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This article introduces a new numerical method for the minimization under constraints of a discrete energy modeling multicomponents rotating Bose-Einstein condensates in the regime of strong confinement and with rotation. Moreover, we consider both segregation and coexistence regimes between the components. The method includes a discretization of a continuous energy in space dimension 2 and a gradient algorithm with adaptive time step and projection for the minimization. It is well known that, depending on the regime, the minimizers may display different structures, sometimes with vorticity (from singly quantized vortices, to vortex sheets and giant holes). In order to study numerically the structures of the minimizers, we introduce in this paper a numerical algorithm for the computation of the indices of the vortices, as well as an algorithm for the computation of the indices of vortex sheets. Several computations are carried out, to illustrate the efficiency of the method, to cover different physical cases, to validate recent theoretical results as well as to support conjectures. Moreover, we compare this method with an alternative method from the literature.

We develop a theory for the representation of opaque solids as volumes. Starting from a stochastic representation of opaque solids as random indicator functions, we prove the conditions under which such solids can be modeled using exponential volumetric transport. We also derive expressions for the volumetric attenuation coefficient as a functional of the probability distributions of the underlying indicator functions. We generalize our theory to account for isotropic and anisotropic scattering at different parts of the solid, and for representations of opaque solids as stochastic implicit surfaces. We derive our volumetric representation from first principles, which ensures that it satisfies physical constraints such as reciprocity and reversibility. We use our theory to explain, compare, and correct previous volumetric representations, as well as propose meaningful extensions that lead to improved performance in 3D reconstruction tasks.

An extension of Transformers is proposed that enables explicit relational reasoning through a novel module called the Abstractor. At the core of the Abstractor is a variant of attention called relational cross-attention. The approach is motivated by an architectural inductive bias for relational learning that disentangles relational information from object-level features. This enables explicit relational reasoning, supporting abstraction and generalization from limited data. The Abstractor is first evaluated on simple discriminative relational tasks and compared to existing relational architectures. Next, the Abstractor is evaluated on purely relational sequence-to-sequence tasks, where dramatic improvements are seen in sample efficiency compared to standard Transformers. Finally, Abstractors are evaluated on a collection of tasks based on mathematical problem solving, where consistent improvements in performance and sample efficiency are observed.

We consider the bit complexity of computing Chow forms and their generalization to multiprojective spaces. We develop a deterministic algorithm using resultants and obtain a single exponential complexity upper bound. Earlier computational results for Chow forms were in the arithmetic complexity model, and our result represents the first bit complexity bound. We also extend our algorithm to Hurwitz forms in projective space, and explore connections between multiprojective Hurwitz forms and matroid theory. The motivation for our work comes from incidence geometry where intriguing computational algebra problems remain open.

We introduce and study swap cosystolic expansion, a new expansion property of simplicial complexes. We prove lower bounds for swap coboundary expansion of spherical buildings and use them to lower bound swap cosystolic expansion of the LSV Ramanujan complexes. Our motivation is the recent work (in a companion paper) showing that swap cosystolic expansion implies agreement theorems. Together the two works show that these complexes support agreement tests in the low acceptance regime. Swap cosystolic expansion is defined by considering, for a given complex $X$, its faces complex $F^r X$, whose vertices are $r$-faces of $X$ and where two vertices are connected if their disjoint union is also a face in $X$. The faces complex $F^r X$ is a derandomizetion of the product of $X$ with itself $r$ times. The graph underlying $F^rX$ is the swap walk of $X$, known to have excellent spectral expansion. The swap cosystolic expansion of $X$ is defined to be the cosystolic expansion of $F^r X$. Our main result is a $\exp(-O(\sqrt r))$ lower bound on the swap coboundary expansion of the spherical building and the swap cosystolic expansion of the LSV complexes. For more general coboundary expanders we show a weaker lower bound of $exp(-O(r))$.

Consider the quotient of a Hilbert space by a subgroup of its automorphisms. We study whether this orbit space can be embedded into a Hilbert space by a bilipschitz map, and we identify constraints on such embeddings.

This study is dedicated to assessing the capabilities of large language models (LLMs) such as GPT-3.5-Turbo, GPT-4, and GPT-4-Turbo in extracting structured information from scientific documents in materials science. To this end, we primarily focus on two critical tasks of information extraction: (i) a named entity recognition (NER) of studied materials and physical properties and (ii) a relation extraction (RE) between these entities. Due to the evident lack of datasets within Materials Informatics (MI), we evaluated using SuperMat, based on superconductor research, and MeasEval, a generic measurement evaluation corpus. The performance of LLMs in executing these tasks is benchmarked against traditional models based on the BERT architecture and rule-based approaches (baseline). We introduce a novel methodology for the comparative analysis of intricate material expressions, emphasising the standardisation of chemical formulas to tackle the complexities inherent in materials science information assessment. For NER, LLMs fail to outperform the baseline with zero-shot prompting and exhibit only limited improvement with few-shot prompting. However, a GPT-3.5-Turbo fine-tuned with the appropriate strategy for RE outperforms all models, including the baseline. Without any fine-tuning, GPT-4 and GPT-4-Turbo display remarkable reasoning and relationship extraction capabilities after being provided with merely a couple of examples, surpassing the baseline. Overall, the results suggest that although LLMs demonstrate relevant reasoning skills in connecting concepts, specialised models are currently a better choice for tasks requiring extracting complex domain-specific entities like materials. These insights provide initial guidance applicable to other materials science sub-domains in future work.

The implication problem for conditional independence (CI) asks whether the fact that a probability distribution obeys a given finite set of CI relations implies that a further CI statement also holds in this distribution. This problem has a long and fascinating history, cumulating in positive results about implications now known as the semigraphoid axioms as well as impossibility results about a general finite characterization of CI implications. Motivated by violation of faithfulness assumptions in causal discovery, we study the implication problem in the special setting where the CI relations are obtained from a directed acyclic graphical (DAG) model along with one additional CI statement. Focusing on the Gaussian case, we give a complete characterization of when such an implication is graphical by using algebraic techniques. Moreover, prompted by the relevance of strong faithfulness in statistical guarantees for causal discovery algorithms, we give a graphical solution for an approximate CI implication problem, in which we ask whether small values of one additional partial correlation entail small values for yet a further partial correlation.

In recent years, the surge in unstructured data analysis, facilitated by advancements in Machine Learning (ML), has prompted diverse approaches for handling images, text documents, and videos. Analysts, leveraging ML models, can extract meaningful information from unstructured data and store it in relational databases, allowing the execution of SQL queries for further analysis. Simultaneously, vector databases have emerged, embedding unstructured data for efficient top-k queries based on textual queries. This paper introduces a novel framework SSQL - Semantic SQL that utilizes these two approaches, enabling the incorporation of semantic queries within SQL statements. Our approach extends SQL queries with dedicated keywords for specifying semantic queries alongside predicates related to ML model results and metadata. Our experimental results show that using just semantic queries fails catastrophically to answer count and spatial queries in more than 60% of the cases. Our proposed method jointly optimizes the queries containing both semantic predicates and predicates on structured tables, such as those generated by ML models or other metadata. Further, to improve the query results, we incorporated human-in-the-loop feedback to determine the optimal similarity score threshold for returning results.

Differential abundance analysis is a key component of microbiome studies. It focuses on the task of assessing the magnitude and statistical significance of differences in microbial abundances between conditions. While dozens of methods for differential abundance analysis exist, they have been reported to produce remarkably discordant results. Currently, there is no consensus on the preferred methods. While correctness of results in differential abundance analysis is an ambiguous concept that cannot be evaluated without employing simulated data, we argue that consistency of results across datasets should be considered as an essential quality of a well-performing method. We compared the performance of 13 differential abundance analysis methods employing datasets from multiple (N = 54) taxonomic profiling studies based on 16S rRNA gene or shotgun sequencing. For each method, we examined how the results replicated between random partitions of each dataset and between datasets from independent studies. While certain methods showed good consistency, some widely used methods were observed to make a substantial number of conflicting findings. Overall, the highest consistency without unnecessary reduction in sensitivity was attained by analyzing total sum scaling (TSS) normalized counts with a non-parametric method (Wilcoxon test or ordinal regression model) or linear regression (MaAsLin2). Comparable performance was also attained by analyzing presence/absence of taxa with logistic regression. In conclusion, while numerous sophisticated methods for differential abundance analysis have been developed, elementary methods seem to provide more consistent results without unnecessarily compromising sensitivity. We therefore suggest that the elementary methods should be preferred in microbial differential abundance analysis when replicability needs to be emphasized.