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Let $D: \mathbb{Z}_m^n \to \mathbb{Z}_m^n$ be defined so that \[D(x_1, x_2, ..., x_n)=(x_1+x_2 \; \text{mod} \; m, x_2+x_3 \; \text{mod} \; m, ..., x_n+x_1 \; \text{mod} \; m).\] We call $D$ the Ducci function and the sequence $\{D^{\alpha}(\mathbf{u})\}_{\alpha=0}^{\infty}$ the Ducci sequence of $\mathbf{u}$ for $\mathbf{u} \in \mathbb{Z}_m^n$. Every Ducci sequence enters a cycle, so we can let $\text{Per}(\mathbf{u})$ be the number of tuples in the Ducci cycle of $\mathbf{u}$, or the period of $\mathbf{u}$. In this paper, we will look at what different possible values of $\text{Per}(\mathbf{u})$ we can have and some conditions that if $\mathbf{u}$ meets at least one of them, $\mathbf{u}$ will generate a period smaller than the maximum period.

Let $D$ be an endomorphism on $\mathbb{Z}_m^n$ so that \[D(x_1, x_2, ..., x_n)=(x_1+x_2 \; \text{mod} \; m, x_2+x_3 \; \text{mod} \; m, ..., x_n+x_1 \; \text{mod} \; m).\] We call the sequence $\{D^{\alpha}(\mathbf{u})\}_{\alpha=0}^{\infty}$ the Ducci sequence of $\mathbf{u} \in \mathbb{Z}_m^n$, which always enters a cycle. Now let $H$ be an endomorphism on $\mathbb{Z}_m^n$ such that \[H(x_1, x_2, ..., x_n)=(x_2, x_3, ..., x_n, x_1).\] In this paper, we will talk about a few cases when $\mathbf{u}$ and $H^{\beta}(\mathbf{u})$ have the same Ducci cycle for $\beta > 0$, as well as prove a few cases of $n,m$ where this is guaranteed for every $\mathbf{u} \in \mathbb{Z}_m^n$.

The Simons Observatory (SO) is a ground-based cosmic microwave background (CMB) survey experiment that currently consists of three 0.42m small-aperture telescopes (SATs) and one 6m large-aperture telescope (LAT), located at an elevation of 5200m in the Atacama Desert in Chile. At the LAT's focal plane, SO will install >62,000 transition-edge sensor detectors across 13 optics tubes (OTs) within the Large Aperture Telescope Receiver (LATR), the largest cryogenic camera ever built to observe the CMB. Here we report on the validation of the LATR in the laboratory and the subsequent dark testing and validation within the LAT. We show that the LATR meets cryogenic, optical, and detector specifications required for high-sensitivity measurements of the CMB. At the time of writing, the LATR is installed in the LAT with six OTs (corresponding to >31,000 detectors), and the LAT mirrors and remaining seven OTs are undergoing development.

We analyze data from multiple flybys by the Solar Orbiter, BepiColombo, and Parker Solar Probe (PSP) missions to study the interaction between Venus' plasma environment and the solar wind forming the induced magnetosphere. Through examination of magnetic field and plasma density signatures we characterize the spatial extent and dynamics of Venus' magnetotail, focusing mainly on boundary crossings. Notably, we observe significant differences in boundary crossing location and appearance between flybys, highlighting the dynamic nature of Venus' magnetotail. In particular, during Solar Orbiter's third flyby, extreme solar wind conditions led to significant variations in the magnetosheath plasma density and magnetic field properties, but the increased dynamic pressure did not compress the magnetotail. Instead, it is possible that the increased EUV flux at this time rather caused it to expand in size. Key findings also include the identification of several far downstream bow shock (BS), or bow wave, crossings to at least 60 Rv (1 Rv = 6,052 km is the radius of Venus), and the induced magnetospheric boundary to at least 20 Rv. These crossings provide insight into the extent of the induced magnetosphere. Pre-existing models from Venus Express were only constrained to within ~5 Rv of the planet, and we provide modifications to better fit the far-downstream crossings. The new model BS is now significantly closer to the central tail than previously suggested, by about 10 Rv at 60 Rv downstream.

Let $D: \mathbb{Z}_m^n \to \mathbb{Z}_m^n$ be defined so \[D(x_1, x_2, ..., x_n)=(x_1+x_2 \; \text{mod} \; m, x_2+x_3 \; \text{mod} \; m, ..., x_n+x_1 \; \text{mod} \; m).\] $D$ is known as the Ducci function and for $\mathbf{u} \in \mathbb{Z}_m^n$, $\{D^{\alpha}(\mathbf{u})\}_{\alpha=0}^{\infty}$ is the Ducci sequence of $\mathbf{u}$. Every Ducci sequence enters a cycle because $\mathbb{Z}_m^n$ is finite. In this paper, we aim to establish an upper bound for how long it will take for a Ducci sequence in $\mathbb{Z}_m^n$ to enter its cycle when $n$ is even.

We describe the cryogenic system being developed for Taurus: a super-pressure balloon-borne microwave polarimeter scheduled to fly in 2027. The Taurus cryogenic system consists of a 660L liquid helium cryostat which achieves a base temperature of <100mK with the help of a capillary-fed superfluid tank and a closed cycle dilution refrigerator. The main tank is supported with fiberglass flexures and is encased in two layers of vapor-cooled shields which allow Taurus to make full use of the extended flight time offered by the super-pressure balloon platform. The Taurus cryostat is projected to hold for over 50 days while weighing under 1000lbs. We present the design, testing, and thermal analysis of the Taurus cryogenic systems.

Significant advancements have been made in understanding the physics of transition-edge sensors (TESs) over the past decade. However, key questions remain, particularly a detailed understanding of the current-dependent resistance of these detectors when biased within their superconducting transition. We use scanning superconducting quantum interference device (SQUID) microscopy (SSM) to image the local diamagnetic response of aluminum-manganese alloy (Al-Mn) transition-edge sensors (TESs) near their critical temperature of approximately 175 mK. By doing so, we gain insights into how the device dimensions influence TES transition width, which in turn affects device operation and informs optimal device design. Our images reveal that the Al-Mn thin film near the niobium (Nb) leads exhibits an excess diamagnetic response at temperatures several milli-Kelvin (mK) higher than the bulk of the film farther from the contacts. A possible origin of this behavior is a longitudinal proximity effect between the Nb and Al-Mn where the TES acts as a weak link between superconducting leads. We discuss how this effect shapes the temperature dependence of the resistance as the spacing between the leads decreases. This work demonstrates that magnetic imaging with SSM is a powerful tool for local characterization of superconducting detectors., Comment: 5 pages, 4 figures, Applied Superconductivity Conference (ASC 2024) proceedings, submitted to IEEE Transactions on Applied Superconductivity (IEEE TAS)

The Simons Observatory (SO) is a cosmic microwave background experiment composed of three 0.42 m Small Aperture Telescopes (SATs) and one 6 m Large Aperture Telescope (LAT) in the Atacama Desert of Chile. The Large Aperture Telescope Receiver (LATR) was integrated into the LAT in August 2023; however, because mirrors were not yet installed, the LATR optical chain was capped at the 4K stage. In this dark configuration we are able to characterize many elements of the instrument without contributions from atmospheric noise. Here we show this noise is below the required upper limit and its features are well described with a simple noise model. Maps produced using this noise model have properties that are in good agreement with the white noise levels of our dark data. Additionally, we show that our nominal scan strategy has a minimal effect on the noise when compared to the noise when the telescope is stationary

Taurus is a balloon-borne cosmic microwave background (CMB) experiment optimized to map the E-mode polarization and Galactic foregrounds at the largest angular scales ($\ell$ $\lt$ 30) and improve measurements of the optical depth to reionization ($\tau$). This will pave the way for improved measurements of the sum of neutrino masses in combination with high-resolution CMB data while also testing the $\Lambda CDM$ model on large angular scales and providing high-frequency maps of polarized dust foregrounds to the CMB community. These measurements take advantage of the low-loading environment found in the stratosphere and are enabled by NASA's super-pressure balloon platform, which provides access to 70% of the sky with a launch from Wanaka, New Zealand. Here we describe a general overview of Taurus, with an emphasis on the instrument design. Taurus will employ more than 10,000 100 mK transition edge sensor bolometers distributed across two low-frequency (150, 220 GHz) and one high-frequency (280, 350 GHz) dichroic receivers. The liquid helium cryostat housing the detectors and optics is supported by a lightweight gondola. The payload is designed to meet the challenges in mass, power, and thermal control posed by the super-pressure platform. The instrument and scan strategy are optimized for rigorous control of instrumental systematics, enabling high-fidelity linear polarization measurements on the largest angular scales.

We simulate a variety of optical systematics for Taurus, a balloon-borne cosmic microwave background (CMB) polarisation experiment, to assess their impact on large-scale E-mode polarisation measurements and constraints of the optical depth to reionisation {\tau}. We model a one-month flight of Taurus from Wanaka, New Zealand aboard a super-pressure balloon (SPB). We simulate night-time scans of both the CMB and dust foregrounds in the 150GHz band, one of Taurus's four observing bands. We consider a variety of possible systematics that may affect Taurus's observations, including non-gaussian beams, pointing reconstruction error, and half-wave plate (HWP) non-idealities. For each of these, we evaluate the residual power in the difference between maps simulated with and without the systematic, and compare this to the expected signal level corresponding to Taurus's science goals. Our results indicate that most of the HWP-related systematics can be mitigated to be smaller than sample variance by calibrating with Planck's TT spectrum and using an achromatic HWP model, with a preference for five layers of sapphire to ensure good systematic control. However, additional beam characterization will be required to mitigate far-sidelobe pickup from dust on larger scales., Comment: 21 pages, 6 tables, 7 figures, this version as published in JCAP after minor revisions to v1

The Simons Observatory is a new ground-based cosmic microwave background experiment, which is currently being commissioned in Chile's Atacama Desert. During its survey, the observatory's small aperture telescopes will map 10% of the sky in bands centered at frequencies ranging from 27 to 280 GHz to constrain cosmic inflation models, and its large aperture telescope will map 40% of the sky in the same bands to constrain cosmological parameters and use weak lensing to study large-scale structure. To achieve these science goals, the Simons Observatory is deploying these telescopes' receivers with 60,000 state-of-the-art superconducting transition-edge sensor bolometers for its first five year survey. Reading out this unprecedented number of cryogenic sensors, however, required the development of a novel readout system. The SMuRF electronics were developed to enable high-density readout of superconducting sensors using cryogenic microwave SQUID multiplexing technology. The commissioning of the SMuRF systems at the Simons Observatory is the largest deployment to date of microwave multiplexing technology for transition-edge sensors. In this paper, we show that a significant fraction of the systems deployed so far to the Simons Observatory's large aperture telescope meet baseline specifications for detector yield and readout noise in this early phase of commissioning., Comment: 10 pages, 5 figures, 1 table. To be presented at SPIE Astronomical Telescopes + Instrumentation 2024

The Simons Observatory will map the temperature and polarization over half of the sky, at millimeter wavelengths in six spectral bands from the Atacama Desert in Chile. These data will provide new insights into the genesis, content, and history of our Universe; the astrophysics of galaxies and galaxy clusters; objects in our solar system; and time-varying astrophysical phenomena. This ambitious new instrument suite, initially comprising three 0.5 m small-aperture telescopes and one 6 m large aperture telescope, is designed using a common combination of new technologies and new implementations to realize an observatory significantly more capable than the previous generation. In this paper, we present the pre-deployment performance of the first mid-frequency "optics tube" which will be fielded on the large aperture telescope with sensitivity to the 90 and 150 GHz spectral bands. This optics tube contains lenses, filters, detectors, and readout components, all of which operate at cryogenic temperatures. It is one of seven that form the core of the large aperture telescope receiver in its initial deployment. We describe this optics tube, including details of comprehensive testing methods, new techniques for beam and passband characterization, and its measured performance. The performance metrics include beams, optical efficiency, passbands, and forecasts for the on-sky performance of the system. We forecast a sensitivity that exceeds the requirements of the large aperture telescope with greater than 30% margin in each spectral band, and predict that the instrument will realize diffraction-limited performance and the expected detector passbands.

The Simons Observatory (SO) is a cosmic microwave background (CMB) survey experiment that includes small-aperture telescopes (SATs) observing from an altitude of 5,200 m in the Atacama Desert in Chile. The SO SATs will cover six spectral bands between 27 and 280 GHz to search for primordial B-modes to a sensitivity of $\sigma(r)=0.002$, with quantified systematic errors well below this value. Each SAT is a self-contained cryogenic telescope with a 35$^\circ$ field of view, 42 cm diameter optical aperture, 40 K half-wave plate, 1 K refractive optics, and $<0.1$ K focal plane that holds $>12,000$ TES detectors. We describe the nominal design of the SATs and present details about the integration and testing for one operating at 93 and 145 GHz.

The Simons Observatory (SO) is a cosmic microwave background instrumentation suite in the Atacama Desert of Chile. More than 65,000 polarization-sensitive transition-edge sensor (TES) bolometers will be fielded in the frequency range spanning 27 to 280 GHz, with three separate dichroic designs. The mid-frequency 90/150 GHz and ultra-high-frequency 220/280 GHz detector arrays, fabricated at NIST, account for 39 of 49 total detector modules and implement the feedhorn-fed orthomode transducer (OMT)-coupled TES bolometer architecture. A robust production-level fabrication framework for these detector arrays and the monolithic DC/RF routing wafers has been developed, which includes single device prototyping, process monitoring techniques, in-process metrology, and cryogenic measurements of critical film properties. Application of this framework has resulted in timely delivery of nearly 100 total superconducting focal plane components to SO with 88% of detector wafers meeting nominal criteria for integration into a detector module: a channel yield > 95% and Tc in the targeted range., Comment: 10 pages, 6 figures. Proceedings of the 20th International Conference on Low Temperature Detectors (LTD20). Submitted to JLTP

Define the Ducci function $D: \mathbb{Z}_m^n \to \mathbb{Z}_m^n$ so \[D(x_1,x_2, ...,x_n)=(x_1+x_2 \;\text{mod} \; m, x_2+x_3 \; \text{mod} \; m, ..., x_n+x_1 \; \text{mod} \; m).\] Call $\{D^{\alpha}(\mathbf{u})\}_{\alpha=0}^{\infty}$ the Ducci sequence of $\mathbf{u}$. Because $\mathbb{Z}_m^n$ is finite, every Ducci sequence will enter a cycle. In this paper, we will prove that if $n$ is odd and $m=2^lm_1$ where $m_1$ is odd, then the longest it will take for a Ducci sequence to enter its cycle is $l$ iterations. Furthermore, we will prove the set of all tuples in a cycle for $\mathbb{Z}_m^n$ is $\{(x_1, x_2, ..., x_n) \in \mathbb{Z}_m^n \; \mid \; x_1+x_2+ \cdots +x_n \equiv 0 \; \text{mod} \; 2^l\}$.

Impacts are critical to producing the aqueous environments necessary to stimulate prebiotic chemistry on Titan's surface. Furthermore, organic hazes resting on the surface are a likely feedstock of biomolecules. In this work, we conduct impact experiments on laboratory-produced organic haze particles and haze/sand mixtures and analyze these samples for life's building blocks. Samples of unshocked haze and sand particles are also analyzed to determine the change in biomolecule concentrations and distributions from shocking. Across all samples, we detect seven nucleobases, nine proteinogenic amino acids, and five other biomolecules (e.g., urea) using a blank subtraction procedure to eliminate signals due to contamination. We find that shock pressures of 13 GPa variably degrade nucleobases, amino acids, and a few other organics in haze particles and haze/sand mixtures; however, certain individual biomolecules become enriched or are even produced from these events. Xanthine, threonine, and aspartic acid are enriched or produced in impact experiments containing sand, suggesting these minerals may catalyze the production of these biomolecules. On the other hand, thymine and isoleucine/norleucine are enriched or produced in haze samples containing no sand, suggesting catalytic grains are not necessary for all impact shock syntheses. Uracil, glycine, proline, cysteine, and tyrosine are the most unstable to impact-related processing. These experiments suggest that impacts alter biomolecule distributions on Titan's surface, and that organic hazes co-occurring with fine-grained material on the surface may provide an initial source for further prebiotic chemistry on Titan., Comment: Accepted to PSJ. 13 pages, 3 tables, 7 figures

Let $D(x_1, x_2, ..., x_n)=(x_1+x_2 \;\text{mod} \; m, x_2+x_3 \; \text{mod} \; m, ..., x_n+x_1 \; \text{mod} \; m)$ where $D \in End(\mathbb{Z}_m^n)$ be the Ducci function. The sequence $\{D^k(\mathbf{u})\}_{k=0}^{\infty}$ will eventually enter a cycle. If $n=3$, we aim to establish the longest a cycle can be for a given $m$.

Let the Ducci function $D: \mathbb{Z}_m^n \to \mathbb{Z}_m^n$ be defined as \[D(x_1, x_2, ..., x_n)=(x_1+x_2 \; \text{mod} \; m, x_2+x_3 \; \text{mod} \; m, ..., x_n+x_1 \; \text{mod} \; m)\] and let the Ducci sequence of $\mathbf{u}$ be the sequence $\{D^{\alpha}(\mathbf{u})\}_{\alpha=0}^{\infty}$. %In this paper, we will prove that if $n,m$ are powers of $2$, then repeatedly applying $D$ will eventually result in $(0,0,...,0)$, as well as establish an upper bound for how many iterations it will take for this to happen. In this paper, we will provide another proof that for $n=2^k$ and $m=2^l$, that all Ducci sequences will end in $(0,0,...,0)$ and additionally prove that this will happen in at most $2^{k-1}(l+1)$ iterations of $D$.

While machine learning (ML) includes a valuable array of tools for analyzing biomedical data, significant time and expertise is required to assemble effective, rigorous, and unbiased pipelines. Automated ML (AutoML) tools seek to facilitate ML application by automating a subset of analysis pipeline elements. In this study we develop and validate a Simple, Transparent, End-to-end Automated Machine Learning Pipeline (STREAMLINE) and apply it to investigate the added utility of photography-based phenotypes for predicting obstructive sleep apnea (OSA); a common and underdiagnosed condition associated with a variety of health, economic, and safety consequences. STREAMLINE is designed to tackle biomedical binary classification tasks while adhering to best practices and accommodating complexity, scalability, reproducibility, customization, and model interpretation. Benchmarking analyses validated the efficacy of STREAMLINE across data simulations with increasingly complex patterns of association. Then we applied STREAMLINE to evaluate the utility of demographics (DEM), self-reported comorbidities (DX), symptoms (SYM), and photography-based craniofacial (CF) and intraoral (IO) anatomy measures in predicting any OSA or moderate/severe OSA using 3,111 participants from Sleep Apnea Global Interdisciplinary Consortium (SAGIC). OSA analyses identified a significant increase in ROC-AUC when adding CF to DEM+DX+SYM to predict moderate/severe OSA. A consistent but non-significant increase in PRC-AUC was observed with the addition of each subsequent feature set to predict any OSA, with CF and IO yielding minimal improvements. Application of STREAMLINE to OSA data suggests that CF features provide additional value in predicting moderate/severe OSA, but neither CF nor IO features meaningfully improved the prediction of any OSA beyond established demographics, comorbidity and symptom characteristics., Comment: 23 pages, 7 figures, 1 table, 1 supplemental information document (77 pages), and 7 ancillary files