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1. Multi‐Frequency Ultrasound Directed Self‐Assembly

   https://doi.org/10.1002/adfm.202400193  

   Presley, Christopher Tre; Guevara_Vasquez, Fernando; Raeymaekers, Bart (April 2024, Advanced Functional Materials)

   Abstract Ultrasound‐directed self‐assembly (DSA) utilizes the acoustic radiation force associated with a standing ultrasound wave field to organize particles dispersed in a fluid medium into specific patterns. State‐of‐the‐art ultrasound DSA methods use single‐frequency ultrasound wave fields, which only allow organizing particles into simple, periodic patterns, or require a large number of ultrasound transducers to assemble complex patterns. In contrast, this work introduces multi‐frequency ultrasound wave fields to organize particles into complex patterns. A method is theoretically derived to determine the operating parameters (frequency, amplitude, phase) of any arrangement of ultrasound transducers, required to assemble spherical particles dispersed in a fluid medium into specific patterns, and experimentally validated for a system with two frequencies. The results show that multi‐frequency compared to single‐frequency ultrasound DSA enables the assembly of complex patterns of particles with substantially fewer ultrasound transducers. Additionally, the method does not incur a penalty in terms of accuracy, and it does not require custom hardware for each different pattern, thus offering reconfigurability, which contrasts, e.g., acoustic holography. Multi‐frequency ultrasound DSA can spur progress in a myriad of engineering applications, including the manufacturing of multi‐functional polymer matrix composite materials that derive their structural, electric, acoustic, or thermal properties from the spatial organization of particles in the matrix. 

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2. End‐periodic homeomorphisms and volumes of mapping tori

   https://doi.org/10.1112/topo.12277  

   Field, Elizabeth; Kim, Heejoung; Leininger, Christopher; Loving, Marissa (March 2023, Journal of Topology)

   Abstract Given an irreducible, end‐periodic homeomorphism of a surface with finitely many ends, all accumulated by genus, the mapping torus, , is the interior of a compact, irreducible, atoroidal 3‐manifold with incompressible boundary. Our main result is an upper bound on the infimal hyperbolic volume of in terms of the translation length of on the pants graph of . This builds on work of Brock and Agol in the finite‐type setting. We also construct a broad class of examples of irreducible, end‐periodic homeomorphisms and use them to show that our bound is asymptotically sharp. 

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3. Representing Hyperbolic Space Accurately using Multi-Component Floats

   Yu Tao; De Sa, Christopher (December 2021, Advances in neural information processing systems)

   Ranzato, M.; Beygelzimer, A.; Dauphin Y.; Liang, P.S.; Wortman Vaughan, J. (Ed.)

   Hyperbolic space is particularly useful for embedding data with hierarchical structure; however, representing hyperbolic space with ordinary floating-point numbers greatly affects the performance due to its \emph{ineluctable} numerical errors. Simply increasing the precision of floats fails to solve the problem and incurs a high computation cost for simulating greater-than-double-precision floats on hardware such as GPUs, which does not support them. In this paper, we propose a simple, feasible-on-GPUs, and easy-to-understand solution for numerically accurate learning on hyperbolic space. We do this with a new approach to represent hyperbolic space using multi-component floating-point (MCF) in the Poincar{\'e} upper-half space model. Theoretically and experimentally we show our model has small numerical error, and on embedding tasks across various datasets, models represented by multi-component floating-points gain more capacity and run significantly faster on GPUs than prior work. 

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4. HELM: Hyperbolic Large Language Models via Mixture-of-Curvature Experts

   He, Neil; Anand, Rishabh; Madhu, Hiren; Maatouk, Ali; Krishnaswamy, Smita; Tassiulas, Leandros; Yang, Menglin; Ying, Rex (December 2025, Neural Information Processing Systems (NeurIPS) Foundation, Inc.)

   Large language models (LLMs) have shown great success in text modeling tasks across domains. However, natural language exhibits inherent semantic hierarchies and nuanced geometric structure, which current LLMs do not capture completely owing to their reliance on Euclidean operations. Recent studies have also shown that not respecting the geometry of token embeddings leads to training instabilities and degradation of generative capabilities. These findings suggest that shifting to non-Euclidean geometries can better align language models with the underlying geometry of text. We thus propose to operate fully in Hyperbolic space, known for its expansive, scale-free, and low-distortion properties. We thus introduce HELM, a family of HypErbolic Large Language Models, offering a geometric rethinking of the Transformer-based LLM that addresses the representational inflexibility, missing set of necessary operations, and poor scalability of existing hyperbolic LMs. We additionally introduce a Mixture-of-Curvature Experts model, HELM-MICE, where each expert operates in a distinct curvature space to encode more fine-grained geometric structure from text, as well as a dense model, HELM-D. For HELM-MICE, we further develop hyperbolic Multi-Head Latent Attention (HMLA) for efficient, reduced-KV-cache training and inference. For both models, we develop essential hyperbolic equivalents of rotary positional encodings and RMS normalization. We are the first to train fully hyperbolic LLMs at billion-parameter scale, and evaluate them on well-known benchmarks such as MMLU and ARC, spanning STEM problem-solving, general knowledge, and commonsense reasoning. Our results show consistent gains from our HELM architectures -- up to 4% -- over popular Euclidean architectures used in LLaMA and DeepSeek, highlighting the efficacy and enhanced reasoning afforded by hyperbolic geometry in large-scale LM pretraining. 

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5. Assessment of Models for Extracting Thermal Conductivity From Frequency Domain Thermoreflectance Experiments

   https://doi.org/10.1115/HT2020-8920  

   Saurav, Siddharth; Mazumder, Sandip (July 2020, Proceedings of the ASME Summer Heat Transfer Conference, SHTC2020)

   Abstract The Fourier heat conduction and the hyperbolic heat conduction equations were solved numerically to simulate a frequency-domain thermoreflectance (FDTR) experimental setup. Numerical solutions enable use of realistic boundary conditions, such as convective cooling from the various surfaces of the substrate and transducer. The equations were solved in time domain and the phase lag between the temperature at the center of the transducer and the modulated pump laser signal were computed for a modulation frequency range of 200 kHz to 200 MHz. It was found that the numerical predictions fit the experimentally measured phase lag better than analytical frequency-domain solutions of the Fourier heat equation based on Hankel transforms. The effects of boundary conditions were investigated and it was found that if the substrate (computational domain) is sufficiently large, the far-field boundary conditions have no effect on the computed phase lag. The interface conductance between the transducer and the substrate was also treated as a parameter, and was found to have some effect on the predicted thermal conductivity, but only in certain regimes. The hyperbolic heat conduction equation yielded identical results as the Fourier heat conduction equation for the particular case studied. The thermal conductivity value (best fit) for the silicon substrate considered in this study was found to be 108 W/m/K, which is slightly different from previously reported values for the same experimental data. 

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