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1. A Gallery of Gaussian Periods

   Eischen, Ellen ; Garcia, Stephan Ramon ( January 2020 , Bridges 2020 Conference Proceedings)

   null (Ed.)

   Gaussian periods are certain sums of roots of unity whose study dates back to Gauss’s seminal work in algebra and number theory. Recently, large scale plots of Gaussian periods have been revealed to exhibit striking visual patterns, some of which have been explored in the second named author’s prior work. In 2020, the first named author produced a new app, Gaussian periods, which allows anyone to create these plots much more efficiently and at a larger scale than before. In this paper, we introduce Gaussian periods, present illustrations created with the new app, and summarize how mathematics controls some visual features, including colorings left unexplained in earlier work. 

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2. On the α-q-Mutual Information and the α-q-Capacities

   https://doi.org/10.3390/e23060702  

   Ilić, Velimir M. ; Djordjević, Ivan B. ( June 2021 , Entropy)

   null (Ed.)

   The measures of information transfer which correspond to non-additive entropies have intensively been studied in previous decades. The majority of the work includes the ones belonging to the Sharma–Mittal entropy class, such as the Rényi, the Tsallis, the Landsberg–Vedral and the Gaussian entropies. All of the considerations follow the same approach, mimicking some of the various and mutually equivalent definitions of Shannon information measures, and the information transfer is quantified by an appropriately defined measure of mutual information, while the maximal information transfer is considered as a generalized channel capacity. However, all of the previous approaches fail to satisfy at least one of the ineluctable properties which a measure of (maximal) information transfer should satisfy, leading to counterintuitive conclusions and predicting nonphysical behavior even in the case of very simple communication channels. This paper fills the gap by proposing two parameter measures named the α-q-mutual information and the α-q-capacity. In addition to standard Shannon approaches, special cases of these measures include the α-mutual information and the α-capacity, which are well established in the information theory literature as measures of additive Rényi information transfer, while the cases of the Tsallis, the Landsberg–Vedral and the Gaussian entropies can also be accessed by special choices of the parameters α and q. It is shown that, unlike the previous definition, the α-q-mutual information and the α-q-capacity satisfy the set of properties, which are stated as axioms, by which they reduce to zero in the case of totally destructive channels and to the (maximal) input Sharma–Mittal entropy in the case of perfect transmission, which is consistent with the maximum likelihood detection error. In addition, they are non-negative and less than or equal to the input and the output Sharma–Mittal entropies, in general. Thus, unlike the previous approaches, the proposed (maximal) information transfer measures do not manifest nonphysical behaviors such as sub-capacitance or super-capacitance, which could qualify them as appropriate measures of the Sharma–Mittal information transfer. 

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3. Latent variable Gaussian process models: A rank‐based analysis and an alternative approach

   https://doi.org/10.1002/nme.6690  

   Tao, Siyu ; Apley, Daniel W. ; Plumlee, Matthew ; Chen, Wei ( April 2021 , International Journal for Numerical Methods in Engineering)

   Gaussian process (GP) models have been extended to emulate expensive computer simulations with both qualitative/categorical and quantitative/continuous variables. Latent variable (LV) GP models, which have been recently developed to map each qualitative variable to some underlying numerical LVs, have strong physics‐based justification and have achieved promising performance. Two versions use LVs in Cartesian (LV‐Car) space and hyperspherical (LV‐sph) space, respectively. Despite their success, the effects of these different LV structures are still poorly understood. This article illuminates this issue with two contributions. First, we develop a theorem on the effect of the ranks of the qualitative factor correlation matrices of mixed‐variable GP models, from which we conclude that the LV‐sph model restricts the interactions between the input variables and thus restricts the types of response surface data with which the model can be consistent. Second, following a rank‐based perspective like in the theorem, we propose a new alternative model named LV‐mix that combines the LV‐based correlation structures from both LV‐Car and LV‐sph models to achieve better model flexibility than them. Through extensive case studies, we show that LV‐mix achieves higher average accuracy compared with the existing two.

    

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4. Extending Global Continuous Geomagnetic Field Reconstructions on Timescales Beyond Human Civilization

   https://doi.org/10.1029/2018GC007966  

   Panovska, S. ; Constable, C. G. ; Korte, M. ( December 2018 , Geochemistry, Geophysics, Geosystems)

   Study of the late Quaternary geomagnetic field contributes significantly to understanding the origin of millennial‐scale paleomagnetic secular variations, the structure of geomagnetic excursions, and the long‐term shielding by the geomagnetic field. A compilation of paleomagnetic sediment records and archeomagnetic and lava flow data covering the past 100 ka enables reconstruction of the global geomagnetic field on such long‐term scales. We use regularized inversion to build the first global, time‐dependent, geomagnetic field model spanning the past 100 ka, namedGGF100k(GlobalGeomagneticField over the past100 ka). Spatial parametrization of the model is in spherical harmonics and time variations with cubic splines. The model is heavily constrained by more than 100 continuous sediment records covering extended periods of time, which strongly prevail over the limited number of discrete snapshots provided by archeomagnetic and volcanic data. Following an assessment of temporal resolution in each sediment's magnetic record, we have introduced smoothing kernels into the forward modeling when assessing data misfit. This accommodates the smoothing inherent in the remanence acquisition in individual sediment paleomagnetic records, facilitating a closer fit to both high‐ and low‐resolution records in regions where some sediments have variable temporal resolutions. The model has similar spatial resolution but less temporal complexity than current Holocene geomagnetic field models. Using the new reconstruction, we discuss dipole moment variations, the time‐averaged field, and paleomagnetic secular variation activity. The new GGF100k model fills the gap in the geomagnetic power spectrum in the frequency range 100–1,000 Ma⁻¹.

    

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5. Characterizing Charge Structure in Central Argentina Thunderstorms During RELAMPAGO Utilizing a New Charge Layer Polarity Identification Method

   https://doi.org/10.1029/2021EA001803  

   Medina, Bruno L. ; Carey, Lawrence D. ; Lang, Timothy J. ; Bitzer, Phillip M. ; Deierling, Wiebke ; Zhu, Yanan ( August 2021 , Earth and Space Science)

   A new automated method to retrieve charge layer polarity from flashes, named Chargepol, is presented in this paper. Using data from the NASA Lightning Mapping Array (LMA) deployed during the Remote sensing of Electrification, Lightning, And Mesoscale/microscale Processes with Adaptive Ground Observations (RELAMPAGO) field campaign in Cordoba, Argentina, from November 2018 to April 2019, this method estimates the polarity of vertical charge distributions and their altitudes and thicknesses (or vertical depth) using the very‐high frequency (VHF) source emissions detected by LMAs. When this method is applied to LMA data for extended periods of time, it is capable of inferring a storm's bulk electrical charge structure throughout its life cycle. This method reliably predicted the polarity of charge within which lightning flashes propagated and was validated in comparison to methods that require manual assignment of polarities via visual inspection of VHF lightning sources. Examples of normal and anomalous charge structures retrieved using Chargepol for storms in Central Argentina during RELAMPAGO are presented for the first time. Application of Chargepol to five months of LMA data in Central Argentina and several locations in the United States allowed for the characterization of the charge structure in these regions and for a reliable comparison using the same methodology. About 13.3% of Cordoba thunderstorms were defined by an anomalous charge structure, slightly higher than in Oklahoma (12.5%) and West Texas (11.1%), higher than Alabama (7.3%), and considerably lower than in Colorado (82.6%). Some of the Cordoba anomalous thunderstorms presented enhanced low‐level positive charge, a feature rarely if ever observed in Colorado thunderstorms.

    

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