. TRAPs are equipped with a horizontal concatenation and partial trace maps. Continuous morphisms on an infinite dimensional topological space and smooth kernels (resp. smoothing operators) on a closed manifold form a TRAP but not a wheeled PROP. We build the free objects in the category of TRAPs as TRAPs of graphs and show that a TRAP can be completed to a unitary TRAP (or wheeled PROP). We further show that it can be equipped with a vertical concatenation, which on the TRAP of linear homomorphisms of a vector space, amounts to the usual composition. The vertical concatenation in the TRAP of smooth kernels gives rise to generalised convolutions. Graphs whose vertices are decorated by smooth kernels (resp. smoothing operators) on a closed manifold form a TRAP. From their universal properties we build smooth amplitudes associated with the graph., Comment: 52 pages, many figures. arXiv admin note: substantial text overlap with arXiv:2005.02115

Within a large database G containing graphs with labeled nodes and directed, multi-edges; how can we detect the anomalous graphs? Most existing work are designed for plain (unlabeled) and/or simple (unweighted) graphs. We introduce CODETECT, the first approach that addresses the anomaly detection task for graph databases with such complex nature. To this end, it identifies a small representative set S of structural patterns (i.e., node-labeled network motifs) that losslessly compress database G as concisely as possible. Graphs that do not compress well are flagged as anomalous. CODETECT exhibits two novel building blocks: (i) a motif-based lossless graph encoding scheme, and (ii) fast memory-efficient search algorithms for S. We show the effectiveness of CODETECT on transaction graph databases from three different corporations, where existing baselines adjusted for the task fall behind significantly, across different types of anomalies and performance metrics., Comment: 24 pages

Recent releases of open-source research codes and solvers for numerically solving partial differential equations in Python present a great opportunity for educators to integrate these codes into the classroom in a variety of ways. The ease with which a problem can be implemented and solved using these codes reduce the barrier to entry for users. We demonstrate how one of these codes,FiPy, can be introduced to students through a short course using progression as the guiding philosophy. Four exercises of increasing complexity were developed. Basic concepts from more advanced numerical methods courses are also introduced at appropriate points. To further engage students, we demonstrate how an open research problem can be readily implemented and also incorporate the use of ParaView to post-process their results. Student engagement and learning outcomes were evaluated through a pre and post-course survey and a focus group discussion. Students broadly found the course to be engaging and useful with the ability to easily visualise the solution to PDEs being greatly valued. Due to the introductory nature of the course, due care in terms of set-up and the design of learning activities during the course is essential. This course, if integrated with appropriate level of support, can encourage students to use the provided codes and improve their understanding of concepts used in numerical analysis and PDEs., Comment: 12 pages, 9 figures. Fixed references

We assign generalised convolutions (resp. traces) to graphs whose edges are decorated by smooth kernels (resp. smoothing operators) on a closed manifold. To do so, we introduce the concept of TraPs (Traces and Permutations), which roughly correspond to ProPs (Products and Permutations) without vertical concatenation and equipped with families of generalised partial traces. They can be equipped with a ProP structure in deriving vertical concatenation from the partial traces and we relate TraPs to wheeled ProPs first introduced by Merkulov. We further build their free object and give precise proofs of universal properties of ProPs and TraPs., Comment: 64 pages